mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 4, december 2020, pp.274 https://doi.org/10.5206/mase/13502 preface: thematic issue in mathematical biology and applied evolutionary equations naveen k. vaidya and dhruba r. adhikari we are pleased to present the thematic issue entitled “mathematical biology and applied evolutionary equations” in mathematics in applied sciences and engineering (mase). this issue includes some selected articles from invited authors who presented their research work during the second international conference on applications of mathematics to nonlinear sciences (http://anmaweb.org/amns-2019/) held on june 27–30, 2019, in pokhara, nepal. the primary focus of the amns conference series is to provide a platform for discussion about current advancement of interdisciplinary research on the applications of mathematics to nonlinear sciences. this thematic issue attempts to highlight some of the latest research trends in evolutionary equations and their applications in natural and life sciences. in this regard, the current issue contains a good mixture of research articles that include the development of theories and formulation of computational methods as well as applications of differential equations to various fields, including mathematical biology. the interesting results and open questions presented in this issue may offer opportunities for further development of advanced evolutionary models and advanced methods for their analysis and computations. we are very much thankful to the contributing authors for sharing their research and the referees for ensuring high quality of this publication. all submissions have gone through a regular peer review process, leading to the acceptance of these ten papers. we thank all reviewers of these submissions for their valuable evaluations and comments. we also thank dr. xingu zou for his help and support during the review process of these articles. finally, we sincerely thank mase for publishing this special issue on it. we look forward to the continuous support from the global mathematical community for the future amns conferences. guest editor, department of mathematics and statistics, san diego state university, usa. e-mail address: nvaidya@sdsu.edu guest editor, department of mathematics, kennesaw state university, usa. e-mail address: dadhikar@kennesaw.edu 1 mathematics in applied sciences and engineering https://doi.org/10.5206/mase/10267 volume 1, number 1, march 2020, pp.85-90 https://ojs.lib.uwo.ca/mase viewpoints on modelling: comments on “achilles and the tortoise: some caveats to mathematical modelling in biology” jinzhi lei abstract. mathematical modelling has been proven to be useful in understanding some problems from biological science, provided that it is used properly. however, it has also attracted some criticisms as partially presented in a recent opinion article [1] from biological community. this note intends to clarify some confusion and misunderstanding in regard to mathematically modelling by commenting on those critiques raised in [1], with a hope of initiating some further discussion so that both applied mathematicians and biologist can better use mathematical modelling and better understand the results from modelling. mathematical models have been widely used in the biological science in the recent decades. meanwhile, there are quite a lot of debates and discussions on the roles that mathematical modelling can play in biology, including the application ranges and limitations of models. in a recent opinion paper published in the journal progress in biology and molecular biology [1], dr. s. f. gilbert raised five caveats for mathematical models in biology, and they are 1) mathematical models are limited by the science known at the time. 2) mathematical models can tell what can happen, not what does or did happen. 3) real-world models can provide a better explanation than the mathematical model. 4) in abstracting reality, the things left behind can be very important. 5) mathematics models can be platonic rather than evolutionary. these caveats are somehow reasonable and suggestive, and we need to be aware of these issues when we try to explain or apply the results obtained from mathematical models. however, the above paper contains many inaccurate descriptions about mathematical modelling, and these inaccuracies may mislead readers and cause confusion, and therefore, they should be clarified. it is unfair and unscientific to expand observations from some models to all mathematical models without solid and convincing arguments. the purpose of this viewpoint note is to clarify the above five caveats, by commenting on each of them. i hope these comments, together with possible following-up discussions, will help us better understand the roles and limitations of mathematical models in biology. first of all, “mathematical modeling” is not “a set of technologies”. in scientific researches, mathematical modeling is a way of thinking and language of describing the hypotheses according to the observed phenomena. the roles of mathematics in scientific research should mainly be reflected in two aspects: (i) mathematics is a way of logical thinking, which is an ability of helping people to understand the logical relationship behind the experimental facts in biology (the same for other disciplines); (ii) mathematics is a language of science, which can accurately express human’s understanding of the laws of nature. received by the editors 12 february 2020; revised 23 february 2020; accepted 23 february 2020; published online 29 february 2020. 85 86 jinzhi lei the charm of mathematics for biology (and other sciences) lies in that it is the best way to express people’s understanding of natural laws and their underlying mechanisms, and infer possible results based on these understandings through logical deductions. the understanding of a mathematical model and its application in biology should be put into such a frame to see more essential connotation. it is with the above considerations that i will comment on each of the above critiques below. on critique 1): mathematical models are limited by the science known at the time. this sentence should be modified as “mathematical models are limited by our understanding of the science known at the time”, to be more appropriate. mathematical models are not only descriptions of the world we can see, but also descriptions of our understanding of natural laws behind the observed phenomena. the word “understanding” here should be the core of the model. sometimes, there are different understandings based on the same phenomenon, and hence different mathematical models can be established based on different hypotheses. some of them are wrong, and some are more reasonable. our understandings of natural laws (e.g., the basic laws in life science) are usually gradual and often depend on new discoveries. therefore, the mathematical models for describing the same phenomenon can also evolve; the limitation of mathematical models does not come from the limitation of mathematics itself, but the limitation of human’s understanding of the natural phenomena at the time. for biology, biologists observe new phenomena, discover the facts, explore the processes of life, and often engage in the work of discovering the known sciences. scientists from other fields, such as physics, mathematics, chemistries, etc., can play important roles in understanding these known sciences, and discovering the underlying laws behind the phenomena that biologists cannot see clearly. for example, in the discovery of dna double helix structure, physicists played important roles in several aspects: from discovering the genetic material and the dna diffraction image, to the discovering of the double helix structure. another example is the establishment of the mathematical model that describes the mechanism of action potential in neurons, that is, the hodgkin-huxley equation [2]. this equation was established based on a series of experiments of dynamic information of the neuronal electrical signals under various stimulations. in order to understand the underlying mechanism behind these experimental phenomena, hodgkin and huxley proposed the gating mechanism of ion channels, and proposed the possible rules for the gating mechanism of sodium and potassium ion channels based on the experimental data. the hodgkin-huxley equation was then established by combining the gating mechanism with the equivalent circuit model of cell membrane. the hodgkin-huxley equation nicely describes the mechanism of action potential and predicts a series of experimental results, and has become a basic equation in computational neuron science. nevertheless, at the time when this equation was established, people knew nothing about the structure of ion channels, not to mention the occurrence of gating mechanism. hodgkin and huxley understood the mechanism of ion transportation before they see the ion channels, and the proposed mathematical model is a tool to describe their hypothesis. if hodgkin and huxley did not speculate about the gating mechanism behind the changes of the electrical signals they have seen, but tried to explore the structure of the ion channels and the detailed processes of ion transportation, the understanding of the mechanism of action potential would be postponed to at least decades later. the hodgkin-huxley equation serves as an excellent example demonstrating that we can establish a reasonable and useful mathematical model through a better understanding of science known at the time. more interestingly, people found that similar mechanisms can also be applied to describe the electrical signal behaviour of cardiomyocytes. on critique 2): mathematical models can tell what can happen, not what does or did happen. comments on mathematical modelling in biology 87 it is one-sided or biased to criticize that “mathematical models can tell what can happen, but not what did happen”. it may be true for a statistical models that usually tell the probabilities of the occurrence of some events/phenomena. many mathematical models are given in a form of deterministic formulations, such as the hodgkin-huxley equation mentioned above, and the reactiondiffusion equation model for turing patterning. the deterministic models can somehow tell us what can happen and what cannot happen, and help us to understand why something did happen. hence, for most deterministic models established based on our understanding of the underlying mechanisms, the above criticism should be rephrased as “mathematical models can tell what can happen, and explain why some phenomena may or may not happen”. as for the relationships between mathematical models and experiments, if one can predict what will happen through simple experiments, it may not be significant to introduce mathematical models, such as the example of of x-linked disease mentioned in gilbert’s article. however, for many experiments with more complex dynamics and multiple scale interactions, it is not trivial, if not impossible, to get insights behind the experimental observations. in this case, a proper mathematical model can help us to understand the experimental results based on different hypotheses, and to select a more reasonable mechanism. for example, in the field of developmental biology, morphogens are important signaling molecules that govern the tissue pattern development, however there were debates on how the morphogen gradients are formed. biologists proposed that morphogen gradients can be established by a process of repeated cycles exocytosis and endocytosis, that is, so-called planar transcytosis. later, a mathematical model with reaction-diffusion equations shows that diffusive mechanisms of gradient formation may be more convincing [3, 4]. in this example, the mathematical model can not only tell us what did happen, but also help us understand why other mechanisms are unreasonable. a good mathematical model should provide reasonable explanations in accordance with experimental facts, and the rationality of the mathematical model needs to be able to withstand the verification of the experimental facts. when different models can explain the same experimental results, we may need to design new experiments to distinguish these models. therefore, mathematical models should be constantly revised with the discovery of new experimental facts, which is the only way for us to understand the laws of nature. however, we should not deny the role of mathematical modelling merely because they are inconsistent with some experimental facts; instead, the deviation between model predictions and experimental facts can sometime stimulate us to explore the ignored factors or wrong hypothesis in the model, corrections to the model hypotheses can eventually lead us to a better understanding of the experimental findings. on critique 3): real-world models can provide a better explanation than the mathematical model. in [1], the author introduced two examples to illustrate the limitations of the mathematical models: kepler’s law in planetary motion and gravitational interpretation, and binary code model in drosophila embryo development. through these two examples, the author tried to criticize that “mathematics can give a model that explains phenomenon, but which does not work as nature actually does”. i think this point is basically correct, and is also a reason why many biologists do not trust mathematical models. however, any model is an approximate description of the real world, especially for those mathematical models in the field of biology. it is impossible to consider all the details. the question is, do we need to consider all details? are different models helpful for the problems under consideration? even in physical sciences, many models are perfect and can stand the test of experiments, but they are also only realworld approximations. when we study the motion of objects with conventional scales (such as the flight of rockets, the motion of planets, etc.), it is sufficient for us to employ the newtonian mechanics models. 88 jinzhi lei when we study problems in the cosmic scale, such as the movement near the black holes or super stars, we need mathematical models based on general relativity. when we study quantum behaviour, we need to consider mathematical models derived from quantum mechanics. these models are very good descriptions of the problems at a certain scale. however, there is no unified theory that can describe all the basic interactions from quantum to gravitation force. even if there is such a theory, it is impossible (and unnecessary) to apply it to study most practical problems. for a better understanding of our world, we need a balance between a workable mathematical model and the real-world models. in fact, it is not appropriate to use kepler’s law and gravitation theory to discuss the point here. kepler’s law is the law of planetary motion induced by astronomical observation data. the law of universal gravitation can be strictly deduced according to newton’s second law and kepler’s three laws, and it is a theoretical improvement of the observed laws. on the basis of recognition of newton’s second law, the law of universal gravitation is equivalent to kepler’s laws. however, kepler’s laws are only a summary of the observed data of planetary orbits. these laws cannot be judged in a wider range, and the predictability of these laws are very limited for the planets that are not easy to be observed. when we obtain the law of gravitation, we can not only explain kepler’s three laws, but also predict the orbits of planets, such as the occurrence of comets, the discovery of pluto, the orbit of the moon landing rocket, and so on. in [1], the author said that the law of universal gravitation cannot explain why planets choose the orbit we can see. this involves another problem of the formation process of planets that may lead to an extended mathematical model that includes the change of material distribution during the birth of the star systems. in the example of the binary code model in drosophila embryo development in [1], the simple model of binary code model may be over simplified for the real world problem. however, in latter studies, more related genes and regulatory factors were found, different types of mathematical models were constructed based on these regulatory relationships which provided good explanations to the relevant experimental phenomena [5]. to better understand the gap between mathematical models and the real world, we need to look back to the first point about the limitation of mathematical model: mathematical models are limited by our understanding of the science known at the time. the gap between the mathematical model and the real world mainly due to our understanding of the real world. peoples may propose different hypotheses and hence different models when they try to formulate the experimental facts. an important ability of experienced applied mathematicians is that they can put forward appropriate explanations and great guesses about the underlying mechanism through their insightful observations into the problems being studied. a successful mathematical model often includes a nice guess on the mechanism. there are many examples in this respect, such as hodgkin and huxley’s guess on the mechanism of gated channel [2], turing’s hypothesis on the mechanism of pattern formation caused by morphogenesis[6], heisenberg’s conjecture on the mechanism of energy cascade in turbulence spectrum, lin and shu’s density-wave structure of disk galaxies [7], etc. for mathematical models in biology, we may need to limit our research to a certain range under specific conditions due to the complexity of biological systems. to work out with a good mathematical model, close cooperations and discussions between mathematicians and biologists are required to make appropriate guesses/hypotheses, and a good model can only be established through iterative model predictions, experimental verifications, and corrections. in this process, mutual trust and cooperation between mathematicians and biologists are very important. each mathematical model has its scope of application. it is not appropriate to deny a mathematical model with facts beyond the scope of application of the model. the applicability of a mathematical model should be considered from several aspects: (1) is the hypothesis proposed in the mathematical comments on mathematical modelling in biology 89 model reasonable? (2) what is the scope of application of the mathematical model? (3) how can we improve/remove the unreasonable assumptions in the model or extend the scope of the model? on critique 4): in abstracting reality, the things left behind can be very important. this point of view is partially correct, especially for complex biological systems, we have to make simplification and abstraction when we establish a mathematical model. hence, there are some factors that cannot be included in the model. the issue is that whether or not these excluded factors are important for the problem being studied? in developing mathematical models, mathematicians often focus on some main factors that are most important for the problem, and ignore the unimportant or less important effects. this process is somehow subject to the judgement on the importance of the factors, and hence different factors may be included in different mathematical models developed by different people. in this case, the things left behind can be very important. when we established mathematical models, we often point to a main problem to be answered, and hence make assumptions according to the problem. in this way, we need to focus on the main interactions that are important for the issue under consideration, and unfortunately neglect other less important factors. for example, in modelling cell behaviour, we can establish mathematical models for a motif of signalling pathways if our goal is to understand the role of a specific gene in the pathway; or we can model the dynamical response of a more complex signalling pathway if we want to know the cellular response to external stimuli; we may also need to include cell-to-cell interactions if we want to study the interactions between cells. there are models of different scales, which may depend on the questions we try to answer. moreover, some important factors may be missing from a mathematical model because of the limitation of current technologies. these issues are common to all researchers in all areas in natural sciences. on the other hand, when important factors are found missing in a mathematical model (or theories), it may motivate us to modify the model and/or even develop new theories. on critique 5): mathematical models can be platonic rather than evolutionary. for this point, the author stated that mathematical models can only be used to describe the ideal state, but cannot describe the complex and changeable environment and long-term evolution. as the author stated: “but one must remember, that mathematics can only model regulations, and that evolution has a large component of containment of contingencies”. from the perspective of mathematical methods, mathematics can describe more complex situations and regularities, including the changing of environment, random process, and the evolution of dynamics systems, etc. there are various mathematical methods that can be used to describe these complex processes. furthermore, in terms of our understanding of the biological world, mathematical models cannot be created out of nothing, and cannot provide good models for facts or phenomena that people do not understand. for example, if we do not know the mechanism behind complex behaviours in biological phenomena and cannot make reasonable assumptions either, we will not be able to establish appropriate mathematical models. mathematical models are not magic, it cannot be used to study the process that we cannot provide reasonable understanding. above are comments on gilbert’s five caveats on mathematical models in biology. from the perspective of applied mathematics, there is no essential difference between the mathematical models in biology and the mathematical models in physics. despite the various aspects of the problems we may face, there is no essential difference in research methods, ideas and processes in mathematical modelling. i would conclude this note by quoting prof. c. c. lin’s general themes of establishing mathematical models in researches of applied mathematics, which are essential to understand the role of mathematical models in different disciplines. as these themes are abstracted from a private communication, i provide them in the form of a scanned figure (fig. 1). 90 jinzhi lei figure 1. general themes of of applied mathematics proposed by prof. c. c. lin in [8]. references [1] gilbert, s. f., achilles and the tortoise: some caveats to mathematical modeling in biology, progress in biophysics and molecular biology 137(2018), 37-45. [2] hodgkin, a. l. and huxley, a. f. , a quantitative description of membrane current and its application to conduction and excitation in nerve, journal of physiology 117(1952), 500-544. [3] lander, a. d., nie, q. and wan, f. y. m., do morphogen gradients arise by diffusion? dev. cell 2(2002), 785?796. [4] matthew, f., morphogen gradients in theory, dev cell 2 (2002), 689-690. [5] morelli, l. g. l., uriu, k. k., ares, s. s. and oates, a. c. a., computational approaches to developmental patterning, science 336 (2012), 187-191. [6] turing, a.m., the chemical basis of morphogenesis, phil tran roy soc london b 237(1952), 37-72. [7] lin, c. c. and shu, f. h., on the spiral structure of disk galaxies, astrophysical journal 140(1964), 646. [8] lin, c.c., private communications. school of mathematical sciences, tiangong university, tianjin 300387, china e-mail address: jzlei@tiangong.edu.cn mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 4, december 2020, pp.403-411 https://doi.org/10.5206/mase/10874 high-order mimetic difference simulation of unsaturated flow using richards equation angel boada, johnny corbino, and jose castillo abstract. the vadose zone is the portion of the subsurface above the water table and its pore space usually contains air and water. due to the presence of infiltration, erosion, plant growth, microbiota, contaminant transport, aquifer recharge, and discharge to surface water, it is crucial to predict the transport rate of water and other substances within this zone. however, flow in the vadose zone has many complications as the parameters that control it are extremely sensitive to the saturation of the media, leading to a nonlinear problem. this flow is referred as unsaturated flow and is governed by richards equation. analytical solutions for this equation exists only for simplified cases, so most practical situations require a numerical solution. nevertheless, the nonlinear nature of richards equation introduces challenges that causes numerical solutions for this problem to be computationally expensive and, in some cases, unreliable. high-order mimetic finite difference operators are discrete analogs of the continuous differential operators and have been extensively used in the fields of fluid and solid mechanics. in this work, we present a numerical approach involving high-order mimetic operators along with a newton root-finding algorithm for the treatment of the nonlinear component. fully-implicit time discretization scheme is used to deal with the problem’s stiffness. 1. introduction in numerical analysis, classical numerical differentiation approaches begin by discretizing the corresponding system of partial differential equations of the specific problem to solve. this process leads to an approximation, at some order of accuracy, that may not result in a stable numerical scheme. in contrast, mimetic or compatible methods are formulated in such a way that allow them to discretely mimic and preserve the physical properties of the vector calculus operators used to describe continuous problems. these discrete operators are then employed to discretize the given problem. typically, the mimetic differentiation scheme will achieve stability if the continuous problem is also stable. castillo-grone’s (cg) mimetic differential operators [4] are discrete approximations of their continuous counterparts and have been broadly used with success in multiple fields of physics and engineering. some of the applications include: wave propagation, fluid dynamics, seismic studies, electromagnetism and image processing [2, 4, 9, 10, 11]. compared to common numerical analysis techniques such as finite elements and discontinuous galerkin (dg), cg mimetic methods are computationally less expensive while achieving the same order of accuracy at the domain interior as well as at the boundary, which is received by the editors 4 july 2020; accepted 23 december 2020; published online 28 december 2020. 2000 mathematics subject classification. primary 35a99, 65m06; secondary 76s05. key words and phrases. unsaturated flow, richards equation, nonlinear partial differential equations, mimetic operators. 403 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/10874 404 a. boada, j. corbino, and j. castillo something the staggered summation by parts methods, for instance, cannot achieve. recent developments have pushed the operators into solving for more realistic problems with complex coefficients [3] and non-trivial geometries [1]. derived from cg operators, castillo-corbino’s (cc) high-order mimetic difference operators provide the same features of the original operators with a more intuitive approach as they do not rely on free parameters which need to be selected for their construction. this results in discrete operators that have an optimal bandwidth and generally performed better, in terms of accuracy, when compared to cg counterparts [8]. the new operators are capable of achieving a uniform high-order of accuracy (up to eight order) in one-, two-, and three-dimensional space. unsaturated flow in porous media is a common physical phenomenon. one instance of this phenomenon is the flow of water in dry soils, which is of high importance in the fields of hydrology and agriculture. the richards equation [6] is a nonlinear partial differential equation (pde) that describes the flow of water in unsaturated soils. most analytical solutions to richards equation are impractical since they rely on simplifications and are limited to lower dimensions. there are multiple numerical approaches to tackle this problem, yet they all turn out to be computationally expensive and in certain circumstances, unreliable. the purpose of this paper is to explain the use of a relatively new numerical approach (mimetic finite differences) to a well known and highly nonlinear problem (richards equation). the work here presented describes and verifies the employment and accuracy of high-order mimetic approximations in conjunction with newton’s method to simulate flow in unsaturated porous media. the advantage of using our approach is that it allows for higher order approximations that are not only computationally cheaper but also intuitive to implement. this paper is organized as follows: in section 2, we talk about the problem’s governing equation and its parameters. in section 3, we give a brief description of mimetic operators and staggered grids. section 4 is about time discretization, and combining the operators presented in section 3 with the equation shown in section 2. in section 5, we solve multiple test cases, and we compare against results obtained by [7]. finally, in section 6 we present our conclusions and proposed future work. 2. richards equation richards equation describes the flow of water through unsaturated or partially saturated porous media. it is derived by combining darcy’s law [6] with the equation of mass conservation. there are two different forms of richards equation that will be considered in this paper, they differ on how to deal with the non-linearity in the transient term. • head-based form: c(ψ) ∂ψ ∂t −∇·k(ψ)∇ψ − ∂k(ψ) ∂z = 0 (2.1) • mixed form: ∂θ ∂t −∇·k(ψ)∇ψ − ∂k(ψ) ∂z = 0 (2.2) where c(ψ) = ∂θ ∂ψ is the specific moisture capacity function, k is the hydraulic conductivity, ψ is the pressure head, θ is the water content, and z is the depth. there are many empirical formulations for the hydraulic conductivity (k) and water content (θ) functions. the most popular model is the van genuchten [6], which describes these functions without discontinuities. it is worth noting that both, the hydraulic conductivity and the water content depend on the pressure head (ψ). indeed, both functions richards equation and mimetic operators 405 are highly nonlinear since they can dramatically change over a small range of ψ [6]. a version of the van genuchten model that can be found at celias paper [6] is: k(ψ) = ks a a + |ψ|γ , θ(ψ) = α(θs −θr) α + |ψ|β + θr (2.3) where ks corresponds to the saturated hydraulic conductivity, and θs and θr represent the saturated and residual moisture content, respectively. richards equation typical use is to simulate infiltration experiments (in both lab and field scale). these experiments begin with a dry soil and then water is poured on top of the ground surface, making the connection with darcy’s law pretty obvious. 3. mimetic operators the gradient (∇) and divergence (∇·) are vector calculus operators that perform linear transformations (differentiation), and as such, they have matrix representation. these matrices can be constructed using standard finite differences [6] to approximate the derivative of the unknown function (in this case ψ). a special subset of these matrices that is closer to the continuum counterpart is known as mimetic. mimetic finite-differences (mfd) have been widely and effectively used in many fields of applied mathematics and physics [5] and [8]. mfd provide high-order uniform accuracy without compromising physical coherence. represented by sparse matrices, the main idea behind the construction of these operators is to find high-order approximations that satisfy the extended gauss divergence theorem in the discrete sense [5]: 〈d~v,f〉q + 〈gf,~v〉p = 〈b~v,f〉i (3.1) where g ≡ ∇ and d ≡ ∇· are the mimetic gradient and divergence operators, respectively, b is a mimetic boundary operator, and the weight matrices p , q and i are self-adjoint. in particular, the q inner product accounts for the scalar inner product in cell centers, the p inner product accounts for a vector-field inner product at the cell faces, and the i inner product is at the boundaries. notice that weighted inner products are defined in the standard form, 〈x,y〉a = ytax (3.2) finally, as discrete counterparts, mimetic operators “mimic” the following vector calculus identities of their continuous analogs making them more faithful to the physics: gfconst = 0 (3.3) d~vconst = 0 (3.4) cgf = 0 (3.5) dc~v = 0 (3.6) dgf = lf (3.7) with l ≡∇2 and c ≡∇× as the mimetic laplacian and curl operators, respectively. 3.1. staggered grids. mimetic operators are defined over staggered grids. scalar variables such as density, pressure, and temperature are stored at the cells’ centers, while vector variables (velocity components, electric conductivity, hydraulic conductivity, etc.) are stored at the edges (or faces in 3d domains). 406 a. boada, j. corbino, and j. castillo 3.2. one-dimensional mimetic operators. considering improvements with respect to the original castillo-grone mimetic operators in terms of accuracy and optimal bandwidth, we follow the castillo and corbino [8] approach for the construction of our operators. we illustrate the second-order onedimensional mimetic divergence (d) and gradient (g) operators, which are the foundations of mimetic operators in higher dimensions and higher order. figure 1. one-dimensional, uniform staggered grid (m = 5). in our one-dimensional staggered grid discretization (depicted in figure 1), the mimetic divergence operator acts on vector components (v-values) defined at m + 1 nodes, with xi = i∆x, i = 0, 1, ...,m. these v-values are regarded as an (m + 1)-tuple. conversely, the mimetic gradient operator acts on u-values defined at both boundary nodes (x0 on the left, and xm on the right), as well as the m cellcenters, xi+1/2 = (i + 1/2)∆x, i = 0, 1, ...,m− 1. therefore, u-values are regarded as (m + 2)-tuple. d is then an (m + 2) × (m + 1) sparse matrix with first and last row as zero vectors (required since the divergence is calculated at cell-centers). the gradient operator, g is a (m + 1) × (m + 2) matrix. the one-dimensional mimetic divergence operator is given by: d = 1 ∆x   0 0 . . . 0 −1 1 . . . . . . −1 1 0 0 . . . 0   (m+2)×(m+1) (3.8) and the mimetic gradient: g = 1 ∆x   −8/3 3 −1/3 −1 1 . . . . . . −1 1 1/3 −3 8/3   (m+1)×(m+2) (3.9) note there is a minimum number of cells needed to construct these mimetic operators. the gradient requires at least 2k cells, whereas the divergence requires at least 2k + 1, where k is the desired order of accuracy. finally, these operators can also be extended to higher dimensions as seen in [8]. 3.3. compact operators. an important feature of mimetic operators is that they provide uniform order of accuracy all the way to the boundary. high-order (m ≥ 4) mimetic operators can be represented in a “compact way” by factoring the original matrices (i.e. the 2nd-order matrix). by doing this, higher orders of accuracy can be attained using only the minimum number of points from the mesh. for instance, a kth-order mimetic gradient operator can be constructed as: gkth = lkthg2nd (3.10) richards equation and mimetic operators 407 where lkth represents the left factor matrix of k th-order. conversely, for a kth-order mimetic divergence operator: dkth = d2ndrkth (3.11) here rkth is the right factor matrix of k th-order. finally, we write the kth-order mimetic laplacian operator in compact form as: lkth = d2ndrkthlkthg2nd (3.12) our implementation uses compact representations of the fourth-order mimetic gradient and divergence for equation (2.1). 4. temporal discretization it is important to mention that mimetic operators are only for spatial discretization. considering that we want to solve a time-dependent problem, a time discretization scheme has to be chosen. both, celia [6] and [7] use a first order forward scheme for time discretization: θ(ψn+1) −θ(ψn) ∆t ≈ ∂θ ∂t (4.1) since we want to replicate [6] and [7] results, we opted for equation (4.1) as our time discretization scheme, and the mixed form of richards equation (2.2). putting all together: θ(ψn+1) −θ(ψn) ∆t −d(k(ψn+1))gψn+1 −dz(k(ψn+1)) = 0 (4.2) where d and g are the mimetic divergence and gradient, respectively. the derivative in the z-direction is written dz, which in one-dimension is the same as the divergence matrix. equation (4.2) is the equation we want to solve, and since it involves multiple ψn+1 we must solve it using a root-finding method, we opted for newton’s method: ψn+1,m+1 = ψn+1,m + αδψ (4.3) where the superscript m indicates the newton’s iteration, α is the step length of the descent, and δψ (update) is obtained by solving the system of linear equations, jψn+1,mδψ = −f(ψn+1,m,ψn) (4.4) where jψn+1,m is the jacobian of the system, and f(ψ n+1,m,ψn) = rn+1,m is the newton’s residual (equation (4.2)). recall that: jψn+1,m = ∂f(ψn+1,m,ψn) ∂ψn+1,m (4.5) 5. test case in order to verify the accuracy of our scheme, we make a direct comparison with results obtained by celia et al. [6] (they only implemented a picard’s iteration model of the head-based formulation), and cockett in [7] (with both, picard and newton’s iteration implementation of the mixed formulation). for this, we use a set of secondand fourth-order mimetic operators based on castillo-corbino’s formulation, while employing a mixed formulation of the richards equation (using a newton’s iteration model). the van genuchten model (equation (2.3)) for this experiment has the following parameters: α = 1.611 × 106, θs = 0.287, θr = 0.075, β = 3.96, a = 1.175 × 106, γ = 4.74, ks = 9.44 × 10−5 m/s. 408 a. boada, j. corbino, and j. castillo for this experiment, the initial condition is a pressure head of ψ0(x, 0) = −61.5cm for a 40cm high one dimensional soil column (initially dry). inhomogeneous dirichlet boundary conditions are considered. here, the bottom of the soil column is consistent with the initial condition: ψ(0cm, t) = −61.5cm. on the other hand, the top of the soil column is given by ψ(40cm, t) = −20.7cm. naturally, this irregularity causes a boundary layer and a steep gradient in the pressure head at early times [7]. although it can be expected that the newton’s iteration method will fail to converge at early times, experimentally, we found success with the method by using the initial condition as the starting guess at every time step. as in [6, 7], the spatial grid resolution is a fix value of 1cm long (note that ours is a staggered grid), and we chose the same time steps to make a fair comparison. figure 2 and figure 3 depict the numerical solution of the richards equation obtained by [6, 7]. furthermore, figure 4 shows our numerical solution with high-order (k = 4) mimetic differences for the mentioned equation using a mixed formulation. visually, our numerical approximation is congruent with the ones produced by [6] and [7]. we estimated the order of convergence of our implementation by comparing successively finer grids around the point x = 0.7cm, and using ∆t = 0.25s. table 1 displays the numerical results for this experiment, based on this we can confirm a second-order convergence for our set of second-order mimetic operators while attaining better than o(h3.5) for our set of fourth-order operators. these results corroborate better convergence for our operators when compared to the first order convergence obtained by [7], while correctly capturing the physics of the nonlinear problem. figure 2. head-based formulation at t = 360s. reprinted from [7]. richards equation and mimetic operators 409 figure 3. mixed formulation at t = 360s. reprinted from [7]. figure 4. mimetic solution of the mixed formulation for richards equation at t = 360s. 410 a. boada, j. corbino, and j. castillo m u2 u4 order(u2) order(u4) 80 -4.1762e+01 -4.2017e+01 160 -4.4207e+01 -4.4659e+01 320 -4.5337e+01 -4.5095e+01 1.11 2.60 640 -4.5703e+01 -4.5146e+01 1.63 3.10 1280 -4.5803e+01 -4.5150e+01 1.87 3.50 2560 -4.5829e+01 -4.5151e+01 1.95 3.60 table 1. numerical convergence for the mixed formulation of the richards equation using our second-, and fourth-order mimetic operators. where uk ≈ ψ with a k-order of accuracy. 6. conclusions and future work in this work we solved richards equation in its mixed formulation by using secondand fourth-order mimetic versions of the gradient and divergence operators. the main reason why we study this equation is due to the highly nonlinear nature of the problem as well as the functions involved (hydraulic conductivity and water content) which are of great importance in mathematical biology. the discretization scheme (first order in time, secondand fourth-order in space) was fully-implicit and involved the use of newtons iteration to deal with the non-linearity. experimentally, we have found that the best results are produced by having the initial condition as starting guess at each time step (it proved a faster convergence with respect to a random guess). numerical results are coherent with those obtained in [6, 7] via standard finite difference approaches. furthermore, the implementation presented here not only allows closer approximations, but also a computationally cheaper and intuitive way to solve the same problem. future work should include: • extension of the model to higher dimensions. • use of a high-order scheme for time discretization. • include scenarios where the physical domain is irregular or non trivial (can be solved with overlapping grids). references 1. m. abouali and j. e. castillo, solving poisson equation with robin boundary condition on a curvilinear mesh using high order mimetic discretization methods, mathematics and computers in simulation 139 (2017), 23–36. 2. c. bazan, m. abouali, j. e. castillo, and p. blomgren, mimetic finite difference methods in image processing, computational & applied mathematics 30 (2011), no. 3, 701–720. 3. a. boada, c. paolini, and j. e. castillo, high-order mimetic finite differences for anisotropic elliptic equations, computers & fluids 213 (2020), 104746. 4. j. e. castillo and r. d. grone, a matrix analysis approach to higher-order approximations for divergence and gradients satisfying a global conservation law, siam journal on matrix analysis and applications 25 (2003), no. 1, 128–142. 5. j. e. castillo and g. f. miranda, mimetic discretization methods, chapman and hall/crc, 2013. 6. m. a. celia, e. t. bouloutas, and r. l. zarba, a general mass-conservative numerical solution for the unsaturated flow equation, water resources research 26 (1990), no. 7, 1483–1496. richards equation and mimetic operators 411 7. r. cockett, simulation of unsaturated flow using richards equation, department of earth and ocean science, university of british columbia (preprint, https://row1.ca/pdf/richards-equation-simulation.pdf). 8. j. corbino and j. e. castillo, high-order mimetic finite-difference operators satisfying the extended gauss divergence theorem, journal of computational and applied mathematics 364 (2020), 112326. 9. l. j. córdova, o. rojas, b. otero, and j. e. castillo, compact finite difference modeling of 2-d acoustic wave propagation, journal of computational and applied mathematics 295 (2016), 83–91. 10. j. de la puente, m. ferrer, m. hanzich, j. e. castillo, and j. m. cela, mimetic seismic wave modeling including topography on deformed staggered grids, geophysics 79 (2014), no. 3, t125–t141. 11. o. rojas, b. otero, j. e. castillo, and s. m. day, low dispersive modeling of rayleigh waves on partly staggered grids, computational geosciences 18 (2014), no. 1, 29–43. corresponding author, computational science research center, san diego state university, 5500 campanile dr, san diego, 92182 e-mail address: aboadavelazco@sdsu.edu computational science research center, san diego state university, 5500 campanile dr, san diego, 92182 e-mail address: jcorbino@sdsu.edu computational science research center, san diego state university, 5500 campanile dr, san diego, 92182 e-mail address: jcastillo@sdsu.edu 1. introduction 2. richards equation 3. mimetic operators 3.1. staggered grids 3.2. one-dimensional mimetic operators 3.3. compact operators 4. temporal discretization 5. test case 6. conclusions and future work references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 3, number 1, march 2022, pp.50-59 https://doi.org/10.5206/mase/14496 dynamics of a discrete predator-prey system with fear effect and density dependent birth rate of the prey species debasis mukherjee abstract. this paper analyses a discrete predator-prey system with fear effect and density dependent birth rate of the prey species. the fixed points of the system are determined and their stability is examined. the criterion for neimark-sacker bifurcation and flip bifurcation is developed. the chaotic orbit at an unstable fixed point can be stabilized by applying the state feedback control method. numerically, we illustrate our analytical findings and observe the complex behaviour of the system that leads to stable state to chaotic one. 1. introduction in recent years, it is observed that the predator-prey interaction is not only governed by direct killing of prey by the predator, but also the indirect effect such as fear caused by the predator. the fear factor influences the birth rate of prey [1]. based on the fact of fear effect on the prey’s growth rate, several research works are explored [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. in predator-prey interaction, it is commonly assumed that the birth rate of the prey species is constant. but in a real ecological system, a birth rate of the prey species is dependent on the density of prey. in [11], the authors considered density dependent birth rate of the prey species and discussed the dynamical behaviour of the predator-prey system. aforesaid studies are mainly confined on continuous predator-prey models with two variables. although, discrete time models are more appropriate than the continuous system when the populations have nonoverlapping generations and virtually remain constant over a generation. from a biological point of view, a discrete time model is applied to investigate the taxonomic group of organisms and species with the passage of time. there are some biological situations where a discrete time system is applicable. for example, fish populations reproduce at specific time moments or for insect populations, for which nonoverlapping generations are occurring in real ecosystems. other examples include monocarpic plants and semelparous animals which have nonoverlapping populations and their births take place in usual breeding seasons. moreover, dynamics of discrete time predator-prey system can exhibit a richer set of patterns than those found in continuous systems [12, 13, 14, 15]. also discrete time models can exhibit chaotic dynamics [12, 13]. so chaos control becomes an interesting topic of research in discrete dynamical system. we will show chaos control by the state feedback control strategy. in [15], the authors observed flip bifurcation and neimark-sacker bifurcation in a discrete predator prey system with holling type iii functional response. santra et al. [16] analysed a discrete predatorprey model with crowley-martin functional response where prey population takes refuge. they showed the effects of refuge on the stability of the system in the discrete domain. furthermore, they obtained period doubling bifurcation and neimark-sacker bifurcation. elettreby et al. [14] addressed the complex behaviour of a discrete prey-predator model considering mixed functional response of holling received by the editors 25 november 2021; accepted 15 january 2022; published online 6 february 2022. 2020 mathematics subject classification. 39a28, 39a30, 92d25. key words and phrases. predator-prey system; fear effect; density dependent birth rate; bifurcation; chaos control. 50 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14496 dynamics of a discrete predator-prey system 51 type i and iii. din [13] investigated the complex nature and chaos prevention in a discrete model of prey-predator interaction and found period doubling and neimark-sacker bifurcation for larger range of bifurcation parameter. seno [17] remarked that the dynamics of discrete prey-predator system is consistent with continuous counterpart. agiza et al. [12] discussed the dynamics of a discrete-time prey-predator model with holling type ii response function. they derived bifurcation diagrams, phase portraits and lyapunov exponents for various system parameters. they also computed the fractional dimension of a strange attractor of the model. in this paper, we propose a discrete predator-prey system with fear effect and density dependent birth rate of the prey species. we study the existence and stability of different fixed points. after then, we identify the system parameters that give neimark-sacker bifurcation and flip bifurcation. the paper is formatted as follows. in section 2, we present a discrete model of predator-prey interaction with fear effect and the density dependent birth rate of the prey species. dynamical analysis of different fixed points is described in section 3. chaos control is shown in section 4. in section 5, the behaviour of the system is demonstrated when values of parameters are changed. a short discussion is given in section 6. 2. discrete model now, we present the following discrete time predator-prey model:  xn+1 = xn [ r (b + cxn)(1 + kyn) −α−βxn − pyn 1 + hxn ] , yn+1 = yn [ −d + pqxn 1 + hxn ] (2.1) where xn and yn represent population densities of prey and predator respectively, and r,b,c,k,α,β,p,q, h,d are positive constants. here r denotes the birth rate of the prey which is affected by the fear factor 1/(1 + kyn) where k is the level of fear factor. the birth rate r is modified by the density of prey in the form of beverton-holt function [18] as r/(b + cxn) where, b and c are positive parameters. α is the natural mortality rate of prey. β represents the intraspecific competition among the prey species. p,h,q,d represent consumption rate, handling time, q conversion efficiency and the death rate of the predator respectively. 3. fixed points and their nature in this section, we determine the fixed points and their dynamics. evidently, system (2.1) has three fixed points e0 = (0, 0),e1 = (x̄, 0) and e ∗ = (x∗,y∗) where x̄ = −{c(1 + α) + βb} + √ {c(1 + α) + βb}2 − 4βc{b(1 + α) −r} 2βc , and   x∗ = 1 + d pq −h(1 + d) , y∗ = −d + √ d2 − 4pk(b + cx∗)(1 + hx∗){(b + cx∗)(α + βx∗ + 1) −r} 2p(b + cx∗) (3.1) where d = (b + cx∗){p + k(1 + hx∗)(α + βx∗ + 1)}. now e1 is feasible if b(1 + α) < r; e∗ is feasible if pq > h(1 + d) and (b + cx∗)(α + βx∗ + 1) < r. 52 d. mukherjee to determine the nature of the fixed points, we compute the jacobian matrix at each fixed point. the jacobian matrix at an arbitrary fixed point (x,y) is given by j(x,y) =   br (b+cx)2(1+ky) −α− 2βx− py (1+hx)2 −x{ rk (b+cx)(1+ky)2 + p 1+hx } pqy (1+hx)2 −d + pqx 1+hx   proposition 1. the fixed point e0 = (0, 0) of system (2.1) is stable if b(α − 1) < r < b(α + 1) and 0 < d < 1. proof. the characteristic equation at e0 is∣∣∣∣ rb −α−λ 00 −d−λ ∣∣∣∣ = 0. thus the eigenvalues are λ1 = r b −α and λ2 = −d. then the fixed point e0 is locally asymptotically stable if |λi| < 1, i = 1, 2. now |λ1| = ∣∣r b −α ∣∣ < 1 then α− 1 < r b < α + 1. also, |λ2| = |−d| < 1 then −1 < d < 1. as d is the death rate of predator, this implies that 0 < d < 1. this completes the proof. proposition 2. assume that b(1 + α) < r holds. then, the fixed point e1 = (x̄, 0) of system (2.1) is stable if the following inequalities are fulfilled: α + 2βx̄− 1 < br (b + cx̄)2 < α + 2βx̄ + 1 and d− 1 < pqx̄ 1 + hx̄ < d + 1. proof. the characteristic equation at e1 is∣∣∣∣∣∣∣ br (b+cx̄)2 −α− 2βx̄−λ −x̄( rk b+cx̄ + p 1+hx̄ ) 0 −d + pqx̄ 1+hx̄ −λ ∣∣∣∣∣∣∣ = 0. hence, the eigenvalues are λ1 = br (b + cx̄)2 −α− 2βx̄, λ2 = −d + pqx̄ 1 + hx̄ . the fixed point e1 is locally stable if |λi| < 1, i = 1, 2. now |λ1| < 1 is equivalent to α + 2βx̄− 1 < br (b + cx̄)2 < α + 2βx̄ + 1 and |λ2| < 1 is equivalent to d− 1 < pqx̄ 1 + hx̄ < d + 1. this completes the proof. we remark that the above stability conditions imply that the predator goes to extinction while prey is there. proposition 3. assume that pq > h(1 + d) and (b + cx∗)(α + βx∗ + 1) < r hold. then, the fixed point e∗ = (x∗,y∗) of system (2.1) is stable if the following inequalities are fulfilled: pqx∗y∗ (1 + hx∗)2 [ rk (b + cx∗)(1 + ky∗)2 + p 1 + hx∗ ] + br (b + cx∗)2(1 + ky∗) < 1 +α+ 2βx∗ + py∗ (1 + hx∗)2 , (3.2) α+2βx∗+ py∗ (1 + hx∗)2 − br (b + cx∗)2(1 + ky∗) < 1+ pqx∗y∗ 2(1 + hx∗)2 [ rk (b + cx∗)(1 + ky∗)2 + p 1 + hx∗ ] . (3.3) dynamics of a discrete predator-prey system 53 proof. the characteristic equation at e∗ is∣∣∣∣∣∣∣ br (b+cx∗)2(1+ky∗) −α− 2βx∗ − py ∗ (1+hx∗)2 −λ −x∗ [ rk (b+cx∗)(1+ky∗)2 + p 1+hx∗ ] pqy∗ (1+hx∗)2 1 −λ ∣∣∣∣∣∣∣ = 0, which is now written in the form λ2 −ηλ + γ = 0, where η = br (b + cx∗)2(1 + ky∗) −α− 2βx∗ − py∗ (1 + hx∗)2 + 1, and γ = br (b + cx∗)2(1 + ky∗) −α− 2βx∗ − py∗ (1 + hx∗)2 + pqx∗y∗ (1 + hx∗)2 [ rk (b + cx∗)(1 + ky∗)2 + p 1 + hx∗ ] . for stability of e∗, we use jury criterion which is given by |η| < 1 + γ < 2. this condition has two parts, namely (i) γ < 1, and (ii) −1 − γ < η < 1 + γ. by the formula for γ given above, part (i) is precisely (3.2). for part (ii), the left inequality is nothing by (3.3), while the right inequality reduces to 0 < pqx∗y∗ (1 + hx∗)2 [ rk (b + cx∗)(1 + ky∗)2 + p 1 + hx∗ ] which is always true when the interior fixed point exists. from the above analysis, we infer that e∗ is stable under the conditions of the theorem, and the proof is completed. 3.1. bifurcation around the interior fixed point. in discrete context, the neimark-sacker bifurcation is the counterpart of the hopf bifurcation that takes place in continuous systems. it was explored by neimark [19] and alone by sacker [20]. hopf bifurcation generates limit cycles in the phase plane in the continuous models. alternately, neimark-sacker bifurcation produces dynamically invariant cycles. subsequently, we may get isolated periodic orbits as well as trajectories that cover the invariant circle densely. biologically, neimark-sacker bifurcation implies that all the populations can oscillate around some mean values. flip bifurcation is another type of bifurcation which is also recognized as period doubling bifurcation and it occurs when a small changes in bifurcation parameters give rise to a new system that bifurcate twice the period as the original system. this bifurcation indicates the loss of stability of a periodic orbit. system (2.1) has at most one unique fixed point e∗, hence the system does not admit fold bifurcation. so we are interested in examining the neimark-sacker bifurcation and flip bifurcation in the sequel. proposition 4. system (2.1) admits neimarck-sacker bifurcation at e∗ if the following conditions are satisfied: br (b + cx∗)2(1 + ky∗) + pqx∗y∗ (1 + hx∗)2 [ rk (b + cx∗)(1 + ky∗)2 + p 1 + hx∗ ] = α+ 2βx∗ + py∗ (1 + hx∗)2 + 1, (3.4) and pqx∗y∗ (1 + hx∗)2 [ rk (b + cx∗)(1 + ky∗)2 + p 1 + hx∗ ] < 4. (3.5) proof.. if the jacobian matrix j(e∗) has two complex conjugate eigenvalues with modulus 1, neimark-sacker bifurcation appears [21]. this requires that det(j(e∗)) = γ = 1 and −2 < tr(j(e∗)) = η < 2. replacing η and γ (see the proof of proposition 3), the first condition (γ = 1) is precisely (3.4); and the second condition on η is (3.5). this completes the proof. 54 d. mukherjee proposition 5. system (2.1) admits a flip bifurcation at e∗ if the following conditions are satisfied: 2 [ 1 + br (b + cx∗)2(1 + ky∗) ] + pqx∗y∗ (1 + hx∗)2 [ rk (b + cx∗)(1 + ky∗)2 + p 1 + hx∗ ] = 2 [ α + 2βx∗ + py∗ (1 + hx∗)2 ] . (3.6) proof. system (2.1) admits flip bifurcatiopn when a single eigenvalue is −1. thus the condition for flip bifurcation can be written in the form 1 +η +γ = 0 where γ and η are as in the proof of proposition 3. this condition is precisely (3.6), and hence completes the proof. 4. chaos control chaos control is a technique of stabilization by means of small perturbation which are used to unstable periodic orbits for a given system. sometimes bifurcation and chaotic behaviour are really undesirable phenomena in discrete dynamical systems, because there may be an extinction of population due to chaos. so controlling chaos is an important issue. there are different methods for controlling chaos, e.g., feedback control strategy, hybrid control technique and pole-placement method. by applying these methods, one can retard or remove the chaotic behaviour due to appearance of bifurcation in the dynamical systems and rebuilt the stability of the system. in this section, we use mainly the state feedback control technique [13] to stabilize a chaotic orbit at an unstable fixed point of system (2.1). consider the following controlled system related to (2.1):  xn+1 = xn [ r (b + cxn)(1 + kyn) −α−βxn − pyn 1 + hxn ] −u(xn,yn), yn+1 = yn [ −d + pqxn 1 + hxn ] (4.1) where u(xn,yn) = c1(xn − x∗) + c2(yn − y∗) is a feedback controlling force with c1 and c2 being the feedback gains and (x∗,y∗) being the unique fixed point of system (2.1). the jacobian matrix of system (4.1) evaluated at (x∗,y∗) is given by j(x∗,y∗) = ( m11 − c1 m12 − c2 m21 m22 ) where m11 = br (b + cx∗)2(1 + ky∗) −α− 2βx∗ − py∗ (1 + hx∗)2 , m12 = −x∗( rk (b + cx∗)(1 + ky∗)2 + p 1 + hx∗ ), m21 = pqy∗ (1 + hx∗)2 , m22 = 1. the characteristic equation of the variational matrix j(x∗,y∗) is λ2 − (m11 + m22 − c1)λ + m22(m11 − c1) −m21(m12 − c2) = 0 (4.2) suppose λ1 and λ2 are the roots of equation (4.2), then we have λ1 + λ2 = m11 + m22 − c1, λ1λ2 = m22(m11 − c1) −m21(m12 − c2). (4.3) the lines of marginal stability can be derived by the equations λ1 = ±1 and λ1λ2 = 1. these restrictions ensure that the eigenvalues λ1 and λ2 have moduli equal to 1. dynamics of a discrete predator-prey system 55 first suppose that λ1λ2 = 1. then from (4.3), we find l1 : c1m22 − c2m21 = m11m22 −m12m21 − 1. next assume that λ1 = 1. then from (4.3), we obtain l2 : c1(1 −m22) + c2m21 = m11 + m22 −m11m22 + m12m21. lastly, assume that λ1 = −1 and from. (4.3), we get l3 : c1(1 + m22) − c2m21 = m11 + m22 + 1 + m11m22 −m12m21. the stable eigenvalues lie within a triangular region bounded by the lines l1, l2 and l3 in the c1-c2 plane. 5. numerical simulation in this section, we present some numerical simulation to illustrate the usefulness of the obtained results as well as for giving direction to find desirable bifurcations and chaos of the discrete time system (2.1). in fig. 1, we select the parameter values r = 4.5,k = 1,b = 1,c = 1,p = 4,q = 1,h = 1,α = 0.1,β = 0.1. we draw the bifurcation diagram with respect to the parameter d in the interval (1.5, 2.8). as d increases, we observe a transition phase from stability to bifurcation within a limit cycle, to a periodic window and ultimately to chaos. in fig. 2, we select the parameter values r = 4.5,k = 1,b = 1,c = 1,p = 4,q = 1,h = 1,α = 0.1,β = 0.01. here β is decreased from 0.1 to 0.01 from previously chosen parameters in fig. 1. we draw the bifurcation diagram with respect to the parameter d in the interval (1.5, 2.8). as d increases, we observe a transition phase from stability to bifurcation within a limit cycle, to a periodic window and ultimately to chaos. here we observe flip bifurcation. in fig. 3, we select the parameter values r = 4.5,b = 1,c = 1,p = 1.8,q = 1,h = 1,α = 0.1,β = 0.01. we draw the bifurcation diagram with respect to the parameter k in the interval (2.1, 2.8). as k increases, we observe a transition phase from stability to bifurcation within a limit cycle, to a periodic window and ultimately to chaos. in fig. 4, we select the parameter values r = 4.5,k = 1,b = 1,c = 1,p = 4,q = 1,h = 1,d = 1,α = 0.1,β = 0.01. and the initial value is (0.1, 0.1). with the above choice of parameters, we find chaotic behaviour of the system. to avoid the chaotic dynamics, feedback gains c1 = 0.3 and c2 = −1.2. are chosen. the chaotic orbit is stabilized at the fixed point (1, 0.306246). 6. discussion in this article, we have studied the qualitative behaviour of a discrete predator-prey model with fear effect and density dependent birth rate of the prey species. the predator functional response is taken as holling type ii. prey’s birth rate is assumed to be as beverton-holt type function [18]. we have mainly identified the system parameters that affect the dynamics of the system. we have observed two boundary fixed points and a unique interior fixed point. stability analysis of these fixed points is examined by jury technique. the criterion for neimark-sacker bifurcation and flip bifurcation are used for examining the bifurcations around the positive fixed point. it is identified that the parameter d, the death rate of predator in the system is more relevant for the appearance of flip bifurcation and neimark-sacker bifurcation whenever it is varied in some appropriate interval. we have also found neimark-sacker bifurcation by varying the parameter k. in investigating bifurcation, we have noted 56 d. mukherjee figure 1. bifurcation diagram for prey and predator populations with d for fixed values r = 4.5,k = 1,b = 1,c = 1,p = 4,q = 1,h = 1,α = 0.1 and β = 0.1. that β, the intraspecific competition among the prey species has an important role. it is checked that smaller values of β may result flip bifurcation while larger values for β may result for neimark-sacker bifurcation. in [1], the authors studied system (2.1) with b = 1, and c = 0 and remarked that the cost of fear affect the existence of hopf bifurcation as well as the direction of hopf bifurcation in the continuous model. but in our discrete time model (2.1), we observed neimark-sacker bifurcation and chaotic behaviour of the system varying the fear factor. recently, kundu et al. [22] analysed similar type of system with b = 1,c = 0 and h = 0 without obtaining different type of bifurcations and they also observed that the system with fear effect becomes stable from chaotic dynamics by increasing fear factor which is not so in our system (see fig. 3). the conditions of proposition 1, shows that when the intrinsic growth rate of prey lies in a certain interval and the death rate of predator remains below a certain threshold value, both the populations go to extinction. if the restrictions of proposition 2 are satisfied, the predator population goes to extinction while prey population can sustain there. the stable coexistence of all the populations are possible when all the conditions of proposition 3 hold. the conditions of proposition 4 suggests that neimark-sacker bifurcation is possible for system (2.1). but it is difficult to interpret biologically these conditions. numerical simulations indicates that e∗ is stable for d < 2.2 and loses its stability at d = 2.2 and the system undergoes neimark-sacker bifurcation when the death rate d exceeds the value 2.2 (see fig. 1). we have observed that when intraspecific competition among the prey species is low and the death rate of the predator exceeds the value 2.05, system admits flip bifurcation (see fig. 2) dynamics of a discrete predator-prey system 57 figure 2. bifurcation diagram for prey and predator populations with d for fixed values r = 4.5,k = 1,b = 1,c = 1,p = 4,q = 1,h = 1,α = 0.1 and β = 0.01. follows from the proposition 5. we also note that when the fear factor k crosses the critical value 2.15, the system undergoes neimark-sacker bifurcation (see fig. 3). the chaotic nature of the system is nicely controlled by the state feedback control strategy (see fig. 4). numerical simulation exhibits that feedback control mechanism can dominate chaos to unstable fixed point strongly and ultimately stability of the system is achieved. acknowledgment. the author is grateful to the anonymous reviewer and the associate editor for their helpful comments for improving this paper. 58 d. mukherjee figure 3. bifurcation diagram for prey and predator populations with k for fixed values r = 4.5,d = 1.6,b = 1,c = 1,p = 1.8,q = 1,h = 1,α = 0.1 and β = 0.01. 0 0.5 1 1.5 2 2.5 3 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 prey p re d a to r 0 0.5 1 1.5 2 2.5 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 prey p re d a to r figure 4. phase diagram of system (2.1) for r = 4.5,k = 1,b = 1,c = 1,d = 1,p = 4,q = 1,h = 1,α = 0.1 and β = 0.01. with initial values (x0,y0) = (0.1, 0.1) in the left panel and controlled system (4) for c1 = 0.3 and c2 = −1.2 in the right panel. dynamics of a discrete predator-prey system 59 references [1] x. wang, l. y. zanette and x. zou, modelling the fear effect in predator-prey interactions, journal of mathematical biology 73 (2016), 1179-1204. 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[22] k. kundu, s. pal, s. samanta, a. sen and n. pal, impact of fear effect in a discrete-time predator-prey system, bulletin of calcutta mathematical society. 110 (2018), 3-12. department of mathematics, vivekananda college, thakurpukur, kolkata-700063, india. email address: mukherjee1961@gmail.com 1. introduction 2. discrete model 3. fixed points and their nature 3.1. bifurcation around the interior fixed point 4. chaos control 5. numerical simulation 6. discussion references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 3, september 2020, pp.224-235 https://doi.org/10.5206/mase/10794 global stability of a predator-prey model with ivlev-type functional response yinshu wu and wenzhang huang abstract. a predator-prey model with ivlev-type functional response is studied. the main purpose is to investigate the global stability of a positive (co-existence) equilibrium, whenever it exists. a recently developed approach shows that for certain classes of models, there is an implicitly defined function which plays an important rule in determining the global stability of the positive equilibrium. by performing a detailed analytic analysis we demonstrate that a crucial property of this implicitly defined function is governed by the local stability of the positive equilibrium, which enable us to show that the global and local stability of the positive equilibrium, whenever it exists, is equivalent. we believe that our approach can be extended to study the global stability of the positive equilibrium for predator-prey models with some other types of functional responses. 1. introduction the predator-prey systems described by differential equations have served as important models in the studies of dynamical interaction of predator-prey species in ecological systems. the mathematical results on the predator-prey systems provide very useful information and insight of the mechanism that govern the evolution of ecological systems. hence the research on the predator prey systems has been one of long and lasting efforts in the theoretical studies of ecology. consider the following type of predator-prey systems  u ′ = ru ( 1 − u k ) −f(u)v, v′ = v [ −d + βf(u) ] , (1.1) where u and v are respectively the prey and predator populations; ru(1 −u/k) is a logistic growth of the prey population with r the intrinsic growth rate and k the carrying capacity; f(u) is the functional response of predator species on prey population; d stands for the predator death rate and β is the predator’s conversion rate after its consumption of prey species. the functional response considered here is assumed to depend on the prey population. there are other models in literature where the functional response can be a function depending on both the prey and predator populations. many forms of the functional response f have been introduced in literature to study the predator-prey interactions under variety of environment and circumstance. one of main interests in the studies of predator-prey models is to understand the long time, or asymptotical dynamical behavior so to gain the key information for the establishment of right strategies for the control and management of the ecological systems. it is apparent that the system (1.1) has always a boundary equilibrium point ek = (k, 0), the equilibrium of predator extinction. and it is easy to see that all positive solutions of (1.1) converge to ek if the system (1.1) does not have any positive (interior/co-existence) equilibrium points, equivalently, −d+βf(u) < 0 for all u ∈ [0,k). however, when the system (1.1) possesses a positive equilibrium point received by the editors 18 june 2020; accepted 12 august 2019; published online 31 august 2020. 2000 mathematics subject classification. primary 37n28; secondary 74g20, 74g25. key words and phrases. ivlev-type functional response; positive equilibrium; local and global stability. 224 global stability of an ivlev-type 225 e∗, it has been shown that for certain types of functional response, (1.1) can exhibit a complicated dynamical behavior, such as heteroclinic orbits, multiple limit cycles, etc. [6, 8, 10, 11, 13]. on the other hand, it has also been confirmed that, for a large class of functional responses, the global dynamics of the predator-prey system (1.1) can be classified as (p1) the system (1.1) has a unique positive equilibrium e∗, and e∗ is globally stable whenever it is locally stable. (p2) the system (1.1) has a unique positive equilibrium e∗, and every positive, non-constant solution of the system (1.1) converges to a limit cycle if the positive equilibrium e∗ is unstable. the systems that share the above properties include the lotka-volterra models, the model with the general holling type functional response [3], the model with the sigmoid functional response [15]. even the model which contains a complicated toxin-determined functional response can share the above properties [1]. some models, such as the model with t holling types ii and iii functional response, actually share the following property stronger than property (p2) [2, 4, 9]: (p3) the system (1.1) has a unique positive equilibrium e∗, and all positive, non-constant solutions converge to a unique limit cycle, a global attract , whenever e∗ is unstable. it is interesting to mention that, the model with ivlev type of functional response has the property p3 [5, 12]. however, the problem on whether the system possesses property (p1) remains open. that is, it is unclear in general whether the locally stability of the positive equilibrium e∗ implies its global stability, except under certain additional condition where the liapunov function can be applied to show the global stability of the positive equilibrium e∗ [14]. in the attempt to fill the above gap, our main interest of this paper is to investigate the global stability of the positive equilibrium e∗ of the model (1.1) with an ivlev type of functional response. that is f(u) = 1 −e−αu. (1.2) by using a recently introduced approach [1], we shall establish the following main theorem about the global stability of the positive equilibrium of the system (1.1) with the ivlev-type functional response. theorem 1.1. suppose that the predator-prey model (1.1) with the ivlev-type functional response given by (1.2) has a positive equilibrium. then it is globally asymptotical stable if and only if it is locally stable. this paper is organized as follows. the local stability of the positive equilibrium is studied in section 2. in section 3 we present a detailed studies of the crucial properties of the function g and an implicitly defined function n and its inverse n−1. the property of the composition g(n−1) is used in the section 4 to complete the proof of our main theorem 1.1 of this paper. 2. preliminary consider the predator-prey model with ivlev-type functional response u ′ = ru ( 1 − u k ) − (1 −e−αu)v, v′ = [ β(1 −e−αv) −d]v. (2.1) let x = u k , y = v k , a = αk. then the system (2.1) is transformed to a system of x and y as{ x′ = rx(1 −x) − (1 −e−ax)y, y′ = [ β(1 −e−ax) −d]y. (2.2) 226 x. one and x. two it is easy to verify that (i) the system (2.2) has always two boundary equilibria e0 = (0, 0), e1 = (1, 0). (ii) the system (2.2) has a positive (interior) equilibrium e∗ = (x∗,y∗) if and only if β(1−e−a) > d. moreover, since the function 1 − e−ax is monotone increasing, the positive equilibrium e∗ is unique if it exists. in this paper, we are interested in the necessary and sufficient condition for the global stability of the positive equilibrium e∗. hence throughout the rest of paper we suppose that β(1 − e−a) > d so that the system (2.2) has a unique positive equilibrium e∗ = (x∗,y∗), where x∗ ∈ (0, 1) is a unique solution to the equation β(1 −e−ax) −d = 0 and y∗ = rx∗(1 −x∗) 1 −e−ax∗ . let us further rewrite the system (2.2) as{ x′ = f(x) [ g(x) −y ] , y′ = [ βf(x) −d ] y (2.3) where f(x) = 1 −e−ax, g(x) = rx(1 −x) 1 −e−ax . (2.4) by the calculation of the jacobin matrix associated with the positive equilibrium e∗, one is able to verify the following lemma (or see proposition 2.1 in [3]). lemma 2.1. the positive equilibrium e∗ is locally asymptotically stable if g ′(x∗) < 0, and is unstable if g′(x∗) > 0. although the local stability of the positive equilibrium e∗ can not be determined by the linearization of the system (2.3) at e∗ when g ′(x∗) = 0, in this paper we shall establish the following global stability criterion which includes the case when g′(x∗) = 0. theorem 2.2. the positive equilibrium e∗ is globally stable if and only if g ′(x∗) ≤ 0, which implies that the local and global stability of the positive equilibrium e∗ is equivalent in the light of lemma 2.1. to prove theorem 2.2 we need first to introduce a couple of functions. let h(x) = β − d f(x) , for x ∈ (0,∞), and θ = n(x) = ∫ x x∗ |h(s)|ds. (2.5) since |h(x)| > 0 for all x 6= x∗, the function n is a strictly monotone increasing function of x for x ∈ (0, 1]. moreover, it is obvious that lim x→0+ n(x) = −∞ since limx→0+ h(x) = −∞. hence n has an inverse function x = n−1(θ) defined for θ ∈ (−∞,m), where m = ∫ 1 x∗ |h(x)|dx. with the definition of n−1(θ), we are able to establish the following global stability of an ivlev-type 227 proposition 2.3. suppose that g ( n−1(−θ) ) > g ( n−1(θ) ) for all θ ∈ (0,m]. (2.6) then the system 2.3 has no closed orbit in the region ir2+ = {(x,y) : x ≥ 0,y ≥ 0}. proof. suppose in opposite that the system (2.3) has a closed orbit γ = {(x(t),y(t)) : t ∈ [0,ω]} of period ω in ir2+. then it is obvious that x(t) ≤ 1 for all t ∈ [0,ω] and the positive equilibrium e∗ = (x∗,y∗) must be an interior point of the region enclosed by γ. by using the property of the vector field associated with the system (2.3), without loss of generality, we can suppose that there is an ω1 ∈ (0,ω) such that   x(0) = x(ω1) = x(ω) = x∗, x(t) > x∗, t ∈ (0,ω1), x(t) < x∗, t ∈ (ω1,ω). (2.7) moreover, y(t) is strictly increasing for t ∈ (0,ω1) and is strictly decreasing for t ∈ (ω1,ω) (see fig. 2.1). let θ(t) = n(x(t)). then   θ′(t) = ∣∣∣h(x(t))∣∣∣f(x(t))[g(x(t))−y(t)], y′(t) = f(x(t)) [ β − d f(x(t)) ] y(t) = f(x(t))h(x(t))y(t). (2.8) notice that h(x(t)) > 0 for t ∈ (0,ω1) and h(x(t)) < 0 for t ∈ (ω1,ω). it follows that  θ′(t) y′(t) = g ( x(t) ) −y(t) y(t) , t ∈ (0,ω1), θ′(t) y′(t) = − g ( x(t) ) −y(t) y(t) , t ∈ (ω1,ω). (2.9) let y0 = y(0), y1 = (ω1). 228 x. one and x. two since y(t) is strictly monotone in the intervals [0,ω1] and [ω1,ω], respectively, there are two continuously differential functions ξ1(y) and ξ2(y) defined for y ∈ [y0,y1] such that t = ξ1(y) for t ∈ [0,ω1] if and only if y = y(t) ∈ [y0,y1], t = ξ2(y) for t ∈ [ω1,ω] if and only if y = y(t) ∈ [y0,y1]. now let θ1(y) = θ(ξ1(y)) and θ2(y) = −θ(ξ2(y)) for y ∈ (y0,y1). then by equalities (2.9) and x = n−1(θ), and following a straight forward computation, we obtain that  dθ1 dy = g(n−1(θ1)) −y y , dθ2 dy = g(n−1(−θ2)) −y y . (2.10) moreover, the definitions of θ1(y) and θ2(y) give that θ1(y0) = θ2(y0) = θ1(y1) = θ2(y1) = 0, θi(y) > 0 for y ∈ (y0,y1), i = 1, 2. (2.11) on the other hand, the assumption (2.6) implies that g(n−1(−θ)) −y y > g(n−1(θ)) −y y (2.12) for all y ∈ [y0,y1] and all θ > 0. from the equations in (2.10), and with the use of the inequality (2.12) and the comparison argument, it follows that θ2(y1) > θ1(y1), which leads to a contradiction to the equality in (2.11). � remark the above proposition 2.3 and its proof are essentially the same as the proposition 3.2 and its proof introduced in [1]. we provide a complete proof of our proposition 2.3 since the functional response in the model studied in [1] is different from the functional response in this paper. as a straight consequence of the proposition 2.3 we have the following theorem, which will be used in this paper to prove our theorem 2.2. theorem 2.4. the positive equilibrium e∗ is globally stable provided that for all θ ∈ (0,∞), g ( n−1(−θ) ) > g ( n−1(θ) ) . (2.13) proof. we shall only give a brief proof here. for more detailed proof we refer readers to the proof of theorem 2.1 in [1]. first it is easy to verify that any positive solution (x(t),y(t)) of the system (2.3) is bounded. therefore its ω-limit set ω ⊂ ir2+ exists. by poncaré-bendixson theorem ω must be one of (a) a closed orbit in ir2+; (b) a cyclic chain, i.e, a closed curve consists of equilibrium points and global orbits in a certain order; (c) an equilibrium point. the proposition 2.3 has excluded the existence of a closed orbit in ir2+. recall that, besides the positive equilibrium e∗ the system (2.3) has another two equilibria e0 and e1. the existence of e∗ implies that the stable manifold of e1 is in the positive x-axis. moreover the stable manifold of e0 is in the y-axis. it follows that ω can not be a cyclic chain that contains either e0 or e1. therefore, one must have ω = {e∗}. therefore (x(t),y(t)) converges to e∗ as t →∞. � global stability of an ivlev-type 229 3. the properties of functions g(x) and n−1(θ) from the statement of theorem 2.4 we see that the property of the composition g ( n−1 ) plays an important role in governing the global stability of the positive equilibrium e∗. hence in this section we shall explore in detail about the properties of g and n−1, which will enable us to establish the inequality (2.6). let us begin with the following lemma. lemma 3.1. let a > 0. if a ≤ 2, then g′(x) < 0 for all x > 0. that is, the function g(x) is strictly decreasing for x > 0. proof. by the definition of g(x) [see (2.4)] we have g′(x) = r [ (1 − 2x)(1 −e−ax) −ax(1 −x)e−ax ] (1 −e−ax)2 = re−axg1(x) (1 −e−ax)2 , (3.1) where g1(x) = (1 − 2x)(eax − 1) −ax(1 −x). the assumption of a ≤ 2 implies that for all x > 0, g′1(x) = −2(eax − 1) + a(1 − 2x)eax −a(1 − 2x) = −(2 −a + 2ax)(eax − 1) < 0 hence g1(x) is strictly decreasing for x > 0. combining the fact that g1(0) = 0 we conclude that g1(x) < 0, and so that, by the equality (3.1), g′(x) < 0 for all x > 0. � next we turn to investigate the property of the function g(x) for a > 2. lemma 3.2. define the function η(x) = rx 1 −e−ax for x > 0. then for all x > 0, η′(x) > 0, η′′(x) > 0, η′′′(x) < 0. proof. let ζ(x) = x 1 −e−x . then η(x) = r a ζ(ax). hence to confirm lemme 3.2 it is sufficient to show that ζ′(x) > 0, ζ′′(x) > 0 and ζ′′′(x) < 0 for x > 0. first we have ζ′(x) = 1 −e−x −xe−x (1 −e−x)2 = ex − (1 + x) ex(1 −e−x)2 > 0 (3.2) for all x > 0 since ex − (1 + x) > 0 for x > 0. noticing that ζ′(x) = ex − (1 + x) ex(1 −e−x)2 = ex − (1 + x) ex + e−x − 2 , a straight forward calculation yields that ζ′′(x) = (ex − 1)(ex + e−x − 2) − (ex − 1 −x)(ex −e−x) (ex + e−x − 2)2 = ζ1(x) (ex + e−x − 2)2 (3.3) 230 x. one and x. two with ζ1(x) = 4 − 2(ex + e−x) + x(ex −e−x). expanding ζ1(x) in the power series we obtain ζ1(x) = 4 − 4 ∞∑ k=0 x2k (2k)! + 2x ∞∑ k=1 x2k−1 (2k − 1)! = −4 ∞∑ k=1 x2k (2k)! + 2 ∞∑ k=1 x2k (2k − 1)! = 2 ∞∑ k=1 [ 1 (2k − 1)! − 2 (2k)! ] x2k = 2 ∞∑ k=2 (2k − 2) (2k)! x2k > 0 for all x > 0. (3.4) so that ζ′′(x) = ζ1(x) (ex + e−x − 2)2 > 0 for x > 0. (3.5) next, we have ζ′′′(x) = ζ2(x) (ex + e−x − 2)3 , (3.6) where ζ2(x) = ζ ′ 1(x)(e x + e−x − 2) − 2ζ1(x)(ex −e−x) = [ − (ex −e−x) + x(ex + e−x) ] (ex + e−x − 2) −2 [ 4 − 2(ex + e−x) + x(ex −e−x) ] (ex −e−x) = 3(e2x −e−2x) + 6x− [ 6(ex −e−x) + x(e2x + e−2x) + 2x(ex + e−x) ] = 6 ∞∑ k=0 (2x)2k+1 (2k + 1)! + 6x− − [ 12 ∞∑ k=0 x2k+1 (2k + 1)! + 2x ∞∑ k=0 (2x)2k (2k)! + 4x ∞∑ k=0 x2k (2k)! ] = 6x + ∞∑ k=0 6 · 22k+1 (2k + 1)! x2k+1 − ∞∑ k=0 [ 12 (2k + 1)! + 22k+1 + 4 (2k)! ] x2k+1 = ∞∑ k=0 a(k)x2k+1, (3.7) global stability of an ivlev-type 231 where a(0) = 6 + 12 − [12 + 6] = 0, a(1) = 6 · 23 3! − [ 12 3! + 23 + 4 2! ] = 1 3! ( 48 − [12 + 3(23 + 4)] ) = 0, a(2) = 6 · 25 5! − [ 12 5! + 25 + 4 4! ] = 1 5! ( 6 · 25 − [12 + 5(25 + 4)] ) = 0. (3.8) for k ≥ 3, one has a(k) = 1 (2k + 1)! ( 6 · 22k+1 − [ 12 + (2k + 1)(22k+1 + 4) ]) < 1 (2k + 1)! ( 6 · 22k+1 − (2k + 1)22k+1 ) < 0. (3.9) from the equalities (3.6), (3.8) and the inequality (3.9) it follows that ζ2(x) < 0, so that ζ ′′′(x) < 0 for all x > 0 by (3.6). � corollary 3.3. g′′′(x) < 0 for all x ∈ (0, 1], so that g′′(x) is strictly decreasing. proof. by the definition of g and η one has g(x) = rx 1 −e−ax [1 −x] = η(x) [ 1 −x ] . it follows that g′(x) = η′(x)[1 −x] −η(x), g′′(x) = η′′(x)[1 −x] − 2η′(x), g′′′(x) = η′′′(x)[1 −x] − 3η′′(x). (3.10) since η′′′(x) < 0 and η′′(x) > 0 for all x > 0 by lemma 3.2, the last equality in (3.10) yields that g′′′(x) < 0 for all x ∈ (0, 1]. � it is easy to verify that g(0+) = lim x→0+ η(x) = lim x→0 rx 1 −e−ax = r a . moreover, a direct computation yields that lim x→0+ η′(x) = lim x→0+ r eax − (1 + ax) eax(1 −e−ax)2 = r 2 . it follows that g′(0+) = lim x→0 g′(x) = lim x→0 ( η′(x)[1 −x] −η(x) ) = r 2 − r a . (3.11) lemma 3.4. suppose a > 2. then there is an x0 ∈ (0, 1) such that g′(x0) = 0 and g′(x) > 0 for x ∈ (0,x0), g′(x) < 0 for x ∈ (x0, 1]. 232 x. one and x. two proof. suppose a > 2. then (3.11) implies that g′(x) > 0 for small x ≥ 0. define the set s by s = {x ∈ (0, 1] : g′(s) > 0, s ∈ [0,x]} . then it is obvious that (0,�] ⊂ s for a small number � > 0. moreover, g′(1) = −η(1) < 0 implies that there is a small number δ > 0 such that x /∈ s for x ≥ 1 − δ. it therefore follows that the number x0 = sup{x ∈ (0, 1] : g′(s) > 0, s ∈ [0,x]} is well defined and 0 < � < x0 ≤ 1 − δ < 1. by the definition of x0 it is apparent that g′(x) > 0 for x ∈ (0,x0) and g′(x0) = 0. consequently one also deduces that g′′(x0) ≤ 0. this yields that since g′′(x) < 0 for all x > x0, for g ′′(x) is strictly decreasing by corollary 3.3. therefore, g′(x) is strictly decreasing for x ∈ (x0, 1] and so that g′(x) < g′(x0) = 0 for x ∈ (x0, 1]. � new let us turn to study the property of the function n−1(θ), the inverse function of n defined in (2.5). by the definition of θ = n(x) and x = n−1(θ), we have θ = n(n−1(θ)) = ∫ n−1(θ) x∗ |h(s)|ds. (3.12) lemma 3.5. for each θ > 0, n−1(−θ) < x∗ < n−1(θ) and x∗ −n−1(−θ) < n−1(θ) −x∗. proof. note that n(x) is strictly increasing and so is for its inverse n−1(θ). the definition of n(x) yields that n(x∗) = 0, i.e. x∗ = n −1(0). it follows that n−1(−θ) < n−1(0) = x∗ < n−1(θ) for θ > 0. moreover, we have h(x∗) = 0 and h′(x) = d dx [ β − d 1 −e−ax ] = ade−ax (1 −e−ax)2 > 0, h′′(x) = − a2de−ax [ (1 −e−ax)2 + 2e−ax(1 −e−ax) ] (1 −e−ax)4 < 0. hence h(x) is monotone increasing and concave downward. notice that θ > 0 is the area of the region between the x-axis and the graph of h(x) on the interval [x∗,n −1(θ)] and −θ is the area of the (negative) region between the graph of h(x) and the x-axis on the interval [n−1(−θ),x∗]. by the downward concavity of h(x) one therefore easily concludes that for each θ > 0, x∗−n−1(−θ) < n−1(θ)−x∗ (see fig. 3.1). � global stability of an ivlev-type 233 4. proof of theorem 2.2 we shall prove theorem 2.2 by the use of theorem 2.4. theorem 4.1. suppose that g′(x∗) ≤ 0. then for all θ ∈ (0,m], g ( n−1(−θ) ) > g ( n−1(θ) ) . proof. let us first suppose a ≤ 2. note that n−1(−θ) < n−1(θ) for θ ∈ (0,m] since n−1(θ) is strictly monotone increasing. it immediately follows that g ( n−1(−θ) ) > g ( n−1(θ) ) for θ ∈ (0,m] because g(x) is strictly decreasing by lemma 3.1. [remark: the global stability of e∗ under the condition a ≤ 2 also was proved in [14] by the liapunov direct method.] next we suppose that a > 2. for θ ∈ (0,m] let x− = n−1(−θ) and x+ = n−1(θ). then by lemma 3.5 we have x− < x∗ = n −1(0) < x+ and x∗ −x− < x+ −x∗. hence there is a number x̄+ ∈ (x∗,x+) such that x∗ −x− = x̄+ −x∗. (4.1) now we have g ( n−1(−θ) ) = g(x−) = g(x∗) + ∫ x− x∗ g′(x)dx = g(x∗) + ∫ x− x∗ g′(x∗)dx + ∫ x− x∗ [g′(x) −g′(x∗)]dx. (4.2) by the equality (4.1) and the assumption of g′(x∗) ≤ 0 we obtain∫ x− x∗ g′(x∗)dx = −g′(x∗)(x∗ −x−) ≥ g′(x∗)(x∗ −x−) = ∫ x̄+ x∗ g′(x∗)dx. (4.3) moreover, recall by corollary 3.3 that g′′(x) is strictly decreasing. thus we arrive at 234 x. one and x. two ∫ x− x∗ [g′(x) −g′(x∗)]dx = − ∫ x− x∗ [∫ x∗ x g′′(s)ds ] dx = ∫ x∗ x− [∫ x∗ x g′′(s)ds ] dx > g′′(x∗) ∫ x∗ x− ∫ x∗ x dsdx = g′′(x∗) ∫ x̄+ x∗ ∫ x x∗ dsdx > ∫ x̄+ x∗ [∫ x x∗ g′′(s)ds ] dx = ∫ x̄+ x∗ [g′(x) −g′(x∗)]dx. (4.4) from (4.2) (4.4) it follows that g ( n−1(−θ) ) > g(x∗) + ∫ x̄+ x∗ g′(x∗)dx + ∫ x̄+ x∗ [g′(x) −g′(x∗)]dx = g(x̄+). (4.5) the assumption g′(x∗) ≤ 0 and lemma 3.4 yeild that g(x) is decreasing for x ∈ [x∗, 1]. hence by the inequality (4.5) and the fact that x̄+ < x+ = n −1(θ) we deduce that g ( n−1(−θ) ) > g(x̄+) > g( ( n−1(θ) ) . � proof of theorem 1.1 is apparent that theorem 2.2, and hence the main theorem 1.1, is a direct consequence of lemma 2.1, theorems 2.4 and 4.1. references [1] c. castillo-chavez, z. feng, and w. huang, global dynamics of a plant-herbivore model with toxin-determined functional response, siam appl. math. 72(2012), 1002-1020. [2] s.h. ding, on a kind of predator-prey system, siam j. math. anal. 20 (1989),1426-1435. [3] w. ding and w. huang, global dynamics of a predator-prey model with general holling type functional responses, j. dyn. diff. eqns. 32 (2020), 965-978. 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[10] g. seo and g.s.k. wolkowicz, existence of multiple limit cycles in a predator-prey model with arctan (ax) as functional response, comm. math. anal. 18(2015), 64-68. [11] g. seo and g.s.k. wolkowicz, sensitive of the dynamics of the general rosenzweig-macarthur model to the mathematical form of the functional response: a bifurcation theory approach, j. math. biol. 76 (2018), 1873-1906. [12] j. sugie, two-parameter bifurcation in a predator-prey system of ivlev type, j. math. anal. appl. 217 (1998), 349-371. global stability of an ivlev-type 235 [13] g.s.k. wolkowicz, bifurcation analysis of a predator-prey system involving group defense, siam appl. math. 48 (1988), 592-606. [14] x. wang and h. ma, a lyapunov function and global stability for a class of predator-prey models discrete dynamics in nature and society 2012, 4(2017), doi:10.1155/2012/21875. [15] y. wu and w. huang, global stability of the predator-prey model with a sigmoid functional response, disc. cont. dyn. syst. b 3(2020), 1159-1167. corresponding author, department of mathematics, alabama a&m universitynormal, al, usa 35762 current address: same e-mail address: yinshu.wu@aamu.edu department of mathematical sciences, university of alabama in huntsville, huntsville, al, usa 35899 e-mail address: huangw@uah.edu mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 2, number 3, september 2021, pp.149-160 https://doi.org/10.5206/mase/14031 a robust phenomenological approach to investigate covid-19 data for france quentin griette, jacques demongeot, and pierre magal dedicated to the memory of professor mayan mimura. abstract. we provide a new method to analyze the covid-19 cumulative reported case data based on a two-step process: first, we regularize the data by using a phenomenological model which takes into account the endemic or epidemic nature of the time period, then we use a mathematical model which reproduces the epidemic exactly. this allows us to derive new information on the epidemic parameters and to compute the effective basic reproductive ratio on a daily basis. our method has the advantage of identifying robust trends in the number of new infectious cases and produces an extremely smooth reconstruction of the epidemic. the number of parameters required by the method is parsimonious: for the french epidemic between february 2020 and january 2021 we use only 11 parameters in total. 1. introduction modeling endemic and epidemic phases of the infectious diseases such as smallpox which by the 16th century had become a predominant cause of mortality in europe until the vaccination by e. jenner in 1796, and present covid-19 pandemic outbreak has always been a means of describing and predicting disease. d. bernoulli proposed in 1760 a differential model [2] taking into account the virulence of the infectious agent and the mortality of the host, which showed a logistic formula [2, p.13] of the same type as the logistic equation by verhulst [9]. the succession of an epidemic phase followed by an endemic phase had been introduced by bernoulli and for example appears clearly in the figures 9 and 10 in [5]. the aim of this article is to propose a new approach to compare epidemic models with data from reported cumulative cases. here we propose a phenomenological model to fit the observed data of cumulative infectious cases of covid-19 that describe the successive epidemic phases and endemic intermediate phases. this type of problem dates back to the 1970s with the work of london and york [8]. more recently, chowell et al. [3] have proposed a specific function to model the temporal transmission speeds τ(t). in the context of covid-19, a two-phase model has been proposed by liu et al. [7] to describe the south korean data with an epidemic phase followed by an endemic phase. in this article, we use a phenomenological model to fit the data (see figure 1). the phenomenological model is used in the modeling process between the data and the epidemic models. the difficulty here is to propose a simple phenomenological model (with a limited number of parameters) that would give a meaningful result for the time-dependent transmission rates τ(t). many models could potentially received by the editors 31 may 2021; accepted 7 july 2021; published online 10 july 2021. 1991 mathematics subject classification. 92d30, 34c60, 65d10, 93c15. key words and phrases. covid-19 outbreak, epidemic modeling, new case data regularization, endemic phase fitting, epidemic reproduction number calculation. q. griette was supported by anr grant mpcuii. p. magal was supported by anr grant mpcuii. 149 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14031 150 q. griette, j. demongeot, and p. magal be used as phenomenological models to represent the data (ex. cubic spline and others). the major difficulty here is to provide a model that gives a good description of the tendency for the data. it has been observed in our previous work that it is difficult to choose between the possible phenomenological models (see figures 12-14 in [4]). the phenomenological model can also be viewed as a regularization of data that should not fluctuate too much to keep the essential information. an advantage in our phenomenological model is the limited number of parameters (5 parameters during each epidemic phase and 2 parameters during each endemic phase). the last advantage of our approach is that once the phenomenological model has been chosen, we can compute some explicit formula for the transmission rate and derive some estimations for the other parameters. data phenomenological model (limited number of parameters) (good description of the tendency) epidemic model (mechanistic) parameters of estimations basic reproduction number forecasting etc . . . figure 1. we can apply statistical methods to estimate the parameters of the proposed phenomenological model and derive their average values with some confidence intervals. the phenomenological model is used at the first step of the modelling process, providing regularized data to the epidemic model and allowing the identification of its parameters. 2. material and methods 2.1. phenomenological model. in this article, the phenomenological model is compared with the cumulative reported case data taken from who [1]. the phenomenological model deals with data series of new infectious cases decomposed into two types of successive phases, 1) endemic phases, followed by 2) epidemic ones. endemic phase. during the endemic phase, the dynamics of new cases appear to fluctuate around an average value independently of the number of cases. therefore the average cumulative number of cases is given by cr(t) = n0 + (t− t0) ×a, for t ∈ [t0, t1], (2.1) where t0 denotes the beginning of the endemic phase. a is the average value of cr(t0) and n0 the average value of the daily number of new cases. in other words, we assume that the average daily number of new cases is constant. therefore the daily number of new cases is given by cr′(t) = a. (2.2) epidemic phase. in the epidemic phase, the new cases contribute to produce secondary cases. therefore the daily number of new cases is no longer constant but varies with time as follows cr(t) = nbase + n(t), for t ∈ [t0, t1], (2.3) phenomenological model for covid-19 151 where n(t) = eχ(t−t0)n0[ 1 + nθ0 nθ∞ ( eχθ(t−t0) − 1 )]1/θ . (2.4) in other words, the daily number of new cases follows the bernoulli-verhulst [2, 9] equation. namely, by setting n(t) = cr(t) −nbase we obtain n′(t) = χn(t) [ 1 − ( n(t) n∞ )θ] (2.5) with the initial value n(t0) = n0. (2.6) in the model nbase + n0 corresponds to the value cr(t0) of the cumulative number of cases at time t = t0. the parameter n∞ + nbase is the maximal increase of the cumulative reported case after the time t = t0. χ > 0 is a malthusian growth parameter, and θ regulates the speed at which the cr(t) increases to n∞ + nbase. regularized model. because the formula for τ(t) involves derivatives of the phenomenological model regularizing cr(t) (see equation (2.13)), we need to connect the phenomenological models of the different phases as smoothly as possible. we let c̃r(t) be the model obtained by placing phenomenological models side by side for different phases. outside of the time window where phenomenological models are used, we consider that the function c̃r(t) is constant. we define the regularized model by using the convolution formula: cr(t) = ∫ +∞ −∞ c̃r(t−s) × 1 σ √ 2π e − s 2 2σ2 ds = (c̃r ∗g)(t), (2.7) where g(t) := 1 σ √ 2π e − t 2 2σ2 is the gaussian function with variance σ2. the parameter σ controls the trade-off between smoothness and precision: increasing σ reduces the variations in cr(t) and decreasing σ reduces the distance between cr(t) and c̃r(t). in any case the resulting function cr(t) is very smooth (as well as its derivatives) and close to the original model c̃r(t) when σ is not too large. in numerical applications, we take σ = 2 days. procedure to fit the phenomenological model to the data. to fit the model to the data, we used the regularized model (2.7) where the periods of the different phases are fixed as in table 1. we use a standard curve-fitting algorithm to find the parameters of the regularized model. in numerical applications we used the levenberg–marquardt nonlinear least-squares algorithm provided by the matlab c© function fit. our 95% confidence intervals are the ones provided as an output of this algorithm. the best-fit parameters and the corresponding confidence intervals are provided in table 1. 2.2. si epidemic model. the si epidemic model used in this work is the same as in [4]. it is summarized by the flux diagram in figure 2. 152 q. griette, j. demongeot, and p. magal (s)usceptibles (i)nfectious (r)eported (u)nreported dead or recovered asymptomatic symptomatic figure 2. schematic view showing the different compartments and transition arrows in the epidemic model. the goal of this article is to understand how to compare the si model to the reported epidemic data and therefore the model can be used to predict the future evolution of epidemic spread and to test various possible scenarios of social mitigation measures. for t ≥ t0, the si model is the following{ s′(t) = −τ(t)s(t)i(t), i′(t) = τ(t)s(t)i(t) −νi(t), (2.8) where s(t) is the number of susceptible and i(t) the number of infectious at time t. this system is supplemented by initial data s(t0) = s0 ≥ 0, i(t0) = i0 ≥ 0. (2.9) in this model, the rate of transmission τ(t) combines the number of contacts per unit of time and the probability of transmission. the transmission of the pathogen from the infectious to the susceptible individuals is described by a mass action law τ(t) s(t) i(t) (which is also the flux of new infectious). the quantity 1/ν is the average duration of the infectious period and νi(t) is the flux of recovering or dying individuals. at the end of the infectious period, we assume that a fraction f ∈ (0, 1] of the infectious individuals is reported. let cr(t) be the cumulative number of reported cases. we assume that cr(t) = cr0 + ν f ci(t), for t ≥ t0, (2.10) where ci(t) = ∫ t t0 i(σ)dσ. (2.11) assumption 2.1 (given parameters). we assume that • the number of susceptible individuals when we start to use the model s0 = 67 millions; • the average duration of infectious period 1 ν = 3 days; • the fraction of reported individuals f = 0.9; are known parameters. parameters estimated in the simulations. as described in [4] the number of infectious at time t0 is i0 = cr′(t0) ν f (2.12) phenomenological model for covid-19 153 the rate of transmission τ(t) at time t is given by τ(t) = νf ( cr′′(t) cr′(t) + ν ) νf (i0 + s0) − cr′(t) −ν (cr(t) − cr0) . (2.13) parameters estimated in the endemic phase. the initial number of infectious is given by i0 = a ν f , and the transmission rate is given by the explicit formula τ(t) = ν2f νf (i0 + s0) −a−ν(t− t0) ×a ,∀t ∈ [t0, t1]. parameters estimated in the epidemic phase: the initial number of infectious is given by i0 = χn0 [ 1 − ( n0 n∞ )θ] ν f , and the transmission rate is given by the explicit formula τ(t) = νf ( n′′(t) n′(t) + ν ) νf (i0 + s0) −n′(t) −ν (n(t) −n0) , and since (recall (2.5)) n′(t) = χn(t) [ 1 − ( n(t) n∞ )θ] and n′′(t) = χn′(t) [ 1 − (1 + θ) ( n(t) n∞ )θ] , (2.14) we obtain an explicit formula τ(t) = νf ( χ [ 1 − (1 + θ) ( n(t) n∞ )θ] + ν ) νf (i0 + s0) −χn(t) [ 1 − ( n(t) n∞ )θ] −ν (n(t) −n0) , (2.15) where n(t) is given by (2.4): n(t) = eχ(t−t0)n0[ 1 + nθ0 nθ∞ ( eχθ(t−t0) − 1 )]1/θ . by using the bernoulli-verhulst model to represent the data, the daily number of new cases is nothing but the derivative n′(t) (whenever the unit of time is one day). the daily number of new cases reaches its maximum at the turning point t = tp, and by using (2.14), we obtain n′′(tp) = 0 ⇔ n(tp) = ( 1 1 + θ )1/θ n∞. 154 q. griette, j. demongeot, and p. magal therefore by using (2.5), the maximum of the daily number of cases equals n′(tp) = χn(tp) [ 1 − ( n(tp) n∞ )θ] . by using the above formula, we obtain a new indicator for the amplitude of the epidemic. theorem 2.2. the maximal daily number of cases in the course of the epidemic phase is given by χ×n∞ ×θ × ( 1 1 + θ )1 θ +1 . (2.16) 2.3. parameter bounds. the epidemic model (2.8) with time-dependent transmission rate is consistent only insofar as the transmission rate remains positive. this gives us a criterion to judge if a set of epidemic parameters has a chance of being consistent with the observed data: since we know the parameters n0, n∞, χ and θ from the phenomenological model, the formula (2.15) allows us to compute a criterion on ν and f which decides whether a given parameter values are compatible with the observed data or not. that is to say that, a set of parameter values is compatible if the transmission rate τ(t) in (2.15) remains positive for all t ≥ t0, and it is not compatible if the sign of τ(t) in (2.15) changes for some t ≥ t0. we refer to [4, proposition 4.3] for more results. the value of the parameter ν is compatible with the model (2.15) if and only if 0 ≤ 1 ν ≤ 1 χθ , (2.17) and the value of the parameter f is compatible with the model (2.15) if and only if f ≥ n∞ −n0 i0 + s0 . (2.18) therefore, we obtain an information on the parameters ν and f, even though they are not directly identifiable (two different values of ν or f can produce exactly the same cumulative reported cases). 2.4. computation of the basic reproduction number. in order to compute the reproduction number in figure 5 we use the algorithm 2 in [4] and the day-by-day values of the phenomenological model. 3. results 3.1. phenomenological model compared to the french data. in figure 3 we present the best fit of our phenomenological model for the cumulative reported case data of covid-19 epidemic in france. the yellow regions correspond to the endemic phases and the blue regions correspond to the epidemic phases. here we consider the two epidemic waves for france, and the chosen period, as well as the parameters values for each period, are listed in table 1. in table 1 we also give 95% confidence intervals for the fitted parameters values. phenomenological model for covid-19 155 figure 3. the red curve corresponds to the phenomenological model and the black dots correspond to the cumulative number of reported cases in france. figure 4 shows the corresponding daily number of new reported case data (black dots) and the first derivative of our phenomenological model (red curve). figure 4. the red curve corresponds to the first derivative of the phenomenological model and the black dots correspond to daily number of new reported cases in france. 3.2. si epidemic model compared to the french data. some parameters of the model are known as s0 = 67 millions for france (this is questionable). some parameters of the epidemic model can not be precisely evaluated [4]. 156 q. griette, j. demongeot, and p. magal period parameters value method 95% confidence interval period 1: endemic phase jan 03 feb 27 n0 = −4.368 a = 1.099 × 10−1 computed fitted a ∈ [−8.582 × 101, 8.604 × 101] period 2: epidemic phase feb 27 may 17 nbase = 0 n0 = 1.675 n∞ = 1.445 × 105 χ = 1.263 θ = 6.315 × 10−2 fixed fitted fitted fitted fitted n0 ∈ [−3.807 × 101, 4.142 × 101] n∞ ∈ [1.367 × 105, 1.523 × 105] χ ∈ [−1.171 × 101, 1.424 × 101] θ ∈ [−6.086 × 10−1, 7.349 × 10−1] period 3: endemic phase may 17 jul 05 n0 = 1.405 × 105 a = 3.11 × 102 computed computed period 4: epidemic phase jul 05 nov 18 nbase = 1.403 × 105 n0 = 1.517 × 104 n∞ = 1.953 × 106 χ = 3.671 × 10−2 θ = 7.679 fitted fitted fitted fitted fitted nbase ∈ [1.367 × 105, 1.439 × 105] n0 ∈ [1.427 × 104, 1.607 × 104] n∞ ∈ [1.92 × 106, 1.986 × 106] χ ∈ [3.62 × 10−2, 3.722 × 10−2] θ ∈ [6.256, 9.102] period 5: endemic phase nov 18 jan 04 n0 = 4.45 × 10−84 a = 1.099 × 10−1 computed fitted a ∈ [1.222 × 104, 1.265 × 104] table 1. fitted parameters and computed parameters for the whole epidemic going from january 03 2020 to january 04 2021. result by using (2.17) we obtain the following conditions for the average duration of infectious period • 0 < 1 ν ≤ 1/(χθ) = 12.5 days during the first epidemic wave; • 0 < 1 ν ≤ 1/(χθ) = 3.5 days during the second epidemic wave. we obtain no constraint for the fraction f ∈ (0, 1] of reported new cases (between 0 and 1 for france). moreover by using the formula (2.16) we deduce that the maximal daily number of cases is • 4110 during the first epidemic wave; • 47875 during the second epidemic wave. importantly, by combining the phenomenological model from section 3.1 and the epidemiological model from section 2.2, we can reconstruct the time-dependent transmission rate given by (2.13) and the corresponding time-dependent basic reproduction number r0(t) = τ(t)s(t)/ν (sometimes called “effective basic reproductive ratio”). the obtained basic reproduction number is presented in figure 5. we observe that r0(t) is decreasing during each epidemic wave, except at the very end where it becomes increasing. this is not necessarily surprising since the lockdown becomes less strictly respected towards the end. during the endemic phases, the r0(t) becomes effectively equal to one, except again near the end. the variations observed close to the transition between two phases may be partially due to the smoothing method, which has an impact on the size of the “bumps”. however, they remain very limited in number and size. phenomenological model for covid-19 157 figure 5. in this figure we plot the time dependent basic reproduction number r0(t) := τ(t)s(t)/ν. we fix the average length of the asymptotic infectious period to 3 days. notice that, contrary to figure 3 and 4, we do not plot to first endemic phase because the basic reproduction number is meaningless before the first wave. 4. discussion in our paper, we use a phenomenological model to reduce the number of parameters necessary for summarizing observed data without loss of pertinent information. the process of reduction consists of three stages: qualitative or quantitative detection of the boundaries between the different phases of the dynamics (here endemic and epidemic phases), choice of a reduction model (among different possible approaches: logistic, regression polynomials, splines, autoregressive time series, etc.) and smoothing of the derivatives at the boundary points corresponding to the breaks in the model. in figure 3, we have a very good agreement between the data and the phenomenological model, for both the original curve and its derivative. the relative error in figure 3 is of order 10−2, which means that the error is at most of the order of 100 000 individuals. in figure 4 the red curve also gives a good tendency of the black dots corresponding to raw data. in figure 5, the phenomenological models are necessary to derive a significant basic reproduction number. otherwise the resulting r0(t) is not interpretable and even not computable after sometime. similar results were obtained in figures 12-14 in [4]. the method to compute r0 can also be applied directly to the original data. we did not show the result here because the noise in the data is amplified by the method and the results are not usable. this shows that it is important to use the phenomenological model to provide a good regularization of the data. in figure 5, the major difficulty is to know how to make the transition from an epidemic phase to an endemic phase and vice versa. this is a non-trivial problem that is solved by our regularization approach (using a convolution with a gaussian). as we can see in figure 5, the number of oscillations is very limited between two phases. without regularization, there is a sharp corner at the transition between two phases which leads to infinite values in τ(t). the choice of the convolution with a gaussian kernel for the regularization method is the result of an experimental process. we tried several different regularization methods, including a smooth explicit interpolation function and hermite polynomials. eventually, the convolution with a gaussian kernel gives the best results. 158 q. griette, j. demongeot, and p. magal to minimize the variations of the curve of r0(t), the choice of the transition dates between two phases is critical. in figure 5 we choose the transition dates so that the derivatives of the phenomenological model do not oscillate too much. other choices lead to higher variations or increase the number of oscillations. finally, the qualitative shape of the curve presented in figure 5 is very robust to changes in the epidemic parameters, even though the quantitative values of r0(t) are different for other values of the parameters ν and f. in figure 5, we observe that the quantitative value of the r0(t) during the first part of the second epidemic wave (second blue region) is almost constant and equals 1.11. this value is significantly lower than the one observed at the beginning of the first epidemic wave (first blue region). yet the number of cases produced during the second wave is much higher than the number of cases produced during the first wave. we observe that the values of the parameters of the phenomenological model are quantitatively different between the first wave and the second wave. several phenomena can explain this difference. the population was better prepared for the second wave. the huge difference in the number of daily reported cases during the second phase can be partially attributed to the huge increase in the number of tests in france during this period. but this is only a partial explanation for the explosion of cases during the second wave. we also observe that the average duration of the infectious period varies between the first epidemic wave (12.5 days) and the second epidemic wave (3.5 days). this may indicate a possible adaptation of the virus sars cov-2 circulating in france during the two periods, or the effect of the mitigation measures, with better respect of the social distancing and compulsory mask-wearing. the huge difference between the initial values of r0(t) in the first and the second waves is an apparent paradox which shows that r0(t) has a limited explanatory value regarding the severity of the epidemic: even if the quantitative value of r0(t) is higher at the start of the first wave, the number of cases produced during an equivalent period in the second wave is much higher. this paradox can be partially resolved by remarking that the r0(t) behaves like an exponential rate and the number of secondary cases produced in the whole population is therefore very sensitive to the number of active cases at time t. in other words, r0(t) is blind to the epidemic state of the population and cannot be used as a reliable indicator of the severity of the epidemic. other indicators have to be found for that purpose; we propose, for instance, the maximal value of the daily number of new cases, which can be forecasted by our method (see equation (2.16)), although other indicators can be imagined. in figure 6 we present an exploratory scenario assuming that during the endemic period preceding the second epidemic wave (may 17 jul 05) the daily number of cases is divided by 10. the resulting cumulative number of cases obtain is five lower the original one. we summarize this observation into the following statement. result • the level of the daily number of cases during an endemic phase preceding an epidemic phase strongly influences the severity of this epidemic wave. • in other words, maintaining social distancing between epidemic waves is essential. phenomenological model for covid-19 159 aug sep oct nov 2020 200 000 400 000 600 000 800 000 1 000 000 1 200 000 1 400 000 1 600 000 1 800 000 2 000 000 data original simulation modified simulation figure 6. cumulative number of cases for the second epidemic wave obtained by using the si model (2.8) with τ(t) given by (2.13), the parameters from table 1. we start the simulation at time t0 = july 05 with the initial value i0 = cr′(t2) νf for the red curve and with i0 = 1 10 cr′(t2) νf for yellow curve. the remaining parameters used are ν = 1/3, f = 0.9, s0 = 66841266. we observe the number is five times lower than then the original number of cases. in figure 4, there is two order of magnitude in the daily number of cases in between cr′(t1) ≈ 1 (with t1 = feb 27) at the early beginning of the first epidemic wave and cr ′(t2) = 422 (with t2 = july 05) at the early beginning of the second epidemic wave. that confirms our result. after the second wave, the average daily number of cases cr′(t) in france is stationary and approximately equal to 12440. therefore, if the above observation remains true and if a third epidemic wave occurs, the third epidemic wave is expected to be more severe than the first and second epidemic waves. our study can be extended in several directions. a statistical study of the parameters obtained by using our phenomenological model with data at the regional scale could be interesting. we could in particular investigate statistically the correlations existing between the parameters changes and the variations with demographic parameters as the median age and the population density, as well as geoclimatic factors as the elevation and temperature, etc. we also plan to extend our method to more realistic epidemic models, like the seiur model from [6], which includes the possibility of transmission from asymptomatic unreported patients. author contributions: qg, jd and pm conceived and designed the study. qg and pm analyzed the data, carried out the analysis and performed numerical simulations, jd and pm conducted the literature review. all authors participated in writing and reviewing of the manuscript. references 1. data from who: https://covid19.who.int/who-covid-19-global-data.csv, accessed june 9, 2021. 2. d. bernoulli, essai d’une nouvelle analyse de la petite vérole, & des avantages de l’inoculation pour la prévenir, histoire de l’académie royale des sciences avec les mémoires de mathématique et de physique tirés de cette académie (1766), 1–45. 3. g. chowell, n. w. hengartner, c. castillo-chavez, p. w. fenimore, and j. m. hyman, the basic reproductive number of ebola and the effects of public health measures: the cases of congo and uganda, j. theor. biol. 229 (2004), no. 1, 119–126. 4. j. demongeot, q. griette, and p. magal, si epidemic model applied to covid-19 data in mainland china, roy. soc. open sci. 7 (2020), no. 12, 201878. https://covid19.who.int/who-covid-19-global-data.csv 160 q. griette, j. demongeot, and p. magal 5. k. dietz and j. a. p. heesterbeek, daniel bernoulli’s epidemiological model revisited, math. biosci. 180 (2002), 1–21. 6. z. liu, p. magal, o. seydi, and g. webb, predicting the cumulative number of cases for the covid-19 epidemic in china from early data, math. biosci. eng. 17 (2020), no. 4, 3040–3051. 7. , understanding unreported cases in the covid-19 epidemic outbreak in wuhan, china, and the importance of major public health interventions, biology 9 (2020), no. 3, 50. 8. w. p. london and j. a. yorke, recurrent outbreaks of measles, chickenpox and mumps: i. seasonal variation in contact rates, am. j. epidemiol. 98 (1973), no. 6, 453–468. 9. p. f. verhulst, notice sur la loi que la population poursuit dans son accroissement, correspondance mathématique et physique 10 (1838), 113–121. univ. bordeaux, imb, cnrs umr 5251, f-33400 talence, france. e-mail address: quentin.griette@u-bordeaux.fr univ. grenoble alpes, ageis ea7407, f-38700 la tronche, france e-mail address: jacques.demongeot@univ-grenoble-alpes.fr corresponding author, univ. bordeaux, imb, cnrs umr 5251, f-33400 talence, france e-mail address: pierre.magal@u-bordeaux.fr 1. introduction 2. material and methods 2.1. phenomenological model regularized model procedure to fit the phenomenological model to the data 2.2. si epidemic model 2.3. parameter bounds 2.4. computation of the basic reproduction number 3. results 3.1. phenomenological model compared to the french data 3.2. si epidemic model compared to the french data 4. discussion references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 2, number 1, march 2021, pp.1-9 https://doi.org/10.5206/mase/11101 an empirical forecasting method for epidemic outbreaks with application to covid-19 bo deng abstract. in this paper we describe an empirical forecasting method for epidemic outbreaks. it is an iterative process to find possible parameter values for epidemic models to best fit real data. as a demonstration of principle, we used the logistic model, the simplest model in epidemiology, for an experiment of live forecasting. short-term forecasts can last for five or more days with relative errors consistently kept below 5%. the method should improve with more realistic models. 1. introduction it would be beneficial for planning if we were able to forecast covid-19 outbreak like we do for weather forecasting. unfortunately such practice is non-existent. surely, infectious diseases modeling has been and still is very active among theorists. but there are no known reliable ways for short-term forecasting, loosely defined to be no more than one week in time, and by extension long-term forecasting, one month or more into the future, remains even more elusive. the main problem lies in a reality that for reported cases of infection over a period of time there can be infinitely many parameter combinations in any epidemic model that can just fit the data similarly. compounding a hard problem still, all possible future trajectories are driven apart by the inherent exponential growth in viral transmission dynamics. long-term forecasting is more of a guessing game than a science, even with reasonable epidemic models. the aim of this paper is to explore ways for short-term forecasting with the hope that it may shed some light on medium-term forecasting and long-term forecasting. more accurately, the rest of the paper was a contemporaneous account on a live forecasting experiment. as a result, the presentation below is kept in the present tense when it was written. every epidemic will come to an end, by which time there will be a total tally of infected for a given region which may or may not be chosen for some arbitrary reasons. for lack of a word, let us call the infected total epidemity, and denote it by its initial, e. e is a function of many variables. among them includes the virus or diseases intrinsic ability to transmit among the host population, the intrinsic infectivity parameter, call it r throughout. it is also determined by how vigilant we are about the threat and what we do to keep it at bay. e.g., the practice of social distancing, which is mostly hidden but affects e greatly. epidemity is real, and there will be an e. it can be as large as the whole population, or as small as zero. most likely it is somewhere in between simply because a portion of population will avoid infection once the population has reached herd immunity. received by the editors 23 october 2019; accepted 14 december 2020; published online 22 december 2020. 2010 mathematics subject classification. 92d30, 92-05, 92-08, 92-10. key words and phrases. covid-19, logistic model, forecasting, gradient search. 1 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/11101 2 b. deng to know e is to require the condition that everyone infected is tested (ignoring the efficacy of a test) and counted. in reality, we only get an approximation of e, called e∗. if there is no limitation on testing, then e∗ = e. for most cases, e∗ < e, or vastly under estimated, e∗ � e. in the model below, we will use e throughout, but with the caveat that we are forecasting e∗ in most cases. similarly, let i(t) denote the cumulative total of infected at time t, measured in days. then what we are actually estimating is i∗(t) instead, with the same test caveat as for e∗. also, there is a time delay between the actual infection and the reported infection, denoted by c(t) for confirmed total cases at time t. but, assuming most infections undergo a mean time delay between infection and confirmation of infection, both variables slide on the time line with a mean translation in length. therefore, if we can forecast c(t) into the future, for all practical purposes, we are predicting i∗ or i if testing is not an issue. to forecast c(t) is to approximate it by a variable, x(t), of an epidemiological model. let t1, t2, ..., t` be the days on which c(t) is used to fit the parameters of the model. here, the time sequence {tk} does not need to be every day, nor consecutive days, nor all reported days. it should be chosen by the forecaster who deems the data c(t) reliable on those {tk} days. for example, pretty much all south koreas data for the first 30 reported days can be used for model fitting because they are not limited by testing according news report. for another example, on february 12, 2020, there was a huge jump in c for the wuhan, china, outbreak. for forecasters, that was a piece of good news, because it could be an accurate count for a long stretch in days, and it should be included to calibrate their forecasts. on the other hand, if we used most of the daily-case numbers prior to february 12, 2020, we would end up forecasting i∗ and e∗, grossly distorting the reality. a similar problem took place for the new york state outbreak, a wuhan lesson not learned and repeated. to reliably forecast a disease, just like for weather, a few accurate readings is a necessity. for our method to work, a few data points of c(tk) are required to be usable because of their close approximations to i(tk). equivalently, one can forecast the daily case numbers, which are simply the difference of the cumulative total, ∆c(t) = c(t) − c(t− 1). more effectively though, we can best-fit both the total c and the daily ∆c to models in combination to find the best forecasting trajectories. 2. model the model we will use is the most basic one for infectious diseases, dx dt = rx ( 1 − x e ) , (2.1) where e is the epidemity and r is the intrinsic infectivity rate. this is the same logistic equation named by pierre franois verhulst, a belgian mathematician who introduced it in the 19th century for population studies ([7, 4, 1]). this model can be justified for infectious diseases in a few ways as follows. biologically, we assume that virus has evolved to sweep through a host population in a rapid and simple way. it aims to spread exponentially, ẋ = rx, the exponential growth model with a constant per spore growth rate ẋ/x = r, with r the intrinsic rate for infectivity. but, it can’t grow exponentially indefinitely. it must run into the epidemity buffer e. therefore, its per spore infection rate is modulated, say, to be ẋ/x = r(1 − x/e). that is, when it is near e, the spread must slow down to a halt. the an empirical forecasting method for epidemic outbreaks 3 factor is not a square of 1−x/e or a square root of 1−x/e, which may or may not take some delicate evolutionary trickeries to do so, but for simplicity, the logistic buffering is a reasonable choice. a second justification goes as follows. treat e as a constant for the moment. then p(t) = x(t)/e can be treated as the fraction of infected, and (1 −p(t)) the fraction of uninfected. the rate of change for the infected depends on social contact between the infected and the uninfected, in product with an intrinsic rate r. the intrinsic rate r depends on the virus’ ability to transmit and the host population’s social defensive measures against the viral transmission, which only applies to the human population. the third justification is similar to the second, except that we assume x(t) directly to be the product of the epidemity e and the infection probability p(t). in this formulation, we have to treat the time t as a random variable. unlike a usual random variable, we have to experience the time t only in the forward direction, and every instance of it. in this view, the probability density function in the time random variable is assumed to take the simplest bionomial distribution dp dt = rp(1 −p). this probabilistic formulation conforms to our view that the viral transmission is fundamentally a random and probabilistic process. all justifications above are based on a common hypothesis that all infected, x, acquire immunity and are removed from the susceptible pool, e − x, and a proportion of which are infectious and the proportionality is absorbed into the intrinsic infectivity rate r. the last justification is to explain why the epidemity e can be treated as a constant rather than a time-evolving variable, because as we have been gradually increasing our social distance, e must be decreasing. the explanation lies in the time scales in which the two variables, x and e, change. the transmission in x takes place at a much faster time scale, measured in perhaps minutes, or hours. but the epidemity e changes at a slow time scale, at least in days, and delayed. because of this fast-slow time asymmetry, for all modeling and computational purposes e can be treated as a constant relative to x(t). thus, each new forecast on x(t) can forecast a slowly evolving e by the model. as as result, the forecasting curve is the logistic solution x(t) = x0e x0+(e−x0) exp(−rt) (2.2) where x0 is the initial condition x(0) = x0, with t = 0 setting at any day the forecaster chooses to be the start of an outbreak. a forecaster’s task is to determine the parameter values in r, e, and the initial value x0 so that the daily prediction x(t) can be made. the most basic requirement is to have x(tk) as close to c(tk) as possible. this requires to minimize the error between the reported data c and the predicted function x. for definitiveness, we will use a common daily relative error function denoted as h(x0,r,e) = 1 n ∑` k=1 (x(tk)−c(tk))2 c(tk)2 . (2.3) it is a per data relative error. we also add one consistency constraint that x(tk) ≥ c(tk), for k = 1, 2, . . . ,` (2.4) because the fitted counts must be no less than the data reported. the problem therefore becomes to minimize the error function h in the parameter space d = {q : x0 > 0,r > 0,e > 0} subject to the constraint above. 4 b. deng at face value, this looks like a straightforward nonlinear optimization problem to solve. this is very similar to energy minimization and is easy to solve by undergraduate students. this basic minimization approach is insufficient and will fail every time for a number of reasons. first, h does not have a unique minimum. it is an empirical fact that there seems to be no limit on the number of local minimums for h. for all practical purposes, one may just think there are infinitely many. second, there are vastly different values in the epidemity e in combination with reasonable r that make h to be similarly small. we can call such a minimizer a critical point. a forecaster can’t just pick any critical point to broadcast. this is why all attempts of forecasting the spread of diseases fall short here. it is only when an outbreak ends that everyone can find the same or similar critical values of e and r, fitting the full outbreak trajectory c(t) posterior, [3]. 3. method we now describe an algorithm in pseudo code to find the forecasting curve. (1) start a reasonable initial guess q = (x0,r,e). (2) use a nonlinear regression algorithm to find its corresponding critical point or minimizer, denoted by q̃ = (x̃0, r̃, ẽ). such an algorithm is usually based on newton’s gradient search idea ([5, 2]). (3) choose an integer, m ≥ 2, call it a multiplier, and another number s > 1, call it a range scale. pick m − 1 many parameter points at random. specifically, for parameter e, randomly pick m−1 many values from the interval [ẽ/s,sẽ], using uniformly distributed random numbers for definitiveness. do the same for x̃0 and respectively for r̃. this creates m− 1 many new initial guesses. run the nonlinear minimization program to create m − 1 many new critical points, which are offsprings of the mother minimizer q̃. denote by m the set of all minimizers. notice that the choice of the lower end of the interval [ẽ/s,sẽ], for example, is just a simple way to avoid the needless choice 0 and we use only one parameter to fix two ends of the searching interval for simplicity. (4) choose a number 0 < b ≤ 100, call it a breeder percentage parameter. let b be a set of minimizers that comprises b% of all minimizers from m. a minimizer is selected to be in b if its h error is inside the better b-percentile of m. it is for the purpose of selecting minimizers with smaller errors which are hard to find because of their small basins of attraction for newton’s fastest-descend searching algorithm. this step may be referred as selection. (5) choose an integer n. repeat step 2 and step 3 for the breeder minimizer set b to obtain a new generation of m and b. in each iteration, the parameter m, s, b may stay the same or vary. by the nth iteration, denoted by mn or just m the set of all minimizers for simpler notation. (6) choose a small error tolerance, � > 0. denote by m� the family of all minimizers whose errors are no greater than �, and the corresponding parameter families by m�,x0, m�,r, m�,e. denote the forecasting parameter values with respect to this error tolerance by f�,x0, f�,r, f�,e. then each value is the median of its respective family: f�,x0 = median(m�,x0 ), f�,r = median(m�,r), f�,e = median(m�,e). (7) for a sequence of � = �1 < �2 < · · · < �k, if f�k stabilizes, the one with the smallest fit error h is used as the forecasting parameters, f∗ = f�i for some 1 ≤ i ≤ k. an empirical forecasting method for epidemic outbreaks 5 we refer to this iterative method the median-path method. notice that, if � is chosen to be the smallest error h for the last set m, then the median path method simply yields the least error of the minimizers. this variation is referred to as the the least-error method, which has been used in combination with the median-path protocol for comparison purposes. the main justification for the least-error method is as follows. it was used to test to see if it can keep the global minimizer if the global minimizer is known. specifically, we first generated a data sequence by the outbreak logistic function for a given parameter value q. we used this q as an initial guess for the least error method. for whatever searching parameter values in m, s, n, b, the method expectedly returned the known global minimizer q with least error h = 0. we also tested the case that initial guesses were chosen away from the known global minimizer q. all simulations had resulted in least-error minimizers near the known global minimizer q. the median-path method had also resulted in forecasting values close to q for small enough �. because of the selective nature of median values, some of the simulation runs landed on the true global minimizer q. as a result, one used both the median-path method and the least-error method to approximate the true global minimizer solution. 4. experiment i started an open experiment, first in when the wuhan outbreak started, and then on twitter (@bodeng17567961) when the us outbreak started. it was intended to simulate live forecasting. i first did it for the epicenter, wuhan, and then the hubei province of which wuhan is the provincial capital, and the whole mainland of china. i did one or two forecasts for the outbreaks of south korea, iran, and germany. the forecasts for china gave reasonably good predictions on the e numbers, the inflection date, from which the daily cases start to decrease, except before the big spike on c on february 12, 2020. these exercises demonstrated two points. one, when covid-19 testing was limited for the population, our forecast could only predict i∗ and e∗. two, one can bypass problematic data and use only a few reliable c to do forecasting for i(t) and e. the reason is because the logistic curve has only three free parameters, and if c(t) is the ideal logistic curve, one only needs three data points in theory to determine the parameters. the reported data c for south korea was almost a perfect logistic curve when i stopped the forecasting exercise. i only needed one repeating forecasting run to capture the outbreak curve well enough. it can be used as a textbook data set in the future for the logistic model for the early phase of disease outbreaks. i did two forecasts for italy for a short period in the early phase of its outbreak. the short term match to real data was very good. i started the exercise for united states on march 9, 2020. all short term matches went well until march 18, 2020. that was the time when i realized there were at least two major outbreaks taking place in us, with vastly different onset date, inflection date, and epidemity. one should be for the new york state and the other for the rest. ideally, each state should have its own forecasting when an outbreak occurs. all reported case data c were taken from wikipedia for its transparency and checkability by crowdsourcing. from march 20, 2020, i focused my experiment exclusively on the outbreak of new york state. all experiments were time-stamped by the contemporaneous online posts. figure 1 shows two sample online posts, the first post and the last post for the nys experiment. for the algorithm parameters, after tried a few values such as s = 2, i used s = 5 consistently for the posted predictions. i used typical values m = 10, 50, 100, and b = 10, 20, 50, 100, for variable m and b, respectively. the size of m was usually greater than 5,000. after sorting, i usually kept the first 4,000 of m to see if the median https://www.linkedin.com/in/bo-deng-180b96 6 b. deng 0.5 1 1.5 2 2.5 3 3.5 d ia g n o s e d c o v id -1 9 10 5 new york state (model a, forecast date: april 25, 2020) mar. 1 mar. 11 mar. 21 mar. 31 apr. 10 apr. 20 apr. 30 forecasted: 3-20 7944 3-21 11050 3-22 15285 3-23 20988 3-24 28535 3-25 38299 3-26 37460 3-27 42000 3-28 52132 3-29 61023 3-30 67605 3-31 73304 4-1 86582 4-2 94843 4-3 102298 4-4 114773 4-5 125247 4-6 135645 4-7 145813 4-8 155612 4-9 164922 4-10 173649 4-11 181726 4-12 189115 4-13 195803 4-14 201797 4-15 207123 inflection date: april 4 start of new forecast made the previous day. new data high est. low est. forecasted: 4-16 211820 4-17 215932 4-18 236048 4-19 242021 4-20 245962 4-21 248744 4-22 250886 4-23 252626 4-24 266531 4-25 276811 4-26 289616 4-27 297991 4-28 303036 4-29 305771 4-30 307253 reported (wiki): 8402 -5.5% 10356 6.7% 15168 0.8% 20875 0.5% 25665 11.2% 30811 24.3% 37258 0.5% 44635 -5.9% 52318 -0.4% 59513 2.5% 66497 1.7% 75795 -3.3% 83727 3.4% 92381 2.7% 102863 -0.5% 113704 0.9% 122031 2.6% 130689 3.8% 138836 5.0% 149316 4.2% 159937 3.1% 170512 1.8% 180458 0.7% 188694 0.2% 195031 0.4% 202208 -0.2% 210001 -1.4% reported: 218506 -3.1% 227450 -5.1% 232954 1.3% 239008 1.3% 243734 0.9% 247912 0.3% 253438 -1.0% 259682 -2.7% 267812 -0.5% 278365 -0.6% 284267 1.9% 288218 3.4% 291328 4.0% 295913 3.3% 300594 2.2% 0 2000 4000 6000 8000 10000 12000 d a il y c a s e s mar. 1 mar. 11 mar. 21 mar. 31 apr. 10 apr. 20 apr. 30 figure 1. all forecasts were made when the algorithm seems to converge to a fixed f�,q. under-predicted are shown in red and over-predicated are shown in blue. relative percentage errors are also shown. usually, a new forecast was made when an underpredication occurred, or a grossly over-predication occurred such as the 3-26 forecast. an empirical forecasting method for epidemic outbreaks 7 values stabilized for the first 1,000, 2,000, 3,000, and 4,000 of m if the median-path protocol was used, each of which corresponds to an m� for some increasing value �. if it did, a convergence was called and a forecasting was made. for practice, i sorted the set m according to the h error from low to high. i would drop a percentage of the high end of m because they were usually not good fits to the data c visually. as an example, the top plot in fig.1 includes the forecasting curve in green and searching curves of the first 2000 parameter values from m. for the last forecasting plot from the bottom graph of fig.1, only the forecasting curve in green was shown, and all other searching curves were omitted. the exercise ended on april 30, 2020, when it became apparent that the logistic model had exhausted its usefulness. more specifically, notice that after the inflection point (around april 4), the daily data exhibited an oscillation of about every seven to ten days. this secondary oscillation, mini-outbreaks on top of the underlining outbreak which the logistic model is aimed to model, was present in the data even before the logistic curve reaches the inflection point, but became increasingly pronounced afterwards. for forecasts made on april 18 and thereafter, the method was modified as follows. we used the difference between the real data and the forecasted data for the secondary mini-outbreak. we used the same method to fit the logistic model to this data difference. we then added the forecasted data difference back to the underlining logistic curve for the new forecast. this is why the forecasted daily curve exhibited the saw-tooth feature after the highest inflection point, or the inflection point for the underlining logistic curve. that is, after that point, the forecasting curve is the superposition of two or more logistic curves, the first for the underlining outbreak, the second for the first mini-outbreak after the highest inflection point, and the third for the second mini-outbreak superimposed onto the superposition of the first two, and so on. from april 4 on, i also included two new curves to the forecasting. one was label “high est.” and the other “low est.”, for an upper estimate and a lower estimate for c. the former was obtained by using only local maximal values of c, and the latter was a curve from the search curves having the lowest e value from the forecasting pool, usually the first 4000 searches ranked by their search errors h from the lowest to the highest. from march 31 on, i also used the method to best-fit the daily case numbers. it became apparent that after the inflection date around april 4, there have been a persistent oscillation in daily case number of a period between 7 and 12 days, exposing the major shortcoming of the logistic model. 5. discussion our method is similar to most artificial intelligence algorithms because both are based on newton’s gradient search method together with large training sets with the goal to recognize an input and to output accordingly. our method has to generate its own training sets, which are generated from a mathematical model from an iterative process rather than existing patterns. in other words, the model carries infinitely many training sets and we have to judiciously select relevant ones. for example, if i included some obviously irrelevant logistic curves, the top plot of fig.1 would look very different, and unorganized. the purpose of our search algorithm is to match a live data set c to some curves from the training set m�. obviously the outcome varies with the size of m� and how uniformly the set m� is distributed. alternative ways were tried to generate training sets, but no convergence was observed, and therefore no forecasting could be made. averaging m� was also tried without success. 8 b. deng the importance of making realistic and reasonable forecasting on epidemic outbreaks is obvious, by which we can estimate other epidemiological variables and parameters, such icu required, mortality rate, etc. that is, a better planning can be made. perhaps more importantly, it may help prevent mass hysteria because when people can see what is to expect they tend to act more rationally. there are many rooms for improvement and standardization. the obvious one is on the model. the simple logistic model used for the experiment is good enough for the outbreak phase of an epidemic. once it passes the inflection, the symmetric nature of the logistic curve becomes inadequate for human population because unlike animal populations we will actively adopt counter measures against the viral spread. more sophisticated models should be used. however, the forecasting method used in this paper should be extended in principle to such models. although more sophisticated model, such as the basic sir model or modifications with more compartments ([6]), can do better in general, removing the symmetry artificiality of the logistic curves, i have not seen in literature that such a model has exhibited the saw-tooth like feature along the outbreak data c. our ad hoc treatment, using superimposed logistic curves, may be used as a patchy work, but the model becomes cumbersome and forecasting becomes passive, reactive, and laborious. a worthy project for an immediate future would be to find a variant of sir model capable of secondary-outbursts riding on a primary trajectory. with or without such a more realistic model, our method presented here should also be tested and refined for other epidemic models. given the urgent nature of outbreaks across the globe, the method should be put to a wider test, and hopefully it can be added to a toolset to fight the covid-19 pandemic. the method present here is imperfect but it is demonstrated in principle that it can work for shortterm forecasting. however, because the epidemity e is changing, though at a slower time scale, and it varies in a wide range, apparent further into future times, (c.f. fig.1), long-term forecasting on e becomes unattainable. it is doubtful this situation can be improved significantly with higher dimensional models. to summarize, a short-term forecasting method was used on real data that can find good approximations of known parameter values of a model if the data are generated by the model. the forecasting seems to work well on the outbreak phase of the epidemic. however, it is difficult to assign a theoretical degree of certainty to the empirical method because we don’t know at the more basic level how the model or any model is measured against a process, natural or human-influenced or both, other than the posterior measure on the error between the real and forecasted data. nevertheless, we believe the method can do better with more sophisticated epidemic models, empirically. acknowledgement: the author thanks prof. j.-y. cai of university of wisconsin-madison, o.x. deng of university of nebraska-lincoln, and the anonymous reviewers for their insightful comments on the manuscript. references [1] f. brauer, c. castillo-chavez, mathematical models in population biology and epidemiology, springer science and business media, 2013. [2] b. deng, an inverse problem: trappers drove hares to eat lynx, acta biotheor, 66(2018), 213-242. [3] e. massad, m.n. burattini, l.f. lopez and f.a. coutinho, forecasting versus projection models in epidemiology: the case of the sars epidemics, medical hypotheses, 65(2005), 17-22. [4] j. r. miner and pierre-franois verhulst, the discoverer of the logistic curve, human biology, 5(1933), 673-689. an empirical forecasting method for epidemic outbreaks 9 [5] a.p. ruszczynski, nonlinear optimization, princeton university press, princeton, 2006. [6] p. van den driessche and j. watmough, reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, mathematical biosciences, 180(2002), 29-48. [7] p.-f. verhulst, recherches mathématiques sur la loi d’accroissement de la population [mathematical researches into the law of population growth increase], nouv. mém. de l’academie royale des sci. et belles-lettres de bruxelles, 18(1845), 1-41. supporting materials • all linkedin p osts (https://www.linkedin.com/in/bo-den}-180b96). • on-line supp ortin} materials: the median-path/least-error searchin} al}orithm in matlab m�les. set m of 10,000 minimizers for march-19 forecast on new york state outbreak: (http://www.math.unl.edu/ bdeng1/research/open experiment covid-19 supporting materials/) • new york state data {rom wikipedia. data {or other countries and re}ions were also taken {rom wikipedia: (https://en.wikipedia.or}/wiki/covid-19 pandemic in new york (state)) department of mathematics, university of nebraska-lincoln, lincoln, ne 68588 e-mail address: bdeng@math.unl.edu https://www.linkedin.com/in/bo-deng-180b96 http://www.math.unl.edu/~bdeng1/covid-19 forecasting https://en.wikipedia.org/wiki/covid-19_pandemic_in_new_york_(state) 1. introduction 2. model 3. method 4. experiment 5. discussion references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 4, december 2020, pp.373-382 https://doi.org/10.5206/mase/10852 transient dynamics of the kidney disease epidemic among hiv-infected individuals dylan hull-nye*, bhawna malik*, ravikiran keshavamurthy*, and elissa j. schwartz abstract. the prevalence of end stage kidney disease (eskd) is rising among hiv-infected populations in several regions worldwide. we used an ordinary differential equation model of the dynamics of the aids and hiv+ eskd populations to investigate the effect of antiretroviral therapy (art) on the transient dynamics of the epidemic. we considered art that blocks the entry to each population, by preventing individuals from joining the aids population and by reducing the development from aids to hiv+ eskd, as well as the combined effects together. numerical simulation of our model revealed that at certain levels of art below 100%, the prevalence of hiv+ eskd drops, but then increases again due to the recovery in the aids population. we then examined the dip in hiv+ eskd seen with art analytically by calculating the minimum hiv+ eskd level and the length of time to achieve this minimum. we also evaluated the minimum hiv+ eskd level and its dependence on art parameters, both singly and in combination. we conclude that our model predicts that the drop in hiv+ eskd prevalence seen after increased art will be followed by an increase, unless art is sufficiently high enough to eradicate hiv/aids. 1. introduction acquired immunodeficiency syndrome (aids) is caused by human immunodeficiency virus (hiv1), a retrovirus that affects the immune system by reducing cd4+ t cells, subsequently increasing the risk of opportunistic infections. by 2018, 37.9 million people were living with hiv/aids worldwide and more than 77 million people had died of the disease [28]. the black population is disproportionately affected by hiv/aids. africa has more than two-thirds of global hiv/aids cases and has seen seven in ten deaths [14]. similarly, african americans, while comprising only 12% of the u.s. population, accounted for 44% of new hiv diagnoses in 2016 [3]. kidney damage or reduced renal function that persists for more than 3 months is called chronic kidney disease (ckd) [7]. ckd is a significant non-infectious complication observed in hiv-infected patients [1, 2]. race is an important risk factor for ckd, which is more prevalent in black patients with hiv [26]. the population of those living with hiv/aids is at risk of developing end stage renal disease (esrd, or also known as end stage kidney disease, eskd), which is a result of impairment of renal function and kidney failure. studies have shown a high rate of african americans among those received by the editors 30 june 2020; accepted 17 december 2020; published online 24 december 2020. 2000 mathematics subject classification. primary 92-08; secondary 37n25. key words and phrases. continuous ode epidemic model, transient dynamics. portions of this research were undertaken as part of the cimpa summer school in mathematical biology in kathmandu, nepal, june 2019 with support from the centre international de mathmatiques pures et appliques (cimpa), the international centre for theoretical physics (ictp) and the society for mathematical biology (smb). *these authors contributed equally to this work. 373 374 dylan hull-nye, bhawna malik, ravikiran keshavamurthy, and elissa j. schwartz progressing from hiv+ to hiv+ eskd [5]. the development of eskd among individuals with aids has been increasing in recent years, and about 90% of those affected are of african descent [15]. figure 1. flow diagram of aids and hiv+ eskd dynamics. individuals with aids (a) progress to nephropathy (n), or hiv+ eskd. this occurs at rate (1−h)s, where h represents the effectiveness of antiretroviral therapy (art) in blocking progression to nephropathy. recruitment to a occurs at rate (1−�)b, where � represents the effectiveness of art in blocking the development of aids. the availability of antiretroviral therapy (art) and scaling up of these lifesaving drugs against hiv1 infection over the past 20 years is heralded as one of the biggest public health achievements of the century [8, 27]. art reduces the hiv viral load by inhibiting viral replication, transforming a highly fatal disease into a treatable, chronic infection [19]. according to the world health organization, in 2018, 62% of people living with hiv were receiving antiretroviral treatment [28]. even though the rollout of art has remained a challenge in low and middle-income countries, including in sub-saharan africa [27], it has substantially improved the prognosis of hiv-1 infection in recent years [24]. relevant to kidney disease, art has been associated with both short-term and long-term toxicities including nephrotoxicity [12, 13]. however, when art is employed along with the close monitoring of kidney function, it has been shown to effectively manage hiv-associated kidney disease. importantly, studies have shown that effective treatment with art decreases the morbidity and mortality in patients with hiv-associated kidney diseases and also the progression to eskd [9, 17, 20], though the exact mechanisms by which art ameliorates kidney disease are not yet clear [10], . here, we present a new epidemiological model of kidney disease development among hiv-infected individuals. our goal is to analyze the transient dynamics of the model in order to predict the shortterm prevalence of disease. specifically, we aim to understand the effects of art on the hiv-infected population with kidney disease. previous work [22] predicted that disease prevalence would rise due to the increase in the hiv-infected population, but the model utilized did not allow for the possibility to assess the effect of art on this population. in this work, we add greater complexity to the population transient dynamics of hiv-infected kidney disease epidemic 375 dynamics to permit this assessment. we also analyze the transient dynamics more closely, and we determine which conditions affect them. 2. model the model of kidney disease dynamics among hiv-infected individuals tracks two populations, aids (a) and aids nephropathy (n) (also called hiv+ eskd). the schematic diagram of the model is shown in figure 1, and the model equations are as follows: da dt = (1 − �)b − (1 − h)sa − ρa − µaa (2.1) dn dt = (1 − h)sa − δn − µnn, (2.2) with a(0) = a0 and n(0) = n0. briefly, new individuals with aids enter the a population at a constant rate, b. these individuals die due to aids at rate ρ, or due to non-aids causes at rate µa. alternatively, they develop hiv+ eskd at rate s. for now, we neglect individuals who develop hiv+ eskd without developing aids. individuals with hiv+ eskd die from the condition at rate δ or due to non-hiv+ eskd causes at rate µn . antiretroviral therapy (art) is the standard of care for all individuals with hiv/aids, so we model its effects on populations a and n: art that blocks individuals from developing aids is modeled by �; art that blocks progression from aids to hiv+ eskd is modeled by h. both � and h assume values between 0 and 1 to indicate art efficacy between 0% and 100%, respectively. our model improves upon the work of schwartz et al. [22], in which populations grew without bound. this expansion was accomplished by separating the aids growth term into distinct entry and exit terms. we exclude the effects of art on mortality in this study, in order to focus more specifically on the dynamics revealed by art affecting b and s. the list of model parameters is shown in table 1, along with the values for parameters and initial conditions used in numerical simulations. these values have been taken from the literature or fitted to cdc data, as described. 3. analytical results 3.1. analytical solution. solving the model equations yields a(t) = (1 − �)b k + ( a0 − (1 − �)b k ) e−kt, (3.1) n(t) = n0e −k′t + (1 − h)s k′ − k ( e−kt − e−k ′t )( a0 − (1 − �)b k ) + (1 − �)b(1 − h)s kk′ ( 1 − e−k ′t ) ,(3.2) where k = (1 − h)s + ρ + µa and k′ = δ + µn . 3.2. stability analysis. there is one endemic equilibrium point (a∗,n∗), a∗ = (1 − �)b [(1 − h)s + ρ + µa] (3.3) n∗ = (1 − h)s(1 − �)b [δ + µn ][(1 − h)s + ρ + µa] . (3.4) 376 dylan hull-nye, bhawna malik, ravikiran keshavamurthy, and elissa j. schwartz table 1. model parameters variables description initial conditions source a(t) population with aids 31500 [4, 22] n(t) population with aids nephropathy (hiv+ eskd) 166 [22] parameters description values (year−1) source � art effectiveness on b between 0 and 1 varies s rate of progression from aids to hiv+ eskd 0.01 [22] h art effectiveness on s between 0 and 1 varies b rate of entry to aids population 21970 fitteda ρ rate of death due to aids 1/6 based upon a 6 year lifespan after aids diagnosis without treatment [6, 11, 23]b δ rate of death due to hiv+ eskd 0.67 based upon a 1.5 year lifespan after diagnosis with aids and hiv+ eskd with no treatment [25] µa natural mortality rate, aids population 1/55 based upon a 55 year lifespan after infection [16] µn natural mortality rate, hiv+ eskd population 1/5 based upon a 5 year lifespan after eskd diagnosis [18, 21]c a fitted from the equation a(t) and data [4] using other model values under the condition of no therapy. b after aids diagnosis, individuals with no therapy survive approximately 6 years [6, 11], with survival generally lower among black individuals [23]. c average life expectancy on dialysis is 5-10 years; the lower bound was taken due to the additional condition of hiv infection and higher mortality seen in black individuals [18, 21]. the model has no basic reproductive number because there is no disease-free equilibrium. eigenvalues of the jacobian matrix are λ1 = − ( (1 − h)s + ρ + µa ) (3.5) λ2 = − ( δ + µn ) . (3.6) since the eigenvalues are negative, the equilibrium (a∗,n∗) is stable. moreover, for any initial conditions, the analytical solution approaches the equilibrium lim t→∞ a(t) = a∗ lim t→∞ n(t) = n∗, which gives us that (a∗,n∗) is a globally asymptotically stable equilibrium point. transient dynamics of hiv-infected kidney disease epidemic 377 3.3. calculation of nmin and tmin. some art efficacy values give rise to declining numbers in hiv+ eskd followed by recovery to a steady state. we calculate the minimum value reached, nmin, as well as the length of time in which this minimum is reached, tmin. this is accomplished by taking the derivative of the analytical solution for n(t) (equation 3.2), setting it to 0, and solving, yielding tmin = ln (m) k − k′ m = (1 − h)s(a0k − b(1 − �)) a0k′s(1 − h) + n0k′(k − k′) − bs(1 − �)(1 − h) (3.7) nmin = m k′ k′−k ( s(1 − h) k − k′ ( a0 − b(1 − �) k ) − sb(1 − �)(1 − h) kk′ + n0 ) + m k k′−k ( s(1 − h) k′ − k ( a0 − b(1 − �) k )) + sb(1 − �)(1 − h) kk′ . 4. numerical results we performed numerical simulations of the epidemic trajectories under conditions representing various efficacies of antiretroviral therapy (art). values for parameters and initial conditions are shown in table 1. first we show the epidemic trajectories with various levels of art that decrease the progression from aids to hiv+ eskd. we investigated how much hiv+ eskd (n) would decrease initially, and then at steady state, with art that decreased the progression from aids to hiv+ eskd (h) by 0%, 70%, 80%, 90%, or 100% (figure 2). in the absence of art (h = 0), both populations, aids (a) and hiv+ eskd (n), show epidemic trajectories that steeply grow and then level off to a plateau. when art completely blocks the progression from aids to hiv+ eskd, h = 1. in this case, the aids (a) population is unaffected (not shown), while the hiv+ eskd (n) population decreases to 0. when the degree to which art prevents progression is 70% (h = 0.7), hiv+ eskd decreases below 150 individuals in under 1 year, but then rises again to a steady state of 400 individuals by year 30. with 80% effective art (h = 0.8), hiv+ eskd decreases to 120 individuals in 1.5 years, and then rises again to a steady state of 270 individuals by year 15. therefore, just 10% more effective therapy can reduce the steady state level by 2/3. when preventing progression is 90% effective (h = 0.9), hiv+ eskd decreases to a minimum just under 75 individuals in 2.5 years, rising to a steady state of 135 individuals. thus, when art effectiveness increases from 70% to 90%, the steady state level of hiv+ eskd decreases to 1/3 of its previous level. we observed that the level of hiv+ eskd (n) decreased when art was initially given, but then n recovered and grew to a new steady state. this resurgence was due to the increase in the aids (a) population, which fuels the hiv+ eskd population (see inset, which shows the aids population increasing as the hiv+ eskd population decreases). in other words, art can block the development of hiv+ eskd from aids, but this effect is not sustained if the risk pool that drives the hiv+ eskd epidemic (i.e., aids population (a)), is sufficient. thus, we next examined the effect of driving down the steady state of hiv+ eskd by reducing the risk pool (aids) with art that blocks the aids entry rate (i.e., �). we simulated a block to the aids entry rate (�) by 0%, 40%, 60%, 70%, or 100% (figure 2). in the absence of art (� = 0), both populations, aids (a) on the top right and hiv+ eskd (n) on the bottom left, show epidemic trajectories that grow and then level off to a plateau. when art completely blocks the entry to aids, 378 dylan hull-nye, bhawna malik, ravikiran keshavamurthy, and elissa j. schwartz 0 10 20 30 40 50 time (years) 0 200 400 600 800 1000 1200 1400 n 0 10 20 30 time (years) 4 6 8 10 a 10 4 50 100 150 n a n 0 10 20 30 40 50 time (years) 0 2 4 6 8 10 12 a 10 4 0 10 20 30 40 50 time (years) 0 200 400 600 800 1000 1200 1400 n 0 10 20 30 40 50 time (years) 0 200 400 600 800 1000 1200 1400 n figure 2. model simulations showing the effects of antiretroviral therapy (art). top left: n over time with h = 0 (red), h = 0.7 (green), h = 0.8 (black), h = 0.9 (blue), and h = 1 (cyan). other values are given in table 1 with � = 0. inset: when h = 0.9, n is shown in blue and a is shown in pink. top right: a over time with � = 0 (black), � = 0.4 (blue), � = 0.6 (cyan), � = 0.7 (red), � = 1 (green). other values are given in table 1 with h = 0. bottom left: n over time with � = 0 (red), � = 0.4 (green), � = 0.6 (black), � = 0.7 (blue), � = 1 (cyan). other values are given in table 1 with h = 0. bottom right: n over time with h = � = 0 (red), h = � = 0.2 (green), h = 0.7 and � = 0.1 (black), h = 0.7 and � = 0.4 (blue), h = 0.9 and � = 0.7 (cyan). other values are given in table 1. � = 1. in this case, both the aids (a) population and the hiv+ eskd (n) population decrease to 0. interestingly, even with 100% effective art (� = 1), the hiv+ eskd (n) population increases transiently before declining. with intermediate � values, aids and hiv+ eskd increase less but still maintain positive steady states. we then examined the potential effect on hiv+ eskd if art blocks not only progression from aids to hiv+ eskd (h > 0), but simultaneously also blocks entry into the aids population (� > 0). the combined effect of art acting at multiple points in the epidemic dynamics is seen here: when blocking progression is 90% (h = 0.9) and blocking entry to aids is 70% (� = 0.7), the steady state transient dynamics of hiv-infected kidney disease epidemic 379 level of hiv+ eskd is reduced to approximately 40 individuals (figure 2, bottom right). thus, in lieu of stronger therapy that effectively blocks a single mechanism, strong gains can be achieved when art can simultaneously reduce entry into the aids population as well as reduce progression to hiv+ eskd, even if its effectiveness at either is lower. 0.5 0.6 0.7 0.8 0.9 1 h 0 1 2 3 4 5 t m in 0 1 2 3 4 a 0 10 4 0 1 2 3 4 5 t m in 100 200 300 n 0 0 1 2 3 4 5 t m in 0.5 0.6 0.7 0.8 0.9 1 h 0 50 100 150 200 n m in 1 2 3 4 a 0 10 4 0 50 100 150 200 n m in 100 200 300 n 0 0 50 100 150 200 n m in figure 3. tmin (top) and nmin (bottom) as a function of h (left), a0 (middle), and n0 (right). left, � = 0 and other values are as in table 1. middle, h = 0.9, � = 0, and other values are as in table 1. right, h = 0.9, � = 0, and other values are as in table 1. we note that different values of h affect not only the steady state n∗ but also the values of tmin and nmin. the closer h is to 1, the lower the minimum value of n, as would be expected, and the greater the minimum value of t. we show these minima across a range of values of h from 0.5 to 1 (figure 3, left). how the values of tmin and nmin vary with initial conditions a0 and n0 are also shown (figure 3, middle and right, respectively). the initial aids population, a0, has a negligible effect on tmin, and a larger value of a0 gives a mildly higher nmin. the larger the initial size of the hiv+ eskd population, n0, the later the tmin is reached, and modestly, the higher the nmin. therefore, low initial n0 and h around 0.6 give the earliest tmin values. for nmin, lowest levels are seen with low n0 values and high h. finally, we investigated the effects of art and initial conditions on nmin more comprehensively. figure 4 shows nmin given a range of values for h and a0 (top left), h and n0 (top right), and h and � (bottom left). the lowest nmin values occur when h is high, particularly when n0 is low. for all values across the surveyed ranges of a0 and �, very high values of h are needed to lower nmin. these figures show that to achieve a low nmin, h has a stronger effect than n0, a0, and also �. in contrast, when we examined the effect of h and � on the steady state n∗, we found that h and � have equivalent efficacies on lowering n∗ (figure 4, bottom right). thus the differential effect of parameters h and � applies only to the transient dynamics. 380 dylan hull-nye, bhawna malik, ravikiran keshavamurthy, and elissa j. schwartz figure 4. nmin values (from 0 in blue to 160 in yellow) shown according to how they vary with h and a0 (top left), h and n0 (top right), and h and � (bottom left). n ∗ values (from 0 in blue to 1200 in yellow) shown according to how it varies with h and � (bottom right). 5. discussion this study demonstrates that when art lowers hiv+ eskd by blocking the progression from aids to hiv+ eskd (via h), small improvements in h efficacy can result in large reductions in prevalence. blocking entry to aids with art (via �) lowers hiv+ eskd prevalence as well, even though its effect is indirect: it drives down hiv+ eskd prevalence by reducing aids prevalence. the best outcomes occur with art that targets entry to aids and progression to hiv+ eskd simultaneously, even if the efficacy through each mechanism is lower. our simulations revealed that a dip in the prevalence of hiv+ eskd is seen with some art values (e.g., h = 0.9), followed by a rise to a lowered steady state. the minimum occurs earliest with values of h around 0.6 and low n0, and the smallest minima occur with high h and low n0. the value of h has a stronger effect on the minimum n than n0, a0 or �. however, practically speaking, it is important to note that the prevalence will rise again (unless art is sufficiently high); this dip should not be viewed as an indication that the disease prevalence will continue to decrease, unless hiv/aids is eradicated. transient dynamics of hiv-infected kidney disease epidemic 381 in future work, the model can be fitted with data sets from specific countries or regions affected by hiv+ eskd and used to estimate treatment levels needed to slow the increase of the epidemic in that region. also, the model can be expanded to include additional complexity in the population dynamics, such as the early stages of hiv infection and the effects of art 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[28] world health organization, report on the global hiv/aids epidemic. http://www.who.int/hiv/en/, accessed june 29, 2019. department of mathematics & statistics, washington state university, pullman, wa, usa e-mail address: dylan.hull-nye@wsu.edu department of mathematics, shiv nadar university, dadri, india e-mail address: bm650@snu.edu.in paul g. allen school for global animal health, washington state university, pullman, wa, usa e-mail address: r.keshavamurthy@wsu.edu corresponding author, department of mathematics & statistics and school of biological sciences, washington state university, pullman, wa, usa e-mail address: ejs@wsu.edu mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 3, number 1, march 2022, pp.1-11 https://doi.org/10.5206/mase/13511 on approximating initial data in some linear evolution equations involving fractional laplacian ramesh karki abstract. we study an inverse problem of recovering initial datum in a one-dimensional linear evolution equation with the dirichlet boundary conditions when the solution to the equation is known only at a suitably fixed space location and suitably chosen finitely many later time instances. to be more explicit, we consider a one-dimensional linear evolutionary equation involving a dirichlet fractional laplacian and an unknown initial datum f which is assumed to be in a suitable subset of a sobolev space, and then construct n future times so that from the known values of the solution at a suitably fixed space location and at these n future times, we recover f with a desired accuracy. 1. introduction consider the initial-boundary value problem ut = −a(t)(−∆)1/2u, u(0, t) = u(π,t) = 0, u(x, 0) = f(x) (1.1) where a is a positive continuous function of t > 0 such that ∫ t 0 a(s)ds exists, ∆ = ∂2/∂x2, and at the moment, f is in l2[0,π]. the problem of finding a solution u(x,t) to (1.1) is quite common when the initial datum f in a suitable function space is known. however, we are interested in studying an inverse type problem, meaning a problem of recovering the initial datum f with a desired accuracy if the solution u(x,t) is known only at a fixed point x0 in [0,π] and suitably selected n later time instances tj, j = 1, 2, 3, . . . ,n. this type of inverse problem is not well-posed in general. for the well-posed, we further assume that f is in the closed subset br of the sobolev space hr[0,π], r > 0, given by br := { f ∈ hr[0,π] : ||f||hr[0,π] ≤ 1 } . (1.2) we are indeed motivated to study this problem from similar problems studied in [1, 9]. in [9], the authors have considered the temperature distribution of a thin uniform one-dimensional body of finite length represented by the one-dimensional heat equation with dirichlet boundary conditions and an initial condition. then they have studied the recovery of the initial temperature measurement with a near optimal rate when temperature measurements taken at a fixed point of the body and at finitely many later times are known. also, they have asked some questions, one of which is whether their method can be extended to the case of involving a diffusion equation with a diffusion coefficient depending continuously on time. this question has been addressed in [1]. moreover, the authors in [1] have also studied the problem of recovering initial data in an initial-boundary value problem involving a parabolic pde with constant coefficients and even order partial differential coefficients with respect to received by the editors 31 december 2020; revised 6 april 2021 and 5 january 2022; accepted 10 january 2022; published online 19 january 2022. 2020 mathematics subject classification. primary 35s11. key words and phrases. evolution equations, initial data, fourier sine series, samples, later time instances, measurement algorithm and optimal error. 1 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/13511 2 r. karki spacial variable. in this paper, we study the case, given in (1.1), that involves a fractional diffusion equation with a diffusion coefficient depending continuously on time. it could be a base for generalizing the method in other factional evolution equations, some of which are mentioned in section 5. these problems are of particular interest as they arise in different areas such as stochastic control theory and mathematical finance, classical mechanics in the context of heat propagation, population dynamics, the theory of water waves, quantum mechanics and phase transition problems [2, 5, 10, 13]. we know that the dirichlet laplacian −∆ on l2[0,π] has eigenvalues λn = n2, n = 1, 2, . . . with the corresponding eigenfunctions en(x) := sin nx, n = 1, 2, . . . that form an orthonormal basis for l2[0,π] when normalized with respect to the inner product 〈u,v〉l2 := 2 π ∫ π 0 u(x)v(x) dx. then each u ∈ l2[0,π] has the following fourier sine series representation u = ∞∑ k=1 ûkek, where the equality has to be understood in the l2-sense or in the sense of almost every x ∈ [0,π] and ûk := 〈u,ek〉l2 , the kth fourier sine coefficient of u. referring to [3, 6, 12, 15], the fractional power operator (−∆)s/2, s > 0 on the dense subset d((−∆)s/2) := {u ∈ l2[0,π] : ∑∞ k=1 k 2s|ûk|2 < ∞} = hr[0,π] of l2[0,π] is given by (−∆)s/2u = ∞∑ k=1 ksûkek (1.3) and has the eigenvalues λ s/2 n = n s, n = 1, 2, . . . with the corresponding eigenfunctions en, n = 1, 2, . . . . thus we can equip hr[0,π] with the norm ||f||hr = ∑∞ k=1 k 2r|f̂k|2. the existence and uniqueness theory guarantees the existence of a (strong or l2-) solution u(x,t) to (1.1), which then has the fourier sine series representation u(x,t) = ∞∑ k=1 ûk(t)ek(x) (1.4) for almost every x ∈ [0,π]. however, solutions to more general evolutionary problems than (1.1) can be expressed in the form of general fourier series representations, which can be obtained by employing tools from the semigroup theory and the spectral theory of self-adjoint operators on hilbert spaces, and have been discussed in [12, 14, 15, 16]. these representations can be used when extending our problem to more general evolutionary equations. under the assumptions we made above, the regularity theory guarantees that the solution u(x,t) to (1.1) is in d((−∆)1/2)). from (1.1), (1.3) and (1.4), we obtain that each time dependent fourier sine coefficient ûk(t) satisfies the initial value problem d dt ûk(t) = −ka(t)ûk(t), ûk(0) = f̂k whose solution is ûk(t) = f̂ke −kt(t), k = 1, 2, . . . , (1.5) where t(t) = ∫ t 0 a(s)ds. from (1.4) and (1.5), we have u(x,t) = ∞∑ k=1 f̂ke −kt(t)ek(x) (1.6) for t ≥ 0 and for almost every x ∈ [0,π]. the reason why the last equality holds in the a. e. sense is because we have used the strong (not the classical) solution to (1.1). since we are going to deal with on approximating initial data 3 the l2-error of approximation to f, it will be sufficient to have this type of solution. throughout the rest of the paper, we will have referred to this solution whenever we call the solution to (1.1). now we summarize what we are going to do in the rest of the paper. in section 2, we will discuss the selection of the space location x0 and prove that for an increasing sequence 0 < t1 < t2 < ... of future times, the values u(x0, tj), j = 1, 2, . . . are enough to determine f uniquely. this consistency result allows us to approximate f from the first n values u(x0, tj), j = 1, 2, . . . ,n, called n samples. in section 3, we will discuss the existence of a lower bound for an optimal error of approximation to f. for this, we will use a measurement algorithm discussed in [1, 9] as an encoder or a continuous map a from a subset b of l2[0,π] into rn together with a decoder m or a continuous map from rn into l2. using this measurement algorithm, we will find an approximation to f as m(a(f)) and also discuss the existence of a lower bound for the optimal error of approximation to f in l2[0,π]. like obtaining a lower bound for the optimal error of approximation to f, one may expect the existence of an upper bound for the optimal error of the approximation. we will not address this in this paper. however, in section 4, we will prove that there exists a sequence of future times 0 < t1 < t2 < ... such that from the first n samples u(x0, tj), j = 1, 2, . . . ,n, f can be approximated in l 2[0,π] with an accuracy of order n−r. in section 5, we will discuss possibilities of extending this method to a few other evolutionary equations and also possibilities of applying it to evolutionary equations with other boundary conditions. 2. choice of x0 and consistency of recovery we need to select x0 ∈ [0,π] in such a way that the samples u(x0, tj), j = 1, 2, . . . , determine f uniquely provided the time sequence 0 < t1 < t2 < ... . observing (1.6), we see that the position x0 in [0,π] have to be chosen so that sin kx0 6= 0 for all k = 1, 2, . . . . so, as in [9], we consider x0/π to be an algebraic number of second order, that is, d ( x0 π , { 0, 1 m , 2 m ... , m m = 1 }) ≥ c0 m2 , m = 1, 2, . . . (2.1) where c0 is a constant. then we have d (kx0,{0,π, . . . ,kπ}) ≥ c0kπ−1, k = 1, 2, . . . , and hence |sin kx0| ≥ d0k−1, k = 1, 2, . . . (2.2) for some constant d0. now the following consistency result. lemma 2.1. for a sequence {tj}∞j=1 of later time instances satisfying t1 < t2 < t3 < ... and the choice of x0 ∈ [0,π] described by (2.2), suppose u(x0, tj), j = 1, 2, 3, . . . are known. then f can be determined uniquely. proof: consider the function f0(z) := ∞∑ k=1 ckz k, (2.3) where ck := f̂kek(x0), k = 1, 2, 3, . . . (2.4) since the sequence {ck}∞k=1 is in l 2, f0 is holomorphic in the unit complex disk d := {z ∈ c : |z| < 1}. since the sequence of points zj = e −t(tj), j = 1, 2, 3, . . . in d converges in d and f0(zj) = u(x0, tj), j = 1, 2, 3, . . . , the identity principle of one-complex variable implies that f0 can be determined uniquely. 4 r. karki this together with (2.3) and (2.4) implies that ck, k = 1, 2, 3, . . . can be determined uniquely and hence f̂k, k = 1, 2, 3, . . . . in this way, f can be determined uniquely. 3. lower bound on optimal error by following the techniques of [7, 9] (one may also see [8, 11]), we obtain a measurement algorithm to determine a lower bound for the optimal error of recovery of f. first, we consider two continuous mappings a and m, where a maps each f in a compact subspace b of l2 := l2[0,π] into a point in rn and m maps each point y ∈ rn into a function m(y) in l2. we view the map a as an encoder or sensor, whereas the map m as a decoder. the set {m(y) ∈ l2 : y ∈ rn} can be viewed as an n-dimensional manifold. an encoder a together with a decoder m forms our measurement algorithm. using this algorithm, we obtain point m(a(f)) =: f̄ in this manifold, which we consider as an approximation to f, and define the manifold width δn(b,l2) as the best of optimal l2-errors δa,m (b,l2) := sup f∈b ||f − f̄||l2, (3.1) or more precisely, δn(b,l2) := inf a,m δa,m,n(b,l2) (3.2) where n is fixed and the infimum is taken over all continuous maps a and m as described above. in particular, for our approximation problem, we consider an encoder a as a map f 7→ (u1,u2, . . . ,un) mapping br into rn, which extracts n samples uj := u(x0, tj), j = 1, 2, 3, . . . using the information about f, and denote this map by an. this map is indeed continuous. lemma 3.1. in our measurement algorithm, the map an : f 7→ (u1,u2, . . . ,un) mapping br into rn is continuous. proof: if f̄ ∈ br with the fourier sine coefficients ˆ̄fk, then for each f ∈ br with the fourier sine coefficients f̂k and for each j = 1, 2, 3 . . . ,n, we have |uj − ūj| =| ∞∑ k=1 f̂ke −kt(tj)ek(x0) − ∞∑ k=1 ˆ̄fke −kt(tj)ek(x0)| ≤ ∞∑ k=1 |f̂k − ˆ̄fk|e−kt(tj) ≤ ( ∞∑ k=1 |f̂k − ˆ̄f2k )1/2 ( ∞∑ k=1 |e−kt(tj)|2 )1/2 ≤||f − f̄||l2||{e−kt(t1)}∞k=1||l2, from which the proof of the lemma follows. in particular, for our approximation problem, we consider a decoder m as a map (u1,u2, . . . ,un) 7→ f̄n mapping each n-tupple of n samples into an approximation f̄n ∈ l2 to f, and denote this map by mn. thus δan,mn,n(br,l2) = supf∈br ||f − f̄n||l2 , where f̄n = mn(an(f)). now we deduce the following. theorem 3.2. for a measurement algorithm with an encoder an : f 7→ (u1,u2, . . . ,un) and a decoder mn : (u1,u2, . . . ,un) 7→ f̄n, we have δan,mn,n(br,l 2) ≥ cn−r (3.3) where c is a constant depending on r only. on approximating initial data 5 proof: for br, the idea discussed in section 3 of [9] implies δn(br,l2) ≥ c(r)n−r (or see [1, 7, 8]). therefore, for the measurement algorithm discussed above, we have δan,mn,n(br,l 2) ≥ δn(br,l2) ≥ c(r)n−r, establishing (3.3). due to some technical challenges, we will not obtain an upper bound for the optimal error of approximation to f. however, we will particularly prove in the next section that we can construct a sequence of future times 0 < t1 < t2 < ... such that from the first n samples u(x0, tj), j = 1, 2, . . . ,n, f can be approximated in l2[0,π] with an error that has an upper bound of order n−r. this is the main outcome of this paper. 4. optimal approximation to initial data as we discussed at the end of the last section, our main goal is to select n future time instances tj, j = 1, 2, 3, . . . ,n so that from the known n samples u(x0, tj), j = 1, 2, 3, . . . ,n, we can construct an approximation to f in l2[0,π] with an accuracy of order n−r. theorem 4.1. let br be as described in (1.2), let f ∈ br, r > 0, let a be as described in (1.1) and let u(x,t) denote the solution to the problem (1.1). fix x0 ∈ [0,π] such that (2.2) holds. additionally, fix t1 > 0 and ρ ≥ 2. there exists a sequence {tj}∞j=1 such that t(tj+1) = ρ jt(t1), j = 1, 2, 3, . . . . if u(x0, tj), j = 1, 2, . . . ,n are known, then there exists f̄n in l 2[0,π] such that ||f − f̄n||l2 ≤ cn−r, (4.1) where c is a constant that depends on d0, r, t1 and ρ. we begin with considering an increasing sequence t1 < t2 < ... of later times. set u(x0, tj) := u(tj), j = 1, 2, 3, . . . . from (1.6), we have u(tj) = ∞∑ k=1 cke −kt(tj), j = 1, 2, 3, . . . , (4.2) where ck = f̂kek(x0), k = 1, 2, 3, . . . . we use u(tn) to compute c1 and recursively, u(tn−k+1) to compute each ck, k = 2, 3, 4, . . . . so, from (4.2) we obtain c1 = e t(tn)u(tn) − ∞∑ j=2 cje (1−j)t(tn) (4.3) and for each k = 2, 3, 4, . . . ck = e kt(tn−k+1)u(tn−k+1) − k−1∑ j=1 cje (k−j)t(tn−k+1) − ∞∑ j=k+1 cje (k−j)t(tn−k+1). (4.4) suppose we have n samples u(tj), j = 1, 2, . . . ,n. from these n samples, we construct an approximation c̄1 to c1 as c̄1 := e t(tn)u(tn) (4.5) and an approximation c̄k to each ck, k = 2, 3, 4, . . . ,n as c̄k := e kt(tn−k+1)u(tn−k+1) − k−1∑ j=1 c̄je (k−j)t(tn−k+1). (4.6) these ck’s and c̄k’s satisfy an important estimate which is described in the next lemma. 6 r. karki lemma 4.2. let f, a, x0 and t1 and ρ be as in theorem 4.1. there exists a sequence {tk}∞k=1 such that t(tk+1) = ρ kt(t1), k = 1, 2, 3, . . . . for this sequence, |ck − c̄k| ≤ s(t1)2ke−t(tn−k+1), k = 1, 2, 3, . . . . (4.7) proof: notice that t is a strictly increasing positive continuous function of t > 0. for each k = 1, 2, 3, . . . , we have t(t1) < ρ kt(t1) and thus ρ kt(t1) is in the range of t . therefore, for each k = 1, 2, 3, . . . , we can choose tk+1 > 0 such that t(tk+1) = ρ kt(t1). we use the method of induction to prove the second part of the lemma. since f ∈ br, we have |cj|2 ≤ |f̂(j)|2 ≤ j−2r ∑∞ k=j k 2r|f̂(k)|2 ≤ j−2r for all j = 1, 2, 3, . . . . then from 4.4 and 4.6, |c1 − c̄1| ≤ ∞∑ j=2 j−re(1−j)t(tn) ≤ e−t(tn) ∞∑ j=0 e−jt(t1) ≤ 2s(t1)e−t(tn), (4.8) where s(t1) := ∑∞ j=0 e −jt(t1) = 1 1 −e−t(t1) . thus we have proved that (4.7) holds true for k = 1. assume that (4.7) holds true for each j ∈ {1, 2, . . . ,k − 1}, where k ≥ 2. for each k ≥ 2, we obtain from (4.4) and (4.6) that |ck − c̄k| ≤ k−1∑ j=1 e(k−j)t(tn−k+1)|cj − c̄j| + ∞∑ j=k+1 j−re(k−j)t(tn−k+1). (4.9) using the induction hypothesis and the formula for t(tj), j = 1, 2, 3, . . . , we have k−1∑ j=1 e(k−j)t(tn−k+1)|cj − c̄j| ≤s(t1) k−1∑ j=1 2je(k−j)t(tn−k+1)−t(tn−j+1) =s(t1) k−1∑ j=1 2je(k−j)t(tn−k+1)−ρ k−jt(tn−k+1) =s(t1) k−1∑ j=1 2je(k−j−ρ k−j)t(tn−k+1). (4.10) also, we have ∞∑ j=k+1 j−re(k−j)t(tn−k+1) ≤(k + 1)−r ∞∑ j=k+1 e(k−j)t(tn−k+1) =(k + 1)−r ∞∑ j=0 e(−j−1)t(tn−k+1) =(k + 1)−re−t(tn−k+1) ∞∑ j=0 e−jt(tn−k+1) ≤(k + 1)−re−t(tn−k+1) ∞∑ j=0 e−jt(t1) ≤s(t1)(k + 1)−re−t(tn−k+1). (4.11) on approximating initial data 7 from (4.9), (4.10) and (4.11), we have |ck − c̄k| ≤s(t1) k−1∑ j=1 2je(k−j−ρ k−j)t(tn−k+1) + s(t1)(k + 1) −re−t(tn−k+1) =s(t1)e −t(tn−k+1)  k−1∑ j=1 2je(k−j−ρ k−j+1)t(tn−k+1) + (k + 1)−r   since ρ ≥ 2 and x + 1 ≤ 2x for x ≥ 1, we have k − j + 1 ≤ 2k−j ≤ ρk−j for all j = 1, 2 . . . ,k − 1. so, |ck − c̄k| ≤s(t1)e−t(tn−k+1)  k−1∑ j=1 2j + (k + 1)−r   ≤s(t1)2ke−t(tn−k+1), proving that (4.7) holds true for k ≥ 2. this completes the proof of the lemma. now we are ready to prove theorem 4.1. under the assumptions of theorem 4.1 and in the view of ck = f̂kek(x0), we use the relation c̄k = ˆ̄fkek(x0) to determine approximate fourier sine coefficients ˆ̄fk to f̂k, k = 1, 2, . . . ,n. so, we define ˆ̄fk := c̄k ek(x0) , k = 1, 2, . . . ,n and then define an approximation to f as f̄n(x) := m∑ k=1 ˆ̄fkek(x), where m = ⌈n 2 ⌉ . (4.12) then the l2-error of approximation to f satisfies ||f − f̄n||2l2 ≤ m∑ k=1 |f̂k − ˆ̄fk|2 + ∞∑ k=m+1 |f̂k|2 ≤ m∑ k=1 |f̂k − ˆ̄fk|2 + ∞∑ k=m+1 ( k m )2r |f̂k|2 ≤ m∑ k=1 |f̂k − ˆ̄fk|2 + m−2r ∞∑ k=m+1 k2r|f̂k|2 ≤ m∑ k=1 |f̂k − ˆ̄fk|2 + m−2r||f||2hr ≤ m∑ k=1 |f̂k − ˆ̄fk|2 + m−2r (4.13) using (2.2) and lemma 4.2, we have |f̂k − ˆ̄fk| = |ck − c̄k| |ek(x0)| ≤ c0k2ke−t(tn−k+1), k = 1, 2, . . . ,n, (4.14) where c0 := s(t1)/d0. since 0 ≤ m − n 2 < 1, we have n − m ≤ n 2 < n − m + 1 and, therefore, t(tn−k+1) = ρ n−kt(t1) ≥ ρn/2−1t(t1) for all k = 1, 2, 3, . . . ,m. also ln k ≤ k for all k = 1, 2, 3, . . . ,m. 8 r. karki with these inequalities, we obtain from (4.13) and (4.14) that ||f − f̄n||2l2 ≤ m∑ k=1 c20k 222ke−2t(tn−k+1) + m−2r ≤c20e −2t(tn−m+1) m∑ k=1 k222k + m−2r ≤c20e −2t(t1)ρn/2−1m ·m222m + m−2r ≤c20e −2t(t1)ρn/2−1n322n + 22rn−2r. (4.15) notice that j2r+322j e2t(t1)ρ j/2−1 ≤ e(2r+3+2 ln 2)j e2t(t1)ρ j/2−1 → 0 as j →∞. so, for each set of choice of r > 0, t1 > 0 and ρ ≥ 2, there exists a constant c1 depending on r, t1 and ρ such that j2r+322j e2t(t1)ρ j/2−1 ≤ c1 for all j = 1, 2, 3, . . . . in particular, we have n2r+322n e2t(t1)ρ n/2−1 ≤ c1. (4.16) form (4.15) and (4.16), we obtain ||f − f̄n||l2 ≤ cn−r, where c is a constant depending on d0, t1, r and ρ. in this way, we have established (4.1) and completed the proof of theorem 4.1. finally, we consider the initial-boundary value problem ut = −a(−∆)1/2u, u(0, t) = u(π,t) = 0, u(x, 0) = f(x), (4.17) where a is positive real number and f ∈br. as a special case of theorem 4.1, we obtain the following. corollary 4.3. let br be as described in (1.2), let f ∈br, r > 0 and let u(x,t) denote the solution to the problem (4.17). fix x0 ∈ [0,π] such that it satisfies (2.2). also, fix t1 > 0 and let ρ ≥ 2. consider a sequence {tj}∞j=1 with tj+1 := ρ jt1, j = 1, 2, 3, . . . . if u(x0, tj), j = 1, 2, . . . ,n are known, then there exists f̄n in l 2[0,π] such that ||f − f̄n||l2 ≤ cn−r, (4.18) where c is a constant that depends on d0, r, t1 and ρ. proof: set t(t) = at. we can see that all the assumptions of theorem 4.1 are satisfied. therefore, the proof of the corollary follows from theorem 4.1. in the next example, we will particularly choose a function f and illustrate the accuracy of an approximation to f versus tk, ρ and n of theorem 4.1. example 4.4. consider r = 2 and define f : [0,π] → r by f(x) = 1 4 x(π −x). then a straightforward calculation gives us the kth fourier since coefficient f̂k = 2 π ∫ π 0 f(x) sin kx dx = 1 πk3 ( 1 + (−1)k+1 ) and also ||f||2hr = ∞∑ k=1 k2r|f̂k|2 ≤ ∞∑ k=1 k2r · 4 π2k6 = 4 π2 ∞∑ k=1 1 k2 = 4 π2 · π2 6 ≤ 1. on approximating initial data 9 thus f ∈ hr([0,π]). consider the following problem ut = −2t(−∆) 1 2 u, u(x, 0) = u(x,π) = 0, u(x, 0) = f(x) (4.19) then t(t) = t2, t ≥ 0. due to (1.6), the solution to this problem is u(x,t) = ∞∑ k=1 e−kt 2 f̂kek(x). (4.20) fix x0, t1 and ρ as in theorem 4.1 and consider the time sequence {tj}∞j=1 as t 2 j+1 = ρ jt21. consider n values u(x0, tj), j = 1, 2, 3, . . . ,n. using these n values, we will determine an approximation f̄n to f and will demonstrate the desired accuracy of f̄n versus tk, ρ and n. as in the proof of theorem 4.1, we will use u(x0, tn) to obtain an approximation c̄1 to c1 and u(x0, tn−k+1) to obtain an approximation c̄k to ck for k = 2, 3, . . . ,n. more precisely, from u(x0, tn) =∑∞ k=1 cke −kt2n where ck = f̂kek(x0), we have c1 = e t2nu(x0, tn) − ∞∑ j=2 cje (1−j)t2j and from u(x0, tn−k+1) = ∑∞ k=1 cke −kt2n−k+1 , k = 2, 3, . . . ,n, we have ck = e kt2n−k+1u(x0, tn−k+1) − k−1∑ j=1 cje (k−j)t2n−k+1 − ∞∑ j=k+1 cje (k−j)t2n−k+1. set c̄1 = e t2nu(x0, tn) and c̄k = e kt2n−k+1u(x0, tn−k+1) − k−1∑ j=1 c̄je (k−j)t2n−k+1, k = 2, 3 . . . ,n. using the method of the proof of lemma 4.2, we get |ck − c̄k| ≤ s(t1)2ke−t 2 n−k+1, k = 1, 2, 3, . . . ,n where s(t1) = ∑∞ j=1 e −jt21 . thus the kth fourier coefficient f̂k and its approximation ˆ̄fk = c̄k/ek(x0) give |f̂k − ˆ̄fk| ≤ |ck − c̄k| |ek(x0)| ≤ s(t1) d0 k2ke−t 2 n−k+1 where d0 as in (2.2). then an approximation to f defined by f̄n = ∑m k=1 ˆ̄fkek where m = dn2e satisfies ||f − f̄n||2l2 ≤ m∑ k=1 |f̂k − ˆ̄fk|2 + m−2r ≤ m∑ k=1 s(t1) 2 d20 k222ke−2t 2 n−k+1 + m−2r ≤ s(t1) 2 d20 e−2t 2 n−m+1 m∑ k=1 k222k + m−2r ≤ s(t1) 2 d20 e−2t 2 n−m+1n222n ·n + (n 2 )−2r =n−4 ( s(t1) 2 d20 e−2ρ n/2−1t21n722n + 4 ) . 10 r. karki but e−2ρ j/2−1t21j722j = e−2ρ j/2−1t21+7 ln j+2j ln 2 ≤ c0 for some constant c0 depending on t1 and ρ because{ e−2ρ j/2−1t21+7 ln j+2j ln 2 }∞ j=1 is a convergent sequence. using this into the last inequality, we get ||f − f̄n||l2 ≤ cn−2 for some constant c depending on (r = 2,) t1 and ρ, thereby verifying (4.1). 5. remarks we may ask whether our approximation method studied in the preceding sections work for more general problems like the following ut = −a(−∆)ηu, u(0, t) = u(π,t) = 0, u(x, 0) = f(x), (5.1) where a is a positive real number, η ∈ (0, 1] and f ∈br, and ut = −a(t)(−∆)ηu, u(0, t) = u(π,t) = 0, u(x, 0) = f(x), (5.2) where a is a positive continuous function of t > 0, η ∈ (0, 1] and f ∈br. even the method has worked for various special cases of these problems such as when a = 1 and η = 1 (see [9]); when η = 1 (see [1]) and when η = 1 2 , the method may require more advanced analytical tools related to spectral properties of unbounded self-adjoint operators on hilbert spaces (see [12, 15, 16]) to address these general cases. we may also ask whether it is possible for the current method to be applied for evolutionary equations with other boundary conditions such as the neumann and robin boundary conditions. it would be worth answering any of the above questions. references [1] r. aceska, a. arsie, r. karki, on near-optimal time samplings for initial data best approximation, matematiche (catania) 74 (2019), no. 1, 173-190. 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[16] g. sell, y. you dynamics of evolutionary equations, applied mathematical sciences, vol. 143, springer-verlag, 2002. corresponding author, natural science and mathematics, indiana university east, richomond, in 47374, u.s.a. email address: rkarki@iue.edu 1. introduction 2. choice of x0 and consistency of recovery 3. lower bound on optimal error 4. optimal approximation to initial data 5. remarks references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 2, number 1, march 2021, pp.10-21 https://doi.org/10.5206/mase/10093 a leader-followers game of emergency preparedness for adverse events myles nahirniak, monica cojocaru, and tangi migot abstract. natural disasters occur across the globe, resulting in billions of dollars of damage each year. effective preparation before a disaster can help to minimize damages, economic impact, and loss of human life. this paper uses a game theory framework to set up a leader-followers model for resource distribution to several geographic zones before an adverse event. the researchers model population members who may choose to prepare in advance of an event by acquiring supplies, whereas others may wait until the last minute. failure to prepare in advance could result in a significant loss due to the chance that supplies may no longer be available. numerical simulations are run to determine how the leader should distribute supplies to maximize the preparedness of the overall population. it was found that population size is a significant factor for supply distribution, but the behaviour of individuals within a zone is also important. much of the current resource allocation research focuses on the logistics and economics of supply distribution, but this paper demonstrates that social aspects should also be considered. 1. introduction throughout 2019 in the united states alone, there were 14 natural disasters each causing 1 billion usd or more of damage [13]. one example of such a disaster is hurricane dorian, which in august and september affected several caribbean nations, most notably the bahamas, as well as the eastern united states and provinces in atlantic canada. effective preparation in advance of a disaster is necessary to mitigate damages, financial costs, and loss of human life. a review by altay and green [1] identified a lack of cooperation between humanitarian agencies. as a response, galindo and batta [5] recommended modelling a leader agency that is responsible for overall coordination. muggy and heier stamm [10] agreed that cooperative models are lacking, and determined that game theory is an appropriate tool for determining an optimal allocation of resources. disaster relief is divided into four stages, known as mitigation, preparedness, response, and recovery [8]. our motivation for this paper is to use a bilevel game to model the distribution of supplies among various geographical zones in preparation for a disaster, while also considering how members of the population choose to prepare. many existing environmental and disaster relief models which use a game theory framework consider cases where relief agencies or suppliers either compete or cooperate. it is known that two parties cooperating to implement an environmental project is more effective than each party working independently [3]. previous studies have examined socio-cognitive reasons that members of a population may or may not prepare for adverse events, however these factors have not been included in mathematical models. even with advance knowledge, many individuals will choose not to prepare for a natural disaster [15]. the level of preparation amongst a population can depend on the ease of acquiring supplies, as well their ease of implementation [14]. another issue is the public’s compliance with recommended safety received by the editors 20 october 2020; accepted 12 january 201; published online 22 january 2021. 2000 mathematics subject classification. 91a40, 91b32, 91a65. 10 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/10093 a leader-followers game of emergency preparedness for adverse events 11 measures put forward by authorites; some population members may not have trust in the authority while others believe the adverse event will be less severe than predicted [2]. finally, lopes [7] determined that individuals overestimate their level of preparedness compared to their actual level of preparedness. in our paper, we posit that social aspects are significant, and incorporate these factors in our model. there is much literature with the focus on optimizing the cost and transport of supplies. nagurney and flores [12] develop an equilibrium model to ensure consistent flow of supplies under cost constraints. similar research was conducted to specifically analyze and reduce the transportation costs of supplies to different sites [6]. these models, however, ignore the problem of how to effectively allocate resources to multiple players. nagurney et al. [11] adapted a generalized nash game model for post-disaster relief, focusing on the distribution of funds. their model assumes multiple organizations that have all decided in advance the amount of relief to be provided at each point. in contrast, we propose an alternative approach. in our model, we consider how resources should be allocated to multiple geographic zones in advance of a disaster, to ensure that the maximum overall population is prepared. additionally, we consider the human behaviour aspect of how a population prepares in advance of a disaster. we model the situation as a leader-followers game in which an overseeing party is responsible for distributing supplies among several geographical zones. in each zone, the population is divided into two groups: one who chooses to acquire supplies in advance of a disaster and the other who waits until the last minute, knowing there is a risk that supplies will run out. we consider two other important factors in our analysis: first, we differentiate the populations in each zones not only by their size (total number of inhabitants per zone), but also by their cost of acquiring preparedness supplies in advance. the cost here should not be regarded as a dollar amount, but rather as a personal cost, in time, money and effort to acquire appropriate supplies. second, we differentiate the zones by assuming different probabilities of running out of distributed supplies and that costs of acquiring supplies may be reduced (by incentives) to nudge individuals to consider supplies in advance, rather than last minute. reductions in personal costs would be for instance: convenience of locations for supplies distribution centers, distribution of supplies ”at residence” for people of higher risk (such as senior citizens or long-term care homes), information campaigns, and ”drive-through” or delivery options, etc. this paper provides an overview of the model, and sets up the problems for the followers and the leader. we provide two types of analyses, based on parameter sensitivity analysis techniques. to begin the analysis in section 3, we first establish a “base case” scenario of the followers problem, then we complete a sensitivity analysis on their possible best strategies. we continue in section 4 with a further sensitivity analysis of the leader’s best decision based on models parameters and leader’s constraints. in section 5 we link the probability of running out of supplies in each zone to the choices of individuals in that zone. we re-evaluate the model under this new assumption, and contrast the results with our previous analyses. we conclude that population density and advance incentives are the most important factors for a leader to consider when allocating supplies. 2. game theoretic resource allocation model in diverse populations in this section, we develop a model for the distribution of resources among several geographical zones. we formulate the model as a leader-followers game. each zone’s followers is described by a symmetric two-player bimatrix game. the leader’s problem is formulated as an optimization problem, given the behaviour of the players in each zone. we will explain the parameters and assumptions used in the model below. consider a leader to be an authority designated to distribute supplies among n zones, labelled as n = 1, . . . ,n. this could represent the government, or a charitable organization providing disaster 12 m. nahirniak, m. cojocaru, and t. migot relief. for some total quantity of supplies, qtot, denote the amount distributed to each of the zones as qn, and any leftover supplies not distributed to be qremain. the total supplies are qtot = ∑n n=1 qn + qremain. the fraction of total supplies provided to a given zone is wn = qn qtot which implies ∑n n=1 wn ≤ 1. represent the population of each zone by pn. the population spread among the zones is given by pn = pn∑ n n=1 pn , where pn is the fraction of the total population in zone n. 2.1. followers’ game. we now assume that the players in each of the zones play the same game. their two pure strategies are represented by: 1 = ”stocking up on supplies before an adverse event”, or 2 = ”waiting until the last minute to purchase supplies”. advance purchasing may be viewed as carrying a lower personal cost to a player, whereas last minute search has a risk that supplies may have run out and the player may not be able to acquire needed supplies. we model the players’ choices as a symmetric bimatrix game in each zone, where the two players share the same matrix, a, of payoffs given by yn 1 −yn xn 1 −xn ( tn −cn tn −γcn tn −γcn tn − (qnln + (1 −qn)2cn) ) . in the above bimatrix, xn represents the frequency that player 1 in zone n plays strategy 1 = ”supply in advance”. because each player may only choose between one of two strategies, the value 1 − xn is the frequency that player 1 in zone n plays strategy 2 = ”supply last minute”. similarly, yn and 1−yn are the frequencies that player 2 in zone n plays strategies 1 and 2, respectively. the “budget” available to each player in zone n is denoted tn, and the cost of acquiring supplies is cn. there is a probability of running out of supplies, qn, which leads to the potential for either a large loss, ln, or for acquiring supplies at a larger personal cost. we assume that at the last minute, the remaining supplies require double the effort/cost to be acquired. there is an incentive factor for the cost of supplies if individuals from a single zone supply in advance, denoted by γ. with no incentive (γ = 0), individuals pay full price; a high value of γ provides an incentive to supply in advance, thus lowering the effective cost of supplies. this can signify, for example, that the time they spent for acquiring supplies is very short, or that supply distribution centres were conveniently located and well-stocked. the parameters in the model, tn,cn,ln,qn, and γ are normalized to be between 0 and 1. we assume further that a player must have enough funds to purchase supplies if desired, and the potential loss is greater than the cost of supplies, so 0 ≤ cn < ln ≤ tn. the expected payoff en for a player in zone n is given by en(xn,yn) = (xn, 1 −xn)a(yn, 1 −yn)t . a strategy (x∗n,y ∗ n) in zone n is a nash equilibrium if x ∗ n is the best response to y ∗ n and vice versa, that is en(x ∗ n,y ∗ n) ≥ en(x ∗ n,yn), ∀yn ∈ [0, 1], and since the game is symmetric, then x∗n = y ∗ n. direct computation allows us to determine the equilibrium points for the game. we find that the mixed equilibrium in the zone n is given by: x∗n = cnγ + 2cnqn − 2cn −lnqn 2cnγ + 2cnqn − 3cn − 2qn . we verify that 0 ≤ x∗n ≤ 1 in section 3. 2.2. leader’s problem. the leader’s goal is to supply as much of the population as possible, in advance of an event. this is done by optimizing the allocation of supplies to each zone and choosing appropriate incentives in the distribution process that would amount to a lesser personal cost cn for a a leader-followers game of emergency preparedness for adverse events 13 player to supply in advance. we assume that the probability of running out of supplies qn is different for each zone, depending on the behaviour of the population in that zone and the amount of supplies, wn, distributed. the leader’s problem then is given by: max (w,γ,x) θl := n∑ n=1 pnxn s.t. (w1, ...,wn,γ) ∈ [0, 1]n+1, 0 ≤ n∑ n=1 wn ≤ 1, xn ∈ arg max xn {en : 0 ≤ xn ≤ 1},∀n. (2.1) 3. sensitivity analysis of followers’ optimal strategies in a 2-zone allocation problem with fixed costs and losses for followers for ease of presentation, let us consider the leader’s problem (2.1), with n = 2 (i.e. for two zones). the leader’s function for two zones now becomes: θl(w1,w2,γ,x ∗) = p1x ∗ 1(p1,w1) + (1 −p1)x ∗ 2(p2,w2). using the solution points from the followers’ games: x∗1 = c1γ + 2c1q1 − 2c1 −l1q1 2c1γ + 2c1q1 − 3c1 − 2q1 x∗2 = c2γ + 2c2q2 − 2c2 −l2q2 2c2γ + 2c2q2 − 3c2 − 2q2 . to analyze the leader’s problem, we assume here an equal population distribution between the two zones (p1 = p2 = 0.5) and an equitable distribution of resources of w1 = w1 = 0.5, then the leader’s function becomes θl = 0.5x ∗ 1 + 0.5x ∗ 2. the leader’s function is maximized when the followers’ best strategy is maximal. thus we will strive to analyze the optimal strategy values for the followers. 3.1. base case. to better understand the problem, we now define a base case scenario, that is: l1 = l2 and c1 = c2, with c1 < l1. we then first compute the followers’ best strategy x ∗ 1 depending on possible values of γ,q1. we draw the evolution of x∗1 as a function of γ and q1 in the base case where c1 = c2 = 0.5 and l1 = l2 = 0.75 in figure 1. we see that, indeed, the values of the follower’s best strategy lie in [0, 1]. assuming an equal population distribution between the two zones (p1 = p2 = 0.5) and an equitable distribution of resources of w1 = w1 = 0.5, then the leader’s function θl is maximized when the followers best strategy is x∗1 = x ∗ 2 = 1 given in figure 1. we can exactly compute it to be θl = ∑2 n=1 0.5x ∗ n with a maximal value of θl = 1 when γ = 1, qn = 0, n = {1, 2} (i.e. all will supply in advance if incentive γ becomes maximal (γ = 1) and there is no possibility of running out of supplies (q1 = q2 = 0). 3.2. deviating from the base case. in this subsection we start to differentiate between the two zones, in the way we setup the follower’s input parameters cn,ln, n = {1, 2}. we look at the following two scenarios: • c2 = 1.5c1 and l1 = l2 = 0.75, where cost in zone 2 is 1.5 times higher than in zone 1, but losses are comparable; • c2 = c1 and l2 = l1 + 0.15, where losses in zone 2 are 20% higher than in zone 1, but costs are comparable. 14 m. nahirniak, m. cojocaru, and t. migot figure 1. evolution of x∗n as a function of γ and qn in the base case where cn = 0.5 and ln = 0.75, n = {1, 2}. in these cases, the plots of the followers’ best strategies are given in figure 2 and 3 as: we see that figure 2. evolution of x∗n as a function of γ and qn where c2 = 1.5c1 and l1 = l2 = 0.75, n = {1, 2}. the leaders’ function, under the base case scenario conditions (p1 = p2 = 0.5 and w1 = w2 = 0.5), is maximized whenever x∗n values are maximal for both populations. in both figure 2 and figure 3 we see that the leader’s function is maximized in the best of circumstances, i.e., all will supply in advance if incentive γ becomes maximal (γ = 1) and there is no possibility of running out of supplies (q1 = q2 = 0). what is interesting to look at is the least favourable scenario, i.e.: the case where there are no incentives (γ = 0) and there is a certainty of running out of supplies (q1 = q2 = 1). here, when losses are the same between zones (figure 2), individuals in the zone with higher supply cost will not supply in advance at all; on the other hand, when costs are the same but losses differ (figure 3) then both groups will supply in advance in some measure. this will lead to higher values for the leader’s function in the latter case. a leader-followers game of emergency preparedness for adverse events 15 figure 3. evolution of x∗n as a function of γ and qn where c1 = c2 = 0.5 and l2 = l1 + 0.15, n = {1, 2}. 4. sensitivity analysis of leader’s function in a 2-zone allocation problem with fixed costs and losses for followers we are now interested to examine a case where costs, as well as losses, are differentiated between zones, but are fixed throughout the analyses. we then present a full sensitivity analysis of the followers’ game in table 1, where the varying parameters are presented in table 2. c1 0.5 c2 0.75 l1 0.75 l2 0.9 table 1. table listing the fixed values of input parameters used in this case γ discounting factor [0, 1] qn prob. supplies run out in zone n [0, 1] wn supplies fraction to zone n [0, 1] p1 population density in zone 1 [0, 1] p2 population density in zone 2 p1 = 1 −p1 table 2. table listing the parameters we vary in the sensitivity analyses below in this problem. here n ∈{1, 2}. all simulations are conducted by selecting 500 randomly distributed points (γ,w1,w2,p1) ∈ [0, 1]4 that satisfy the conditions of the problem, and then computing the corresponding values for the leader’s objective function. we note that there is interdependence between the probability of running out of supplies in a zone, qn, its population density pn, and the allocation of resources wn. specifically, with higher population density in one zone, this probability may increase; on the other hand, with higher allocation of supplies, this probability may decrease. let us consider that the probability qn is proportional to pn and inversely proportional to wn. to ensure that 0 ≤ qn ≤ 1, we choose q1 = min(1, p1√w1 −p1) and q2 = min(1, 1−p1√ w2 − (1 −p1)). we show 16 m. nahirniak, m. cojocaru, and t. migot the plot of this functional dependency in figure 4 below, and note that qn saturates at 1 for large populations if supplies are not sufficiently allocated. figure 4. plot of qn as a function of pn and wn, n = {1, 2}. we use the following subcases to present our results: (1) we can observe the effect of varying the incentive factor, γ. (2) we consider how the population distribution pn between zones affects the allocation weights w1,w2. (3) finally, we distribute all parameters freely to determine the optimal values of the leader’s function. case 1. using a 30-70% population split, we show two scenarios: figure 5 shows the effect of γ while freely distributing supplies (left panel), and with all supplies distributed using w1 + w2 = 1 (right panel). figure 5. three parameters are freely distributed: (w1,w2 ≤ 1−w1,γ) while p1 = 0.3, q1 = p1√ w1 −p1 and q2 = 1−p1√w2 − (1 −p1). in the right panel we depict the case where supplies are exhausted, i.e. w1 + w2 = 1 a leader-followers game of emergency preparedness for adverse events 17 the incentive factor has a strong effect for large γ, and it can be seen that the objective function is much higher towards the top of the plot in the yellow zone (where γ = 1). this indicates that incentivizing the purchase of supplies at a lower effective cost increases the value of the leader’s objective function and allows more of the population to prepare in advance. additionally, below γ = 0.6, there is very little variation in the objective function values. hence values of γ ≥ 0.6 affect supply distribution. case 2. to observe the effect of varying population density between the two zones, we consider two scenarios: figure 7 shows the effect of p1 while freely distributing supplies, as well as distributing all supplies using w1 + w2 = 1. the highest objective values occur where the proportion of supplies distributed to a zone is comparable to the fraction of the population in that zone. for the purposes of illustration, a constant value of γ = 0.5 was used in both simulations, but the same conclusions can be drawn for any other value of γ ∈ [0, 1]. figure 6. two parameters are freely distributed: (w1 and p1) while γ = 0.5, q1 = p1√ w1 −p1 and q2 = 1−p1√w2 − (1 −p1). in the left panel, w2 is freely distributed with the constraint w1 + w2 ≤ 1, whereas the right panel depicts the case where supplies are exhausted, with w1 + w2 = 1. case 3. here the parameters (w1,w2,p1,γ) are freely distributed, with p2 = 1 −p1, and results are shown in figure 8. the weights and population size are plotted along the axes, with γ being represented by the size of the marker at each point. the optimal points occur in the top right (p1 = 1 and w1 = 1) and lower left (p1 = 0 and w1 = 0) regions of figure 8a. these correspond to the extreme cases where all of the population is in a single zone, and all supplies distributed to that zone, suggesting that higher weights are beneficial to the zone with higher population density. for mixed population distributions, the objective function is optimal when all supplies are distributed, i. e., w1 + w2 = 1 we also notice that higher values of γ increase the objective function, as large-sized points (high γ) close to small-sized points (low γ) have higher objective values. with all supplies distributed, as shown in figure 8b, higher γ values are preferred. 18 m. nahirniak, m. cojocaru, and t. migot figure 7. two parameters are freely distributed: (w1 and p1) while γ = 0.5, q1 = p1√ w1 −p1 and q2 = 1−p1√w2 − (1 −p1). in the left panel, w2 is freely distributed with the constraint w1 + w2 ≤ 1, whereas the right panel depicts the case where supplies are exhausted, with w1 + w2 = 1. figure 8. three parameters are freely distributed (w1,p1,γ) with p2 = 1 −p1, q1 = min ( p1√ w1 −p1, 1), and q2 = min ( 1−p1√w2 − (1 −p1), 1). in the left panel, w2 is freely distributed with the constraint w1 + w2 ≤ 1. the size of each point corresponds to the value of γ. the right panel shows the case where w1 + w2 = 1. 5. interplay of supplies between zones in this section, we further assume that the probability of running out of supplies depends in general on the supply strategies in both zones. we can write functions for q1 and q2 in terms of the parameters a leader-followers game of emergency preparedness for adverse events 19 x1,x2,w1, and w2, noting w1 + w2 = 1. let q1 = x21 w1 + x1x2 2w1w2 − x1 2w1 and q2 = x22 w22 + 2x1x2 5w1w2 − x2 2w2 + 1 10 by inserting the above expressions for q1 and q2 into the followers’ games and making the choice γ = 0.5, the payoffs in each zone are: e1 = x1(− x1 4 + 3 4 ) + (1 −x1)( 3x1 4 + ( x21 4w21 + x1x2 8w1w2 − x1 8w1 )(1 −x1)) and e2 = x2(− 3x2 8 + 5 8 ) + (1 −x2)( 5x2 8 + (− 11 25 + 3x22 5w22 + 6x1x2 25w1w2 − 3x2 10w2 )(1 −x2)). the computations are a two-step process. first we distribute w1 ∈ [0, 1] with a step size of 0.05 and using the constraint w1 +w2 = 1, then solve the followers’ game for each pair of values. it is worth noting that different choices of γ significantly modify the followers’ payoffs. using the equilibrium points x∗1 and x∗2 from these games, we can simultaneously solve for the values of q1 and q2 in the above formulae. once q1 and q2 are known, we again solve the followers’ game using those probabilities to gain x1 and x2. this finally allows us to optimize the leader’s function to determine the optimal supply distribution between the two zones. the solutions to the game are given in figure 9. figure 9. solutions x∗1 (left panel) and x ∗ 2 (right panel) of the game with γ = 0.5, varying supply allocation and population between zones . to observe the effects of relative population sizes, we additionally freely distribute the parameter p1, with the constraint p1+p2 = 1, before solving the followers’ game. similarly, to determine the impact of the incentive factor, we can vary γ before solving the followers’ game. this leads to the results in figure 10. 20 m. nahirniak, m. cojocaru, and t. migot figure 10. leader’s function with γ ∈{0, 0.5, 1}, varying supply allocation and population between zones. . it turns out that γ has a large effect on the leader’s function, and in fact, the leader’s values are optimal at the points where γ = 0. this is in contrast to the previous models from sections 3 and 4 where high γ values were preferred. this shows that with the interplay between both zones, larger objective function values are found if no incentive is provided. in examining population size, there is a strong correlation between the size of the population in a zone and the optimal weighting of supplies distributed to that zone. when one zone’s population is double the other’s or greater (i.e., when p1 ≥ 2p2), there is a bias towards distributing most of the supplies to the larger zone. if both zones are of similar size, the leader has flexibility as to the weighting, with minimal impact on the objective function. 6. discussion and conclusion this research examines the problem of resource allocation to multiple geographical zones to prepare for an adverse event using a bilevel game-theoretic approach. the focus is to incorporate the choices of the population on whether or not to prepare in advance, as well as zonal population densities and probabilities of running out of supplies. we use a leader-followers problem to optimise the supplies allocation to each zone for the leader, in order to supply as much of the population in advance as possible. the main factors impacting the leader’s decision and allocation schemes are zonal population densities, incentivizing the acquisition of supplies in advance by population groups (for instance groups “at-risk”), and the probabilities of allocated supplies to be exhausted. in the case of 2 zones with equal populations, the leader’s function is optimal for values of γ approaching 1, as in figures 1, 2, and 3. this indicates that the leader should incentivize the purchase of supplies in order to maximize the objective function. with the additional assumption that available supplies depend on the population size of each zone, again, high γ values are preferred. the functional form of the probability of running out of supplies is highly important: in the situation where supply availability is also dependent on the supply strategies in each zone, γ = 0 is the best choice as in figure 10. with the interdependence of both zones, neither zone should receive an incentive. in the model from section 3, the result is straightforward that larger zones receive more supplies. as soon as the probability of running out of supplies takes a functional form, the optimal supply distribution is less clear, as is evident from figure 8. in general, supplying the larger zone is still preferred. for the a leader-followers game of emergency preparedness for adverse events 21 case in section 5 with q1 and q2 based on the strategies of each player, if the zones are comparably sized, there is a large region in figure 10 in which the objective function is roughly constant. this suggests that the leader has freedom in how to distribute supplies to the zones, without significantly affecting how much of the population supplies in advance. however, if the population sizes are dramatically different, then the best policy is to supply the largest zone. future research will focus on the case where the populations in the different zones are subject to 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[15] k. j. tierney, m. lindell and r. perry, facing the unexpected: disaster preparedness and response in the united states, joseph henry press, washington dc, 2020. https://doi.org/10.17226/983. corresponding author, department of mathematics and statistics, university of guelph, 50 stone rd e., guelph, on, n1g 2w1 e-mail address: mnahirni@uoguelph.ca department of mathematics and statistics, university of guelph, 50 stone rd e., guelph, on, n1g 2w1 e-mail address: mcojocar@uoguelph.ca department of mathematics and statistics, university of guelph, 50 stone rd e., guelph, on, n1g 2w1 current address: department of mathematics and industrial engineering, polytechnique montral, montral, canada e-mail address: tangi.migot@gmail.com https://doi.org/10.1007/978-3-319-97442-2/ https://www.ncdc.noaa.gov/billions/ https://doi.org/10.17226/983 1. introduction 2. game theoretic resource allocation model in diverse populations 2.1. followers' game 2.2. leader's problem 3. sensitivity analysis of followers' optimal strategies in a 2-zone allocation problem with fixed costs and losses for followers 3.1. base case 3.2. deviating from the base case 4. sensitivity analysis of leader's function in a 2-zone allocation problem with fixed costs and losses for followers 5. interplay of supplies between zones 6. discussion and conclusion references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 3, number 1, march 2022, pp.12-23 https://doi.org/10.5206/mase/14018 application of ant colony optimization metaheuristic on set covering problems christian alvin h. buhat, jerson ken l. villamin, and genaro a. cuaresma abstract. ant colony optimization (aco) metaheuristic is a multi-agent system in which the behaviour of each ant is inspired by the foraging behaviour of real ants to solve optimization problem. set covering problems (scp), on the other hand, deal with maximizing the coverage of every subset while the weight nodes used must be minimized. in this paper, aco was adapted and used to solve a case of set covering problem. the adapted aco for solving the scp was implemented as a computer program using scilab 5.4.1. the problem of determining the optimal location of wi-fi access points using the 802.11n protocol in the up los banos math building (mb) was solved using this metaheuristic. results show that in order to have 100% coverage of the mb, 7 access points are required. methodology of the study can be adapted and results of the study can be used by decision makers on related optimization problems. 1. introduction the set covering problem (scp) is a typical optimization problem that serves as a model for many applications in the real world. it is a combinatorial optimization problem (cop) which can be formulated as an integer linear programming (ilp) where the goal is to minimize the number of sets such that every element of the set from the universe will be covered and every set is either in the set cover or not. scp has been used to model problems such as vehicle routing, nurse scheduling, airline crew scheduling, facility location problem, and resource allocation problem (schiff, 2013). according to lessing et al (2004), scp is an np-hard problem with many developed algorithms that can be used to solve it. some algorithms can solve a limited size and is also time consuming that is why metaheuristic is advised for solving such problem. metaheuristic is used to define heuristic methods and it is also a set of algorithmic concepts applicable to a very large set of problems. ren, et al. (2008), reported that list of kind of heuristics were applied for the scp and these are genetic algorithm (ga), simulated annealing algorithm, and tabu search algorithm. they also conducted a research to solve scp based from the recently developed, populationbased metaheuristic named ant colony optimization (aco) algorithm. aco was published in the early 90’s and it is a probabilistic technique about the simulation of real behavior of ants regarding how they function through indirect communication via pheromones (brezina and cickova, 2011). pheromones are chemical substances excreted by ants to attract other ants that are seeking for food. biologists have observed that ants tend to find the path with the shortest distance between a source of food and their colony which leads in the conclusion that it would be useful to study their individual behavior (buezas. 2010). solutions to a given optimization problem are obtained received by the editors 26 may 2021; accepted 11 january 2022; published online 20, january 2022. 2020 mathematics subject classification. primary 90b50, 90b80; secondary 90b10, 90b90. key words and phrases. constrained optimization, decision theory, information technology, optimal placement, metaheuristics. 12 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14018 application of ant colony optimization metaheuristic on set covering problems 13 by a set of software agents called artificial ants. in applying aco, the optimization problem must be transformed into the problem of finding the best path on a weighted graph. hereafter, ants incrementally gets solutions as the graph goes on. aco is commonly used in solving traveling salesman problem (tsp), but it is also applicable on set covering problems (dorigo and stutzle, 2004). given a set of facilities and the areas covered by the facility, a number of ants determine what facility to choose using a probability function for aco which is based on pheromone trails and heuristic information, while the area it chooses is randomized. after the chosen areas by the ant satisfies the system constraints or covers all the areas in the system, a next ant will proceed, and the pheromone trails and heuristic information will be updated. this will continue until all ants have passed, and the best solution for the problem is returned and recorded. on 2008, ren et al proposed a novel aco-based approach, called ant-cover. it has three main differences between other existing aco-based approaches: first, for constructing solutions, the approach used single-row-oriented method, next, the use of lagrange dual information for the search in the solution space, lastly, the use of local search procedure is developed to maintain the feasibility and for the solutions to be improved by ants. this study used a customized ant-cover to get a faster near-optimal solution than other metaheuristics. due to the numerous problems about set covering, this paper is to adapt the ant colony optimization algorithm to solve a set cover problem. in line, this paper is concerned in determining the optimal placement of wi-fi access points in the university of the philippines los baños (uplb) math building using aco metaheuristic. 2. theoretical framework the set covering problem is a binary integer programming model that seeks to find the minimum number of facilities to be installed and their locations to each demand node is covered by at least one facility (torregas, 1970). let i = set of all possible locations for installing a facility, j = set of all areas which are to be covered, xi = { 1, if a facility is to be installed at site i, 0, otherwise; yj = { 1, if whenever a facility is installed at site i the facility covers area j, 0, otherwise. the model can be constructed as: minimize z = ∑ i∈i xi, (2.1) subject to ∑ i∈i xiyij ≥ 1, ∀j ∈ j; (2.2) xi = {0, 1}, ∀i ∈ i. (2.3) the objective function (1) determines the goal of the problem which is to minimize the number of facilities used. the first constraint (2) satisfies the condition that each area j should be covered by at least one facility and the second constraint (3) serves as the binary constraint. 14 cah. buhat, jkl. villamin, and ga. cuaresma 3. results and discussion pseudocode this paper adapts a similar pseudocode to ren et al. (2008) which uses aco on solving scp. the following is the procedure of adapting aco to solving scp. pre-process the given scp and initialize results initialize pheromone trails, heuristic information, and related parameters while termination condition is not met do for ant i=1 to n (number of ants) do (a)construct a solution based on srom if system constraints are satisfied then move to next ant else repeat (a) end for update pheromone trails and heuristic information end while return the best solution found preprocessing of scp (general) based on the scp formulation, we transform the problem into an m×n matrix, with m being the areas to be covered and n being the facilities. an element of the matrix will have a value of 1 defining that that area has been covered under that facility, while 0 if it has yet to be covered, i.e., m(i,j) = { 1, if facility j covers area i, 0, otherwise. table 1. example of a preprocessed scp model 1 2 3 4 5 6 . . . . n 1 1 0 0 1 0 1 0 0 0 0 1 2 0 1 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 4 0 0 0 1 0 0 0 0 0 0 1 5 0 0 0 0 0 1 0 0 0 0 0 6 0 1 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 1 0 0 0 . 1 0 0 0 1 0 0 0 0 0 0 . 0 0 0 0 0 0 1 0 0 0 0 . 0 0 0 0 0 0 0 0 0 1 0 m 1 0 0 0 0 0 0 0 0 0 0 in the above matrix, this means that area 1 can be covered by facilities in locations 1, 4, 6, ...,n. parameters used values of parameters used in metaheuristics such as aco are determined by the proponent where a slight change in such values can generate a large difference in the results. in this paper we used values suggested by ren and others (2008) for the value of our β and ρ. meanwhile, values for number of application of ant colony optimization metaheuristic on set covering problems 15 iterations, initial pheromone trails and heuristic information are standard values used in aco. values and definition of variables are given in the table below. table 2. definitions and values of parameters used variable definition value reference β importance of heuristic factor to pheromone 5 ren and others (2008) ρ pheromone persistence 0.99 ren and others (2008) ants number of iterations 1000 assumed tau(1) initial pheromone trail 1 assumed heu(1) initial heuristic information 1 assumed single row-oriented solution construction method is a novel single-row-oriented solution construction method which can generate solutions in a faster way (compared to other branch-and-bound/branchand-cut algorithms) and allows ants to explore broader solution areas (ren and others, 2008). two of the main ideas of srom on how it adapts aco in solving scp are to: • randomly select a row i from the row set i • select a column from the column set j using the probability • randomly select a row i from the row set i • select a column from the column set j using the probability p (st = j|rt = i, i ∈ rt−1) =   τjη β j∑ q∈ji τqη β q if j ∈ j, 0 otherwise, where rt−1 denotes the set of uncovered rows before step t, and rt denotes the row chosen at step t. the time complexity of selecting a column from the set ji is o(|ji|). let d be the density of scp, the average number of columns covering a row can be obtained, ñ = dn. then we can get a complete solution s with k columns in o(kñ) time. this time complexity is linear with the value of ñ. generally, ñ � n holds (ren and others, 2008). pheromone trail update pheromone trails are decreased uniformly in order to simulate evaporation and allow ants to forget part of the history experience. ants then deposit an amount of pheromone on the column contained in the best solution. after all the ants have completed a solution, the trails are updated as follows: τj ← ρτj + ∆τj, ∀j ∈ j, ∆τj =   1∑ q∈sgb cq if j ∈ sgb, 0 otherwise. here, ρ = pheromone persistence (0 ≤ ρ < 1), ∆τj = amount of pheromone put on column j,∑ q∈sgb cq = total amount of cost. 16 cah. buhat, jkl. villamin, and ga. cuaresma pheromone trail updates help the model to adapt to real-time changes, which is one of its advantages over other heuristics such as simulated annealing and genetic algorithm (selvi and umarani, 2010). heuristic information update heuristic information considers the dual information associated with the still uncovered rows. it mines the specific information of the scp and provides a good guideline for ants’ search in the solution space. it is updated through: ϕj = |ij ∩r| , ηj = ϕj cj , where: ϕj = number of new rows that can be covered by column j, r = set of still uncovered rows. these updates help the considered artificial ants in the heuristic to obtain the optimal solution that we need for our lp model (abd-alsabour and others, 2013). program description application of ant colony optimization metaheuristic on set covering problem program generates a near-optimal solution to a preprocessed scp using aco metaheuristic. aco metaheuristic was implemented in scilab 5.4.1 under microsoft windows 7 on intel pentium dual cpu 2.16 ghz, 2.0 gb ram with parameter values, β = 5, ρ = 0.99, ants=1000, tau(1)=heu(1)=1. the user inputs a specified number of facilities and areas associated, or provides a preprocessed m×n matrix which will then be run to provide a near-optimal value, with its associated optimal solution, number of iterations needed before attaining the near-optimal value, and the running time of the program. table 3 summarizes the description of the aco on scp program. table 3. program description of aco on scp program title application of ant colony optimization metaheuristic on set covering problems description this program solves for a nearoptimal solution of a set covering problem by applying aco metaheuristic programming language scilab 5.4.1 input number of facilities and areas associated/ m x n matrix (preprocessed scp) output optimal value of the problem, associated optimal solution, number of iterations, minimum running time 4. application in order to further determine the effectiveness and efficiency of how aco adapts in solving aco, the study was applied to a local set covering problem and used to determine the optimal place of wi-fi routers in math building. data used was provided by the uplb-itc and ovcpd. in this paper, 802.11n wi-fi routers were considered as the facility, while area was based on the division of area/rooms in math building. since math building is a two-storey building, the problem was divided into two parts (1st floor and 2nd floor) for better explanation of the study and application. note however that the problem can also be formulated as one scp model. constraints of the model indicate that an area should be covered by at least one repeater. this will guarantee that the entire math building is covered. application of ant colony optimization metaheuristic on set covering problems 17 as in section 2, the related variables are defined as below in the context of wifi routers now: let i = set of all possible locations for installing a wi-fi router j = set of all rooms/areas which are to be covered xi = { 1, if a wi-fi router is to be installed at room i 0, otherwise yj = { 1, if whenever a wi-fi router is installed at room i the facility covers room j 0, otherwise figure 1 serve as a map where xi’s are the areas in math building being covered, i being the area number, as shown: figure 1. old math building 1st (a) and 2nd (b) floor plan generated through autocad math building first floor: minimize z = 18∑ i=1 xi 18 cah. buhat, jkl. villamin, and ga. cuaresma subject to the contraints x1 + x2 + x3 ≥ 1 x1 + x2 + x3 + x4 + x12 ≥ 1 x1 + x2 + x3 + x4 + x5 + x12 + x13 ≥ 1 x2 + x3 + x4 + x5 + x12 + x13 ≥ 1 x3 + x4 + x5 + x6 + x12 + x13 ≥ 1 x5 + x6 + x7 ≥ 1 x6 + x7 + x8 + x9 + x10 ≥ 1 x6 + x7 + x8 + x9 + x10 ≥ 1 x7 + x8 + x9 + x10 ≥ 1 x7 + x8 + x9 + x10 + x11 ≥ 1 x9 + x10 + x11 + x18 ≥ 1 x3 + x4 + x12 + x13 + x16 ≥ 1 x3 + x4 + x5 + x12 + x13 + x16 + x17 ≥ 1 x14 + x15 ≥ 1 x12 + x13 + x15 + x16 ≥ 1x12 + x13 + x15 + x16 + x17 ≥ 1 x12 + x13 + x16 + x17 + x18 ≥ 1 x10 + x11 + x18 ≥ 1 xi = {0, 1}, i = 1, 2, 3, ..., 18 math building second floor: minimize z = 10∑ i=1 xi subject to x1 + x2 + x3 ≥ 1 x1 + x2 + x3 + x4 + x9 ≥ 1 x1 + x2 + x3 + x4 + x5 + x9 + x10 ≥ 1 x2 + x3 + x4 + x5 + x9 + x10 ≥ 1 x3 + x4 + x5 + x6 + x9 + x10 ≥ 1 x4 + x5 + x6 + x7 + x9 ≥ 1 x5 + x6 + x7 + x8 ≥ 1 x6 + x7 + x8 ≥ 1 x2 + x3 + x4 + x5 + x9 + x10 ≥ 1x3 + x4 + x5 + x9 + x10 ≥ 1 xi = {0, 1}, i = 1, 2, 3, ..., 10 preprocessing of scp (specific) application of ant colony optimization metaheuristic on set covering problems 19 using the definition of the decision variables and a range of 140ft., with xi, i=1,...,18 for the first floor and j=1,...,10 for the second floor, the pre-processed matrix for the scp is given in table 4 and 5 for the first and second floors respectively in the math building. math building 1st floor: table 4. m x n matrix representing scp formulation of mb first floor 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 1 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 4 0 1 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 5 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 10 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 11 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 12 0 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 13 0 0 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 16 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 17 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 18 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 math building second floor: table 5. m x n matrix representing scp formulation of mb second floor 1 2 3 4 5 6 7 8 9 10 1 1 1 1 0 0 0 0 0 0 0 2 1 1 1 1 0 0 0 0 1 0 3 1 1 1 1 1 0 0 0 1 1 4 0 1 1 1 1 1 0 0 1 1 5 0 0 1 1 1 1 1 0 1 1 6 0 0 0 0 1 1 1 1 0 0 7 0 0 0 0 0 1 1 1 0 0 8 0 0 0 0 0 0 1 1 0 0 9 0 1 1 1 1 1 0 0 1 1 10 0 0 1 1 1 0 0 0 1 1 20 cah. buhat, jkl. villamin, and ga. cuaresma program results after 1000 iterations/ants, aco on scp program determined a near-optimal solution of 7 for the uplb math building (5 for the first floor, 2 for the second floor). these wi-fi routers must then be placed at area number: 3, 6, 10, 12, 14 for the first floor, and 3, 6 for the second floor. however, the associated optimal solution given by the program may not be unique and may have other results which still give a near-optimal solution to the problem. optimal wi-fi routers: 1st floor: 5 2nd floor: 2 associated optimal solution: 1st floor: 0-0-1-0-0-1-0-0-0-1-0-1-0-1-0-0-0-0 2nd floor: 0-0-1-0-0-1-0-0-0-0 figure 2. old math building 1st (a) and 2nd (b) floor plan with optimal wi-fi placement comparison to dynamic programming using linear program solver (lips), a software used on solving linear, integer and goal programming problems, results of aco on scp program have been compared to that of lips to determine how their results vary. for the 1sr floor of math building: optimal value run time number of iterations areas covered aco on scp program 5 40 seconds* 160 100% lips 5 .33 seconds 24 100% table 6. comparison between results of aco on scp program and lips on math building 1st floor. * the time it took to finish the whole number of iterations. application of ant colony optimization metaheuristic on set covering problems 21 for the 2nd floor of math building: optimal value run time number of iterations areas covered aco on scp program 2 11 seconds* 4 100% lips 2 .07 seconds 13 100% table 7. comparison between results of aco on scp program and lips on math building 2nd floor. * the time it took to finish the whole number of iterations. both tables show that both programs obtained the same optimal value for each scp although aco on scp program may not be consistent since it only computes for a near-optimal value. it can also be observed that the running time for both programs show that they are quite on par with each other. though acoonscp has a larger running time because it finishes 1000 iterations before producing an optimal solution, expanding the problem to a large number of variables (representative to the real-world problems) will give lips more computing time since it uses simplex to compute for the optimal solution. not to mention, more difficulty in manually inputting values and the limitation on the number of variables and constraints that the lips program have (hood, 2016). aco on scp program is not consistent on number of iterations before producing an optimal result, but it is quite expected since it has a nature of a metaheuristic and operates based mainly on random numbers. still, 100% of areas have been covered by both programs using the associated optimal solution they generated. also, since the ants used in the study are “normal ants” in aco terms, adding more ants/iterations to the acoonscp will not have that much effect on the result, and may even further prolong the iteration time (johansson and pettersson, 2018). comparison to current itc wi-fi set-up according to the data from the university of the philippines los baños information technology center (uplbitc), math building currently has 1 wi-fi router installed on a certain area which covers 20% of the whole math building. meanwhile, based on the result of aco on scp program, 7 are needed to cover the entire math building (1st floor and 2nd floor). number of routers areas covered aco on scp program 7 100% lips (dynamic programming) 1 20% table 8. comparison between proposed number of aco on scp program to current wi-fi placement 5. summary and conclusion in this research, ant colony optimization metaheuristic was applied to the set covering problem. single row oriented method was used to generate solutions. pheromone trails and heuristic information were updated to provide a faster and near-optimal solution. solutions were returned, and the optimal solution was computed through a scilab 5.4.1 program. specifically, aco was applied to a 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[24] n. abd-alsabour, h. hefny and a. moneim, heuristic information for ant colony optimization for the feature selection problem, ieee conference anthology, (2013), 1-5. doi:10.1109/anthology.2013.6784795. http://www.math.clemson.edu/~mjs/courses/mthsc.440/integer.pdf http://www.math.clemson.edu/~mjs/courses/mthsc.440/integer.pdf https://docplayer.net/27853871-vassilis-kostoglou-url-linear-programming-1.html https://www.diva-portal.org/smash/get/diva2:1214402/fulltext01.pdf https://www.diva-portal.org/smash/get/diva2:1214402/fulltext01.pdf doi: 10.1109/anthology.2013.6784795 application of ant colony optimization metaheuristic on set covering problems 23 corresponding author, institute of mathematical sciences and physics, university of the philippines los banos, laguna, philippines, 4030 department of mathematics, university of houston, tx, usa, 77204 email address: chbuhat@uh.edu institute of mathematical sciences and physics, university of the philippines los banos, laguna, philippines, 4030 email address: jersonvillamin@gmail.com institute of mathematical sciences and physics, university of the philippines los banos, laguna, philippines, 4030 email address: gacuaresma@up.edu.ph 1. introduction 2. theoretical framework 3. results and discussion 4. application 5. summary and conclusion references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 2, number 1, march 2021, pp.22-31 https://doi.org/10.5206/mase/13393 traveling waves in cooperative predation: relaxation of sublinearity srijana ghimire and xiang-sheng wang abstract. in this paper, we investigate traveling wave solutions of a diffusive predator-prey model which takes into consideration hunting cooperation. the sublinearity condition is violated for the function of cooperative predation. when the basic reproduction number for the diffusion-free model is greater than one, we find a critical wave speed below which no positive traveling wave solution shall exist. on the other hand, if the wave speed exceeds this critical value, we prove the existence of a positive traveling wave solution connecting the predator-free equilibrium to the unique positive equilibrium under a technical assumption of weak cooperative predation. the key idea of the proof contains two major steps: (i) we construct a suitable pentahedron and find inside it a trajectory connecting the predator-free equilibrium; and (ii) we construct a suitable lyapunov function and use lasalle invariance principle to prove that the trajectory also connects the positive equilibrium. at the end of this paper, we propose five open problems related to traveling wave solutions in cooperative predation. 1. introduction the lotka-volterra system has been widely used in the models of predation ever since lotka and volterra did two independent studies [17, 20] near one century ago. taking spatial diffusion into consideration, a general diffusive lotka-volterra system can be written as ∂tu = d1∂xxu + b(u) −f(u,v), (1.1) ∂tv = d2∂xxv + rf(u,v) −µv, (1.2) where u(t,x) and v(t,x) correspond to the densities of prey and predator, respectively. the constants d1 and d2 are two non-negative diffusion rates. in the absence of the predator, the dynamics of prey is determined by the function b(u), which usually possesses a stable positive equilibrium k. the constant k is also referred to as the carrying capacity in the literature. the only nonlinearity in the model system is the predation rate f(u,v) which is an increasing function of both u and v, and f(0,v) = f(u, 0) = 0. the constants r and µ are rates of energy conversation and predator death, respectively. it is obvious that the model system has a predator-free equilibrium (k, 0). we also assume that it has a positive equilibrium (u,v ), where v = rb(u)/µ and u is a positive root of b(u) − f(u,rb(u)/µ) = 0. a traveling wave solution connecting the predator-free equilibrium to the positive equilibrium is a solution of the special form (u(x + ct),v(x + ct)) with c > 0 such that (u(−∞),v(−∞)) = (k, 0) and (u(∞),v(∞)) = (u,v ). there are plenty of results on the existence received by the editors 6 december 2020; accepted 29 january 2021; published online 1 february 2021. 2000 mathematics subject classification. primary 92d25; secondary 35k57, 34b40. key words and phrases. predator-prey model; cooperative predation; traveling waves; sublinearity. 22 traveling waves in cooperative predation: relaxation of sublinearity 23 theory of traveling wave solutions to the lotka-volterra system of predation; see [5, 10, 11, 12, 13, 16, 22] and references therein. in the aforementioned literature, sublinearity of the predation rate is a crucial condition to prove existence of traveling wave solutions; that is, one should assume that there exists m > 0 such that f(u,v) ≤ mv (1.3) for all u ∈ [0,k] and v ≥ 0. the objective of this paper is to relax this sublinearity condition and establish existence theory of travelling waves for a diffusive model of cooperative predation. cooperative hunting plays an important role in phylogenetics [2] and is necessary for some predators [19]. for example, the yellowstone wolves of larger group size have bigger success rates in capturing bison, their most formidable prey [18]. during the season of nonbreeding, harris’ hawks in new mexico will improve capture success by cooperative hunting [3]. d. discoideum, which is a soil amoeba feeding on bacteria, lives as single cells during most of the time, but will develop social cooperation if it is under starvation conditions [4, 15]. a two-dimensional lotka-volterra system was proposed in [1] which considers logistic growth of the prey and cooperative hunting on the predator. this model was extended in [14] to investigate allee effects on the prey. in this paper, we will introduce spatial diffusion to the predator and study the following diffusive predator-prey model with cooperative predation. ∂tu = α− δu−puv −quv2, (1.4) ∂tv = d∂xxv + r(puv + quv 2) −µv, (1.5) where α and δ are the constant birth rate and per capita death rate of the prey. the nonlinear predation rate is composed of mass-action predation puv and cooperative predation quv2. for simplicity, we neglect the spatial diffusion of the prey and assume that the predator has a constant diffusion rate d > 0. the conversation rate of predation energy is r > 0 and the per capita death rate of the predator is µ > 0. it is obvious that the model system has a predator-free equilibrium (k, 0), where k = α/δ is the carrying capacity of the prey. define r0 = rpk µ = rpα µδ (1.6) as the basic reproduction number for the diffusion-free ordinary differential system. throughout this paper, we will assume that r0 > 1. a simple argument shows that the model system possesses a unique positive equilibrium (u,v ) with α− δu = puv + quv 2 = µv/r. (1.7) we are interested in the traveling wave solution (u(x + ct),v(x + ct)) that connects the predator-free equilibrium (k, 0) to the positive equilibrium (u,v ). as we mentioned earlier, the nonlinear predation rate f(u,v) = puv +quv2 does not satisfy the sublinearity condition (1.3) and the results in the existing literature do not apply. to overcome this difficulty, we will modify the shooting method which was introduced by dunbar [6, 7], and later developed by hosono and ilyas [8, 9] and by huang [12, 13]. especially, we will apply an extension of the technique introduced by huang [12, 13] to prove the existence of a positive traveling wave connecting the predator-prey equilibrium. the rest of this paper is organized as follows. in section 2, we introduce non-dimensional scales for system (1.4)-(1.5) and state our main theorem. in section 3, we convert the traveling wave equations to a three-dimensional dynamical system and construct a suitable pentahedron. in section 4, we analyze 24 srijana ghimire and xiang-sheng wang the unstable manifold of the predator-free equilibrium and find a trajectory which approaches the predator-free equilibrium as ξ → −∞ and stays in the pentahedron for all ξ ∈ r. in section 5, we use lyapunov function technique to prove that this trajectory converges to the positive equilibrium as ξ →∞. in section 6, we give a proof of our main theorem. in section 7, we conclude our paper with a brief discussion and propose some open problems. 2. nondimensionalization and main theorem for convenience, we nondimensionalize the model by scaling the variables: t̃ = δt, x̃ = x√ d/δ , ũ = u k , ṽ = v rk . (2.1) recall that k = α/δ. the system (1.4)-(1.5) is rewritten as ∂t̃ũ = 1 − ũ− p̃ũṽ − q̃ũṽ 2, (2.2) ∂t̃ṽ = ∂x̃x̃ṽ + p̃ũṽ + q̃ũṽ 2 − µ̃ṽ, (2.3) where p̃ = prk δ , q̃ = qr2k2 δ , µ̃ = µ δ . (2.4) hence, we may assume without loss of generality that α = δ = d = r = 1 and drop the tilde in the above system. denote ξ = x + ct. the traveling wave solution is a solution to the boundary value problem cu′ = 1 −u−puv −quv2, (2.5) cv′ = v′′ + puv + quv2 −µv, (2.6) together with the boundary conditions u(−∞) = 1, v(−∞) = 0, u(∞) = u, v(∞) = v, (2.7) where 1 −u = puv + quv 2 = µv. (2.8) the condition r0 > 1 is simplified as p > µ. our main theorem is stated as below. theorem 2.1. let α = δ = d = r = 1. if p > µ, then for any c > 2 √ p−µ, there exists q > 0 such that the boundary value problem (2.5)-(2.7) with q ∈ [0,q] has a positive solution which corresponds to a traveling wave solution to (1.4)-(1.5) connecting the predator-free equilibrium to the positive equilibrium. on the other hand, if p > µ and 0 ≤ c < 2 √ p−µ, then the boundary value problem (2.5)-(2.7) with q ≥ 0 does not have a positive solution. 3. dynamical system approach we convert the boundary value problem (2.5)-(2.6) into a three-dimensional dynamical system cu′ = 1 −u−puv −quv2, (3.1) v′ = cv − cw, (3.2) cw′ = puv + quv2 −µv. (3.3) traveling waves in cooperative predation: relaxation of sublinearity 25 we intend to find a trajectory such that the boundary conditions (2.7) are satisfied. let ε > 0 be a positive constant to be determined later. we obtain from the above equations (cu + εv + cw)′ = 1 −ε(cu + εv + cw) − (1 − cε)u− (µ− cε−ε2)v. (3.4) we will choose ε to be sufficiently small so that 1 − cε > 0, µ− cε−ε2 > 0; (3.5) namely, we require ε < min{ 1 c , c + √ c2 + 4µ 2 }. (3.6) let 0 < k1 < k2 be two positive constants to be determined later. we introduce a pentahedron p with two parallel triangular bases b0 := {u = 0, k1v < w < k2v, cu + εv + cw < 1/ε}, (3.7) b1 := {u = 1, k1v < w < k2v, cu + εv + cw < 1/ε}, (3.8) and three trapezoid sides s0 := {cu + εv + cw = 1/ε, 0 < u < 1, k1v < w < k2v}, (3.9) s1 := {w = k1v, 0 < u < 1, cu + εv + cw < 1/ε}, (3.10) s2 := {w = k2v, 0 < u < 1, cu + εv + cw < 1/ε}. (3.11) we have the following lemma. lemma 3.1. assume p > µ, c > 2 √ p−µ and q < ε2[c2/4 − (p−µ)]. define k1 := 1 − √ 1 − 4(p + q/ε2 −µ)/c2 2 , k2 := 1 + √ 1 + 4µ/c2 2 . (3.12) the direction fields of the ordinary differential system (3.1)-(3.3) point inward on b0 ∪ b1 ∪ s0 and outward on s1 ∪s2. in other words, if (u(ξ1),v(ξ1),w(ξ1)) ∈ b0 ∪b1 ∪s0 for some ξ1 ∈ r, then there exists δ > 0 such that (u(ξ),v(ξ),w(ξ)) ∈ p for all ξ ∈ (ξ1,ξ1 + δ) and (u(ξ),v(ξ),w(ξ)) /∈ p for all ξ ∈ (ξ1 −δ,ξ1); and if (u(ξ1),v(ξ1),w(ξ1)) ∈ s1 ∪s2 for some ξ1 ∈ r, then there exists δ > 0 such that (u(ξ),v(ξ),w(ξ)) /∈ p for all ξ ∈ (ξ1,ξ1 + δ) and (u(ξ),v(ξ),w(ξ)) ∈ p for all ξ ∈ (ξ1 −δ,ξ1). proof. it is obvious that 0 ≤ u ≤ 1, 0 ≤ v ≤ 1/ε2 and 0 ≤ w ≤ 1/(cε) for any (u,v,w) ∈ p . if (u(ξ1),v(ξ1),w(ξ1)) ∈ b0 for some ξ1 ∈ r, then u(ξ1) = 0. it follows from (3.1) that u′(ξ1) = 1/c > 0. hence, there exists δ > 0 such that u(ξ) > 0 for ξ ∈ (ξ1,ξ1 +δ) and u(ξ) < 0 for ξ ∈ (ξ1−δ,ξ1). this implies that the direction field on b0 is inward. if (u(ξ1),v(ξ1),w(ξ1)) ∈ b1 for some ξ1 ∈ r, then u(ξ1) = 1. it follows from (3.1) that u′(ξ1) = −[pv(ξ1) + qv2(ξ1)]/c < 0. hence, there exists δ > 0 such that u(ξ) < 1 for ξ ∈ (ξ1,ξ1 + δ) and u(ξ) > 1 for ξ ∈ (ξ1 − δ,ξ1). this implies that the direction field on b1 is inward. if (u(ξ1),v(ξ1),w(ξ1)) ∈ s0 for some ξ1 ∈ r, then cu(ξ1) + εv(ξ1) + cw(ξ1) = 1/ε. it follows from (3.4) that cu′(ξ1) + εv ′(ξ1) + cw ′(ξ1) = −(1 − cε)u(ξ1) − (µ − cε − ε2)v(ξ1) < 0. hence, there exists δ > 0 such that cu(ξ) + εv(ξ) + cw(ξ) < 1/ε for ξ ∈ (ξ1,ξ1 + δ) and cu(ξ) + εv(ξ) + cw(ξ) > 1/ε for ξ ∈ (ξ1 −δ,ξ1). this implies that the direction field on s0 is inward. 26 srijana ghimire and xiang-sheng wang if (u(ξ1),v(ξ1),w(ξ1)) ∈ s1 for some ξ1 ∈ r, then w(ξ1) = k1v(ξ1). it follows from (3.2) and (3.3) that w′(ξ1) −k1v′(ξ1) c < p + q/ε2 −µ c2 v(ξ1) −k1v(ξ1) + k21v(ξ1) = 0. hence, there exists δ > 0 such that w(ξ) < k1v(ξ) for ξ ∈ (ξ1,ξ1+δ) and w(ξ) > k1v(ξ) for ξ ∈ (ξ1−δ,ξ1). this implies that the direction field on s1 is outward. if (u(ξ1),v(ξ1),w(ξ1)) ∈ s2 for some ξ1 ∈ r, then w(ξ1) = k2v(ξ1). it follows from (3.2) and (3.3) that w′(ξ1) −k2v′(ξ1) c > − µ c2 v(ξ1) −k2v(ξ1) + k22v(ξ1) = 0. hence, there exists δ > 0 such that w(ξ) > k2v(ξ) for ξ ∈ (ξ1,ξ1+δ) and w(ξ) < k2v(ξ) for ξ ∈ (ξ1−δ,ξ1). this implies that the direction field on s2 is outward. � 4. unstable manifold of predator-free equilibrium throughout this section, we assume that the conditions in lemma 3.1 are satisfied; namely, p > µ, c > 2 √ p−µ and q < ε2[c2/4 − (p−µ)]. the predator-free equilibrium (1, 0) of the system (2.5)-(2.6) corresponds to the equilibrium e0 = (1, 0, 0), which is also referred to as the predator-free equilibrium, of the dynamical system (3.1)-(3.3). the jacobian matrix at e0 is calculated as j0 =  −1/c −p/c 00 c −c 0 (p−µ)/c 0   . (4.1) in addition to a negative eigenvalue −1/c, the matrix j0 has two positive eigenvalues λ± = c± √ c2 − 4(p−µ) 2 . (4.2) the eigenvectors associated with λ± are e± =  −p/(cλ± + 1)1 1 −λ±/c   . (4.3) by [21, theorem 3.2.1], the predator-free equilibrium e0 possesses a smooth two-dimensional local invariant unstable manifold wu(e0) tangent to the plane spanned by e+ and e−. note that 1 − λ± c = 1 ∓ √ 1 − 4(p−µ)/c2 2 ∈ (k1,k2), where k1 and k2 are given in (3.12). hence, both vectors e± starting at the equilibrium e0 = (1, 0, 0) point inward the pentahedron p . we then find a smooth curve γ ∈ wu(e0)∩p with two end points on s1 and s2, respectively. for each i = 1, 2, we denote γi to be the point on γ such that the trajectories starting from these points will remain in p until touching si at a finite time. obviously, γ1 and γ2 are disjoint open subsets of γ. since γ is smooth and connected, there exists at least one point on γ \ (γ1 ∪γ2). the trajectory starting from this point will never touch s1 or s2, and hence stays in p all the time. we summarize our argument in the following lemma. traveling waves in cooperative predation: relaxation of sublinearity 27 lemma 4.1. assume p > µ, c > 2 √ p−µ and q < ε2[c2/4 − (p − µ)]. there exists a trajectory (u(ξ),v(ξ),w(ξ)) of the dynamical system (3.1)-(3.3) such that lim ξ→−∞ (u(ξ),v(ξ),w(ξ)) = (1, 0, 0), and (u(ξ),v(ξ),w(ξ)) ∈ p for all ξ ∈ r. 5. heteroclinic orbit and lyapunov function technique let (u(ξ),v(ξ),w(ξ)) be the trajectory which connects to the predator-free equilibrium e0 = (1, 0, 0) as ξ →−∞ and stays in the pentahedron p for all ξ ∈ r; see lemma 4.1. we will prove in this section that this trajectory converges to the positive equilibrium e1 = (u,v,v ) as ξ →∞, where (u,v ) is the positive equilibrium of the original system (1.4)-(1.5). from (2.8), we calculate u = q + pµ− √ (q + pµ)2 − 4qµ2 2q , v = q −pµ + √ (q + pµ)2 − 4qµ2 2qµ . it is easy to verify that 0 < u < 1 and k1 < 1 < k2. moreover, in view of (2.8) and (3.6), we obtain cu + εv + cv < u ε + µv ε = 1 ε . consequently, (u,v,v ) ∈ p . lemma 5.1. assume p > µ, c > 2 √ p−µ and q ≤ q with q := min{ ε4c2(p−µ) 2µ2 , 2ε4c2 µ , ε2[c2 − 4(p−µ)] 5 } (5.1) and ε := min{ 1 2c , −c + √ c2 + 4µ 4 , µ3/4 c[4/c2 + (µ− 1)2/(4µ)]1/4 }. (5.2) let (u(ξ),v(ξ),w(ξ)) be the trajectory given in lemma 4.1 and (u,v,v ) be the positive equilibrium calculated in (2.8). we have lim ξ→∞ (u(ξ),v(ξ),w(ξ)) = (u,v,v ). proof. for any (u,v,w) ∈ p , we construct a lyapunov function as l(u,v,w) := c(u−u ln u + w −v w v −v ln v) + κ 2c [(cu + µ c v + cw) − (cu + µ c v + cv )]2, (5.3) where κ > 0 is a positive constant to be determined later. restricting the lyapunov function on the trajectory (u(ξ),v(ξ),w(ξ)) and taking derivative with respect to ξ yield d dξ l(u(ξ),v(ξ),w(ξ)) =c[(1 − u u(ξ) )u′(ξ) + (1 − v v(ξ) )w′(ξ) + ( v w v(ξ)2 − v v(ξ) )v′(ξ)] + κ c [c(u(ξ) −u) + µ c (v(ξ) −v ) + c(w(ξ) −v )][cu′(ξ) + µ c v′(ξ) + cw′(ξ)]. for simplicity, we will drop the dependence on ξ and write u(ξ), v(ξ) and w(ξ) as u, v and w, respectively.. it then follows from (3.1)-(3.3) that l′ =[ u−u u (1 −u−puv −quv2) + v −v v (puv + quv2 −µv) + v (w −v) v2 c2(v −w)] + κ[(u−u) + µ c2 (v −v ) + (w −v )](1 −u−µw). 28 srijana ghimire and xiang-sheng wang denote ū = u−u, v̄ = v −v and w̄ = w −v . it is readily seen from (2.8) that u−u u (1 −u−puv −quv2) = − 1 uu ū2 −pūv̄ −qūv̄2 − 2qv ūv̄, v −v v (puv + quv2 −µv) = pūv̄ + qūv̄2 + qv ūv̄ + quv̄2, v (w −v) v2 c2(v −w) = − c2v v2 (v̄2 − 2v̄w̄ + w̄2), and [(u−u) + µ c2 (v −v ) + (w −v )](1 −u−µw) = −ū2 − µ c2 ūv̄ − (µ + 1)ūw̄ − µ2 c2 v̄w̄ −µw̄2. note that u ≤ 1, u ≤ 1 and v ≤ 1/ε2. we obtain from the above equations that l′ ≤[−ū2 −qv ūv̄ + quv̄2 −ε4c2v (v̄2 − 2v̄w̄ + w̄2)] + κ[−ū2 − µ c2 ūv̄ − (µ + 1)ūw̄ − µ2 c2 v̄w̄ −µw̄2]. now, we choose κ = 2ε4c4v/µ2 to eliminate v̄w̄ on the right-hand side of the above inequality. it follows that l′ ≤−(1 + κ)ū2 − (qv + κµ c2 )ūv̄ − ( κµ2 2c2 −qu)v̄2 −κ(µ + 1)ūw̄ − ( κµ2 2c2 + κµ)w̄2 ≤ 0, provided q < ε4c2v/u = κµ2/(2c2u) and (qv + κµ/c2)2 2κµ2/c2 − 4qu + κ2(µ + 1)2 2κµ2/c2 + 4κµ ≤ 1 + κ. (5.4) note from (2.8) that µv + u = 1 ≥ pu/µ. especially, v/u ≥ (p − µ)/µ2. for any q ≤ q, we have qv ≤ κµ/c2 and qu ≤ κµ2/(4c2). consequently, (qv + κµ/c2)2 2κµ2/c2 − 4qu + κ2(µ + 1)2 2κµ2/c2 + 4κµ −κ ≤ 4κ c2 + κ(µ + 1)2 2µ2/c2 + 4µ −κ ≤ 2ε4c4 µ3 [ 4 c2 + (µ− 1)2 4µ ], where we have made use of κ = 2ε4c4v/µ2 and v ≤ 1/µ in the last inequality. by the choice of ε in (5.2), the right-hand side of the above inequality is no more than 1. hence, (5.4) is satisfied and l′ ≤ 0 for all q ≤ q. we claim that lim ξ→∞ l(u(ξ),v(ξ),w(ξ)) > −∞. assume to the contrary that l(u(ξ),v(ξ),w(ξ)) → −∞ as ξ → ∞. it follows from the definition of l in (5.3) that v(ξ) → 0 as ξ → ∞. for any small ε0 > 0, there exists ξ0 ∈ r such that pu(ξ)v(ξ) + qu(ξ)v(ξ)2 < ε0 for all ξ > ξ0. in view of (3.1), we have cu ′ > 1 − u − ε0 for all ξ > ξ0. by comparison principle, we obtain lim infξ→∞u(ξ) ≥ 1 − ε0. since ε0 > 0 is arbitrary, letting ε0 → 0+ gives lim infξ→∞u(ξ) ≥ 1. this together with u(ξ) ≤ 1 implies that limξ→∞u(ξ) = 1. note that p > µ and v(ξ) ≥ 0. there exists ξ1 ∈ r such that pu(ξ)v(ξ) −µv(ξ) ≥ 0 for all ξ ≥ ξ1. on account of (3.3), we obtain cw′(ξ) ≥ 0 for all ξ ≥ ξ1. let w∞ = limξ→∞w(ξ) ≥ 0. if w∞ > 0, then there exists ξ2 ∈ r such that v(ξ) −w(ξ) < −w∞/2 for all ξ > ξ2. it follows from (3.2) that v′(ξ) < −cw∞/2 for ξ > ξ2, which contradicts to limξ→∞v(ξ) = 0. hence, w∞ = limξ→∞w(ξ) = 0. since w(ξ) ≥ 0 and cw′(ξ) ≥ 0 for all ξ ≥ ξ1, we obtain w(ξ) = 0 for all ξ ≥ ξ1. choose ξ3 ≤ ξ1 such that w(ξ) = 0 for all ξ ≥ ξ3 and w(ξ) > 0 if ξ is smaller than and close to ξ3. in view of (3.2) and limξ→∞v(ξ) = 0, we obtain traveling waves in cooperative predation: relaxation of sublinearity 29 v(ξ) = 0 for all ξ ≥ ξ3. since the line v = w = 0 is negatively invariant for the system (3.1)-(3.3), we have v(ξ) = w(ξ) = 0 for all ξ ∈ r, a contradiction. the above argument implies that l∞ := lim ξ→∞ l(u(ξ),v(ξ),w(ξ)) > −∞. now, we apply lasalle invariance principle to show that the trajectory (u(ξ),v(ξ),w(ξ)) converges to the positive equilibrium (u,v,v ) as ξ → ∞. let (ũ0, ṽ0, w̃0) be any point in the omega limit set of the trajectory (u(ξ),v(ξ),w(ξ)); namely, there exists a subsequence ξ1 < ξ2 < · · · < ξn → ∞ such that (u(ξn),v(ξn),w(ξn)) → (ũ0, ṽ0, w̃0) as n →∞. let (ũ(ξ), ṽ(ξ), w̃(ξ)) be the solution of (3.1)-(3.3) with (ũ(0), ṽ(0), w̃(0)) = (ũ0, ṽ0, w̃0). for any ξ ∈ r, we then have (u(ξn + ξ),v(ξn + ξ),w(ξn + ξ)) → (ũ(ξ), ṽ(ξ), w̃(ξ)) ∈ p as n →∞. especially, we obtain l(ũ(ξ), ṽ(ξ), w̃(ξ)) = lim n→∞ (u(ξn + ξ),v(ξn + ξ),w(ξn + ξ)) = l∞, and thus d dξ l(ũ(ξ), ṽ(ξ), w̃(ξ)) = 0. from the proof of l′ ≤ 0, we conclude that (ũ(ξ), ṽ(ξ), w̃(ξ)) = (u,v,v ) for all ξ ∈ r. therefore, the omega limit set of the trajectory (u(ξ),v(ξ),w(ξ)) is a singleton (u,v,v ), which implies that (u(ξ),v(ξ),w(ξ)) → (u,v,v ) as ξ →∞. this completes the proof. � 6. proof of theorem 2.1 the existence result follows from a sequence of lemma 3.1, lemma 4.1 and lemma 5.1. to establish the nonexistence result, we assume to the contrary that (u(ξ),v(ξ)) is a positive solution to the boundary value problem (2.5)-(2.7). define w(ξ) = v(ξ) − v′(ξ)/c. then, (u(ξ),v(ξ),w(ξ)) is a solution to the dynamical system (3.1)-(3.3) such that v(ξ) > 0 and (u(ξ),v(ξ),w(ξ)) → (1, 0, 0) as ξ → −∞. recall that the jacobian matrix j0 about the predator-free equilibrium e0 = (1, 0, 0) has three eigenvalues −1/c and λ±; see (4.1) and (4.2). since c < 2 √ p−µ, the eigenvalues λ± are complex with positive real parts and nonzero imaginary parts. there exist two (complex conjugate) constants c± such that u(ξ)v(ξ) w(ξ)   =  10 0   + c+eλ+ξe+ + c−eλ−ξe− + o(ere λ+ξ), as ξ → −∞, where e± are the eigenvectors associated with the eigenvalues λ±; see (4.3). especially, v(ξ) oscillates around 0 as ξ → −∞. this contradicts to the assumption that v(ξ) > 0 for all ξ ∈ r. the proof is complete. 7. discussion in this paper, we have established an existence theory of traveling wave solutions for a diffusive predator-prey system with cooperative hunting. the main challenge lies in the fact that the function of cooperative predation is not dominated by a linear function; namely, the sublinearity condition is not satisfied. we first transform the traveling wave equations to a dynamical system. by using an extension of the approach introduced in [12, 13], we construct a suitable pentahedron, and find inside this pentahedron a trajectory connecting the predator-free equilibrium at one end. lyapunov function technique and lasalle invariance principle are applied to prove that this trajectory connects the positive 30 srijana ghimire and xiang-sheng wang equilibrium at the other end. our result indicates that there exists a critical wave speed c∗ such that no positive traveling wave solution exists when c < c∗. on the other hand, if c > c∗, then a positive traveling wave solution exists for any sufficiently small cooperative predation rate q. there are some open problems related to traveling waves in cooperative predation. (1) strong cooperative predation. in our analysis, we need a technical assumption q < q, where q is a positive constant depending on the model parameters and the wave speed. it is conjectured that when q ≥ q, there still exists a positive traveling wave solution if c > c∗. (2) prey diffusion. in our model, we simply assume that the diffusion rate of the prey is zero. it is reasonable to expect that a similar result holds when the prey has a positive diffusion rate. in this case, one needs to investigate a four-dimensional dynamical system and construct a suitable polychoron (i.e., 4-polytope). we leave this problem as future work. (3) critical case. we have proved the existence of traveling wave solution when c > c∗ and the non-existence of traveling wave solution when c < c∗. we conjecture that the traveling wave solution exists for the critical case c = c∗. (4) bistable equilibria. we assume that the basic reproduction number r0 is greater than one and hence there exists a unique positive equilibrium for the diffusion-free system. however, if r0 < 1, the diffusion-free system may have two positive equilibria with one unstable and the other locally asymptotically stable. it is interesting to study the traveling wave solution connecting the unstable positive equilibrium to the stable positive equilibrium. one may also want to investigate the traveling wave solution connecting the unstable positive equilibrium to the (stable) predation-free equilibrium. (5) general prey growth. in our model, we assume that the prey growth function is linear. it is natural to ask whether one could extend our study to the general case when the growth rate of the prey is a general function such as the logistic (quadratic) function or the allee effect (cubic) function. acknowledgment we would like to thank two anonymous referees for careful reading and helpful suggestions which led to an improvement to our original manuscript. references [1] m. alves and f.m. hilker, hunting cooperation and allee effects in predators, j. theo. biol. 419 (2017), 13–22. [2] g. beauchamp, social predation: how group living benefits predators and prey, academic press, 2014. [3] j. bednarz, cooperative hunting harris’ hawks (parabuteo unicinctus), science 239 (1988), 1525–1527. [4] r. chisholm and r. firtel, insights into morphogenesis from a simple developmental system, nat. rev. mol. cell. biol. 5 (2004), 531–541. 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[22] t. zhang, w. wang, and k. wang, minimal wave speed for a class of non-cooperative diffusion-reaction system, j. differential equations 260 (2016), 2763–2791. department of mathematics, university of louisiana at lafayette, lafayette, la 70503, usa. e-mail address: srijana.ghimire1@louisiana.edu corresponding author, department of mathematics, university of louisiana at lafayette, lafayette, la 70503, usa. e-mail address: xswang@louisiana.edu mathematics in applied sciences and engineering https://doi.org/10.5206/mase/10221 volume 1, number 1, march 2020, pp.39-49 http://www.uwo.ca/lib/mase analysis of solutions and disease progressions for a within-host tuberculosis model wenjing zhang, federico frascoli, and jane m heffernan abstract. mycobacterium tuberculosis infection can lead to different disease outcomes, we analyze a within-host tuberculosis infection model considering interactions among macrophages, t lymphocytes, and tuberculosis bacteria to understand the dynamics of disease progression. four coexisting equilibria that reflect tb disease dynamics are present: clearance, latency, and primary disease, with low and high pathogen loads. we also derive the conditions for backward and forward bifurcations and for global stable disease free equilibrium, which affect how the disease progresses. numerical bifurcation analysis and simulations elucidate the dynamics of fast and slow disease progression. 1. introduction mycobacterium tuberculosis (mtb) is a bacterium that causes an ancient and deadly infectious disease in humans, called tuberculosis (tb) [9]. currently, tb affects approximately one third of the world’s population [10, 6]. in 2018, the world health organization (who) estimated approximately 10 million infections globally, and 1.2 million deaths among hiv-negative people [12]. it has also been found that tb susceptibility and disease are increased in hiv-aids infected individuals, resulting in higher mortality rates [8, 1, 14, 16]. the pathological outcomes of tb infection include clearance, latent infection, and primary disease with fast or slow progression [13]. after initial infection, 5−10% of infected subjects can clear the disease. of the remaining individuals, 5 − 10% will progress to primary disease, and the rest will remain in a latently infected state with no clinical symptoms, with the possibility of re-activation to primary disease later in their life. a large number of mechanisms have been proposed to explain tb disease progression considering individual factors, including bacterial and immune response mechanisms. however, the most influential factors for tb outcomes are not currently known. motivated by this, we analyze a tb host-pathogen model first proposed in ref. [3]. the model incorporates known mechanisms of host-pathogen interaction in tb dynamics, and includes all realistic disease outcomes. analysis is performed to determine the driving factors behind disease progression and outcome, especially fast or slow progression to primary disease. the paper is organized as follows. in section 2, we introduce the established tuberculosis progression model. in section 3, model dynamics are shown through the proofs of the well-posedness of solutions, the existence of equilibrium solutions, and analyses of the disease free equilibrium. the basic reproduction number r0 and the vector field on the center manifold for the disease free equilibrium when r0 = 1 are derived analytically. the conditions for the occurrence of the backward and forward bifurcations received by the editors 13 february 2020; revised 4 march 2020; accepted 5 march 2020; published online 7 march 2020. 2000 mathematics subject classification. primary 37g15; secondary 92-08. key words and phrases. tuberculosis, disease progression, stability, bifurcation. w. zhang was supported in part by texas tech university new faculty startup fund; j. m. heffernan was funded by nserc and supported by a york research chair program. 39 40 w. zhang, f. frascoli, and j. m. heffernan are also derived. in section 4, numerical continuations are carried out for the infected equilibrium to confirm the existence of a backward bifurcation. the corresponding numerical simulations show the fast and slow disease progressions to latency and primary diseases. finally, conclusions are drawn in section 5. 2. model the 4-dimensional model (2.1) includes the mtb ideal target cell population, macrophages (their uninfected mu and infected mi populations). it also includes the mtb bacterial population b, and a population of cd4 t cells, which aid in tb clearance. the model is as follows: dmu dt = sm −µmmu −βmub dmi dt = βmub − bmi −γmi t/mi t/mi + c db dt = δb ( 1 − b k ) + mi ( n1b + n2γ t/mi t/mi + c ) −mub(η + n3β) dt dt = st + cmmit emt + 1 + cbbt ebt + 1 −µtt. (2.1) briefly, uninfected macrophages mu enter the system with constant rate sm , and can die naturally (µm ), or be infected by the pathogen b (βmub). it is assumed that infected macrophages can release new bacteria into the system in two different ways: (1) through cell death and bursting b, producing n1 new bacteria, and (2) through cytotoxic t-lymphocyte killing (represented by the ratio t/mi) with rate γ and saturating factor c, which releases n2 new bacteria into the system. it is assumed that the bacteria population can divide (δb(1 − b/k) and that bacteria can be lost due to interaction with macrophages. this occurs through immune system neutralization ηbmu or macrophage infection βbmu involving, on average, n3 individual bacteria. finally, it is assumed that t-cells are produced at a constant rate st by the thymus, can be stimulated to proliferate through interactions with the infected macrophage cmmit/(emt + 1) and bacteria cbbt/(ebt + 1), and can die naturally, with rate µt . infection is initiated with an initial pathogen load. we refer the reader to du et al. [3] for more detail on the biology and model assumptions. parameters and their values are listed in table 1. in previous work, du et al. [3] found four biologically realistic equilibria and determined the basic reproduction number. note that, in the original contribution, there is no mention of the driving factors behind the different outcomes of disease (namely, clearance, latency, and primary disease with fast or slow progression) and only an asymptotic version of the model that neglects the effects of the cd4 t-cell population is used/analyzed. in the following, we expand and elaborate on the four disease outcomes and other interesting aspects of the model using the full model system eq. 2.1. 3. model dynamics 3.1. well-posedness of solutions. let d = { (mu, mi, b, t) ∈ r4+ : mu + mi ≤ mmax, b ≤ bmax, t ≤ tmax } , where mmax = sm min{µm, b} , tmax = 1 µt ( st + cm em mmax + cb eb bmax ) , bmax = k 2 + √ (4kδmmax(n1b + n2γ) + k2δ2 2δ . (3.1) analysis of solutions and disease progressions for a within-host tuberculosis model 41 proposition 3.1. under the flow of (2.1), there exists a positive invariant set d that attracts all solutions in r4+ as time moves forward. proof. the smoothness of the right hand side of model (2.1) guarantees the local existence and uniqueness of the solution of the initial value problem of model (2.1). the trajectories starting from positive initial values never cross the boundary of r4+, since dmu dt |mu=0 = sm > 0, dmi dt |mi=0 = β mu b ≥ 0, and db dt |b=0 = mi ( n1b + n2γ t t + cmi ) ≥ 0, dt dt |t=0 = st > 0. next, we show that positive solutions are bounded. due to the positiveness, we have d dt (mu + mi) < sm −µmmu − bmi ⇒ lim t→+∞ sup(mu + mi)(t) = sm min{µm, b} := mmax. moreover, db dt < δb ( 1 − b k ) + mi (n1b + n2γ) −mub(η + n3β), tt+cmi ∈ (0, 1) < − δ k b2 + δb + mmax (n1b + n2γ) ⇒ b(t) = k/2 + tanh [√ (4kδmmax(n1b + n2γ) + k2δ2)(c0 + t)/(2k) ] × √ (4kδmmax(n1b + n2γ) + k2δ2/(2δ), where c0 is determined by initial condition and c0 + t > 0 for sufficiently large t. we have b(t) = k 2 + √ (4kδmmax(n1b + n2γ) + k2δ2 2δ := bmax. then, the last equation in (2.1) satisfies dt dt < st + cmmmaxt emt + 1 + cbbmaxt ebt + 1 −µtt < st + cm em mmax + cb eb bmax −µtt. it hence follows that t(t) < 1 µt ( st + cm em mmax + cb eb bmax ) := tmax, and the proposition is proven. � 3.2. equilibrium solutions. denote model (2.1) as m′u = f1, m ′ i = f2, b ′ = f3, t ′ = f4. the corresponding steady states are derived as follows: f1 = 0;⇒ m̄u(b) = sm βb + µm . (3.2) case 1: if (b + γ)mi −βmub 6= 0 or βsmb − (b + γ)(βb + µm )mi 6= 0, we have f2 = 0 ⇒ t̄(m̄u) = cmi [ γmi (b + γ)mi −βm̄ub − 1 ] (3.2) ==⇒ t̄(b) = [βsmb − (βb + µm )bmi]cmi βsmb − (b + γ)(βb + µm )mi , t̄(b) > 0 if βsmb < (βb + µm )bmi or βsmb > (b + γ)(βb + µm )mi. (3.3) 42 w. zhang, f. frascoli, and j. m. heffernan considering the preceding results (3.2) and (3.3), we obtain f3 = cγm 2 i f3af3b = 0, where f3a = (b + γ)(βb + µm )mi −βsmb f3b = kb (βb + µm ) mi + ((βb + µm )δ + sm [(n2 −n3)β −η])kb −b2δ(βb + µm ). the existence of t̄ in (3.3) implies f3a 6= 0. further, mi = 0 induces that f4(mu = m̄u(b), mi = 0, b, t̄(b) = 0) = st 6= 0. this indicates that the equilibrium does not exist. therefore f3 = 0 only implies f3b = 0 followed by m̄i(b) = ( bδ k − δ + sm (n2 −n3)β −smη βb + µm ) b b(n1 −n2) , m̄i(b) > 0 if bδ k > δ + sm (n2 −n3)β −smη βb + µm and n1 > n2. (3.4) the b in (3.4) satisfies f4(m̄u(b), m̄i(b), b, t̄(b)) = 0, the following is true: f(b) = −ebemµt t̄3(b) + [ (cmm̄i(b) + emst −µt )eb + em (cbb −µt ) ) t̄2(b) + [ cbb + cmm̄i(b) + ebst + emst −µt ] t̄(b) + st = 0. (3.5) then, we find the infected equilibrium e∗ = (m̄u(b), m̄i(b), b, t̄(b)). we note that there could be more than one solution, and up to three feasible infected equilibria. case 2: if βsmb − (b + γ)(βb + µm )mi = 0, we have f2 = 0 ⇒ m̄i0 = βsmb (b + γ)(βb + µm ) . (3.6) then substituting m̄u(b) in (3.2) and m̄i0 in (3.6) into f3(m̄u(b), m̄i0) = 0, yields b̄0 = 0. (3.7) this is followed by f4(m̄u(b), m̄i0, b̄0) = 0, which yields t̄0 = st µt . (3.8) we thus find the disease free equilibrium (dfe) e0 = (m̄u(b̄0), m̄i0(b̄0), b̄0, t̄0), where m̄u(b̄0) = sm/µm and m̄i0(b̄0) = 0. 3.3. analysis of the disease free equilibrium. 3.3.1. calculation of the basic reproduction number. following the next-generation matrix approach in ref. [11], the basic reproduction number r0 is the spectral radius of fv −1, where fv −1 =   0 βsmµm n1b + n2γ δ  [b + γ 0 0 sm µm (n3β + η) ]−1 =   0 β n3β + η n1b + n2γ b + γ µmδ sm (n3β + η)   , and r0 = ρ(fv −1) = δµm 2sm (n3β + η) + 1 2 [ δ2µ2m s2m (n3β + η) 2 + 4β(n1b + n2γ) (n3β + η)(b + γ) ]1/2 . (3.9) analysis of solutions and disease progressions for a within-host tuberculosis model 43 the jacobian matrix of model (2.1) at the disease free equilibrium is: j0 =   −µm 0 − βsm µm 0 0 −b−γ βsm µm 0 0 n1b + n2γ δ − sm µm (n3β + η) 0 0 cmst emst + µt cbst ebst + µt −µt   , (3.10) and gives the following characteristic equation (z + µt ) (z + µm ) ( z2 + pz + q ) = 0, (3.11) where p = b + γ −δ + sm µm (n3β + η) , q = [(−n2 + n3)γβ − b(n1 −n3)β + η(b + γ)] sm µm − δ(b + γ). equation (3.11) admits at least two negative roots, z = −µt and z = −µm . the third root, z = δ − b−γ − sm µm (n3β + η), is negative if b + γ + sm µm (n3β + η) > δ. the last root is zero, if [(−n2 + n3)γβ − b(n1 −n3)β + η(b + γ)] sm µm − δ(b + γ) = 0, (3.12) which is equivalent to r0 = 1. theorem 3.1. under the condition b+γ + sm µm (n3β + η) > δ, the disease free equilibrium e0 is locally asymptotically stable if r0 < 1 and unstable if r0 > 1. 3.3.2. existence of a backward bifurcation. following theorem 4.1 in ref. [2], we first shift the disease free equilibrium to the origin by letting x1 = mu − smµm , x2 = mi − 0, x3 = b − 0, x4 = t − st µt , and φ = β −βt . here r0(βt ) = 1 and βt = (−δµm + ηsm )(b + γ) smγ(n2 −n3)γ + smb(n1 −n3) . then we compute the approximated center manifold for the system near the origin with one simple zero eigenvalue at r0 = 1, and three negative eigenvalues. we choose a right eigenvector associated with the simple zero eigenvalue, w, and the left eigenvector, v, satisfying vw = 1 as follows: w = 1 n   (δµm −ηsm )(b + γ) µ2mw̃ δµm −ηsm µmw̃ 1 st {[(w̃cb − δcm )µt + (emw̃cb −δcmeb)st ] µm + ηcmsm (ebst + µt )} (ebst + µt )(emst + µt )µmµtw̃   , v = [ 0, n1b + n2γ b + γ , 1, 0 ] , 44 w. zhang, f. frascoli, and j. m. heffernan where n = (n2 −n3)µmγ + [(n1 + n2 − 2n3)b−n2δ]µmγ + n2smηγ µm (b + γ)w̃ + (n1 −n3)b2µm −n1δbµm + n1smηb µm (b + γ)w̃ and w̃ = (n2 −n3)γ + b(n1 −n3). further, the flow of the center manifold y(t) truncated at the quadratic term is written as ẏ = ay2 + bφy, (3.13) a = v 2 [ w′ ( ∂f1 ∂xi∂xj )∣∣∣ e0 w, w′ ( ∂f2 ∂xi∂xj )∣∣∣ e0 w, w′ ( ∂f3 ∂xi∂xj )∣∣∣ e0 w, w′ ( ∂f4 ∂xi∂xj )∣∣∣ e0 w ]′ = an ad , an = ((ã− [µtcsm (n1 −n2)]bγ)kδ −sm [(n2 −n3)γ + b(n1 −n3)]2(b + γ)st )µ2mδ, −smδkηµmã + kµtγbcs3mη 2(n1 −n2), ã = −st (n2 −n3)γ3 − bst (n1 + 2n2 − 3n3)γ2 − b3st (n1 −n3) − (2n1 + n2 − 3n3)stb2γ + 2(n1 −n2)µtcsmbγ, ad = smµ 2 m [(n2 −n3)γ + b(n1 −n3)] 2(b + γ)stk, and b = v ( ∂fi ∂xi∂β ) e0 w = [(n2 −n3)γ + b(n1 −n3)]sm (b + γ)µm , where i, j = 1 . . . 4. the non-zero terms in ( ∂fk ∂xi∂xj )∣∣∣ e0 , where i, j, k = 1 . . . 4, are ∂f1 ∂x1∂x3 ∣∣∣ e0 = ∂f1 ∂x3∂x1 ∣∣∣ e0 = − ∂f2 ∂x1∂x3 ∣∣∣ e0 = − ∂f2 ∂x3∂x1 ∣∣∣ e0 = (δµm −ηsm )(b + γ) [(n2 −n3)γ + b(n1 −n3)]sm , ∂f3 ∂x1∂x3 ∣∣∣ e0 = ∂f3 ∂x3∂x1 ∣∣∣ e0 = n3 ∂f1 ∂x1∂x3 ∣∣∣ e0 −η, ∂f2 ∂x2∂x2 ∣∣∣ e0 = 2γµtc st , ∂f3 ∂x2∂x2 ∣∣∣ e0 = −n2 2γµtc st , ∂f3 ∂x3∂x3 ∣∣∣ e0 = −2 δ k , ∂f4 ∂x2∂x4 ∣∣∣ e0 = ∂f4 ∂x4∂x2 ∣∣∣ e0 = cmµ 2 t (emst + µt )2 , ∂f4 ∂x3∂x4 ∣∣∣ e0 = ∂f4 ∂x4∂x3 ∣∣∣ e0 = cbµ 2 t (ebst + µt )2 . analysis of solutions and disease progressions for a within-host tuberculosis model 45 theorem 3.2. under the condition b > 0, we have ad > 0. then the model (2.1) at the disease free equilibrium e0, when r0 = 1 undergoes (1) a backward bifurcation if an > 0 and (2) a forward bifurcation if an < 0. furthermore, an(c = 0, β = βt , b = bb) = 0, where bb = γ [(n2 −n3)sm + kδ]µm −ηksm [(n3 −n1)sm −kδ]µm + ηksm . (3.14) 3.3.3. global stability analysis for the disease free equilibrium e0. proposition 3.1 shows that state variables mu, mi, b, and t are bounded for sufficiently large time. that is, there exists a time t > 0 such that mu < mmax, mi < mmax, b < bmax, and t < tmax. applying the “fluctuation lemma” [5], there exists time sequences τn →∞ and σn → +∞ such that m∞i := lim supt→∞mi(t) = limn→+∞mi(τn) and limn→+∞ dmi(τn) dt = 0, b∞ := lim supt→∞b(t) = limn→+∞b(σn) and limn→+∞ db(σn) dt = 0. (3.15) the preceding equations are followed by βmu(τn)b(τn) − bm∞i −γm ∞ i t(τn) t(τn) + cm ∞ i = 0 =⇒ βmu(τn)b(τn) = ( b + γ t(τn) t(τn) + cm ∞ i ) m∞i ≤ βmmaxb ∞ −→ m∞i ≤ βmmaxb ∞ b + γ t(τn) t(τn) + cm ∞ i < β b mmaxb ∞, (3.16) and δb∞ ( 1 − b ∞ k ) + mi(σn) ( n1b + n2γ t(σn) t(σn)+cmi ) −mu(σn)b∞(η + n3β) = 0 =⇒ −δb∞ ( 1 − b ∞ k ) + mu(σn)b ∞(η + n3β) = mi(σn) ( n1b + n2γ t(σn) t(σn)+cmi ) ≤ mi(σn) (n1b + n2γ) =⇒ [mu(σn)(η + n3β) − δ] b∞ ≤ mi(σn) (n1b + n2γ) =⇒ b∞ ≤ mi(σn) n1b + n2γ mu(σn)(η + n3β) − δ ≤ n1b + n2γ mu(σn)(η + n3β) −δ m∞i , (3.17) where t(τn) t(τn)+cm ∞ i ≤ 1. subsequently, m∞i ≤ β b mmaxb ∞ ≤ β b mmax n1b + n2γ mu(σn)(η + n3β) − δ m∞i . (3.18) if β b mmax n1b + n2γ mu(σn)(η + n3β) −δ ≤ 1, or equivalently mu(σn) ≥ 1 η + n3β [ βsm bµm (n1b + n2γ) + δ ] := mmaxu , (3.19) then m∞i = 0, implying b ∞ = 0, and the disease free equilibrium e0 is globally stable. theorem 3.3. if b + γ + sm µm (n3β + η) > δ and r0 < 1, the uninfected macrophage population mu should satisfy mu ≥ mmaxu to completely eliminate tb infection. 46 w. zhang, f. frascoli, and j. m. heffernan symbol description (unites) value (range) source sm recruitment rate of mu (1/ml day) 5000 (0.33, 33) [13] [7] [4] st recruitment rate of t (1/ml day) 6.6 (3300, 7000) [13] [7] [4] µm death rate of mu (1/day) 0.01 (0.01, 0.011) [13] [7] [4] b loss rate of mi (1/day) 0.11 (0.05, 0.5) [13] [7] [4] µt death rate of t (1/day) 0.33 (0.05, 0.33) [13] [7] [4] β infection rate by b (1/day) 2 × 10−7 (10−8, 10−5) [13] [7] [4] η bacteria killing rate by mu rate (1/ml day) 1.25 × 10−8 (1.25 × 10−9, 1.25 × 10−7) [13] [7] [4] γ cell-mediated immunity rate (1/day) 0.5 (0.1, 2) [13] [7] [4] δ proliferation rate of b (1/day) 5 × 10−4 (0, 0.26) [13] [7] [4] cm expansion rate of t induce by mi (1/day) 10 −3 (10−8, 1) estimated cb expansion rate of t induce by b (1/day) 5 × 10−3 (10−8, 1) estimated em saturating factor of t expansion related to mi 10 −4 (10−6, 10−2) estimated eb saturating factor of t expansion related to b 10 −4 (10−6, 10−2) estimated c half-saturation ratio for mi lysis (t/mi) 3 (0.3, 30) estimated k carrying capacity of b (1/ml) 10−8 (106, 1010) estimated n1 max moi of mi (b/mi) 50 (50, 100) [13] [7] [4] n2 max no. of b released by apoptosis (t/mi) 20 (20, 30) [13] [7] [4] n3 n3 = n1/2 (b/mi) 25 (25, 50) [13] [7] [4] mu uninfected macrophages mi infected macrophages b extra and intra-cellular bacteria t cd4 t-cells table 1. parameter symbol, descriptions, values, and sources [3] the occurrence of a backward bifurcation destabilizes the globally stable disease free equilibrium e0 under the condition r0 < 1 and an extra condition to regain stability is needed, i.e. b + γ + sm µm (n3β + η) > δ, as shown in theorem 3.3. in the next section, we verify the existence of a backward bifurcation computationally and investigate the associated dynamical behaviors by numerical simulations. 4. bifurcation analysis and numerical simulations consider the n-dimensional nonlinear system with m parameter values dx dt = f(x,p), x ∈ rn, p ∈ rm, f : rn+m → rn. (4.1) the equilibrium solutions xe = xe(p) are derived from the equilibrium condition f(xe(p),p) = 0, x ∈ rn, p ∈ rm. (4.2) the local stability of the equilibrium points xe(p) is determined by the eigenvalues of the jacobian j(p) = [∂fi(xe(p),p)/∂xj], which are the roots of the corresponding characteristic polynomial equation pn(λ) = det[λi −j(p)] = λ + a1(p)λn−1 + a2(p)λn−2 + · · · + an−1(p)λ + an(p). (4.3) the necessary and sufficient conditions for zero-eigenvalue bifurcation (zero-singularity) are given in ref. [15]. analysis of solutions and disease progressions for a within-host tuberculosis model 47 (a) (b) b b bt1 bt4 b=0.035 b=0.0298 time (days) 0 200 400 600 800 1000 1200 0 2 4 6 8 10 12 14 16 b s iz e ( m m -3 ) b= 0.035, ic h b= 0.035, ic l b = 0.0298, ich b = 0.0298, ic l 1 3 5 7 9 b b 4 1 3 5 7 9 b b bt3 bt2 0 0.5 1 1.5 2 2.5 3 time (days) 10 4 0 2 4 6 8 10 12 b s iz e ( m m -3 ) 10 7 b=0.31 b=0.288 b=0.288 b=0.4 b=0.288 figure 1. bifurcation diagrams of model (2.1) with b vs b and simulations. e0 and e1 are in green and red curves. four zero-eigenvalue bifurcation are denoted as black points as bt1, bt2, bt3, and bt4. simulations are carried out for five fixed b values as b = 0.0298, b = 0.035, b = 0.288, b = 0.31, and b = 0.4. the first three b values shows bistability. the last three b values show different progression speed for the bacterial population b. theorem 4.1. the necessary and sufficient conditions for system (4.1) to have a k-zero singularity at a fixed point (equilibrium), x = xe(p), of the system are given by an(p) = an−1(p) = · · · = an+1−k(p) = 0, (4.4) which ai(p)’s are the coefficients of the characteristic polynomial (4.3). further, if the remaining coefficients a1, a2, . . . an−k still obey the hurwitz conditions for order n − k, then all the remaining eigenvalues of the jacobian have negative real parts. based on the results of the uncertainty and sensitivity analysis in ref. [3], the model is significantly affected by the change of macrophage loss rate b, the infection rate β, cell-mediated immunity rate γ, and bacterial killing rate η. we thus choose the macrophage loss rate b as a bifurcation parameter to verify the analytical result for the backward bifurcation discussed in theorem 3.1. the other parameter values are fixed and shown in table 1. using theorem 4.1, we numerically find four zero-eigenvalue bifurcation critical points at bt1 = 0.0295, with et1 = (497833, 122, 8.7, 40), bt2 = 0.2993, with et2 = (21089, 2664, 45417, 6952168), bt3 = 0.1363, with et3 = (20, 3055, 49998972, 7575684467), and bt4 = 0.3000036, which yields et4 = (500000, 0, 0, 20). the summarized bifurcation, equilibria, and their stability are shown in figure 1. two values, i.e. b = 0.0298, b = 0.035, are chosen near bt1 and time series show that bistability occur for both b values, with solutions landing onto different equilibria depending on their initial conditions (see panel (a)). this is of interest because it means that the disease can die out or persist to latency depending on the initial infection status. interestingly, the progression to latency shows different dynamics and lasts different periods of time for the chosen b values. the red and yellow curves take different time to stabilize at their latency levels. 48 w. zhang, f. frascoli, and j. m. heffernan we then take three b values, i.e. b = 0.288, b = 0.31, and b = 0.4, close to bt2 (see panel (b)). again, bistability occurs when b = 0.288 on the left of bt2. there is an obvious difference in the speed of disease progression for the three different b values, as shown by the curves in the inset of figure 1(b). these examples of fast and slow disease progression dynamics seem to confirm the numerical findings in ref. [3] and are the object of current investigation. 5. conclusion in this paper, we analyze a four-dimensional within-host model (2.1) for tuberculosis infection, which has been previously proposed and studied numerically in ref. [3]. we carry out analyses for the well-posedness and boundedness for solutions, existence of the disease free and infected equilibriums and local and global stability analysis. a bifurcation analysis for the disease free equilibrium is also conducted, and a numerical continuation for the infected equilibrium shows when a backward bifurcation occurs. numerical simulations finally show how fast and slow disease progressions take place close to the bifurcation, with examples of bistability behaviour. this is important because different initial infections can lead to different disease progressions, with considerable differences among latency times. an in-depth analysis of the bifurcation scenario of this model is currently under progress, with the aim of characterising the different, possible behaviours towards infection that tb shows. references [1] m. w. borgdorff and d. van soolingen, the re-emergence of tuberculosis: what have we learnt from molecular epidemiology? clin. microbiol. infect., 9(2013), 889-901. [2] c. castillo-chavez and b. song, dynamical models of tuberculosis and their applications. math. biosci. eng., 1(2004), 361-404. [3] y. du, j. wu and j. m. heffernan, a simple in-host model for mycobacterium tuberculosis that captures all infection outcomes. math. popul. stud., 24(2017), 37-63. [4] d. gammack, s. ganguli, s. marino, j. segovia-juarez and d. e. kirschner, understanding the immune response in tuberculosis using different mathematical models and biological scales. multiscale model sim, 3(2005), 312-345. [5] w. m. hirsch, h. hanisch and j. p. gabriel, differential equation models of some parasitic infections: methods for the study of asymptotic behavior. commun. pur. appl. math., 38(1985), 733-753. [6] p. l. lin and j. l. flynn, understanding latent tuberculosis: a moving target. j. immunol., 185(2010),15-22. [7] s. marino and d. e. kirschner, the human immune response to mycobacterium tuberculosis in lung and lymph node. j. theor. biol., 227(2004), 463-486. [8] j. dh porter and k. pwj mcadam, the re-emergence of tuberculosis. annu. rev. publ. health, 15(1994), 303-323. [9] a. sakula, centenary of the discovery of the tubercle bacillus. the lancet, 319(8274)(1982), 750. [10] d. sud, c. bigbee, j. l. flynn and d. e. kirschner, contribution of cd8+ t cells to control of mycobacterium tuberculosis infection. j. immunol., 176(2006), 4296-4314. [11] p. van den driessche and j. watmough, reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. math. biosci., 180(2002), 29-48. [12] who, golbal tuberculosis report 2019. who, 2019. [13] j. e. wigginton and d. kirschner, a model to predict cell-mediated immune regulatory mechanisms during human infection with mycobacterium tuberculosis. j. immunol., 166 (2001), 951-1967. [14] j. yang, t. kuniya, f. xu and y. chen, evaluation of the tuberculosis transmission of drug-resistant strains in mainland china. j. biol. syst., 26(2018), 533-552. [15] p. yu, closed-form conditions of bifurcation points for general differential equations. int. j. bifurcat. chaos, 15(2005), 1467-1483. [16] k. zaman, tuberculosis: a global health problem. j. health popul. nutr., 28 (2) (2010), 111. analysis of solutions and disease progressions for a within-host tuberculosis model 49 corresponding author, department of mathematics and statistics, texas tech university, broadway and boston, lubbock, tx 79409-1042. e-mail address: wenjing.zhang@ttu.edu department of mathematics, faculty of science, engineering and technology, swinburne university of technology, john st, 3122, hawthorn, vic, australia. e-mail address: ffrascoli@swin.edu.au department of mathematics and statistics, centre for disease modelling, york university. 4700 keele st, toronto, on, canada, m3j 1p3. e-mail address: jmheffer@yorku.ca mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 4, december 2020, pp.425-440 https://doi.org/10.5206/mase/10876 determining the effectiveness of practicing non-pharmaceutical interventions in improving virus control in a pandemic using agent-based modelling steven kyle d.c. villanueva and christian alvin h. buhat abstract. to determine the effectiveness of non-pharmaceutical interventions on an epidemic, we develop an agent-based model that simulates the spread of an infectious disease in a small community and its emerging phenomena. we vary parameters such as initial population, initial infected, infection rate, recovery rate, death rate, and asymptomatic rates, as inputs. our simulations show that (i) random mass testing and quarantines decreases the number of deaths, infections, and time duration; (ii) social distancing lengthen outbreak period to an extent and helps flatten the epidemic curve; (iii) the most effective combination of npis to minimize death, infection, and duration is no mass testing, no social distancing, and a total lockdown but is not ethical; and (iv) the most feasible intervention is to implement an enhanced community quarantine while doing random testing on a maximum of 100 individuals. results of this study can aid policymakers in determining interventions for their communities during a pandemic. 1. introduction in late 2019, an unidentified coronavirus had emerged from wuhan, china, and had initially spread to southeast asian countries. in february 2020, the disease was officially named coronavirus disease 2019 (covid-19) and had been acknowledged by the world health organization as a global pandemic due to its rapid human to human transfer. though it has an estimated 2 5% mortality rate, the death toll has reached half a million people worldwide since june 2020 [23, 24, 2]. in the philippines, the first suspected covid-19 case was confirmed last january 2020 [7, 10]. in early march 2020, the first local transmission was confirmed by the countrys fifth case [11]. due to the quick surge in the number of suspected and confirmed cases, different interventions such as lockdown procedures were initially implemented in metro manila, which eventually expanded to the whole island of luzon and ultimately the whole nation [21, 5]. the house bill bayanihan to heal as one act was written and signed to provide the president further power to combat the pandemic [22]. however, the number of infected still rose due to factors such as difficulty in testing, thus the quarantine protocol continued [4]. then, it was alleviated in may to minimal risk areas [9]. since then, multiple interventions/ variations have been attempted by each lgus (through the support of research) to try and fight the pandemic. one way experts study patterns of certain phenomena and gain insights in these pandemic situations is through modeling. some of the advantages in modeling include lesser risk to subjects being infected compared to experimentation and fieldwork, easier to solve and analyze through computer algorithms, and can be done repeatedly with different conditions [15]. in this study, we use agent-based modeling (abm) to simulate and analyze the underlying patterns and trends of such virus spread. received by the editors 6 july 2020; accepted 23 december 2020; published online 29 december 2020. key words and phrases. agent-based model, covid-19, epidemics, non-pharmaceutical interventions . this study is supported by the university of the philippines resilience institute, university of the philippines. 425 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/10876 426 skdc. villanueva and cah. buhat here, we first determine the effect of a certain epidemic by varying city information and virus dynamics. then, we check if the different non-pharmaceutical interventions (npis) that have been implemented are necessary for controlling the spread of the said virus. after that, we want to determine if simultaneously practice of such interventions is effective in combating the pandemic. we then determine what levels of preventive measures would minimize the number of deaths and the duration of the epidemic. lastly, we investigate other parameters that may have an effect on the outbreak. 2. model framework we use an agent-based model (abm) to simulate the transmission of the virus among individuals and to determine the effects of factors and npis in the process of infection. 2.1. agent-based model. abm is a type of microscale model used to simulate interactions among individuals agents and their complex behavior [3]. it aims to find and study emergent trends and phenomena that may arise on a given set of autonomous agents that follows specific predefined procedures [1, 16]. this has been utilized by different fields ranging from tourism, marketing, and epidemiology due to its flexibility, simplicity and affordability [16, 14, 20, 12]. here, we consider a model that simulates the infection among individuals given various factors in an environment. we then determine the effects of various npi to the number of infections, deaths, etc. the model determines significant developments whenever there are altercations in the population of virus dynamics. it also identifies whenever there is an effect in the the number of cases, deaths if agents practice the npis. the agents in this simulation have the following boolean attributes: infected?, confirmed?, hospitalized?, dead?, and permissible? which is simulated per tick (14-day infectious period). furthermore, parameters are used in the simulation, and it can be easily revised through the use of sliders, switches, or choosers of the program. they are categorized into two groups namely: city information and virus dynamics. 1 tick represents 1 infectious period for covid-19 which is 14 days [6], thus, in the simulation environment, all rates are multiplied by 14 signifying the 14 days per tick effect in the simulation. we also incorporate scientific control in the model for the comparisons between different runs. we assume the control to take on the following default values from table 1. moreover, global variables are also declared to update the monitors and plots for every tick of the simulation. they are as follows: population, healthy, infected, confirmed, unconfirmed, hospitalized, dead, permissible, and max infected. note that the netlogo software has two built-in elements namely patches and ticks, which represent the space and time variables respectively. since we are only concerned with the underlying behavior and trends of the agents including the repercussions after their interactions, we may not specify the variables in actual space and time measurements. however, if researchers and policymakers deemed it necessary for their analysis, they may assign an appropriate conversion table between the simulation and real-time values. lastly, these are the procedures that the simulation does to affect all agents behaviors, attributes, and interactions. with all the variables and procedures declared, we use netlogo programming environment for our abm. the simulation environment of the program and its development is shown in figs 1,2,3. the setup button generates the agents into random locations inside the environment. the number of agents placed depends on the initial population. consequently, a 4 by 4 hospital, which has been colored blue, is placed in the bottom-left corner of the environment. all agents parameters are then varied depending on the initial infected and asymptomatic rate. all those who are infected and confirmed effectiveness of practicing npis in improving virus control in a pandemic using abm 427 table 1. parameters and their descriptions parameter description default value reference initial population the initial number of agents in the simulation environment 1000 assumed initial infected the initial number of agents that are infected 10 assumed testing capacity an integer variable which will be used in the test procedure varied quarantine the type of quarantine the city is implementing, which have 4 options: none, general, enhanced. and total community quarantine which allows 100%, 50%, 25% and 0% of the present agents to move respectively. note that the quarantine cannot be changed in the middle of a run. varied social distancing a boolean variable which will be used in the check sd procedure varied infection rate the chance an agent gets infected if another agent is near them during an infectious period 70/14 [13] recovery rate the chance (in %) an agent don’t get infected provided that they are hospitalized during an infectious period 20/14 [8] death rate the chance (in %) an agent die provided that they are infected during an infectious period 20/14 [8] asymptomatic rate the chance (in %) an infected agent will not be confirmed 18/14 [19] figure 1. netlogo user interface of the agent-based modelling program before a simulation run becomes red in color. agents who are infected but not yet confirmed turns to orange. the remaining agents stay green. monitors will also be updated depending on the simulation. 428 skdc. villanueva and cah. buhat table 2. procedure names and their descriptions procedure name description clear clears the world of agents, plots, monitors, and notes setup runs the setup-hospital, setup-turtles, update-monitors, update-display procedures, and reset the number of ticks setup-hospital sets a small 4 by 4 unit of patches on the lower-left of the world to be colored blue, which will be designated as the hospital setup-turtles creates a number of turtles in the world based on the initial population in random locations, sets the agents attributes based on the initial infected, and to run the check quarantine procedure check-quarantine sets all present agents permission to move based on the chosen quarantine type update-monitors updates all monitors on the program based on the global variables update-display updates all agents to be hidden if dead sets all agents colors based on their attributes (red if infected and confirmed, orange if infected but not confirmed, and green if not infected) and checks whether they are inside the hospital or not start runs the check sd, test , move, infect, recover, death, update-monitors, and update-display procedures, does a tick on the tick counter and runs the update-output once the simulation ends check sd checks the social distancing variable and if true, all agents will move back whenever another agent is near them test randomly picks agents based on the testing capacity variable and confirms the tested agents if they are infected move sets all hospitalized agents movement based on their attributes (all infected towards or within the hospital and all not infected towards outside the hospital), and runs the outside movements procedure outside movements sets all permitted agents movement outside the hospital to walk randomly infect checks all infected agents and other non-infected agents near them and runs the get sick procedure based on the infection rate variable get sick sets agent to be infected and also sets agent to be confirmed or not based on the asymptomatic rate variable recover checks all hospitalized agents and runs the get healthy procedure based on the recovery rate variable death checks all infected agents and sets them dead based on the death rate variable update-output prints the results of the simulation in the output once the setup is done, the start button proceeds to run the simulation. the procedures, which were defined in table 2, for agents movement, testing measures, infections, deaths, and recoveries are done until it hits the stopping criterion. the run stops if (1) an extinction occurs where all the agents are dead or (2) the pandemic had subsided meaning there are no more infected agents. effectiveness of practicing npis in improving virus control in a pandemic using abm 429 figure 2. netlogo user interface of the agent-based modelling program during a running simulation figure 3. netlogo user interface of the agent-based modelling program after a simulation run 3. results and discussion 3.1. simulation outcome. given various sets of variables, we determine if a certain epidemic situation either subside and ease up on the agents infections and deaths, or lead to extinction based on the stopping criterion through our model. here are two runs that show the simulation results based on the same city information but with different virus dynamics, immediately followed by a comparison table of the simulation results. the simulation has three important results after every run: deceased, max infected cases, and ticks taken. deceased pertains to those agents that have died while the ticks taken refers to the duration of the epidemic regardless if it ended in the survival of the agent set or pure extinction. moreover, the max infected cases refer to the peak number of infected cases throughout the process. 430 skdc. villanueva and cah. buhat figure 4. simulation run a of the program (default) figure 5. simulation run b of the program 3.2. behavior of simulation results according to number of runs using default values. consequently, we also observe the minimum number of runs required to observe at which point the simulation results approaches in terms of deceased, infected, and ticks taken. also, we use the same number of runs to forecast all other remaining experiments. after proceeding on the experiments with a effectiveness of practicing npis in improving virus control in a pandemic using abm 431 table 3. a comparative table of runs a and b variables run a run b initial population 1000 1000 initial infected 10 10 testing capacity 0 0 quarantine none none social distancing no no infection rate 70% 95% recovery rate 20% 1% death rate 20% 10% asymptomatic rate 18% 50% deceased 155 983 max infected cases 42 275 ticks taken 40 83 varying number of runs, we need to have at least 1000 runs for every experiment made to achieve a less than 1% error on average from 100000 runs. to ensure the stability of the results from the experiments, we consider 5000 runs for all succeeding simulations. table 4. the effects of increasing testing capacity on different simulation results testing cap deceased % change % sum max infected % change % sum ticks taken % change % sum 0 262.75 60.32 44.58 10 256.67 -2.32 -2.32 59.90 -0.69 -0.69 43.60 -2.20 -2.20 20 250.37 -2.45 -4.71 59.29 -1.01 -1.70 42.46 -2.61 -4.75 30 246.76 -1.44 -6.09 59.12 -0.30 -1.99 41.83 -1.48 -6.17 40 242.95 -1.54 -7.54 59.35 0.39 -1.61 40.91 -2.21 -8.24 50 239.20 -1.54 -8.96 59.17 -0.29 -1.90 40.25 -1.61 -9.71 60 235.25 -1.65 -10.47 58.75 -0.71 -2.60 39.81 -1.11 -10.71 70 232.54 -1.15 -11.50 58.59 -0.27 -2.86 39.16 -1.61 -12.16 80 229.86 -1.15 -12.52 58.27 -0.55 -3.39 38.77 -1.02 -13.05 90 226.63 -1.40 -13.75 58.17 -0.17 -3.56 38.44 -0.85 -13.79 100 225.86 -0.34 -14.04 58.46 0.50 -3.08 37.93 -1.33 -14.93 3.3. effects of increments on testing capacity. note that all deceased, max infected, and ticks taken results have been decreasing whenever we increase the testing capacity of the hospital. in a small community with a population of 1000 where 10 are infected by a virus, by randomly testing a maximum of 100 individuals per tick, we are able to lower the death toll by up to 14%, and the maximum number of infected cases by up to 3%. also, the duration of the epidemic can be shortened by up to 14% as compared to runs without testing. however, as we increase the testing capacity, its effectiveness significantly reduces as well, as we can see in the percent change columns of table 4. 3.4. effects of quarantine types. from table 1, the quarantine types have an irregular increment, which is as follows: 100% of the present agents are permissible to move if there is no quarantine, 50% 432 skdc. villanueva and cah. buhat for general, 25% for enhanced, and 0% for a total quarantine. thus, the percent differences displayed in table 5 are all referenced from the control run, which is no community quarantine is in place. table 5. the effects of quarantine types on different simulation results quarantine deceased % change max infected % change ticks taken % change none (100%) 263.2618 60.562 44.5142 general (50%) 179.8694 -31.677 46.589 -23.072 37.172 -16.494 enhanced (25%) 135.3152 -48.601 38.3046 -36.751 32.5722 -26.827 total (0%) 88.0672 -66.548 29.359 -51.522 26.096 -41.376 the same trend can be seen here as well, as the testing capacity. as we implement stricter quarantine policies, we can reduce the death toll, maximum infected cases, and duration. by implementing a total community quarantine, we may be able to reduce virus-related deaths by up to 66%, infected cases by up to 51%, and the situation period by up to 41%. table 6. the effects of social distancing on different simulation results social distancing deceased max infected ticks taken false 261.0946 60.0242 44.4006 true 379.6116 72.6072 55.1476 3.5. effects of social distancing. interestingly, the advantages of social distancing have increased the number of deaths, the total maximum of infected cases, and the duration of the epidemic. there is a 45% growth in death numbers, 21% in max infected cases, and 24% in ticks taken. this phenomenon might be a result of the stochasticity of the agent-based model and computational programming errors, such as the conditions set by the check sd procedure and zigzagging occurrence. for the procedure, all agents that are near other agents are checked. then, an agent x will face the nearby agent y and take a unit step backward. however, this might result in agent x being near to another agent, say z. this can lead to a recursive problem due to the spatial capacity of the environment. a zigzagging occurrence also happens when a confirmed infected agent does winding movements on its way to the hospital since it also avoids nearby agents, in accordance with the social distancing procedure. this leads to delays in providing medical assistance to infected agents and further infections due to their unnecessary movements. a study on social distancing has also been proven that it can lengthen the duration of virus outbreaks. this intervention was also said to flatten infection curves [18]. however, by flattening curves, it does not directly imply that the peak, or the maximum infected cases, will be lowered as well. furthermore, [17] have mentioned that even if social distancing is observed, the outbreak rebounds once it is removed. the researchers suggested that as much as social distancing can delay the spread of the virus, consequent testing of suspected individuals must also take place, as well as developments for improving healthcare systems. 3.6. effects of simultaneous npis. we now check the different combinations of the non-pharmaceutical interventions and identify which set of the said policies provides us with the minimal average number of fatalities, maximum number of infected cases, and the duration of the epidemic. using various arrangements of such policies might vary its effects as well and might help us optimize on what policies effectiveness of practicing npis in improving virus control in a pandemic using abm 433 figure 6. the effects of simultaneous non-pharmaceutical interventions on deceased we can practice. with all previous simulations that have been done, we simulate 5000 runs on each of the following experiments. for a community, populated with 1000 individuals where 10 of those are infected, no mass testing, no social distancing, and a total lockdown must take place in order to minimize deaths, infections, and overall duration. note that this combination offers the least mobility of agents. thus, the outbreak slows down since there are no crowding situation in the hospital and no unnecessary movements. only those who are infected and confirmed are allowed to be admitted to the hospital. however, this leads to ethical and humanitarian issues since asymptomatic individuals are left off to recover for themselves. moreover, this is unrealistic since the entire population cannot be quarantined. looking for an alternative, the next best option, which is the most ethical, is to implement an enhanced community quarantine with no social distancing while doing random testing on a maximum of 100 individuals. by looking at the figures and on sections 3.3 to 3.5, we can observe that the doing stricter quarantine procedures (excluding total) and having greater testing capacity, the deaths, infections, and duration can be minimized. again, the social distancing protocol may be subject to change provided that the environment space is improved so that there are no zigzagging occurrences. another way we can combat the virus is by delaying its spread or simply flattening the infection curve. by practicing no quarantine, no mass testing, and an active social distancing policy, we are able to allocate the maximum time for our scientists and researchers to study and formulate either a vaccine or cure. if successful, this completely eradicates the virus over a guaranteed period of time without having any rebound or subsequent waves. 434 skdc. villanueva and cah. buhat figure 7. the effects of simultaneous non-pharmaceutical interventions on max infected 3.7. initial population and initial infected cases. let us now determine if there will be significant insights that can be found by gradually increasing the initial number of infected cases on the simulation results. as the initial number of infected cases increases, the number of deaths and maximum infected cases increases as well, while the duration of the outbreak slows down. however, the rate of change in all aspects also decreases as the number of initial infected cases increases. moreover, looking at fig 9, if the number of initial infected cases is at least 70% of the general population, the population retains the same number of infected cases. this would imply that throughout the pandemic, the population has at least 70% infected individuals. thus, quick and effective response against the pandemic must be taken into consideration. 3.8. initial population and spatial capacity of environment. as for this experiment, we improve the program by considering a wider environment and check if there will be changes in the results of the simulation. note that in both cases, we use the default (control) values. the expanded environment is scaled 2 times larger than the original environment and same as before, we simulate this for 5000 runs each. given the same population of 1000 people where 10 are infected, table 7 have shown the drastic effect of the spatial capacity of the environment. since there is a wider place the agents can move around, the chance of interaction, either inside or outside of the hospital, is lesser compared to the previous environment. this may play a very important role in dense cities and capitals. governing bodies may opt to space out individuals by slowly decongesting highly populated areas to wider locations effectiveness of practicing npis in improving virus control in a pandemic using abm 435 figure 8. the effects of simultaneous non-pharmaceutical interventions on ticks taken figure 9. the effects of gradual increments on initial infected cases on deceased and max infected with less population. unfortunately, this may have a detrimental effect on the economic, social, and organizational aspects of society. furthermore, this solution may also contaminate virus-free places. 436 skdc. villanueva and cah. buhat figure 10. the effects of gradual increments on initial infected cases on ticks taken table 7. a comparative table of the original and modified environments variables and results original program modified program % change environment size 30 x 30 60 x 60 hospital size 4 x 4 8 x 8 average deceased 259.6444 25.6146 -90.1347 average max infected 60.1062 11.1038 -81.5264 average ticks taken 44.1326 22.144 -49.8239 3.9. virus dynamics. we now vary these parameters such as infection rate, death rate, recovery rate, and asymptomatic rates to further analyze the effects of each of them in the transmission of the disease. on figs 11, 12, and 13, we can see that the simulation results almost follow similar trends with each other. as the infection rate increases, the number of deaths, maximum infection and ticks taken increases as well. this trend also occurs in increments on asymptomatic rate. however, the growth in deceased and max infected results on infection rate seems to follow an exponential curve, while the rest looks linear. the results in varying asymptomatic rates may be due to the fact that it is harder to trace such patients thus lengthening and worsening the outbreak. on the other hand, there is an inverse relationship on recovery and death rate and in the simulation results. while the recovery rate build up, the deaths, infections and duration lessens in a linear manner. as the virus has an increasing death rate, the number of deaths, infections and duration decreases exponentially. the outcome from varying death rate may be caused by the chance of death overriding the chance of infection, thus reducing transmissions between agents. also, we can consider an epidemic with a 0% death rate as a special case as we can see in fig 14. after 5 runs of the specific experiment, the program does not reach any of the stopping criteria since this situation will not lead to extinction nor the end of the epidemic. it simply reaches an equilibrium effectiveness of practicing npis in improving virus control in a pandemic using abm 437 figure 11. the effects of variations in virus dynamics on deceased figure 12. the effects of variations in virus dynamics on max infected with about 85% of the initial population being infected as seen on the first monitor. however, looking closer to the infected cases, the number of asymptomatic patients becomes greater than the number of confirmed ones as time passes. unfortunately, this case will not be of further significance to the study of the virus since there are already confirmed death cases due to it. thus, the virus has a nonzero death rate. 438 skdc. villanueva and cah. buhat figure 13. the effects of variations in virus dynamics on ticks taken figure 14. a simulation run of the program with default values but with 0% death rate note that the governing bodies and institutions can make no particular policies that can help control the following virus rates. nonetheless, these experiments are very beneficial and valuable to research and development teams as these show how crucial further studies on the virus affects the future of the community. 4. conclusion the use of agent-based modeling has never been more important in our current pandemic situation. without the use of heavy experimentation and costly operations, our model was able to determine how an epidemic/pandemic will behave based on its demographics and the dynamics of the infection/virus. given an assumed control for the simulations, we have considered three vital outcomes of an epidemic, effectiveness of practicing npis in improving virus control in a pandemic using abm 439 namely the death toll, peak infected cases, and outbreak duration. we computed with at least 1000 runs for every experiment made to accurately estimate results from 100000 runs. we then tested non-pharmaceutical interventions individually among 1000 individuals, where 10 of those are infected. mass testing can lower the death toll by up to 14%, the maximum number of infected cases by up to 3%, and the time extent by up to 14%. by practicing community quarantines, we may be able to reduce virus-related deaths by up to 66%, infected cases by up to 51%, and the outbreak period by up to 41%. meanwhile, social distancing has lengthened the duration of the outbreak, by up to 24%, to provide ample time for the epidemic curve to flatten and for healthcare facilities to cope up with the number of cases. by combining these npis, the most optimal intervention was to implement no mass testing, no social distancing, and a total lockdown to be implemented in a small virus-infested community. this arrangement minimized the mobility of agents to its fullest. this resulted in the lowest deaths, infections, and duration in comparison to other combinations. although this is unethical and detrimental to other fields such as, but not limited to, the economy. thus, a more feasible, recommendable, and realistic option will be to implement an enhanced community quarantine with no social distancing while doing random testing on a maximum of 100 individuals. we may also consider another choice that is to have no mass testing, no quarantine but with active social distancing if the goal was to flatten the curve. note that this may lead to the longest time taken by the outbreak, which can stretch the duration long enough for scientists to formulate either a vaccine and further control the infections. additionally, the infected individuals must not reach 70% of the general population as this situation reaches equilibrium. this made the virus very hard to exterminate and we might need to resort to drastic measures. also, a wider spatial capacity of the environment has made a significant development on the simulation results since the chances of interaction are minimized. this lowered deaths by up to 90%, infections by up to 81%, and time took by up to 49%. lastly, increments on virus dynamics have been analyzed as well. an increase in infection and asymptomatic rates also led to an increase in deaths, infections, and duration. conversely, as the recovery and death rate increase, the said results decrease. these results from our simulation model play a vital role in how governing bodies can optimally make guidelines such that we can properly utilize our resources, at the same time mitigating the adverse effects of the virus. this can also be a foundation for policymakers and for all other similar studies that dwell on epidemiology and important insights in mitigating virus spread. 5. limitations the non-pharmaceutical interventions are assumed to be followed and practiced by all of the agents in the simulation environment. furthermore, there is no delay time considered in such npis and no temporal analysis has been made, which may also play an important part in mitigating the outbreak. moreover, the npis that were analyzed are all collective actions, hence, individuals practicing other solitary preventive measures were not considered. finally, all agents are assumed to have the same protection or immunity rate against the virus. 6. recommendations note that assumptions such as movement in the study are random, which should be taken into consideration in considering the study. this study can be further extended by determining the minimum testing capacity needed to achieve minimal infections and deaths, in accordance with various parameter testing. moreover, taking other precautionary measures and physiological factors can be made as procedures in the program as well. among these are wearing of a mask, frequent handwashing, reproduction 440 skdc. villanueva and cah. buhat rates, or limited foreign travels. lastly, having actual and accurate data from a reliable and trustworthy source to assign on variables will make all the simulation results as close to the real-world situation. 7. supplementary files the agent-based model can be accessed by visiting the website: https://github.com/styledcv/epidemicnpissimulations. the program 1.nlogo is the original program that was used in the simulation testings done on the study. meanwhile, program 2.nlogo is the expanded version of program 1 that was used on section 3.8. references [1] a. bazghandi, techniques, advantages and problems of agent based modeling for traffic simulation, international journal of computer science issues (ijcsi) 9(2012), 115-119. 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[24] y. c. wu, c. s. chen, and y. j. chan, the outbreak of covid-19: an overview, journal of the chinese medical association 83(2020), 217-220. institute of mathematical sciences and physics, university of the philippines los baños, laguna 4031, philippines e-mail address: sdvillanueva3@up.edu.ph corresponding author, institute of mathematical sciences and physics, university of the philippines los baños, laguna 4031, philippines university of the philippines resilience insitute, university of the philippines, philippines e-mail address: chbuhat@up.edu.ph https://pcoo.gov.ph/ops-content/on-code-red-sublevel-2/ https://pcoo.gov.ph/ops-content/on-code-red-sublevel-2/ https://www.senate.gov.ph/bayanihan-to-heal-as-one-act-ra-11469.pdf https://www.senate.gov.ph/bayanihan-to-heal-as-one-act-ra-11469.pdf 1. introduction 2. model framework 2.1. agent-based model 3. results and discussion 3.1. simulation outcome 3.2. behavior of simulation results according to number of runs using default values 3.3. effects of increments on testing capacity 3.4. effects of quarantine types 3.5. effects of social distancing 3.6. effects of simultaneous npis 3.7. initial population and initial infected cases 3.8. initial population and spatial capacity of environment 3.9. virus dynamics 4. conclusion 5. limitations 6. recommendations 7. supplementary files references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 3, september 2020, pp.236-248 https://doi.org/10.5206/mase/10828 global dynamics of a two-strain hiv infection model with intracellular delay jin xu abstract. in this paper, we formulate a mathematical model to describe the interaction of two strains of hiv virus and the target cells within a host. the model is in the form of delay differential equations with two discrete delays to account for the average time for replication for the two strains. the model dynamic turns to be generically determined by two composite parameters r1 and r2, the basic reproduction numbers for strain 1 and strain 2, respectively in the absence of the other strain, in the sense that except for the critical case r1 = r2 > 1, the solutions are proved to converge to the corresponding equilibrium globally. the method used is lyapunov functionals. 1. introduction it has been realized that mathematical modelling can provide valuable insight into hiv-1 pathogenesis. these mathematical models are formulated by using differential equations to explore the mechanisms and dynamical behaviors of the viral infection process [3, 8, 17, 18, 19]. such understanding may offer guidance for developing efficient anti-viral drug therapies [14, 15, 10]. most existing mathematical models for hiv virus dynamics are by systems of ordinary differential equations. a standard and classic differential equation model for hiv infection is the following system of odes [16, 14, 18]:   ṫ = λ−dt −ktv, ṫ∗ = ktv −µt∗, v̇ = pt∗ − cv, (1.1) where t(t),t∗(t) and v (t) are the population sizes of uninfected target cells, infected cells and the free virus particles, respectively, at time t. the assumption is that uninfected cells are generated at a constant rate, λ, and die at a rate d. free virus particles infect uninfected target cells at a rate proportional to the product of their abundances, ktv . the rate constant, k, describes the efficacy of this process. infected cells produce free virus particles at a rate proportional to their abundance, pt∗. infected cells die at a rate µt∗ either due to the natural death or the action of the virus and free virus particles are removed from the system at rate cv by the immune system or natural decay. therefore, the average life-time of an infected cell, a free virus particle and an uninfected cell are 1/µ, 1/c and 1/d respectively. the model well predicts the primary phase of hiv infection, showing that during the first weeks of infection there is a peak in viral load with a subsequent decline to a relatively stable steady-state. received by the editors 29 june 2020; accepted 24 august 2020; online first 6 september, 2020. 2000 mathematics subject classification. 34k20, 34k25, 92b05, 92d25. key words and phrases. hiv, virus dynamics, intracellular delay, global stability, lyapunov functional. 236 global dynamics of a two-strain hiv infection model with intracellular delay 237 now, we assume there is an another subtype of virus in within a host which competes with the original virus for host cell resource. assuming that super infection is negligible, an ordinary differential equations can be formulate along the line of (1.1) to describe the interaction between the two subtype viruses and host cells, as given below.  ṫ = λ−dt −k1tv1 −k2tv2, ṫ1 = k1tv1 −µ1t1, ṫ2 = k2tv2 −µ2t2, v̇1 = p1t1 − c1v1, v̇2 = p2t2 − c2v2, (1.2) where t1(t) denotes the population size of cells productively infected by strain-1 virus, whereas t2(t) denotes the population size of cells productively infected by strain-2 virus at time t; v1(t) and v2(t) represent the respective population sizes of subtype-1 and subtype-2 viruses; k1 and k2 represent the rate constants at which uninfected target cells are infected by subtype-1 and subtype-2 viruses, respectively. the two subtypes of infected cells are assumed to have two different death rate µ1 and µ2. once uninfected target cells are infected by subtype-1 (subtype-2) viruses, new subtype-1 (subtype-2) virus particles are produced with constant rate p1 (p2). the new subtypes of virus have the respective clearance rate c1 and c2. all the parameters of the model are assumed to be positive. here we omit the super-infection in host cells. however, in reality, there is a lag between the time target cells are contacted by virus particles and the time the contacted cells become actively affected meaning that the virions have enter cells and started producing new virions [23]. this can be explained by the initial phase of the virus life cycle, which include all stages from viral attachment until the time that the host cell contains the infectious viral particles in its cytoplasm. to account for this lag, models that include time delays have been developed and investigated [8, 15, 23]. one distinct feature of delay differential equation models is that a delay typically destabilizes an stable equilibrium and causes sustained oscillation through hopf bifurctions. by rigorously establishing the global dynamics of the two-strain competitive viral model with intracellular delays, we show that no sustained oscillations are possible in our model. to incorporate the intracellular phase of the virus life-cycle, we assume that subtype-1 virus and subtype-2 virus production occur in average, τ1 and τ2 time units later, after the respective virus enter the host cells. the recruitment of subtype-1 virus producing cells at time t is given by the number of cells that were newly infected by strain-1 at time t−τ1 and are still alive at time t. in the same way, the recruitment of subtype-2 virus producing cells at time t is given by the number of cells that were newly infected by strain-2 at time t− τ2 and are still alive at time t. if we assume two constant death rates s1 and s2 for infected but not yet virus-producing cells for subtype-1 and subtype-2, the probability of subtype-1 surviving the time period from t−τ1 to t is e−s1τ1 , the probability of subtype-2 surviving the time period from t−τ2 to t is e−s2τ2 . the transfer diagram for the transmission of viral infection under such a scenario is shown in figure 1. thus the following delay differential equations model is proposed:  ṫ = λ−dt(t) −k1t(t)v1(t) −k2t(t)v2(t), ṫ1 = k1t(t− τ1)v1(t− τ1)e−s1τ1 −µ1t1(t), ṫ2 = k2t(t− τ2)v2(t− τ2)e−s2τ2 −µ2t2(t), v̇1 = p1t1(t) − c1v1(t), v̇2 = p2t2(t) − c2v2(t), (1.3) 238 j. xu 1 − infected cells that can free subtype-1 virus infected cells by subtype-1 produce subtype-1 uninfected cells d free subtype-2 virus infected cells by subtype-2 infected cells that can produce subtype-21 − figure 1. transfer diagram for model (1.3) delays have been incorporated into virus dynamics models in [8, 23, 12], but only for single strain models. here we consider two strains. many previous in-host models also considered the effects of anti-viral drug therapies such as haart [15, 1, 21], but only local stability were analysed in these works. we note that by renaming the coeffiicients due to the effect of reverse transcriptase inhibitors and protease inhibitors, the model in [15, 1, 21] can be transformed into the form of (1.3). our results on the global dynamics of model (1.3) can apply to these models with anti-viral therapies, and hence can rule out the exitence of periodic solutions. this shows novelty of this work and should benefit other researchers working on similar models. in the present section we analyse model (1.3) including intracellular delays. we establish global asymptotic stability of the infected-free, and single-infected by constructing lyapunov functionals. to this end, we first establishes the well-posedness of (1.3) in section 3.2. then we discuss the existence of equilibria in the feasible region and derive the basic reproductive number r0. it turns out that r1 is a decreasing function of the delay τ1 and r2 is a decreasing function of the delay τ2. these imply that ignoring the intracellular delays will overestimate the basic reproduction number. we show that the basic reproductive number r0 generically determines the global dynamics of model (1.3). more specifically, if r0 ≤ 1, the infection-free equilibrium e0 is globally asymptotically stable, and two subtype viruses will be cleared; if r0 > 1 and r1 6= r2, the single-infected equilibrium arising from the greater basic reproduction number is globally asymptotically stable. the proof utilizes a global lyapunov functional that is motivated by the work in [11, 12]. the global stability of single-infected equlibira rule out any possibility of sustained oscillations. in addition, numerical simulations are also conducted to demonstrate global dynamics of system (1.3). 2. well-posedness in the same way as in the previous section, the system (1.3) is biologically acceptable in the sense that no population goes negative. we expect that starting from non-negative initial values, the corresponding solution remains non-negative. to proceed, we follow the convention to denote by c1 = c([−τ1, 0],r) and c2 = c([−τ2, 0],r) the banach spaces of continuous functions mapping the interval [−τi, 0] into r, i = 1, 2, with norm ‖φi‖ = sup−τi≤θ≤0 |φi(θ)| for φi ∈ ci. let τ = max{τ1,τ2}, denote by global dynamics of a two-strain hiv infection model with intracellular delay 239 c = c([−τ, 0],r) the banach space of continuous functions mapping the interval [−τ, 0] into r, with norm ‖φ‖ = sup−τ≤θ≤0 |φ(θ)| for θ ∈ c. the nonnegative cone of c,c1 and c2 are defined as c+ = c([−τ, 0],r+),c+1 = c([−τ1, 0],r+) and c + 2 = c([−τ2, 0],r+). the initial conditions for system (1.3) are chosen at t = 0 as ϕ ∈ c+ × r+ × r+ ×c+1 ×c + 2 . the well-posedness for our delay differential equation model (1.3) is established by the following theorem. theorem 2.1. under the above initial conditions, all solutions of system (1.3) are positive and ultimately bounded in c ×r×r×c1 ×c2 proof. first, we prove that t(t) is positive for all t ≥ 0. assuming the opposite, let t1 > 0 be the first time such that t(t1) = 0, which means t(t) > 0 as t ∈ [0, t1). since ṫ = λ−dt(t) −k1t(t)v1(t) −k2t(t)v2(t), we get ṫ(t1) = λ > 0,and hence t(t) < 0 for t ∈ (t1 − �,t1) where � > 0 is sufficiently small. this contradicts t(t) > 0 for t ∈ [0, t1). it follows that t(t) > 0 for t > 0. next, we show v1(t) ≥ 0 for all t ≥ 0. assume the opposite and let t2 > 0 be the first time such that v1(t2) = 0. since v̇1(t) = p1t1(t) − c1v1(t), we have v̇1(t2) = p1t1(t2). on the other hand, solving t1(t) by the second equation of (1.3) gives t1(t2) = (t1(0) + ∫ t2 0 k1t(θ − τ1)v1(θ − τ1)e−s1τ1eµ1θdθ)e−µ1t2 > 0 hence v̇1(t2) = p1t1(t2) > 0 implying v1(t) is positive for all t ≥ 0. the positiveness of t(t) and v1(t) and the following formula t1(t) = (t1(0) + ∫ t 0 k1t(θ − τ1)v1(θ − τ1)e−s1τ1eµ1θdθ)e−µ1t > 0. in turn leads to the positiveness of t1(t) for all t ≥ 0. similarly, we can show that v2(t) and t2(t) are positive for t ≥ 0 under positive initial conditions. from the first equation of (1.3), we obtain ˙t(t) ≤ λ−dt(t). hence limsupt→∞t(t) ≤ λd . adding the first three equations of (1.3), it follows (t(t) + t1(t + τ1) + t2(t + τ2)) ′ = λ−dt(t) −µ1t1(t + τ1) −µ2t2(t + τ2) + k1t(t)v1(t)(e −s1τ1 − 1) + k2t(t)v2(t)(e−s2τ2 − 1) ≤ λ− r̃(t(t) + t1(t + τ1) + t2(t + τ2)) where r̃ = min{d,µ1,µ2}. thus, limsupt→∞(t(t) +t1(t+τ1) +t2(t+τ2)) ≤ λr̃ . for any � > 0,∃t ∗ > 0, such that t(t) + t1(t + τ1) + t2(t + τ2) ≤ λr̃ + � for all t ≥ t ∗. thus, t(t),t1(t) and t2(t) are all ultimately bounded by λ r̃ . the fourth equation of (1.3) implies v̇1 = p1t1(t) − c1v1(t) ≤ p1( λ r̃ + �) − c1v1(t), t ≥ t∗ this implies limsupt→∞v1 ≤ p1c1 ( λ r̃ +�). since � > 0 is arbitrary, we attain limsupt→∞v1(t) ≤ p1λc1r̃ . similarly, we can obtain limsupt→∞v2(t) ≤ p2λc2r̃ . therefore, t(t),t1(t),t2(t),v1(t) and v2(t) are ultimately bounded in c ×r×r×c1 ×c2. 240 j. xu 3. equilibria and basic reproduction numbers in system (1.3), without infection (t1,t2,v1,v2) = (0, 0, 0, 0), uninfected target cells stabilizes at the equilibrium t = λ d . the basic reproductive number r1 for in-host models [17, 12, 16] measures the average number virus-producing target cells produced by a single subtype-1 virus-producing target cell during its entire infectious period in an entirely uninfected target-cell population. as illustrated in figure 2, the basic reproduction number r1 for strain-1 is given by r1 = p1 µ1 · k1e −s1τ1 c1 · λ d . (3.1) similarly, the basic reproduction number r2 for strain-2 which is the average number virus-producing target cells produced by a single subtype-2 virus producing target cell during its entire infectious period in an entirely uninfected target-cell population is obtained by r2 = p2 µ2 · k2e −s2τ2 c2 · λ d . (3.2) when no intracellular delay is considered, τ1 = τ2 = 0, our r1 and r2 reduce to the respective basic reproduction number for our previous model (3.1) (i.e. (2.21)). if s > 0, r1 and r2 is the decreasing functions of the delay τ1 and τ2. it shows that the intracellular delays decrease r1 and r2 if cells die during the delay periods. thus, ignoring the intracellular delay in a viral model will overestimate the basic reproduction number. from our system (1.3) and our result (3.1) (3.2), we define the system basic reproduction number r0 = max{r1,r2} . (3.3) 1 − infected cells that can free subtype-1 virus infected cells by subtype-1 produce subtype-1 uninfected cells d free subtype-2 virus infected cells by subtype-2 infected cells that can produce subtype-21 − burst size basic reproduction number= ⋅ ⋅ figure 2. an illustration of the basic reproduction number of model(1.3) model system (1.3) always has the infection-free equilibrium e0 = ( λ d , 0, 0, 0, 0 ) . there are two possible single-infection equilibria e1 = ( t̄, t̄1, 0, v̄1, 0 ) and e2 = ( t̃, 0, t̃2, 0, ṽ2 ) , where t̄ = λ d 1 r1 , t̄1 = dc1 k1p1 (r1 − 1), v̄1 = d k1 (r1 − 1). (3.4) and t̃ = λ d 1 r2 , t̃2 = dc2 k2p2 (r2 − 1), ṽ2 = d k2 (r2 − 1). (3.5) it turns out that the values of r1 and r2 determine the existence of the single-infection equilibria: e1 exists if and only if r1 > 1 and e2 exists if and only if r2 > 1. obviously, e1 and e2 are biologically meaningful under the conditions. global dynamics of a two-strain hiv infection model with intracellular delay 241 it is also possible for our model (1.3) to obtain the double-infection equilibrium which means a equilibrium with all components being positive. denote such a possible equilibrium by e3 = (t∗,t∗1 ,t ∗ 2 ,v ∗ 1 ,v ∗ 2 ), then calculation shows that the components in e3 must satisfy  t∗ = µ1c1e s1τ1 k1p1 (i.e. d λr1 ) = µ2c2e s2τ2 k2p2 (i.e. d λr2 ), t∗1 = c1v ∗ 1 p1 , t∗2 = c2v ∗ 2 p2 , d(r1 − 1) = k1v ∗1 + k2v ∗ 2 , d(r2 − 1) = k1v ∗1 + k2v ∗ 2 . (3.6) by the last two equation in (3.6), it is clear that e3 exists if and only if r1 = r2 > 1. (3.7) if (3.7) holds, there are actually infinitely many co-existence equilibria. summarizing the above results, we have the following conclusion. when r0 ≤ 1, e0 is the only equilibrium; when r1 > 1,r2 ≤ 1, there are e0 and e1; when r2 > 1,r1 ≤ 1, there are e0 and e2; when r1 > 1 and r2 > 1, in addition to e0,e1 and e2, there are infinitely many co-exitence equilibria if r1 = r2 > 1. considering the fact that there are ten model parameters in r1 and r2, the identity r1 = r2 is unlikely to hold in practice (or infeasible), and hence, e3 will not be considered here in this thesis. 4. global stability of equilibria in this section we study the global stability of equilibria by using the lyapunov functionals. we apply lyapunov functionals similar to those recently used by [11, 6, 20]. a useful function is used to construct our lyapunov fuctionals: g(x) = x− ln(x) − 1. this function attains the global minimum at x = 1, g(1) = 0, and remains positive for all other postitive values of x. our lyapunov functionals take advantage of these properties of g(x). in the following theorems we show that the equilibria exhibit global stability under some threshold conditions. theorem 4.1. if r0 ≤ 1, the infection free-equilibrium e0 is globally asymptotically stable. proof. let t0 = λ d and consider the lyapunov functional v (t,t1,t2,v1,v2) = t0g(t(t)/t0) + e s1τ1t1(t) + e s2τ2t2(t) + µ1 p1 es1τ1v1(t) + µ2 p2 es2τ2v2(t) +k1 ∫ 0 −τ1 t(t + θ)v1(t + θ) dθ + k2 ∫ 0 −τ2 t(t + θ)v2(t + θ) dθ. obviously, v (t,t1,t2,v1,v2) is non-negative in the positive cone c +×r+×r+×c+1 ×c + 2 and attains zero at e0. we will show that the derivative of v along the trajectories of our model (1.3) is negatively defininte. differentiation gives 242 j. xu v̇ = ˙t(t) − t0 t(t) ˙t(t) + es1τ1 ˙t1(t) + e s2τ2 ˙t2(t) + µ1 p1 es1τ1 ˙v1(t) + µ2 p2 es2τ2 ˙v2(t) +k1t(t)v1(t) −k1t(t− τ1)v1(t− τ1) + k2t(t)v2(t) −k2t(t− τ2)v2(t− τ2) = λ−dt(t) −k1t(t)v1(t) −k2t(t)v2(t) − t0 t(t) (λ−dt −k1t(t)v1(t) −k2t(t)v2(t)) +es1τ1 ( k1t(t− τ1)v1(t− τ1)e−s1τ1 −µ1t1 ) + es2τ2 (k2t(t− τ2)v2(t− τ2)e−s2τ2 −µ2t2) + µ1 p1 es1τ1 ( p1t1(t) − c1v1(t)) + µ2 p2 es2τ2 (p2t2(t) − c2v2(t) ) +k1t(t)v1(t) −k1t(t− τ1)v1(t− τ1) + k2t(t)v2(t) −k2t(t− τ2)v2(t− τ2) after cancelling terms, using t0 = λ d and rearranging terms, we get v̇ = λ−dt(t) − t0 t(t) λ + dt0 + ( k1t0 − c1µ1 p1 es1τ1 ) v1(t) + ( k2t0 − c2µ2 p2 es2τ2 ) v2(t) = λ ( 2 − t(t) t0 − t0 t(t) ) + c1µ1 p1 es1τ1 ( k1p1λ µ1c1d es1τ1 − 1 ) v1(t) + c2µ2 p2 es2τ2 ( k2p2λ µ2c2d es2τ2 − 1 ) v2(t) = λ ( 2 − t(t) t0 − t0 t(t) ) + µ1c1 p1 es1τ1 (r1 − 1)v1(t) + µ2c2 p2 es2τ2 (r2 − 1)v2(t). since the arithmetic mean is greater than or equal to the geometric mean, if r0 = max{r1,r2}≤ 1, each of the three terms on the right hand side is non-positive. hence v̇ (t,t1,t2,v1,v2) ≤ 0, and v̇ = 0 if and only if (t,t1,t2,v1,v2) = ( λ d , 0, 0, 0, 0 ) = e0 therefore, the globally asymptotical stability of e0 follows from the lyaunov-lasalle invariance principle by [7]. when r0 > 1, then e0 becomes unstable and at least one of the e1 and e2 exists. we now investigate the global stability of these two possible single-strain equilibria. theorem 4.2. assume that e1 exists (i.e. r1 > 1), if r2 < r1, then, e1 is globally asymptotically stable. proof. define a lyapunov functional v : c ×r×r×c1 ×c2 → r by v (t,t1,t2,v1,v2) = t̄g( t(t) t̄ ) + t̄1e s1τ1g( t1(t) t̄1 ) + es2τ2t2(t) + µ1 p1 v̄1e s1τ1g( v1(t) v̄1 ) + µ2 p2 es2τ2v2(t) + k1t̄v̄1 ∫ 0 −τ1 g( t(t + θ)v1(t + θ) t̄v̄1 ) dθ +k2 ∫ 0 −τ2 t(t + θ)v2(t + θ) dθ. by the properties of g(x), the lyapunov functional v (t,t1,t2,v1,v2) is non-negative in the positive cone c+ × r+ × r+ ×c+1 ×c + 2 and attains zero at e1. in order to show v̇ is negatively definite, we differentiate v (t,t1,t2,v1,v2) along the trajectories of (1.3) to get global dynamics of a two-strain hiv infection model with intracellular delay 243 v̇ = ṫ(t) + t̄ t(t) ṫ(t) + es1τ1ṫ1(t) −es1τ1 t̄1 t1(t) ṫ1(t) + e s2τ2ṫ2(t) + µ1 p1 es1τ1v̇1(t) − µ1 p1 es1τ1 v̄1 v1(t) v̇1(t) + µ2 p2 es2τ2v̇2(t) +k1t̄v̄1 d dt ∫ 0 −τ1 g( t(t + θ)v1(t + θ) t̄v̄1 ) dθ +k2t(t)v2(t) −k2t(t− τ2)v2(t− τ2). (4.1) note that k1t̄v̄1 d dt ∫ 0 −τ1 g( t(t + θ)v1(t + θ) t̄v̄1 ) dθ = k1t̄v̄1 ∫ 0 −τ1 d dt g( t(t + θ)v1(t + θ) t̄v̄1 ) dθ = k1t̄v̄1 ∫ 0 −τ1 d dθ g( t(t + θ)v1(t + θ) t̄v̄1 ) dθ = k1t̄v̄1 ( g( t(t)v1(t) t̄v̄1 ) −g( t(t− τ1)v1(t− τ1) t̄v̄1 ) ) = k1t̄v̄1 ( t(t)v1(t) t̄v̄1 − ln t(t)v1(t) t̄v̄1 − t(t− τ1)v1(t− τ1) t̄v̄1 + ln t(t− τ1)v1(t− τ1)) t̄v̄1 ) = k1t(t)v1(t) −k1t̄v̄1lnt(t)v1(t) + k1t̄v̄1lnt̄v̄1 −k1t(t− τ1)v1(t− τ1) +k1t̄v̄1lnt(t− τ1)v1(t− τ1) −k1t̄v̄1lnt̄v̄1 = k1t(t)v1(t) −k1t(t− τ1)v1(t− τ1) +k1t̄v̄1lnt(t− τ1)v1(t− τ1) −k1t̄v̄1lnt(t)v1(t) (4.2) plugging (4.2) and system of (1.3) into equation (4.1), we obtain v̇ =λ−dt(t) −k1t(t)v1(t) −k2t(t)v2(t) − t̄ t(t) λ + dt̄ + k1t̄v1(t) + k2t̄v2(t) + k1t(t− τ1)v1(t− τ1) −µ1es1τ1t1(t) − t̄1 k1t(t− τ1)v1(t− τ1) t1(t) + es1τ1µ1t̄1 + k2t(t− τ2)v2(t− τ2) −µ2es2τ2t2(t) + µ1es1τ1t1(t) − µ1c1 p1 es1τ1v1(t) −µ1es1τ1v̄1 t1(t) v1(t) + µ1c1 p1 es1τ1v̄1 + µ2e s2τ2t2(t) − µ2c2 p2 es2τ2v2(t) + k1t(t)v1(t) −k1t(t− τ1)v1(t− τ1) −k1t̄v̄1lnt(t)v1(t) + k1t̄v̄1lnt(t− τ1)v1(t− τ1 + k2t(t)v2(t) −k2t(t− τ2)v2(t− τ2). (4.3) the components of e1 are related by the equilibrium equation, i.e., 244 j. xu   λ = dt̄ + k1t̄v̄1 k1t̄v̄1 = µ1t̄1e s1τ1 p1t̄1 = c1v̄1 k1t̄ = µ1t̄1e s1τ1 v̄1 = µ1t̄1e s1τ1c1 p1t̄1 = µ1c1 p1 es1τ1. (4.4) making use of these, we can rearrange and simplify the equation (4.3) as v̇ = dt̄ ( 2 − tt t̄ − t̄ t(t) ) − k1t̄ 2v̄1 t(t) +k1t̄v̄1 + k2t̄v2(t) −k1t̄1 t(t− τ1)v1(t− τ1) t1(t) +k1t̄v̄1 − k1t̄v̄1 t̄1 v̄1 t1(t) v1(t) + k1t̄v̄1 − µ2c2 p2 es2τ2v2(t) −k1t̄v̄1lnt(t)v1(t) + k1t̄v̄1lnt(t− τ1)v1(t− τ1) = dt̄ ( 2 − tt t̄ − t̄ t(t) ) −k1t̄v̄1 ( g( t̄1t(t− τ1)v1(t− τ1) t̄v̄1t1(t) ) − ln t̄1t(t− τ1)v1(t− τ1) t̄v̄1t1(t) ) −k1t̄v̄1 ( g( t̄ t(t) ) − ln t̄ t(t) ) −k1t̄v̄1(g( v̄1t1(t) t̄1v1(t) ) − ln v̄1t1(t) t̄1v1(t) ) −k1t̄v̄1 [lnt(t)v1(t) − lnt(t− τ1)v1(t− τ1)] + ( k2t̄ − µ2c2 p2 es2τ2 ) v2(t) = dt̄ ( 2 − tt t̄ − t̄ t(t) ) −k1t̄v̄1g( t̄1t(t− τ1)v1(t− τ1) t̄v̄1t1(t) ) −k1t̄v̄1g( t̄ t(t) ) −k1t̄v̄1g( v̄1t1(t) t̄1v1(t) ) + k2λ d ( 1 r1 − 1 r2 ) v2(t). therefore, by our assumptions, v̇ ≤ 0 with equality holding only at e1. from the lyapunovlasalle inveriance principle [7], the equilibrium e1 is globally asymptotically stable. the proof is completed. parallel to theorem 4.2, we have the following theorem for e2 theorem 4.3. assume that e2 exists (i.e. r2 > 1), if r1 < r2, then e2 is globally asymptotically stable. proof. the proof of this theorem is symmetric to that of theorem 4.2 by considering the following lyapunov functional:v : c ×r×r×c1 ×c2 → r global dynamics of a two-strain hiv infection model with intracellular delay 245 v (t,t1,t2,v1,v2) = t̃g( t(t) t̃ ) + es1τ1t1(t) + t̃2e s2τ2g( t2(t) t̃2 ) + µ1 p1 es1τ1v1(t) + µ2 p2 ṽ2e s2τ2g( v2(t) ṽ2 ) + k1 ∫ 0 −τ1 t(t + θ)v1(t + θ) dθ k2t̃ṽ2 ∫ 0 −τ2 g( t(t + θ)v2(t + θ) t̃ṽ2 ) dθ we omit the details of the proof. 5. numerical simulations in this section, we present some numeric simulations for the dde model (3.2) to confirm and illustrate the theoretic results obtained in section 3.4, which is not significantly different from those for the ode model (2.2), except that some plottings are in logarithmic function for better and clearer displays. first, we chose the following values for the model parameters: λ = 6,d = 1,k1 = 2,p1 = 1,c1 = 3,µ1 = 10,s1 = 2,τ1 = 0.1,k2 = 3,p2 = 2,c2 = 2.5,µ2 = 15,s2 = 1.5,τ2 = 0.15. this give the values two individual basic reproduction numbers r1 = 0.327 and r2 = 0.767. three sets of initial values are used: (i) t(0) = 80,t1(0) = 50,t2(0) = 40,v1(0) = 45,v2(0) = 35; (ii) t(0) = 60,t1(0) = 70,t2(0) = 50,v1(0) = 30,v2(0) = 20; (iii) t(0) = 50,t1(0) = 60,t2(0) = 30,v1(0) = 20,v2(0) = 45. we used a base 10 logarithmic scale for target cells population. the corresponding solutions are presented in figure 3. second, we chose the following values for the model parameters: λ = 6,d = 1,k1 = 5,p1 = 6,c1 = 4,µ1 = 3,s1 = 2,τ1 = 0.1,k2 = 1,p2 = 4,c2 = 3,µ2 = 4,s2 = 1.5,τ2 = 0.15. this give the values two individual basic reproduction numbers r1 = 12.28 and r2 = 1.597. three sets of initial values are used: (i) t(0) = 80,t1(0) = 50,t2(0) = 40,v1(0) = 45,v2(0) = 35; (ii) t(0) = 60,t1(0) = 70,t2(0) = 50,v1(0) = 30,v2(0) = 20; (iii) t(0) = 50,t1(0) = 60,t2(0) = 30,v1(0) = 20,v2(0) = 45. a base 10 logarithmic scale for target cells population, subtype-1 infected cells and subtype-1 virus cells was employed in our figures. the corresponding solutions are presented in figure 4. third, we chose the following values for the model parameters: λ = 6,d = 1,k1 = 4,p1 = 8,c1 = 8,µ1 = 5,s1 = 2,τ1 = 0.1,k2 = 3,p2 = 10,c2 = 5,µ2 = 4, ,s2 = 1.5,τ2 = 0.15. this give the values two individual basic reproduction numbers r1 = 3.93 and r2 = 7.19. three sets of initial values are used: (i) t(0) = 80,t1(0) = 50,t2(0) = 40,v1(0) = 45,v2(0) = 35; (ii) t(0) = 60,t1(0) = 70,t2(0) = 50,v1(0) = 30,v2(0) = 20; (iii) t(0) = 50,t1(0) = 60,t2(0) = 30,v1(0) = 20,v2(0) = 45. a base 10 logarithmic scale for target cells population, subtype-2 infected cells and subtype-2 virus cells was employed in our figures. the corresponding solutions are presented in figure 5. 246 j. xu 0 1 2 3 4 5 6 7 8 9 10 10 −5 10 0 10 5 target uninfected cells time lo g1 0 ta rg et c el ls 0 2 4 6 8 10 0 200 400 infected cells by subtype virus−1 time in fe ct ed c el ls 0 2 4 6 8 10 0 500 1000 infected cells by subtype virus−2 time in fe ct ed c el ls 0 2 4 6 8 10 0 100 200 300 time vi ru s− 1 po pu la tio n subtype virus−1 cells 0 2 4 6 8 10 0 200 400 time vi ru s− 2 po pu la tio n subtype virus−2 cells figure 3. r1 < 1 and r2 < 1: viruses of both strains all die out 0 1 2 3 4 5 6 7 8 9 10 10 −5 10 0 10 5 target uninfected cells time lo g1 0 ta rg et c el ls 0 5 10 10 0 10 2 10 4 infected cells by subtype virus−1 time lo g1 0 in fe ct ed c el ls 0 5 10 0 200 400 infected cells by subtype virus−2 time in fe ct ed c el ls 0 5 10 10 0 10 2 10 4 timelo g1 0 vi ru s− 1 po pu la tio n subtype virus−1 cells 0 5 10 0 100 200 time vi ru s− 2 po pu la tio n subtype virus−2 cells figure 4. r1 > 1 and r2 < r1: subtype-1 wins the competition global dynamics of a two-strain hiv infection model with intracellular delay 247 0 1 2 3 4 5 6 7 8 9 10 10 −5 10 0 10 5 target uninfected cells time lo g1 0 ta rg et c el ls 0 5 10 0 500 1000 infected cells by subtype virus−1 time in fe ct ed c el ls 0 5 10 10 −5 10 0 10 5 infected cells by subtype virus−2 time lo g1 0 in fe ct ed c el ls 0 5 10 0 200 400 600 time vi ru s− 1 po pu la tio n subtype virus−1 cells 0 5 10 10 0 10 2 10 4 timelo g1 0 vi ru s− 2 po pu la tio n subtype virus−2 cells figure 5. r2 > 1 and r1 < r2: subtype-2 wins the competition 6. discussion it is widely recognized that time delays cause sustained oscillations in form of periodic solutions in inhost models with cell divisions and intracellular delays [2]. it is interesting to explore the dynamics of the viral load for two strains with intracellular delays both from mathematical and biological perspective [9]. in this paper, we employ a two-strain mathematical model to study the mechanistic basis of the emergence of the competitive viral strains in host cells. we have carried out complete analysis for two-strain in-host model with intracellular delays system (1.3). the analysis suggests that the basic reproductive ratio palys an important role in predicting viral persistence or eradication. the global dynamics of model (1.3) is rigorously established: if the basic reproduction number r0 ≤ 1, then all solutions converge to the infection-free equilibirum e0; if r0 > 1, then all positive solutions converge to the single chronic-infection equilibrium e1 or e2 which is determined by the relative magnitudes of r1 and r2. the stability results for e0, e1 and e2 are obtained analytically, while the stability of the co-existence equilibrium e3 via numerical simulations. the intracellular delays can reduce the basic reproduction number r0 if cell die during the delay period (3.1) (3.2). as a consequence, ignoring the delay will produce overestimation of r0. our result shows that no sustained oscillation regime exists without cell division even in the presence of intracellular delays. the two-strain hiv model with intracellular delays could provide worthwile information that potentially could allow the design of efficient individual strategies of hiv treatment. 248 j. xu references [1] s. bonhoeffer, r. m. may, g. m. shaw and m. a. nowak, virus dynamics and drug therapy, proc.natl.acad.sci.usa, 94 (1997), 6971–6976. 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[23] h. zhu and x. zou, impact of delays in cell infection and virus production on hiv-1 dynamics, mathematical medicine and biology, 25 (2008), 99–112. school of artificial intelligence, jianghan university, wuhan 430056, p. r. china e-mail address: jxu259@protonmail.com mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 2, number 2, june 2021, pp.123-133 https://doi.org/10.5206/mase/13822 global properties of a virus dynamics model with self-proliferation of ctls cuicui jiang, huan kong, guohong zhang*, and kaifa wang* abstract. a viral infection model with self-proliferation of cytotoxic t lymphocytes (ctls) is proposed and its global dynamics is obtained. when the per capita self-proliferation rate of ctls is sufficient large, an infection-free but immunity-activated equilibrium always exists and is globally asymptotically stable if the basic reproduction number of virus is less than a threshold value, which means that the immune effect still exists though virus be eliminated. qualitative numerical simulations further indicate that the increase of per capita self-proliferation rate may lead to more severe infection outcome, which may provide insight into the failure of immune therapy. 1. introduction outbreaks of viral infection have become a major global health concern. different kinds of virus, such as hepatitis b virus (hbv), hepatitis c virus (hcv), human immunodeficiency virus (hiv), ebola virus and zika virus, have been associated with severe outcomes. a great deal of effort has been put toward to understand the life cycle of these virus. with the development of biomedical research, mathematical models also play an increasingly important role to provide insights into virus infection and dynamics, as well as on how an infection can be reduced or even eradicated. for example, on hbv infection, nowak et al. [13] first proposed a basic three-dimensional viral infection model within-host. note that immune responses play a critical part in the process of viral infections. nowak et al. [14] further proposed the following four-dimensional system with the cytotoxic t lymphocytes (ctls) population based on the basic model:   x′ = s−βxv −d1x, y′ = βxv −d2y −pyz, v′ = ky −uv, z′ = cyz −d3z. (1.1) here x(t),y(t),v(t) and z(t) represent uninfected target cells, infected cells, free virus and ctls, respectively. uninfected cells are produced at a constant rate s, die at rate d1x, and become infected at rate βxv. infected cells are produced at rate βxv and die at rate d2y. free virus are produced from infected cells at rate ky and die at rate uv. ctls are produced at rate cyz due to the stimulation of infected cells, and die at rate d3z. infected cells are eliminated by ctls at rate pyz. after that, based on the basic models, many studies were carried out to analysis of the dynamics of various virus infection within-host, such as [1, 9, 12, 15, 16, 17, 22, 23, 25] and the reference therein. received by the editors 5 april 2021; revised 13 may 2021; 19 may 2021 accepted; published online 24 may 2021. 2000 mathematics subject classification. primary 92d30; secondary 34k20, 34k25. key words and phrases. viral infection; self-proliferation of ctls; lyapunov function; global stability. cuicui jiang’s research was supported by nsfc grant 11901576 and the youth foundation of army medical university 2017xqn05. guohong zhang’s research was partially supported by an nsfc grant 11871403. kaifa wang’s research was partially supported by an nsfc grant 11771448. 123 124 cuicui jiang, huan kong, guohong zhang*, and kaifa wang* recent studies on the production mechanism of immune cells show that its self-proliferation cannot be neglected besides the stimulation of infected cells in [7]. thus, to understand the effect of selfproliferation, based on the system (1.1) and [7], we propose the following new virus infection model:  x′ = s−βxv −d1x, y′ = βxv −d2y −pyz, v′ = ky −uv, z′ = cyz + rz(1 − z m ) −d3z. (1.2) here the logistic proliferation term rz(1 − z/m) describes the self-proliferation of ctls, in which parameter r denotes a per capita self-proliferation rate, and m means the capacity of ctls population. when r = 0, i.e., without self-proliferation of ctls, (1.2) has been completely analyzed in [10] if there is not explicit dynamics of free virus under a plausible quasi steady-state assumption. to explore the effects of the recruitment of immune responses on virus infection, the main contribution of the present paper is to obtain the complete global properties of (1.2) when r > 0. 2. global dynamics analysis since we are interested in the dynamics of viral infection, and not the initial processes of infection, we assume that the initial condition of (1.2) has the form x(0) > 0,y(0) > 0,v(0) > 0 and z(0) > 0. based on the initial conditions, it is easy to show that the solutions of system (1.2) are non-negative and ultimately bounded. the equilibria of (1.2) are the solutions of the following algebraic equations:  s−βxv −d1x = 0, βxv −d2y −pyz = 0, ky −µv = 0, cyz + rz(1 − z m ) −d3z = 0. (2.1) clearly, system (1.2) always has infection-free equilibrium e0 = ( s d1 , 0, 0, 0). according to the definition and algorithm of the basic reproduction number of virus in [4], we can obtain the basic reproduction number of virus r0 = βsk d1d2µ . using the fourth equation of (2.1), we have z = 0 or z = m r (r −d3 + cy). (2.2) when z = 0, based on the first three equations of (2.1), it is easy to obtain that the immunity-inactivated infection equilibrium e1 = ( s d1r0 , d1µ(r0−1) βk , d1(r0−1) β , 0) always exists if r0 > 1. in addition, using the third and first equation of (2.1), we have v = k µ y and x = s βv + d1 = sµ βky + d1µ . (2.3) after that, using the second equation and (2.3), we have βk µ xy −d2y −pyz = 0. (2.4) thus, when y = 0, using (2.2), we know that an infection-free but immunity-activated equilibrium e2 = ( s d1 , 0, 0, m(r−d3) r ) will appear if r > d3. otherwise, using (2.2), (2.3) and (2.4), we have f(y) ≡ βk µ x−d2 −pz = βks βky + d1µ −d2 − mp r (r −d3 + cy) = 0. clearly, f(+∞) < 0 and function f(y) is monotonically decreasing since f′(y) < 0 always valid. as a result, f(y) = 0 has a unique positive root if f(0) = βks d1µ −d2 − mp r (r −d3) = d2[r0 − 1 − mp rd2 (r −d3)] > 0, global properties of a virus dynamics model with self-proliferation of ctls 125 i.e., r0 > 1 + mp rd2 (r −d3). when r < d3, according to (2.2), in order to keep the positive of z, we need f( d3 −r c ) = βksc βk(d3 −r) + d1µ −d2 > 0 is valid, i.e., r0 > 1 + βk(d3−r) cd1µ . in summary, we have the following proposition. proposition 2.1. the following hold. (i) if r0 > 1, the immunity-inactivated infection equilibrium e1 always exists. especially, an infection-free but immunity-activated equilibrium e2 will appear if r > d3. (ii) suppose that 0 ≤ r ≤ d3. if r0 > 1 + βk(d3−r) cd1µ , system (1.2) has a unique immunity-activated infection equilibrium e3 = (x3,y3,v3,z3), where x3 = sµ βky3 + d1µ , v3 = k µ y3, z3 = m r (r −d3 + cy3), and y3 is the unique positive root of f(y) = 0 in this case. (iii) suppose that r > d3. if r0 > 1 + pm(r−d3) rd2 , system (1.2) has a unique immunity-activated infection equilibrium e4 = (x4,y4,v4,z4), where x4 = sµ βky4 + d1µ , v4 = k µ y4, z4 = m r (r −d3 + cy4), and y4 is the unique positive root of f(y) = 0 in this case. in order to obtain the stability of above mentioned equilibria, we first give the jacobian matrix j of system (1.2) at (x,y,v,z), j =   −βv −d1 0 −βx 0 βv −d2 −pz βx −py 0 k −µ 0 0 cz 0 cy + r −d3 − 2rzm   . (2.5) so we have the following results. theorem 2.2. the following hold. (i) when 0 ≤ r ≤ d3, the infection-free equilibrium e0 is globally asymptotically stable if r0 < 1, and it is unstable when r0 > 1. (ii) when r > d3, the infection-free equilibrium e0 is always unstable. proof. according to (2.5), we have the characteristic equation of system (1.2) at e0 (λ + d1)(λ− (r −d3))h0(λ) = 0, (2.6) where h0(λ) = λ 2 + (d2 + µ)λ + d2µ(1−r0). it is easy to show that λ1 = −d1 < 0 and λ2 = r−d3 are the roots of (2.6). further, we can get all roots of h0(λ) are negative real part if r0 < 1, and there is one positive real root if r0 > 1. (i) when 0 ≤ r < d3, we have λ2 < 0. as a result, the infection-free equilibrium e0 is locally asymptotically stable if r0 < 1 and 0 ≤ r < d3, and is unstable if r0 > 1. when r = d3, λ2 = 0. thus, the center manifold is a curve tangent to the z-axis. in this case, settling a transformation 126 cuicui jiang, huan kong, guohong zhang*, and kaifa wang* x̃ = x−x0, ỹ = y, ṽ = v, z̃ = z, we know (1.2) becomes  x′ = −βxv − βs d1 v −d1x, y′ = βxv + βs d1 y −d2y −pyz, v′ = ky −µv, z′ = cyz − r m z2, (2.7) where we substitute x,y,v,z for x̃, ỹ, ṽ, z̃. to obtain the approximative expression of the center manifold, we set x = k2z 2 + k3z 3 + o(z3), y = n2z 2 + n3z 3 + o(z3), v = b2z 2 + b3z 3 + o(z3), (2.8) and obtain dx dt = [2k2z + 3k3z 2 + o(z2)] dz dt , dy dt = [2n2z + 3n3z 2 + o(z2)] dz dt , dv dt = [2b2z + 3b3z 2 + o(z2)] dz dt . (2.9) substituting (2.7) and (2.8) into (2.9), we have  − ( βs d1 b2 + d1k2)z 2 − ( βs d1 b3 + d1k3 − 2r m k2)z 3 + o(z3) = 0, − ( βs d1 n2 + d2n2)z 2 − ( βs d1 n3 + d2n3 − 2r m n2)z 3 + o(z3) = 0, (kn2 −µb2)z2 + (kn3 −µb3 + 2r m b2)z 3 + o(z3) = 0. (2.10) comparing the coefficients of z and z2 in (2.10), we have k2 = k3 = n2 = n3 = b2 = b3 = 0. as a result, substituting (2.8) into the last equation of (2.7), we have dz dt = − r m z2 + o(z3). (2.11) thus, the zero point z = 0 of (2.11) is locally asymptotically stable, then e0 is locally asymptotically stable if r0 < 1 and r = d3. let l0 = x−x0 −x0 ln x x0 + y + d2 k v + p c z. taking the time derivative of l0 along the solution of system (1.2), we have l′0 =(1 − x0 x )(s−βxv −d1x) + βxv −d2y −pyz + d2 k (ky −µv) + p c ( cyz + rz(1 − z m ) −d3z ) =d1x0(2 − x x0 − x0 x ) + d2µ k (r0 − 1)v − prz2 cm + p(r −d3)z c ≤ 0 global properties of a virus dynamics model with self-proliferation of ctls 127 if r0 < 1 and 0 ≤ r ≤ d3, and l′0 = 0 only if x = x0,v = 0 and z = 0 simultaneously, i.e., the maximal invariant subset in {(x,y,v,z) : l′0|(1.2) = 0} is the singleton {e0}. as a result, e0 is globally asymptotically stable based on the lasalle’s invariance principle. (ii) when r > d3, the eigenvalue λ2 = r − d3 > 0. so the infection-free equilibrium e0 is always unstable. � theorem 2.3. suppose that immunity-inactivated infection equilibrium e1 exists, i.e., r0 > 1. (i) when 0 ≤ r < d3, e1 is globally asymptotically stable if 1 < r0 < 1 + βk(d3−r) cd1µ , and it is unstable when r0 > 1 + βk(d3−r) cd1µ . (ii) when r ≥ d3, e1 is always unstable. proof. let x1 = s d1r0 ,y1 = d1µ(r0−1) βk ,v1 = d1(r0−1) β . according to (2.5), we have the following characteristic equation of system (1.2) at e1 (λ− cy1 −r + d3)h1(λ) = 0, where h1(λ) = λ 3 + a1λ 2 + a2λ + a3, and a1 = d2 + µ + d1r0 > 0, a2 = d1(d2 + µ)r0 > 0, a3 = d1d2µ(r0 − 1) > 0. here we use βv1 = d1(r0 − 1),βkx1 = d2µ. further, we have a1a2 −a3 = r20d 2 1(d2 + µ) + d1d2µ(r0 + 1) + r0d1(d 2 2 + µ 2) > 0. clearly, λ1 = r − d3 + cy1 = r − d3 + cd1µ(r0−1) βk is an eigenvalue at e1 of system (1.2), and the real parts of h1(λ) are negative according to routh-hurwitz criterion. (i) when 0 ≤ r < d3, we know λ1 < 0 if r0 < 1 + βk(d3−r) cd1µ , i.e., e1 is locally asymptotically stable in this case. otherwise, it is unstable. let l1 = x−x1 −x1 ln x x1 + y −y1 −y1 ln y y1 + βx1v1 ky1 (v −v1 −v1 ln v v1 ) + p c z. taking the time derivative of l2 along the solution of system (1.2), and using s = βx1v1 +d1x1,βx1v1 = d2y1 and ky1 = µv1, we have l′1 =(1 − x1 x )(s−βxv −d1x) + (1 − y1 y ) ( βxv −d2y −pyz ) + (1 − v1 v ) βx1v1 ky1 (ky −µv) + p c ( cyz + rz(1 − z m ) −d3z ) =d1x1 ( 2 − x1 x − x x1 ) + βx1v1 ( 3 − x1 x − y1xv yx1v1 − v1y vy1 ) + pd1µ βk ( r0 − 1 − βk(d3 −r) cd1µ ) z − prz2 cm ≤ 0 if r0 < 1 + βk(d3−r) cd1µ , and l′1 = 0 only if x = x1, y y1 = v v1 and z = 0. in this case, it is easy to obtain that the maximal invariant subset in {(x,y,v,z) : l′1|(1.2) = 0} is the singleton {e1}. as a result, e1 is globally asymptotically stable based on the lasalle’s invariance principle. (ii) when r ≥ d3, λ1 = r −d3 + cy1 > 0 is always valid. so e1 is always unstable. � theorem 2.4. suppose that r > d3. the infection-free but immunity-activated equilibrium e2 is globally asymptotically stable if r0 < 1 + pm(r−d3) rd2 and unstable if r0 > 1 + pm(r−d3) rd2 . 128 cuicui jiang, huan kong, guohong zhang*, and kaifa wang* proof. according to (2.5), we have the following characteristic equation of system(1.2) at e2 (λ + d1)(λ + r −d3)h3(λ) = 0, where h3(λ) = λ 2 + (d2 + µ + pz2)λ + d2µ(1 + pm(r−d3) rd2 −r0). it is easy to obtain that λ1 = −d1 < 0,λ2 = −r + d3 < 0 are the eigenvalues at e2 of system(1.2), and all roots of h3(λ) are negative real part if r0 < 1 + pm(r−d3) rd2 . thus, e2 is locally asymptotically stable if r0 < 1 + pm(r−d3) rd2 . otherwise, it is unstable. let l2 = x−x2 −x2 ln x x2 + y + d2 k r0v + p c (z −z2 −z2 ln z z2 ), where x2 = s d1 ,z2 = m(r−d3) r . taking the time derivative of l2 along the solution of system (1.2), we have l′2 =(1 − x2 x )(s−βxv −d1x) + βxv −d2y −pyz + d2 k r0(ky −µv) + p c (1 − z2 z ) ( cyz + rz(1 − z m ) −d3z ) =d1x2(2 − x x2 − x2 x ) + d2(r0 − 1 − pm(r −d3) rd2 )y − rp cm (z −z2)2 ≤ 0 if r0 < 1 + pm(r−d3) rd2 , and l′2 = 0 only if x = x2,y = 0 and z = z2, i.e., the maximal invariant subset in {(x,y,v,z) : l′2|(1.2) = 0} is the singleton {e2}. as a result, e2 is globally asymptotically stable based on the lasalle’s invariance principle. � theorem 2.5. the following hold. (i) when 0 ≤ r ≤ d3, the immunity-activated infection equilibrium e3 is globally asymptotically stable as long as it appears, i.e., r0 > 1 + βk(d3−r) cd1µ . (ii) when r > d3, the immunity-activated infection equilibrium e4 is globally asymptotically stable as long as it appears, i.e., r0 > 1 + pm(r−d3) rd2 . proof. for the sake of description, the equilibria e3 and e4 are uniformly denoted as e∗ = (x∗,y∗,v∗,z∗). first, we discuss the local stability of e∗, according to (2.5), we have j(e∗) =   −d1 −σ 0 −αµk 0 σ −α αµ k −py∗ 0 k −µ 0 0 cz∗ 0 − rmz∗   , where σ = βv∗ > 0,α = d2 + pz∗ > 0, αµ k = βx∗ > 0. the characteristic equation of system (1.2) at e∗ is λ4 + â1λ 3 + â2λ 2 + â3λ + â4 = 0, in which â1 = r m z∗ + α + µ + $ > 0, â2 = cpz∗y∗ + r m z∗(α + µ + $) + $(α + mu) > 0, â3 = cpz∗y∗(µ + $) + r m z∗$(α + µ) + αµσ > 0, â4 = cµpz∗y∗$ + αµσ r m z∗ > 0, $ = d1 + σ. global properties of a virus dynamics model with self-proliferation of ctls 129 after some calculations, we have â1â2 − â3 =cpz∗y∗( r m z∗ + α) + (α + µ + $) r2 m2 z2∗ + r m z∗(α + µ + $) 2 + α(α$ + 2d1µ + µσ + $) + µ$(µ + $) > 0, â3(â1â2 − â3) − â21â4 =µαc 2p2z2∗y 2 ∗ + c 2p2µ r m z3∗y 2 ∗ + c 2p2z2∗y 2 ∗$( r m z∗ + α) + cp r2 m2 z3∗y∗ [ α(2$ + µ) + ($ + µ)2 ] + cp r m z2∗y∗ [ (2$ + µ)α2 + µ(3d1 + 2µ + 4σ) + 2$ 2 ] α + (µ + $)($2 + µ2) ] + cαpz∗y∗ [ ($2 + µσ)α + $($ + d1µ)) ] − cαµ2pσz∗y∗ + [ r3 m3 z3∗ + r2 m2 z2∗(α + µ + $) + r m z∗$(α + µ) ][ $α2 + α($2 + 2d1µ + µσ) + µ$(µ + $) ] + αµσ [ $α2 + ($2 + 2d1µ + µσ)α + µ($ 2 + d1µ) ] + αµ3σ2. notice that this only has one negative term. using the first and the last terms, and this negative term, we have αµ(c2p2z2∗y 2 ∗ − cµpσz∗y∗ + µ 2σ2) ≥ αµ(2cpµσz∗y∗ − cµpσz∗y∗) > 0, thus, â3(â1â2− â3)− â21â4 > 0. moreover, it follows from the routh-hurwitz criterion that e∗ is locally asymptotically stable. in order to obtain the global stability of e∗, let l∗ = x−x∗ −x∗ ln x x∗ + y −y∗ −y∗ ln y y∗ + βx∗v∗ ky∗ (v −v∗ −v∗ ln v v∗ ) + p c (z −z∗ −z∗ ln z z∗ ). taking the time derivative of l∗ along the solution of system(1.2), we get l′∗ =(1 − x∗ x )(s−βxv −d1x) + (1 − y∗ y )(βxv −d2y −pyz) + (1 − v∗ v ) βx∗v∗ ky∗ (ky −µv) + p c (1 − z∗ z ) ( cyz + rz(1 − z m ) −d3z ) . using s = βx∗v∗ + d1x∗, βx∗v∗ = d2y∗ + py∗z∗, ky∗ = µv∗ and y∗ = rz∗ + m(d3 −r) mc , we have l′∗ =2d1x∗ −d1x−d1x∗ x∗ x + βx∗v∗ + βx∗v −βx∗v∗ x∗ x −βxv y∗ y + βx∗v∗ −py∗z∗ + py∗z + pyz∗ −βx∗v −βx∗v∗ v∗y vy∗ + βx∗v∗ + p c ( rz(1 − z m ) −d3z ) −pyz∗ − p c ( rz∗(1 − z m ) −d3z∗ ) =d1x∗(2 − x∗ x − x x∗ ) + βx∗v∗(3 − x∗ x − xvy∗ x∗v∗y − v∗y vy∗ ) − pr mc (z −z∗)2 ≤ 0, and l′∗ = 0 only if x = x∗, y y∗ = v v∗ and z = z∗. in this case, it is easy to obtain that the maximal invariant subset in {(x,y,v,z) : l′∗|(1.2) = 0} is the singleton {e∗}. as a result, e∗ is globally asymptotically stable if it exists based on the lasalle’s invariance principle. � for the convenience of reading, we summarize the complete global properties of system (1.2) as shown in figure 1. 130 cuicui jiang, huan kong, guohong zhang*, and kaifa wang* 1 r 0 e 0 is gas e 0 is us e 1 is gas e 3 is gas e 1 is us e 0 is us 0≤ r<d 3 r 1 1 r 0 e 3 is gas e 1 is us e 0 is us r=d 3 e 0 is gas 1 r 0 e 2 is gas r 2r>d 3 e 0 is use0 is us e 1 is us e 2 is gas e 1 is us e 4 is gas e 2 is us e 0 is us figure 1. global properties of system (1.2). here, e0 = ( s d1 , 0, 0, 0), e1 = ( s d1r0 , d1µ(r0−1) βk , d1(r0−1) β , 0), e2 = ( s d1 , 0, 0, m(r−d3) r ), e3 = (x3,y3,v3,z3) and e4 = (x4,y4,v4,z4) are the equilibria of system (1.2), the expression of e3 and e4 is shown in proposition 2.1. r1 = 1 + βk(d3−r) cd1µ and r2 = 1 + pm(r−d3) rd2 . 3. numerical simulations although the complete global properties of system (1.2) have been obtained in figure 1, it is noted that the immunity-activated infection equilibrium e3 or e4 is related to the parameters of selfproliferation of ctls (r and m) from proposition 2.1. when e3 or e4 is globally asymptotically stable, the infected cells (y3 or y4) and corresponding proportion of infected cells y3 x3+y3 or y4 x4+y4 are often used to describe the severity of the infection. in this section, we give numerical simulations to investigate the effect of self-proliferation of ctls, and explore the potential significance in clinical practice. first, we fix s = 1.0 × 104 ml−1 · day−1, β = 3.0 × 10−4 ml−1 · day−1, p = 0.5 ml−1 · day−1, c = 9.6 × 10−6 ml−1 · day−1, d1 = 0.01 day−1, d2 = 1.0 day−1, d3 = 0.035 day−1, k = 8 virions/cell, µ = 2.4 day−1, which are within the similar ranges as those ones employed in [5, 12, 17, 22]. then, let parameters r and m change to qualitatively explore their influence on the values of e3 or e4, and the corresponding proportion of infected cells. figure 2. simulations of e3 and the corresponding proportion of y3 when 0 ≤ r ≤ d3. panel (a)-(e) are the simulated surface with two parameters changing. global properties of a virus dynamics model with self-proliferation of ctls 131 when r ∈ [0.0, 0.035] and m ∈ [100, 1000], i.e., 0 ≤ r ≤ d3, figure 2 shows the simulated surface of e3 and the corresponding proportion of y3 changing with parameters r and m. along with the increase of parameter r, although the infected cells (y3) and virus load (v3) will decrease (figure 2b and figure 2c), the proportion of infected cells ( y3 x3+y3 ) is gradually increase (figure 2e). in particular, the immune cells (z3) also decrease with the increase of r (figure 2d). these qualitatively indicate that the severity of infection may increase with the increase of r in case of 0 ≤ r ≤ d3. along with the increase of parameter m, figure 2a and figure 2d show that uninfected cells (x3) and immune cells (z3) will increase gradually, while infected cells (y3), virus load (v3) and the proportion of infected cells ( y3 x3+y3 ) are decrease gradually. these qualitatively indicate that increasing parameter m can reduce the severity of virus infection. figure 3. simulations of e4 and the corresponding proportion of y4 when 0 ≤ r ≤ d3. panel (a)-(e) are the simulated surface with two parameters changing. when r ∈ [0.0351, 0.07] and m ∈ [100, 1000], i.e., r > d3, figure 3 shows the simulated surface of e4 and the corresponding proportion of y4 changing with parameters r and m. comparing figure 3 with figure 2, we can find that all components of the immunity-activated infection equilibrium e4 will increase with the increase of parameter r, including the proportion of infected cells ( y4 x4+y4 ). along with the increase of parameter m, the variation of all components is similar to that in figure 2. 4. discussion in this paper, a viral infection model with self-proliferation of ctls is proposed and its global dynamic behavior is obtained. from figure 1, comparing with [10], we can find the dynamic behavior of (1.2) will not change if the per capita self-proliferation rate of ctls is insufficient, i.e., 0 < r < d3. however, when r = d3, the immunity-inactivated infection equilibrium e1 is always unstable if it appears. especially, when r > d3, a new steady state (named as infection-free but immunity-activated equilibrium e2) appears and is globally asymptotically stable if the basic reproduction number is less than a threshold, which means that the immune effect still exists though virus be eliminated. this is consistent with the clinical practice of virus infection because immune cells are usually not depleted after a patient recovers. in fact, memory t cells can be maintained in lymphoid and nonlymphoid organs through self-renewal [3], including central memory t cells and effector memory t cells [18]. in particular, recent study on hbv also find that hbv specific tnf-α cd4 t cells may be in the early stage of differentiation rather than depletion of t cells [19], which suggests that the stability of immunity-inactivated infection equilibrium e1 may be impossible in clinical practice. on the effect of self-proliferation of immune cells, qualitative numerical simulations (figure 2 and figure 3) indicate that although there are different shape mode under different intensity of self-proliferation, 132 cuicui jiang, huan kong, guohong zhang*, and kaifa wang* the increase of per capita self-proliferation rate (r) can worsen the infection, while the increase of the capacity of ctls (m) can reduce the severity of infection. thus, inappropriate intensity of per capita self-proliferation rate may lead to more severe infection outcome, which is similar to the effect of covalently closed circular dna (cccdna) self-amplification rate in hbv infection [8]. these may provide insight into the failure of immune therapy [2]. recently, under a plausible quasi steady-state assumption, [6] ignored the direct effect of virus load, but introduced the delayed activation effect of immune cells, and the dynamics of the corresponding virus model was studied. compared with the results of [6], we can find that quai steady-state assumption cannot affect the dynamic behavior, which is consistent with the conclusion of [20]. note that the spatial migration of virus particles is an inherent characteristic of virus infection within-host [21, 28]. we will further 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[28] t. zhang, w. wang and k. wang, minimal wave speed of a bacterial colony model, appl. math. model. 40 (2016), 10419-10436. department of mathematics, college of basic medicine, army medical university, chongqing 400038, p. r. china e-mail address: jiangcc16@foxmail.com school of mathematics and statistics, southwest university, chongqing 400715, p. r. china e-mail address: 1349966363@qq.com co-corresponding author, school of mathematics and statistics, southwest university, chongqing 400715, p. r. china e-mail address: zgh711@swu.edu.cn co-corresponding author, school of mathematics and statistics, southwest university, chongqing 400715, p. r. china e-mail address: kfwang72@163.com mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 3, septembe 2020, pp.207-223 https://doi.org/10.5206/mase/10745 dynamical analysis of a fractional-order model incorporating fear in the disease transmission rate of severe infectious diseases like covid-19 chandan maji and debasis mukherjee abstract. this paper deals with a fractional-order three-dimensional compartmental model with fear effect. we have investigated whether fear can play an important role or not to spread and control the infectious diseases like covid-19, sars etc. in a bounded region. the basic results on uniqueness, non-negativity and boundedness of the solution of the system are investigated. stability analysis ensures that the disease-free equilibrium point is locally asymptotically stable if carrying capacity exceeds a certain threshold value. we have also derived the conditions for which endemic equilibrium is globally asymptotically stable that means the disease persists in the system. numerical simulation suggests that the fear factor has an important role which is observed through hopf-bifurcation. 1. introduction over the past few years, human civilization had repeatedly been plagued by attacks of various coronaviruses. they are a broad family of viruses, some of which can cause several human diseases, varying from cold to sars (severe acute respiratory syndrome). two important coronavirus disease outbreaks have already occurred in the past two decades : sars in 2003 [15] and mers (middle east respiratory syndrome) in 2012 [7]. by the end of 2019, human civilization is once again infected by the coronavirus which has not been previously identified in humans. on february 11, 2020 the world health organization (who) declared the official name coronavirus disease 2019. it is abbreviated as covid-19. the most common symptoms of this disease are fever, tiredness and dry cough. more scarcely, the disease can be serious and fatal. aged person and individuals with other medical constraints (such as asthma, diabetes or heart disease) may be vulnerable to becoming seriously ill. it is observed in december 2019, sars cov-2 has been found as the causal factor in a series of critical cases of pneumonia originating in wuhan hubei province china [48]. since then it spreads rest of the world. as per report of who(2020), 210 countries/territories had confirmed cases of this disease. with the extremely high infection rate and high mortality, individuals naturally began worrying about the covid-19. one distinctive nature of infectious disease related with other conditions is fear. as the disease outbreak is ongoing, a wave of fear is developed in the society [28]. the fear and worry are obvious as peoples concern their health. no one wants to get infected with a virus that has a relatively high risk of death [23]. fear is presently connected with its transmission rate [32, 4, 41]. as the outbreak of covid-19 spreads to more countries and death toll rises, the uncertainty of what lies ahead is concerning. though present treatment on covid-19 throughout the world has mainly confined into infection control, effective vaccine and recovery rate [14], the psychosocial aspect is neglected. as countries worldwide have to work on diminishing the transmission rate of covid-19, they should pay received by the editors 28 may 2020; accepted 24 august 2020; online first 30 august 2020. 2000 mathematics subject classification. 34k20, 92b05. key words and phrases. covid-19; fear effect; fractional order; stability; bifurcation. 207 208 c. maji and d. mukherjee attention on individuals fears to gain a society free of covid-19. from scientific point of view, reactions to fear are normal and potentially beneficial. fear of the present disease outbreak is understandable, not to mention almost universal. with the fact in mind, it is reasonable to develop a mathematical model that incorporate fear factor in the disease transmission rate to control the covid-19. already a lot of works have been done mathematically and numerically to give an efficient prediction on covid 19 outbreaks [46, 34, 49, 39, 35, 44, 43]. the study of fractional order differential equations has received much interest to the researchers during the past few decades due to their ability to provide a good description of certain non-linear phenomena [22]. it is an extension of classical calculus that generalizes the order of derivatives and integrals to a non-integer order. fractional calculus was first proposed by leibnitz and hospitals in 1965 [36]. the fractional-order models are more realistic than integer-order model as the system has nonlocal characteristic which is absent in integer-order model and it has greater degree of freedom. another reason for considering the fractional order system is to address memory which exists in most biological systems but such effects are in fact neglected in integer-order system. apart from, using fractional order differential equations can help us to reduce the errors arising from the neglected parameters in modelling real life phenomena [12]. however, due to progress of fractional calculus many researchers in different fields such as biology, physics, engineering, finance , medicine considered fractional calculus to develop their problems [10, 17, 38, 11, 16, 29, 30, 5, 18]. by using fractional-order derivative a lot of work has done in epidemiology [20, 8, 21, 40, 45]. motivated from the above literature survey, we proposed a fractional order sei model of covid-19 outbreaks. the paper is organized as follows. in section 2, the model is proposed. basic definitions are presented in section 3. existence, uniqueness, non-negativity and boundedness of solutions are shown in section 4. equilibria and their local stability are discussed in section 5. global stability of endemic equilibrium point is derived in section 6. hopf bifurcation phenomena are demonstrated in section 7. numerical simulations are carried out in section 8. a brief discussion follows in section 9. 2. model formulation in epidemic models, the bilinear incidence rate βsi is frequently used. recent study [28] indicates that the fear effect will reduce the disease transmission rate, so we modify βsi by multiplying a factor f(α,i) which leads to f(α,i)βsi. here, the parameter α represents the level of fear. for biological justification of α, i and f(α,i), it is appropriate to consider the following : f(0,i) = 1, f(α, 0) = 1, lim α→∞ f(α,i) = 0, lim i→∞ f(α,i) = 0, ∂f(α,i) ∂α < 0, ∂f(α,i) ∂i < 0. for convenience of analysis, we assume the following form for the fear effect f(α,i) = 1 1 + αi . here, when there is no infected individuals, there is no reduction in the susceptible individuals due to the fear factor i.e f(α, 0) = 1. based on above assumption, in this present paper, we formulate a three dimensional compartmental model with fear effect with the help of fractional-order caputo-type derivative which is given as dynamical analysis of a fractional-order model incorporating fear 209 follows: cdµs(t) = b̂s ( 1 − s k̂ ) − β̂si 1 + α̂i , cdµe(t) = β̂si 1 + α̂i − (ĉ + d̂)e, (2.1) cdµi(t) = ĉe − γ̂i with initial conditions s(0) = s0 ≥ 0,e(0) ≥ 0 and i(0) ≥ 0, where µ ∈ (0, 1) and cdµ is the standard caputo differentiation. here, s(t) is the total density of the susceptible individuals, e(t) is the number of individuals to the infected but not infectious and i(t) denotes the infected individuals who are infected. b̂ is the net per capita growth rate of the susceptible individuals and k̂ is the environmental carrying capacity. α̂ represents the level of fear among the individuals which have the controlling effect not to spread the disease. in this classical endemic model the transmission coefficient for the disease is denoted by β̂. ĉ is the rate per unit time (day) that infected individuals become infectious. d̂ is the natural death rate of the exposed individuals. the infected individuals removed at a rate of γ̂, which include natural death of the infected population and the recovery rate of the hospitalized infectious individuals. the system (2.1) has some defects as regard to the time dimension because the right hand side expressions have dimension (time)−1, whereas the left hand side expressions have dimension (time)−µ. the corrected form of system (2.1) is as follows: cdµs(t) = b̂µs ( 1 − s k̂ ) − β̂µsi 1 + α̂i , cdµe(t) = β̂µsi 1 + α̂i − (ĉµ + d̂µ)e, (2.2) cdµi(t) = ĉµe − γ̂µi with initial conditions s(0) = s0 ≥ 0,e(0) ≥ 0 and i(0) ≥ 0, where µ ∈ (0, 1) and cdµ is the standard caputo differentiation. for convenience, we redefine the parameters as follows: b = b̂µ,k = k̂,β = β̂µ,α = α̂,c = ĉµ,d = d̂µ,γ = γ̂µ. therefore, the modified system is as follows: cdµs(t) = bs ( 1 − s k ) − βsi 1 + αi , cdµe(t) = βsi 1 + αi − (c + d)e, (2.3) cdµi(t) = ce −γi with s(0) = s0 ≥ 0,e(0) ≥ 0 and i(0) ≥ 0. 3. basic definitions fractional calculus is a powerful tool for mathematical modeling and it has a wild application in different field of sciences. throughout this paper, we use a caputo fractional-order derivative as the initial conditions of fractional differential equations with caputo derivatives consider on the identical form 210 c. maji and d. mukherjee as for integer-order ones, which can be used in modelling and analysis. in this section, some basic definitions for fractional calculus have been presented. definition 3.1. [37] the riemann-liouville fractional integral operator of order µ of a continuous function f ∈ l1[0,a], t ∈ [0,a] is presented as iµf(t) = 1 γ(µ) ∫ a 0 (t− τ)µ−1f(τ)dτ, where γ(µ) is the euler’s gamma function. definition 3.2. [37] the definition of caputo’s fractional derivative of order µ for a function f ∈ cn([0, +∞],r) is defined by cdµf(t) = 1 γ(n−µ) ∫ a 0 (t− τ)n−µ−1f(n)(τ)dτ, where γ(.) is the euler’s gamma function and the operator cdµ is known as ”caputo differential operator of order µ”. t ≥ 0 and n is the positive integer such that n− 1 < µ < n,n ∈ n. particularly, when 0 < µ < 1, cdµf(t) = 1 γ(n−µ) ∫ t 0 f′(τ) (t− τ)µ dτ. riemann-liouville (r-l) was first introduced the idea of fractional derivative but in r-l fractional differential equation, initial value is usually taken in the form of fractional derivative, which is not appropriate in real sense whereas in caputo fractional derivative, the derivative is not defined locally at time t, it depends on the total effects of the so called n-order integer derivative on the interval [0,s]. thus it is reasonable to consider the variation of a system in which the instantaneous change rate depends on the past rate, which is known as ”memory effect”[33]. 4. main results in this section we shall discuss about existence, uniqueness, non-negativity and boundedness of the solutions for fractional order system (2.3). 4.1. existence and uniqueness. before we prove the existence and uniqueness of the solution of system (2.3), we need the following lemma. lemma 4.1. [26] define the system cdµx(t) = f(t,x), t > 0 (4.1) with initial condition x0, where µ ∈ (0, 1],f : [0,∞) × ω → rn, ω ∈ rn, then there exists a unique solution of (2.3) whenever f(t,x) follows locally lipschitz condition with respect to x on [0,∞) × ω. theorem 4.2. for any non-negative initial conditions the fractional order system (2.3) admits a unique solution. proof. existence and uniqueness of system (2.3) will be shown in the region ∆ × (0,t] where ∆ = {(s,e,i) ∈ r3 : max(|s|, |e|, |i| ≤ m)}. now, we follow the approach used in [27].we denote x = (s,e,i) and x̄ = (s̄, ē, ī). consider a mapping dynamical analysis of a fractional-order model incorporating fear 211 h(x) = (h1(x),h2(x),h3(x)) and h1(x) = bs ( 1 − s k ) − βse 1 + αi h2(x) = βse 1 + αi − (c + d)e, h3(x) = ce −γi. (4.2) for x,x̄ ∈ d, it follows from equation (4.2) that ‖h(x) −h(x̄)‖ = |h1(x) −h1(x̄)| + |h2(x) −h2(x̄)| + |h3(x) −h3(x̄)| = ∣∣∣∣bs ( 1− s k ) − βsi 1+αi −bs̄ ( 1− s̄ k ) + βs̄ī 1+αī ∣∣∣∣+ ∣∣∣∣ βsi1+αi −(c+d)e− βs̄ī1+αī + (c+d)ē ∣∣∣∣+ ∣∣∣∣ce−γi−cē +γī ∣∣∣∣ = ∣∣∣∣b(s−s̄)− bk (s2−s̄2)−β ( si 1+αi − s̄ī 1+αī )∣∣∣∣+ ∣∣∣∣β ( si 1+αi − s̄ī 1+αī ) −(c+d)(e−ē) ∣∣∣∣+ ∣∣∣∣c(e−ē)−γ(i−ī) ∣∣∣∣ ≤ ( b + 2bm k + 2β α )|s − s̄| + (2m + γ)|i − ī| + (2c + d)|e − ē| ≤ l||x − x̄||, where l = max{b + 2β α + 2bm k , 2c + d, 2m + γ}. hence, lipschitz condition is satisfied for h(x). thus there exist a unique solution x(t) of system (2.3), follows from lemma 4.1. � 4.2. non-negativity and boundedness. theorem 4.3. all the solutions of system (2.3) which are initiating in r3+ are uniformly bounded within a region π = {(s,e,i) ∈ r3+ : v ≤ k(b+λ)2 4λb + �,� > 0}. proof. here we follow an approach which is used in [27]. define a function v (t) = s(t) + e(t) + i(t). then, cdµv (t) =c dµs(t) +c dµe(t) +c dµi(t) now for any positive number λ, we calculate cdµv (t) + λv (t) = bs(1 − s k ) − βse 1 + αi + βse 1 + αi − (c + d)e + ce −γi + λ(s + e + i) = − b k s2 + (b + λ)s + (λ−γ)i + (λ−d)e = − b k (s − k(b + λ) 2b )2 + k(b + λ)2 4b + (λ−γ)i + (λ−d)e ≤ k(b + λ)2 4b , where λ = min{d,γ}. applying the standard comparison theorem for fractional order in chol et al. [9], we get v (t) ≤ v (0)gµ(−λ(t)µ) + ( k(b + λ)2 4b ) tµgµ,µ+1(−λ(t)µ) 212 c. maji and d. mukherjee where gµ is the mittag-leffler function. so application of lemma 5 and corollary 6 in [9] yields v (t) ≤ k(b + λ)2 4bλ ,t →∞. therefore, all solution of fractional order system (2.3) which are initiating in r3+ will enter the region π = {(s,e,i) ∈ r3+ : v ≤ k(b + λ)2 4λb + �,� > 0}. (4.3) � theorem 4.4. all solutions of system (2.3) which start in r3+ are nonnegative in nature. proof. from first equation of system (2.3), we obtain cdµs(t) = bs(1 − s k ) − βsi 1 + µi (4.4) again from equation (4.3), we get s + e + i ≤ k(b + λ)2 4λb = k1(say). (4.5) so from equation (4.4) and (4.5), we have, cdµs(t) ≥ bs ( 1 − k1 k ) −βs = s ( b− bk1 k −β ) = k2s, where k2 = b−β − bk1k . now according to the standard comparison theorem for fractional order in [9] and the positivity of mittag-leffler function gµ,1(t) > 0 for any µ ∈ (0, 1) [47] it follows that s(t) ≥ s0gµ,1(qtµ) =⇒ s(t) ≥ 0. from second equation of system (2.3), we obtain cdµe(t) = βsi 1 + αi − (c + d)e ≥−(c + d)e. therefore, e(t) ≥ e0gµ,1(−(c + d)tµ) =⇒ e ≥ 0. again from the third equation of system (2.3), cdµi(t) = ce −γi ≥−γi. so, i(t) ≥ i0gµ,1(−τtµ) =⇒ i ≥ 0. hence all solution of system (2.3) are non-negative. � 5. equilibria of the fractional order system and their stability to evaluate the equilibrium points of system (2.3), let cdµs(t) = 0, cdµe(t) = 0, cdµi(t) = 0. then we obtain the following equilibrium points: e0(0, 0, 0), e1(k, 0, 0) and e ∗(s∗,e∗,i∗) where e∗ = γi∗ c , s∗ = γ(c + d) cβ (1 + αi∗) and i∗ is a positive root of the equation α2bγ(c + d)i2 + {2bγ(c + d)α− bαkcβ + kβ2c}i + b{γ(c + d) −kcβ} = 0. dynamical analysis of a fractional-order model incorporating fear 213 now,we want to check the stability analysis of the above equilibria based on the standard linearization technique by using the jacobian matrix. the jacobian matrix of system (2.3) around any point (s,e,i) is given by j(s,e,i) =   b− 2bs k − βi 1+αi 0 − βs (1+αi)2 βi 1+αi −(c + d) βs (1+αi)2 0 c −γ   . theorem 5.1. the population free equilibrium point e0 of system (2.3) is always unstable while disease free equilibrium point e1 is stable if β < γ(c+d) kc . proof. according to the mittag-leffler function [31], the equilibrium point ei of system (2.3) is locally stable if all the eigenvalues λi of j(ei) satisfy |arg(λi)| > µπ2 . the jacobian matrix of system (2.3) at the equilibrium point e0 is given by j(e0) =   b 0 00 −(c + d) 0 0 c −γ   . the eigenvalues corresponding to the equilibrium point e0 are: λ1 = b > 0,λ2 = −(c + d) < 0, and λ3 = −γ < 0. we observed that |arg(λ1)| = 0 < µπ2 , |arg(λ2)| = π > µπ 2 , |arg(λ3)| = π > απ2 . hence, the equilibrium e0 is always a saddle point. again, the jacobian matrix of system (2.3) at the equilibrium point e1 is given by: j(e1) =   −b 0 −βk0 −(c + d) βk 0 c −γ   . the eigenvalues corresponding to the equilibrium point e1 are: λ1 = −b > 0 and other two λ2,λ3 are obtained by solving the characteristic equation λ2 + λ(c + d + γ) + γ(c + d) −βkc = 0. (5.1) the eigenvalues corresponding to the equation (5.1) are λi = (c + d + γ) ± √ (c + d + γ)2 − 4{(c + d)γ −βkc} 2 , i = 2, 3. now we see that |arg(λ1)| = π > µπ2 ; and if β < γ(c+d) kc then λi < 0, i = 2, 3 and |arg(λ2,3)| = π > µπ2 . hence, the equilibrium e1 is locally asymptotically stable. � to analyze the stability of the endemic equilibrium point e∗, we compute the jacobian matrix of system (2.3) around e∗ and is given by j(e∗) =   − bs∗ k 0 − βs ∗ (1+αi∗)2 βi∗ 1+αi∗ −(c + d) βs ∗ (1+αi∗)2 0 c −γ   . the eigenvalues of j(e∗) are the roots of the following characteristic equation p(λ) = λ3 + p1λ 2 + p2λ + p3 = 0 (5.2) 214 c. maji and d. mukherjee where p1 = bs∗ k + c + d + γ, p2 = r(c + d) − cβs∗ (1 + αi∗)2 + bs∗ k (c + d + γ), p3 = r(c + d) − cβs∗ (1 + αi∗)2 . let d(p) be the discriminant of the cubic polynomial p(λ), which can be written as d(p) =   1 p1 p2 p3 0 0 1 p1 p2 p3 3 2p1 p2 0 0 0 3 2p1 p2 0 0 0 3 2p1 p2   = 18p1p2p3 + (p1p2) 2 − 4p3p21 − 4p 3 2 − 27p 2 3. then, we have the following results by [2]. proposition 5.2. suppose β > γ(c+d) kc . then the equilibrium e∗ of system (2.3) is asymptotically stable if one of the following conditions are satisfied. (i) d(p) > 0,p1 > 0,p3 > 0 and p1p2 > p3; (ii) d(p) < 0,p1 ≥ 0,p2 ≥ 0,p3 > 0 and µ < 23 ; (iii) d(p) < 0,p1 > 0,p2 > 0,p1p2 = p3 and for all µ ∈ (0, 1). 6. global stability in this section we present global stability of endemic equilibrium point e∗. before stating our theorem we define the matrix a as follows: a =   −b k − β 2α − β 2(1+αi∗)2 β 2α (c + d) −1 2 ( βs∗ α(1+αi∗) + c ) β 2(1+αi∗) −1 2 ( βs∗ α(1+αi∗) + c ) γ   . theorem 6.1. the endemic equilibrium point e∗ of system (2.3) is globally asymptotically stable if 4α2b(c + d) > kβ2 and det a > 0. proof. consider the following positive definite function about e∗ v (s,e,i) = s −s∗ −s∗ ln s s∗ + 1 2 (e −e∗)2 + 1 2 (i − i∗)2. we compute the µ order derivative of v (s,e,i) along the solution of system (2.3) with the help of lemma 3.1 in [42] and lemma 1 in [1]. thus we have cdµv (s,e,i) ≤ ( 1 − s ∗ s ) cdµs + (e −e∗)cdµe + (i − i∗)cdµi = (s −s∗){b(1 − s k ) − βi 1+αi } + (e −e∗){ βsi 1+αi − (c + d)e} + (i − i∗){ce −γi} dynamical analysis of a fractional-order model incorporating fear 215 = (s −s∗){βi(s−s ∗) 1+αi + βs∗(i−i∗) (1+αi)(1+αi∗) − (c + d)(e −e∗)} + (i − i∗){c(e −e∗) −γ(i − i∗)} ≤−b k (s −s∗)2 + β 1+αi∗ |s −s∗||i − i∗| + β α |e −e∗||s −s∗| + ( βs∗ α(1+αi∗) + c ) |e −e∗||i − i∗| −(c + d)(e −e∗)2 −γ(i − i∗)2. consequently, cdµv (s,e,i) ≤ 0 when a is positive definite. the result follows by the application of lemma 4.6 in huo et al. [20]. � 7. hopf-bifurcation in this section, we discuss about the hopf-bifurcation analysis of system (2.3). define, a function with respect to µ by m(µ) = µπ 2 − min 1≤i≤3 |arg(λi)|. theorem 7.1. ([24]) (existence of hopf bifurcation) when bifurcation parameter µ passes through the critical value µ∗ ∈ (0, 1), fractional-order system (2.3) undergoes a hopf bifurcation at the endemic equilibrium point e∗, if the following conditions are satisfied: (i) the corresponding characteristic equation (5.2) of system (2.3) has a pair of complex conjugates λ1,2 = θ + iω (where θ > 0) and one negative real root λ3; (ii) m(µ∗) = µ ∗π 2 − min1≤i≤3 |arg(λi)| = 0; (iii) dm(µ) dµ |µ=µ∗ 6= 0. we now present the conditions for the existence of a hopf bifurcation at the endemic equilibrium point e∗ as the order of derivative passes a critical value. theorem 7.2. suppose the characteristic equation (5.2) of system (2.3) has two complex conjugate eigenvalues λ1,2 = θ + iω. then the fractional-order system (2.3) undergoes a hopf bifurcation at the endemic equilibrium point e∗ when µ passes through the critical value µ∗ = 2 π arctan(ω θ ). proof. from the given assumptions and m(µ) = µπ 2 − min1≤i≤3 |arg(λi)|, we get, m(µ∗) = µ∗π 2 − min 1≤i≤3 |arg(λi)| = µ∗π 2 − arctan ( ω θ ) = arctan ( ω θ ) − arctan ( ω θ ) = 0. furthermore, dm(µ) dµ |µ=µ∗ = π 2 6= 0. therefore, from theorem 7.1, we conclude that system (2.3) undergoes a hopf bifurcation at e∗ when bifurcation parameter µ crosses a critical value µ∗. � 8. numerical simulation in this section we present some numerical simulation to check the dynamics of the fractional order system. although there are different type of numerical method to solve nonlinear fractional differential equations [13, 25, 6], but adams type predictor corrector method is more appropriate and useful to solve the dynamical behaviour of the solutions of fractional differential equations. here we have considered a 216 c. maji and d. mukherjee 0 50 100 150 200 t 0.6 0.8 1 1.2 1.4 1.6 1.8 2 p o p u la ti o n s e i figure 1. unstable behaviour of system (1) for fractional order derivative µ = 1 and parameter values α = 0.125,β = 0.75,c = d = γ = 1 3 ,b = 0.764158,k = 7.8382146. 0 10 20 30 40 50 60 70 80 90 100 t 1 1.2 1.4 1.6 1.8 2 2.2 p op ul at io n s e i figure 2. phase diagram for solution of system (2.3) with µ = 1 and other parameter set β = 0.75,c = d = γ = 1 3 ,b = 0.764158,k = 7.8382146. time series plot and phase diagram for α = 0.5 . set of parametric values α = 0.125,β = 0.75,c = d = γ = 1 3 ,b = 0.764158,k = 7.8382146. with the help of these parameters from figure 1 we observe that when µ = 1 the solution of system (2.3) is unstable in nature. next, we plot the solution of system (2.3) in figure 2 by choosing the same set of parameters except for α = 0.5. here we observed that the system is stable in nature. so for integer order system fear effect α has an interesting role. if we increase the value of α, the system becomes stable that means fear effect can stabilizes the system. influence of fear effect on different population is given in figure 3. again the bifurcation diagram of system (2.3) with respect to the parameter α has been drawn in figure 4. dynamical analysis of a fractional-order model incorporating fear 217 figure 3. influence of fear effect on each population now to see the effect of fractional-order µ on each population we plotted a diagram in figure 5. for the above set of parameters, from figure 6 we have seen that for integer-order system µ = 1, system (2.3) is unstable when α = 0.125. again, for fractional-order derivative µ = 0.99 and µ = 0.98 the system shows unstable behaviour but the system changes its stability when µ = 0.96 and µ = 0.92 and it becomes stable (figure 6(c),6(d)). thus, from the above figures we conclude that fractional order derivative may change the system dynamical behaviour from unstable to stable. hence, fractional-order derivative has an important role on the system stability of our considered system and it may improve system stability. a bifurcation diagram with respect to the fractional-order µ is given in figure 7. 218 c. maji and d. mukherjee figure 4. bifurcation diagram of system (2.3) with respect to α when µ = 1 and other parameters are β = 0.75,c = d = γ = 1 3 ,b = 0.764158,k = 7.8382146. 9. discussion recently, harper et al.[19] investigated psychological predictors of behavior change and fear response to the covid-19 pandemic 2020. in [3], the authors discussed that the fear of covid-19 scale is a seven-item uni-dimensional scale with robust psychometric properties. they also concluded that this approach is convincing and appropriate in examining fear of covid-19 among the peoples. wang et. al. [46] investigated a time-dependent mathematical model of covid-19 to focus on the effects of medical resources on transmission of covid-19 but fear effect and fractional-order is not addressed in their work. at present there is no proper estimate about how long this disease persist, the number of dynamical analysis of a fractional-order model incorporating fear 219 0 10 20 30 40 50 time (t) 0.4 0.6 0.8 1 1.2 1.4 1.6 s u s c e p ti b le =0.9 =0.8 =0.7 =0.6 (a) 0 10 20 30 40 50 time(t) 0.4 0.6 0.8 1 1.2 1.4 e x p o s e d =0.9 =0.8 =0.7 =0.6 (b) 0 10 20 30 40 50 time(t) 0.6 0.8 1 1.2 1.4 1.6 in fe c te d =0.9 =0.8 =0.7 =0.6 (c) figure 5. time series plot of each population for parameter values α = 0.125,β = 0.75,c = d = γ = 1 3 ,b = 0.764158,k = 7.8382146 with different values of µ individuals worldwide who will be infected and how long human lives will be affected. extensive research is going on to control the spread of the disease in different way . so above observation motivate us to find out a way to combat the disease. the significance of communicable disease behavior induces scientists to develop a mathematical model that can examine the spread procedure, rule, and direction etc. its purpose is that, according to the aspect of infectious disease, the model is suitably formed, the relevant parameters are selected and reasonable variables are chosen. we have investigated the three component epidemiological fractional system which considers the fear effect in the disease transmission rate of coronavirus disease. the 220 c. maji and d. mukherjee figure 6. phase diagram for solution of system (1) when α = 0.125 and other parameters are β = 0.75,c = d = γ = 1 3 ,b = 0.764158,k = 7.8382146. with (a) µ = 0.99 (b) µ = 0.98 (c) µ = 0.96 and (d) µ = 0.92. dynamical behavior of the given system is studied. the classical first order time derivative is modelled with the caputo fractional derivative of order µ ∈ (0, 1]. mathematical analysis of existence and uniqueness of solutions are shown. basic stability properties of disease free and endemic equilibrium points are examined. we observed that as long as disease transmission rate remains below a certain threshold value (β < γ(c+d) kc ) the disease free equilibrium point is locally asymptotically stable. if the disease transmission rate increases then endemic equilibrium point appears. stability property of endemic equilibrium point is described in theorem 5. for disease eradication the conditions of theorem 5 should be avoided. from our analytical and numerical studies indicate that a hopf bifurcation due to variation of the fractional order µ ∈ (0, 1]. from figure 1, we observed that fear has an important role on the dynamics of our system. when the level of fear is very low the system exhibits unstable behaviour while increasing the value of fear stabilizes the system. we have plotted a bifurcation diagram choosing α as a bifurcation parameter in figure 4. from this figure we have seen that when the value of α in the range 0.05 ≤ α < 0.127, the system is unstable but in the range 0.125 ≤ α ≤ 0.2 the system dynamical analysis of a fractional-order model incorporating fear 221 figure 7. bifurcation diagram of system (2.3) with respect to µ when α = 0.1251 and other parameters are β = 0.75,c = d = γ = 1 3 ,b = 0.764158,k = 7.8382146. is stable. therefore we conclude that the level of fear may decrease disease transmission rate and the epidemic may be under control. it is also noted that the fractional order model is more stable than the integer order model. acknowledgement: the authors are grateful to the editor and anonymous reviewers for their valuable comments and suggestions for improving the paper. 222 c. maji and d. mukherjee references [1] n. aguila-camacho, m. a. durate-mermol and j. a. galleges, lyapunov functions for fractional order systems, commun. nonlinear. sci. numer. simul. 19(2014), 2951-2957. 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[49] y. yang, q. lu, m. liu, y. wang, a. zhang, n. jalali, n. dean, i. longini, m. elizabeth halloran, b. xu, x. zhang, l. wang, w. liu, and l. fang. epidemiological and clinical features of the 2019 novel coronavirus outbreak in china, medrxiv, 2020. department of mathematics, vivekananda college, thakurpukur, kolkata-700063, india current address: same e-mail address: chandanmaji.ju@gmail.com department of mathematics, vivekananda college, thakurpukur, kolkata-700063, india e-mail address: mukherjee1961@gmail.com mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 2, number 2, june, pp.134-148 https://doi.org/10.5206/mase/13889 a covid-19 epidemic model predicting the effectiveness of vaccination glenn webb abstract. a model of a covid-19 epidemic is developed to predict the effectiveness of vaccination. the model incorporates key features of covid-19 epidemics: asymptomatic and symptomatic infectiousness, reported and unreported cases, and social measures that decrease infection transmission. the model incorporates key features of vaccination: vaccination efficiency, vaccination scheduling, and relaxation of social measures that increase infection transmission as vaccination is implemented. the model is applied to predict vaccination effectiveness in the united kingdom. 1. introduction the objective of this paper is to predict the outcome of vaccine implementation for the mitigation of covid-19 epidemics. currently, vaccine policies are underway throughout the world, and their outcomes offer great hope for curtailment and elimination of the covid-19 pandemic. there is to some extent, controversy about the implementation strategies underway, in terms of the vaccine effectiveness and the consequence of resumption of normal social behaviour as the number of vaccinated people increases. this paper addresses these issues with a mathematical model incorporating key features of covid-19 epidemics and key features of covid-19 vaccination implementation. related works can be found in [1]-[43]. the organization of this paper is as follows: in section 2, an ordinary differential equations model of a covid-19 epidemic is developed to analyse the transmission dynamics of the epidemics, based on current reported cases data and current vaccination data. in section 3, the model is used to predict the outcome of vaccine implementation in the united kingdom, from april, 2021 to january, 2022. the transmission dynamics of the model are advanced forward from the current data in the uk. the projection examines the roles of vaccine efficiency, numbers of people vaccinated, and the restoration of normal social practices in predicting the future of the epidemic in the uk. 2. a general covid-19 model based on current daily data the compartments of the model are s(t) = susceptible individuals at time t, i(t) = asymptomatic infectious individuals at time t, r(t) = symptomatic infectious individuals at time t who will be reported, u(t) = symptomatic infectious individuals at time t who will not be reported, and v (t) = vaccinated received by the editors 23 april 2021; revised 11 may 2021; accepted 11 may 2021, published online 14 may 2021. 2010 mathematics subject classification. primary 92d30; secondary 92c60. key words and phrases. epidemic, cases, transmission, asymptomatic, symptomatic, vaccination. 134 predicting the effectiveness of vaccination 135 individuals at time t. the equations of the model are s′(t) = −τ(t,s(t),i(t),r(t)) −v(t)s(t), t ≥ t0, (2.1) i′(t) = τ(t,s(t),i(t),r(t)) − (ν1 + ν2)i(t), t ≥ t0, (2.2) r′(t) = ν1i(t) −ηr(t), t ≥ t0, (2.3) u′(t) = ν2i(t) −ηu(t), t ≥ t0, (2.4) v ′(t) = v(t)s(t), t ≥ t0. (2.5) figure 1. flow diagram of the model the parameters of the model are as follows: • the transmission rate is time-dependent. before the most recent day of daily reported cases, it is defined in terms of the rolling weekly average of the daily reported cases data. after this time, the transmission rate has the form τ(t) s(t)(i(t) + (1 + f)r(t) ), where f is a fixed ratio of unreported symptomatic cases to reported symptomatic cases. the total number of symptomatic cases at time t is (1 + f) r(t). it is assumed that asymptomatic cases, unreported symptomatic cases, and reported symptomatic cases have equal likelihood of transmission to susceptibles. the function τ(t) incorporates time dependent relaxation of social distancing behavior as vaccination is implemented. • asymptomatic infectious individuals i(t) are infectious for an average period of 1/ν days. • the fraction f of asymptomatic infectious become reported symptomatic infectious at rate ν1 = f ν, and the fraction 1−f become unreported symptomatic infectious at rate ν2 = (1−f) ν, where ν = ν1 + ν2. • a reported symptomatic individual is reported, on average, after 1/η days, and no longer creates transmissions due to isolation. • unreported symptomatic individuals u(t) are infectious for an average period of 1/η days. • susceptible individuals are removed from the possibility of infection as a result of vaccination, at a rate of v(t) per day, where this time dependent rate incorporates the first and second vaccine doses, and their effectiveness in preventing infection. 136 glenn webb the time-dependent transmission rate in the model before the last date of daily reported cases, is obtained from the daily reported cases data. since the daily reported cases data is typically very erratic, a rolling weekly average of the daily reported cases data is used to smooth this data. let dr(t1),dr(t2), . . . be the rolling weekly average number of daily reported cases each day, from the first week up to the last day of daily reported cases, where time t1, t2, . . . is discrete, day by day. in the model, the continuum version dr(t) of dr(t1),dr(t2), . . . , can be assumed to satisfy dr′(t) = ν1 i(t) −dr(t) ⇒ i(t) = ( dr′(t) + dr(t) ν1 ) . (2.6) then, equation (2.2) for i(t) in the model i′(t) = τ(t,s(t),i(t),r(t)) −νi(t), implies the transmission rate τ(t,s(t),i(t),r(t)) satisfies, until the last day of reported cases data, τ(t,s(t),i(t),r(t)) = i′(t) + ν i(t) = dr′′(t) + dr′(t) ν1 + ν ( dr′(t) + dr(t) ν1 ) . similar methods have been used in [9], [16], [17], [28], [29] to relate reported cases data to model dynamics. the discrete smoothing of the daily reported cases data to rolling weekly average values, can be interpolated by a continuum cubic spline curve cs(t). this curve is constructed by defining cubic polynomials on successive pairs of intervals [t1, t2], [t2, t3], [t3, t4], [t4, t5], . . . , where the interpolation agrees with the rolling weekly average daily cases data at the integer values, and is three times differentiable from the first to last day of rolling weekly average daily cases. then, dr(t) in (2.6) for the model can be equated to cs(t), and the derivatives dr′(t) = cs′(t) and dr′′(t) = cs′′(t) can also be obtained. thus, the continuum interpolation cs(t) derived from the rolling weekly average daily data agrees exactly with this data at discrete day by day values, and has continuous first and second derivatives on its domain. the continuum time-dependent transmission rate in the model before the last date of daily reported cases, is thus given by τ(t,s(t),i(t),r(t)) = cs′′(t) + cs′(t) ν1 + ν ( cs′(t) + cs(t) ν1 ) . (2.7) the model with this form for the transmission dynamics provides information about s(t), i(t), r(t), and u(t) up to the last date of daily reported cases. after this date, the transmission dynamics can be extrapolated, based on their most recent history before this last date, and the dynamics of the epidemic can be projected forward in time. the asymptotic behaviour of the solutions of equations (2.1), (2.2) and (2.3) is obtained as follows: add equations (2.1), (2.2) and (2.3) and integrate from t0 to t to obtain s(t) + i(t) + r(t) + ∫ t t0 v(s) s(s)ds + (ν −ν1) ∫ t t0 i(s)ds + η ∫ t t0 r(s)ds = s(t0) + i(t0) + r(t0). then, ∫ t t0 i(s)ds < ∞, and since i′(t) is continuous and bounded, limt→∞i(t) = 0. similarly, limt→∞r(t) = 0. integration of equation (2.6) yields dr(t) = e−t ( dr(t0) + ν1 ∫ t t0 esi(s)ds ) , predicting the effectiveness of vaccination 137 which implies limt→∞dr(t) = 0. 3. application of the covid-19 model to the united kingdom a chronology of the covid-19 epidemic in the united kingdom, starting in february, 2020, is given below [45]: • february: first cases reported. • march: the government imposed stay-at-home order banning all non-essential travel and closing most gathering places. • april: first wave of daily reported cases. • late april, may, june: number of cases slowed, and the government eased the lock-down restrictions. • july and august: cases remained at relatively low levels. • september and october: second wave of daily reported cases and the government re-imposed lock-down measures. • november 25: 696 deaths reported, the highest since may. • late november: number of cases and deaths slowed. • december: third wave of daily reported cases. • december 8: vaccination began with a 2-dose pfizer vaccine regimen. • december 30: the nhs delayed the second dose for the more than 500,000 people receiving the first dose up to that date, in order to provide a first dose to as many people as possible. also on december 30, astrazeneca vaccine was approved, and began implementation in january, with the same policies as the pfizer vaccine. the second dose for both was supposed to be approximately 12 weeks after the first dose. • december 30: new government restrictions imposed across the country. • january, february, march: daily cases subsided. • by january 2021 testing was running at approximately 4,000,000 tests per week, and by midfebruary 2021, approximately 75,000,000 tests had been conducted. • february 21: prime minister boris johnson announced vaccination goal to give first dose to all over the age of 50 by mid-april and all adults by end of july. • february 22: government projected that all lockdown measures would be ended by june 21 at the earliest: schools re-open in mid-march, travel permitted outside of local areas in late march, opening of non-essential retail and personal services in mid-april, outdoor social contact restrictions eased by mid-may, all lockdown limits removed by june 21 or later. the prime minister announced that international travel would resume on may 17, at the earliest, to allow free international travel of vaccinated travellers. the daily reported cases data in the uk is graphed in figure 2, from march 1, 2020 to april 1, 2021. a reported case means at least one positive laboratory test result or a lateral-flow device test result ([44]). the rolling weekly average daily cases data in the uk is also graphed in figure 2, as is the cubic spline interpolation cs(t) of this discrete rolling weekly averaged daily cases data. in figure 3 the daily number of first dose vaccinations and the daily number of second dose vaccinations in the uk, from january 11, 2021 to april 1, 2021, are graphed ([44]). from january 11, 2021 to april 1, 2021, 31, 318, 262 people had received a first dose vaccination, and 4, 958, 874 people had received a second dose vaccination. 138 glenn webb 0 100 200 300 400 10000 20000 30000 40000 50000 60000 70000 m a rc h 7 a p ri l1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 f e b ru a ry 1 m a rc h 1 a p ri l1 figure 2. vertical bars are daily reported cases from march 7, 2020 to april 1, 2021, red dots are discrete rolling weekly averaged daily reported cases, and the green graph is the continuum cubic spline interpolation cs(t) of the red dots. figure 3. vaccinations in the uk, beginning january 11, 2021, through april 1, 2021. daily first dose step function v1(t) (top). daily second dose step function v2(t) (bottom) 3.1. model parameters for the covid-19 epidemic in the united kingdom. set the initial susceptible population s(0) = 68, 000, 000, the current population of the united kingdom. set ν = 1/7, predicting the effectiveness of vaccination 139 100 200 300 400 50000 100000 150000 200000 250000 300000 m a rc h 7 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 f e b ru a ry 1 m a rc h 1 a p ri l 1 figure 4. graph of the model transmission rate from march 1, 2020 to april 1, 2021, as in (2.7). ν2/ν2 = 1.5. which means that asymptomatic infectious individuals remain infectious for 7 days. set η = 1/7, which means that reported symptomatic individuals r(t) are infectious for an average period of 1/η = 7 days, as are unreported symptomatic individuals u(t). the values for ν and η are uncertain, but reasonable for this application. the value of η = 1/7 is consistent with the assumption that the daily reported cases data can be replaced by rolling weekly averaged values. the ratio of unreported cases to reported cases will be assumed as 3 to 2, 2 to 1, and 3 to 1 in simulations of the model. before april 1, 2021, the model transmission rate, based on the rolling weekly average of daily data as in (2.7), is graphed in figure 4 and figure 6 for the case that this ratio is 3 to 2 (ν1 = .4/7, ν2 = .6/7, ν2/ν1 = 1.5). the case that this ratio is 2 to 1 is graphed in figure 7 and the case that this ratio is 3 to 1 is graphed in figure 8. 3.2. simulation of the model covid-19 epidemic in the united kingdom. in figure 5, the graph of symptomatic reported cases r(t) and the graph of cumulative reported cases cr(t), from the model simulation with the above parameters, are shown for march 7, 2020 through april 1, 2021. the graphs are the same for ν1 = .4/7, ν1 = .3333/7, and ν1 = .25/7. 3.3. projecting the model forward from the last date of reported cases data. the last date of daily reported cases data and vaccination data is td = april 1, 2021. from january 11, 2021 to april 1, 2021, the daily vaccination rate is v1(t) per day, as in figure 3 (top) for the first dose, and v2(t) per day, as in figure 3 (bottom) for the second dose. for the model projections forward from april 1, 2021, the vaccination rate is assumed as v1(t) = 200, 000 vaccinations per day for the first dose, and v2(t) = 200, 000 vaccinations per day for the second dose. in equation (2.1), set v(t) = 0.9 v1(t) + 0.05 v2(t) s0 , t > td. vaccination removes individuals from the susceptible population at time t at a rate proportional to the remaining susceptible individuals at time t. in equation (2.1), the loss of susceptibles per day due to 140 glenn webb 100 200 300 400 1×106 2×106 3×106 4×106 data cr(t) r(t) m a rc h 7 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 f e b ru a ry 1 m a rc h 1 a p ri l 1 figure 5. graphs of the daily reported cases r(t) and the rolling weekly averaged cumulative daily reported cases cr(t) from the model simulation. cr(t) = approximately 4,370,000 on april 1, 2021. the graphs are the same for ν2/ν1 = 1.5, 2.0, 3.0. vaccination v(t) s(t) = ( 0.9 v1(t) + 0.05 v2(t) ) s(t) s(0) , t > td involves the fraction s(t)/s(0) of unvaccinated population and prior infected population, still susceptible at time t. the form of v(t) incorporates the effectiveness of vaccination, through both doses. after time td = april 1, 2021, a time t1 = april 15, 2021 is set such that there is an increasing return to normalcy of social distancing behaviour. after april 15, the increase in the transmission rate per day involves a linear scaling factor ω, and lasts until time t2 = july 1, 2021. thus, the transmission rate has the following form: from day t0 = march 1, 2020 until time td = april 1, 2021, the transmission rate is as in (2.7): τ(t,s(t),i(t),r(t)) = dr′′(t) + dr′(t) ν1 + ν ( dr′(t) + dr(t) ν1 ) , t0 ≤ t ≤ td; for td = april 1, 2021 < t ≤ t1 = april 15, 2021: τ(t,s(t),i(t),r(t)) = τ(td,s(td),i(td),r(td)) ( (i(t) + (1 + ν2 ν1 ) r(t)) s(t) (i(td) + (1 + ν2 ν1 ) r(td)) s(td) ) ; for t1 = april 15, 2021 < t ≤ t2 = july 1, 2021: τ(t,s(t),i(t),r(t)) = ( 1.0 + ω (t−t1) ) τ(td,s(td),i(td),r(td)) ( (i(t) + (1 + ν2 ν1 )r(t)) s(t) (i(td) + (1 + ν2 ν1 ) r(td)) s(td) ) ; for t2 = july 1, 2021 < t, the scaling factor ω term, corresponding to social distancing relaxation, is maximized at t2 − t1: τ(t,s(t),i(t),r(t)) = ( 1.0+ω (t2−t1) ) τ(td,s(td),i(td),r(td)) ( (i(t) + (1 + ν2 ν1 )r(t)) s(t) (i(td) + (1 + ν2 ν1 ) r(td)) s(td) ) . predicting the effectiveness of vaccination 141 the transmission rate is continuous, and in particular, continuous at day td = april 1, 2021, day t1 = april 15, 2021, and day t2 = july 1, 2021. the magnitude of the parameter ω, corresponding to resumption of normal social distancing behaviour, is critical for resurgence of the epidemic. in figure 6, the graphs of the model transmission rate τ(t,s(t),i(t),r(t)) are shown from march 7, 2020 to january 1, 2022, with the ratio of unreported cases to reported cases 3 to 2 (ν1 = .4/7, ν2 = .6/7), and ω = .03, .035 and .04. 100 200 300 400 500 600 50000 100000 150000 200000 250000 m a rc h 7 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 f e b ru a ry 1 m a rc h 1 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 figure 6. graphs of the transmission rate τ(t,s(t),i(t),r(t)), with relaxation of social distancing measures beginning on april 15, 2021. ν2/ν1 = 1.5. the relaxation scaling values are ω = .03 (green), ω = .035 (red), ω = .04 (purple). ν2/ν1 = 1.5 ω = .03 ω = .035 ω = .04 τ(t,s(t),i(t),r(t)) 5,713 5,713 5,713 on td = april 1, 2021 max of τ(t,s(t),i(t),r(t)) after td 2,738 10,812 40,709 occurring on aug 19, 2021 sept 18, 2021 sept 26, 2021 τ(t,s(t),i(t),r(t)) on january 1, 2022 274 2,313 6,639 table 1. transmission rate τ(t,s(t),i(t),r(t)) for the ratio of unreported cases to reported cases ν2/ν1 = 1.5, and the social behaviour relaxation parameter ω. in figure 7, the graphs of the model transmission rate τ(t,s(t),i(t),r(t)) are shown from march 7, 2020 to january 1, 2022, with the ratio of unreported cases to reported cases 2 to 1 (ν1 = .3333/7, ν2 = .6667/7), and ω = .03, .035 and .04. 142 glenn webb 100 200 300 400 500 600 50000 100000 150000 200000 250000 300000 350000 m a rc h 7 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 f e b ru a ry 1 m a rc h 1 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 figure 7. graphs of the transmission rate τ(t,s(t),i(t),r(t)), with relaxation of social distancing measures beginning on april 15, 2021. ν2/ν1 = 2.0. the relaxation scaling values are ω = .03 (green), ω = .035 (red), ω = .04 (purple). ν2/ν1 = 2.0 ω = .03 ω = .035 ω = .04 τ(t,s(t),i(t),r(t)) 6,856 6,856 6,856 on day td = april 1, 2021 max of τ(t,s(t),i(t),r(t)) after td 3,230 12,400 44,028 occurring on aug 18, 2021 sept 16, 2021 sept 21, 2021 τ(t,s(t),i(t),r(t)) on january 1, 2022 310 2,399 5,744 table 2. transmission rate τ(t,s(t),i(t),r(t)) for the ratio of unreported cases to reported cases ν2/ν1 = 2.0, and the social behaviour relaxation parameter ω. in figure 8, the graphs of the model transmission rate τ(t,s(t),i(t),r(t)) are shown from march 7, 2020 to january 1, 2022, with the ratio of unreported cases to reported cases 3 to 1 (ν1 = .25/7, ν2 = .75/7), and ω = .03, .035 and .04. ν2/ν1 = 3.0 ω = .03 ω = .035 ω = .04 τ(t,s(t),i(t),r(t)) 9,141 9,141 9,141 on day td = april 1, 2021 max of τ(t,s(t),i(t),r(t)) after td 4,149 15,054 48,395 occurring on aug 16, 2021 sept 12, 2021 sept 13, 2021 τ(t,s(t),i(t),r(t)) on january 1, 2022 363 2,367 4,248 table 3. transmission rate τ(t,s(t),i(t),r(t)) for the ratio of unreported cases to reported cases ν2/ν1 = 3.0, and the social behaviour relaxation parameter ω. predicting the effectiveness of vaccination 143 100 200 300 400 500 600 100000 200000 300000 400000 m a rc h 7 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 f e b ru a ry 1 m a rc h 1 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 figure 8. graphs of the transmission rate τ(t,s(t),i(t),r(t)), with relaxation of social distancing measures beginning on april 15, 2021. ν2/ν1 = 3.0. the relaxation scaling values are ω = .03 (green), ω = .035 (red), ω = .04 (purple). in figure 9, the graphs of the model simulation are shown from march 7, 2020 to january 1, 2022 for daily reported cases, with the ratio of unreported cases to reported cases 3 to 2 (ν1 = .4/7, ν2 = .6/7), and ω = .03, .035, and .04. 100 200 300 400 500 600 10000 20000 30000 40000 50000 60000 m a rc h 7 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 f e b ru a ry 1 m a rc h 1 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 figure 9. graphs of daily reported cases dr(t) for the model simulations with relaxation of social distancing measures beginning on april 15. ν1 = .4/7, ν2 = .6/7. the relaxation scaling values are ω = .03 (green), ω = .035 (red), ω = .04 (purple). 144 glenn webb ν2/ν1 = 1.5 ω = .03 ω = .035 ω = .04 number of susceptibles on january 1, 2022 17,612,000 16,781,000 14,242,000 cumulative reported cases april 1, 2021 january 1, 2022 234,600 600,200 1,953,000 new asymptomatic cases on january 1, 2022 142 1,145 3,438 table 4. model simulation output for the ratio of unreported cases to reported cases ν2/ν1 = 3 to 2, and the social behaviour relaxation parameter ω. 100 200 300 400 500 600 10000 20000 30000 40000 50000 60000 m a rc h 7 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 f e b ru a ry 1 m a rc h 1 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 figure 10. graphs of daily reported cases dr(t) for the model simulations with relaxation of social distancing measures beginning on april 15. ν1 = .3333/7, ν2 = .6667/7. the relaxation scaling values are ω = .03 (green), ω = .035 (red), ω = .04 (purple). in figure 10, the graphs of the model simulation are shown from march 7, 2020 to january 1, 2022 for daily reported cases, with the ratio of unreported cases to reported cases 2 to 1 (ν1 = .3333/7, ν2 = .6667/7), and ω = .03, .035, and .04. in figure 11, the graphs of the model simulation are shown from march 7, 2020 to january 1, 2022 for daily reported cases, with the ratio of unreported cases to reported cases 3 to 1 (ν1 = .25/7, ν2 = .75/7), and ω = .03, .035, and .04. 4. summary a general model of covid-19 epidemics has been developed to predict the effectiveness of vaccination. the model incorporates basic elements of covid-19 dynamics: transmission due to asymptomatic and symptomatic infected individuals, transmission due to reported and unreported cases, and transmission mitigation due to social distancing measures. because the daily reported cases data is typically very erratic, a rolling weekly averaging process is used to provide better consistency with the model dynamics. the model formulation is constructed so that the daily reported cases dr(t) in the model agrees exactly with the rolling weekly averaged daily reported cases data. predicting the effectiveness of vaccination 145 100 200 300 400 500 600 10000 20000 30000 40000 50000 60000 m a rc h 7 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 f e b ru a ry 1 m a rc h 1 a p ri l 1 m a y 1 j u n e 1 j u ly 1 a u g u s t 1 s e p te m b e r 1 o c to b e r 1 n o v e m b e r 1 d e c e m b e r 1 j a n u a ry 1 figure 11. graphs of daily reported cases dr(t) for the model simulations with relaxation of social distancing measures beginning on april 15. ν1 = .25/7, ν2 = .75/7. the relaxation scaling values are ω = .03 (green), ω = .035 (red), ω = .04 (purple). the objective of the modelling process is to project forward in time from the last day td of daily reported cases data, the effectiveness of vaccination in controlling the epidemic. the transmission rate in the model, forward from day td, is based on the model transmission rate on this day. as time proceeds forward from td, the transmission rate is moderated, in correspondence with a restoration of normal social distancing, as the number of susceptible individuals is reduced due to vaccination. two dates t1 and t2 are set, such that td < t1 < t2, and the transmission rate increases from t1 to t2 with a scaling factor ω, that corresponds to the reduction of social distancing measures. the model is applied to the covid-19 epidemic in the united kingdom. the last day of daily reported cases td = april 1, 2021. the restoration of normal social distancing is from t1 = april 15, 2021 to t2 = july 1, 2021. the model outputs are analysed with ratios of unreported case to reported cases (ν2/ν1) as 3 to 2, 2 to 1, and 3 to 1, and with of restoration of normal social distancing scaling parameters ω = .03, .035, and .04. the model output shows the following dependence of the cumulative reported cases cr(t) between april 1, 2021 and january 1, 2022: (1) the cumulative reported cases cr(t) between april 1, 2021 and january 1, 2022 are decreasing as the ratio of unreported to reported cases ν2/ν1 increases; (2) the cumulative reported cases cr(t) between april 1, 2021 and january 1, 2022 are increasing as the scaling factor ω corresponding to relaxation of social behaviour restrictions increases. these results are consistent with the results in [24] and [34], which were based on extensive data input for the covid-19 epidemic in the united kingdom. the general model is applicable to covid-19 epidemics in all locations. the parameterisation of the model is based on daily reported cases data and daily vaccination data, which is readily available in all locations. the model parameters ν,ν1,ν2,η,td, t1, t2 can be adjusted to specific locations, and predictions based on vaccination implementation and restoration of normal social distancing can be provided. in future work the model will be applied to covid-19 epidemics in other locations. 146 glenn webb ν2/ν1 = 2.0 ω = .03 ω = .035 ω = .04 number of susceptibles on january 1, 2022 16,837,000 15,906,000 13,312,000 cumulative reported cases april 1, 2021 january 1, 2022 231,500 631,200 1,751,000 new asymptomatic cases on january 1, 2022 134 999 2,537 table 5. model simulation output for the ratio of unreported cases to reported cases ν2/ν1 = 2.0, and the social behaviour relaxation parameter ω. ν2/ν1 = 3.0 ω = .03 ω = .035 ω = .04 number of susceptibles on january 1, 2022 15,300,000 14,221,000 11,623,000 cumulative reported cases april 1, 2021 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[44] urlcoronavirus.data.gov.uk [45] urlwikipedia.org/wiki/covid-19 mathematics department, vanderbilt university, nashville, tn, usa mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 4, december 2020, pp.412-424 https://doi.org/10.5206/mase/10857 a procedure for deriving new ode models: using the generalized linear chain trick to incorporate phase-type distributed delay and dwell time assumptions. paul j. hurtado and cameron richards abstract. ordinary differential equations (ode) models are used in a wide variety of applications throughout the sciences. despite their widespread use, these models are sometimes viewed as inflexible with respect to time delay and dwell time assumptions. the generalized linear chain trick (glct) provides a theoretical foundation that can help modelers incorporate much more flexible phase-type distributed delay (or dwell time) assumptions into ode models, including traditional exponential and erlang distribution assumptions. the glct serves as a bridge between stochastic processes and dynamical systems theory for odes, opening up opportunities to use concepts and tools from markov chain theory in the development, analysis, and interpretation of ode models. to facilitate the practical application of this theory, in this paper we introduce a new glct-based procedure for deriving new ode models by generalizing or approximating existing ode, dde, or distributed delay equation models. we apply this procedure to multiple models from the literature, using it to derive new models that are generalizations or approximations of those models. 1. introduction ordinary differential equations (ode) models are widely used in the sciences, in part because of the relative ease of formulating and analyzing ode models [35, 25, 3, 7, 1]. however, they are often criticized for their limited capacity to only incorporate a narrow range of delay and dwell time assumptions. such delays are more easily incorporated into models using other mathematical frameworks, e.g., delays can be modeled most generally using integral equations or integro-differential equations to incorporate distributed delays into dynamic models, or using delay differential equations (ddes) to incorporate fixed delays [37, 9, 8, 31, 19, 30]. in an ode framework, the linear chain trick has long been used to incorporate exponential and erlang distributed delays [23, 22, 26, 34, 6]. however, recently this technique has been generalized to a much broader family of delay or dwell time distributions [16]. this generalized linear chain trick (glct) theory [16] allows modelers to incorporate delays and dwell times that obey phase-type distributions. the phase-type family of distributions is made up of the set of all possible absorption time distributions for continuous time markov chains (ctmcs) with one or more transient states and at least one absorbing state. these include exponential, erlang (gamma with an integer shape parameter), hyperexponential, hypoexponential (generalized erlang), and coxian distributions [4, 12, 27, 14]. the glct also permits the use of similar time-varying versions of such distributions [16]. together, these tools and techniques enable modelers to draw from a richer set of ode model assumptions when constructing new models, and a framework for more clearly seeing received by the editors 1 july 2020; accepted 15 december 2020; published online 29 december 2020. 2010 mathematics subject classification. primary 92b99; secondary 37n25, 37m05. key words and phrases. linear chain trick; gamma chain trick; phase-type distribution; coxian distribution; hypoexponential distribution. pjh was supported in part by the sloan scholars mentoring network of the social science research council with funds provided by the alfred p. sloan foundation; and nsf grant no. deb-1929522. 412 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/10857 a procedure to derive new ode models via the glct 413 how underlying stochastic model assumptions are reflected in the structure of mean field ode models. moreover, concepts from markov chain theory, including reward process theory, can be used in the analysis of odes derived using the glct as well as the interpretation of general analytical results [18]. to help modelers put the glct into practice, in this paper we introduce a new glct-based procedure that can be used to quickly formulate new ode models that explicitly incorporate phase-type distributed delays, or that otherwise make use of markov chains to organize complex state transition structures in such models. below, we describe this relatively simple procedure and illustrate it’s application by using it to derive new ode models that are extensions of models taken from the literature. these include ode models with no explicit delays, dde models, and distributed delay models in the form of integro-differential equations. in section 1.1 we review the glct framework for phase-type distributions, and the standard linear chain trick (lct). in section 1.2, we review procedures for deriving ode models as special cases of distributed delay equations, or as approximations of ddes. in section 2.1 we describe the glct-based procedure. we then use it to extend multiple models starting in section 2.2, where we generalize a tumor growth inhibition model by simeoni et al [32]. in section 2.3 we generalize an opioid epidemic model by battista et al [2], then a within-host immune-pathogen interaction model by hurtado [15] in section 2.4, and a model of cell-to-cell hiv spread by culshaw et al [5] in section 2.5. 1.1. generalized linear chain trick. we here provide a statement of the generalized linear chain trick (glct) for phase-type distributions. the glct in its most general form, as detailed in [16], extends lemma 1.1 below to include time-varying parameters (analogous to extending homogeneous poisson process rates to the time-varying rates of inhomogeneous poisson processes). phase-type distributions are a family of continuous, matrix exponential distributions that can be thought of as the absorption time distributions for ctmcs with one or more absorbing states. they are parameterized by a column vector v, which is the initial distribution vector across the set of transient states1, and the transient state block m of the transition rate matrix, which together define the corresponding ctmc. equations for the density function f(t), cumulative distribution function f(t), and jth moments e(y j) of a phase-type distributed random variable y are given in eqs. (1.1) below, where superscript t indicates transposition and 1 is an appropriately long column vector of ones (for more on phase-type distributions, see [17, 18, 4, 14, 12, 27]). f(t) = vt emt (−m1) (1.1a) f(t) = 1−vt emt 1 (1.1b) e(y j) = j! vt (−m)−j1. (1.1c) lemma 1.1 (glct for phase-type distributions [16]). assume individuals enter a state (call it state x) at rate i(t) ∈ r and that the distribution of time spent in state x follows a continuous phase-type distribution given by the length k initial probability vector v and the k ×k matrix m. then partitioning x into k sub-states xi, and denoting the corresponding amount of individuals in state xi at time t by xi(t), then the mean field equations for the quantities xi are given by d dt x(t) = vi(t) + mt x(t) (1.2) where the rate of individuals leaving each of these sub-states of x is given by the vector (−m 1) ◦ x, where ◦ is the hadamard (element-wise) product of the two vectors, and thus the total rate of individuals leaving state x is given by the sum of those terms, i.e., by (−m 1)tx = −1tmtx. 1the full initial distribution vector, for a ctmc with one absorbing state, would be [vt , v0] t . 414 p. hurtado and c. richards the standard linear chain trick (lct) is well known (see [16] and references therein) and is a special case of lemma 1.1 above. however, it is usually stated without the above matrix-vector notation. the following is a formal statement of the standard lct, but here we have slightly generalized it to include generalized erlang distributions (i.e., the sum of k independent exponentially distributed random variables, each with potentially different rates ri) as this only changes a few subscripts in the mean field equations. see [16, 34] for a similar statement of the standard lct (for erlang distributions). corollary 1.1 (linear chain trick for hypoexponential distributions). consider the glct above. assume that the dwell-time distribution is a generalized erlang (hypoexponential) distribution with rates r = [r1, r2, . . . ,rk] t, where ri > 0, or an erlang distribution with rate r (all ri = r) and shape k (or if written in terms of shape k and mean τ = k/r, use r = k/τ). then the corresponding mean field equations are dx1 dt = i(t)−r1 x1 dx2 dt = r1 x1 −r2 x2 ... dxk dt = rk−1 xk−1 −rk xk. (1.3) proof. the phase-type distribution formulation of this generalized erlang distribution is given by v =   1 0 ... 0   and m =   −r1 r1 0 · · · 0 0 −r2 r2 ... 0 ... ... ... ... ... 0 0 ... −rk−1 rk−1 0 0 · · · 0 −rk   . (1.4) substituting these into eq. (1.2) yields eqs. (1.3). if all ri = r then the phase-type distribution is an erlang distribution with rate r and shape k (mean k/r and coefficient of variation 1/ √ k). � 1.2. deriving odes from delay differential equations & distributed delay equations. the first step in our glct-based procedure detailed below is to take an existing ode, dde, or distributed delay equation and derive odes that either approximate (in the dde case) or are a special case of (in the distributed delay equation case) that model. here we briefly review how this can be done. first, for distributed delay equations (e.g., integral equations) the standard linear chain trick can be applied, as detailed in [16]. this involves assuming erlang distributed delays, and from those distributed delay equations deriving an ode model by differentiating integral equations and employing the recursive property of erlang density functions. alternatively, in some cases you may be able to use results such as lemma 1.1 or corollary 1.1 to formulate these equations directly. for ddes, one can use the following (well known) technique to obtain an approximating system of odes. here we summarize this approximation as described in section 7.3 of [34] (for more on this technique, see [10, 11, 24]). consider an equation of the form d dt x(t) = f(x(t),x(t− τ)) (1.5) with solution x(t) that includes the initial data over [−τ,0] given by x(t) = φ(t), t ∈ [−τ,0]. our goal is to obtain a system of odes that approximates eq. (1.5). to do this, first observe that the assumed discrete delay of τ time units is equivalent to assuming a distributed delay model where a procedure to derive new ode models via the glct 415 the delay distribution is a dirac-δ function shifted by τ time units, δ(t− τ). that is, the cdf for this distribution is f(t) = 0 for t < τ and then it jumps to f(t) = 1 for t ≥ τ. this dirac-δ distributional assumption can be approximated with an erlang (gamma) distribution that has mean τ and an arbitrarily small variance. recall that the sum of k independent exponential random variables with rate r is gamma distributed with rate parameter r, shape parameter k, mean k/r, variance k/r2, and coefficient of variation 1/ √ k. thus, fixing the mean at τ by setting the rate r = k/τ, it follows that increasing k can yield an arbitrarily small coefficient of variation. we may therefore use the erlang case of corollary 1.1 to write down a system of odes that approximates eq. (1.5). by the above (and see [34] and references therein), it follows that eq. (1.5) can be approximated by d dt x0(t) = f(x0(t),xn(t)) (1.6a) d dt xi(t) = k τ [xi−1(t)−xi(t)], i = 1, . . . ,k (1.6b) with initial conditions xi(0) = φ(−τ i/k) for i = 0, . . . ,k. here, x0(t) ≈ x(t) and that approximation improves as the value of k increases. 2. results with the above preliminaries in hand, we may now detail our glct-based procedure which serves as a tool for implementing the glct to derive new ode models with phase-type dwell time assumptions. 2.1. glct-based procedure for deriving ode models. many existing mean field state transition models, in the form of an ode, dde, or distributed delay equation can be extended by a system of odes derived using the following glct-based procedure. (1) for an ode model (with exponential dwell times), it is usually straightforward to obtain a generalized model using lemma 1.1 directly. in the case of a distributed delay equation or a dde model, this first step entails generating a system of odes using the standard linear chain trick as detailed in section 1.2 above. (2) the ode model obtained in the first step above can then be written in matrix-form, as suggested by the form of equations in lemma 1.1 and corollary 1.1. the resulting set of equations will then be in a matrix form, where the matrix-vector pairs that were parameterized for the erlang distributions used to derive the model (see eq. (1.4)) can now be replaced by a more general vector and matrix pair that together parameterize an arbitrary phase-type distribution. in the sections below, we will illustrate the application of this glct-based procedure for deriving new ode models by generalizing various biological models taken from the peer reviewed literature. for each model, we first introduce the application context and highlight some of the key model assumptions related to delays and the distribution of times spent in different states. this entails viewing each model as mean field equations corresponding to some unspecified stochastic model. we then derive from each model an ode model that incorporates phase-type distributed delays or dwell times, thereby generalizing or approximating the original model. 2.2. model 1: tumor growth inhibition (tgi) model. simeoni et al [32] introduced a simple model of tumor growth inhibition that employs the standard linear chain trick (lct) to incorporate an erlang distributed time to cell death following tumor cell damage from treatment. that model was subsequently analyzed using standard approaches from dynamical systems [21, 20], and has been used elsewhere in the study of tumor growth and the development of cancer treatments [29, 33]. the simeoni model has also been extended to a ‘‘competing poisson processes” type of assumption (compare fig. 6 416 p. hurtado and c. richards z0 z1 z2 z3 cell death k0 c(t) k1 k1 k1 tumor growth gf(z0,w) figure 1. schematic diagram of the tumor growth inhibition model (tgi) by simeoni et al [32]. see the main text for further details. in [16] to fig. 2 in [29] and fig. 1 in [36]) in order to model tumor cell death arising from the combined effects of two drugs with no pharmacokinetic interaction [29, 36]. the basic tgi model in [32] is given by eqs. (2.1) and (2.2) below. in the absence of pharmacological treatment, the amount of cycling (replicating) tumor cells at time t, z0(t), grows according to the overall growth rate2 gf(z0,w) = λ0 z0(t)[ 1 + ( λ0 λ1 w(t) )ψ] 1 ψ . (2.1) treatment is assumed in [32] to begin at time t0 > 0, and accordingly the effect of that treatment c(t) = 0 for 0 ≤ t ≤ t0. once treatment begins, cells that are damaged by the treatment then progress through a series of states zi, i = 1, . . . ,n, prior to cell death (see fig. 1). together, the full model is given in [32] by dz0(t) dt = gf(z0(t),w(t))−k0 c(t)z0(t) (2.2a) dz1(t) dt = k0 c(t)z0(t)−k1 z1(t) (2.2b) dzi(t) dt = k1 zi−1(t)−k1 zi(t), i = 2, . . . ,n (2.2c) w(t) = n∑ i=0 zi; z0(0) = w0, zi(0) = 0, i = 1, . . . ,n (2.2d) where w is the total amount of tumor cells, and k0 and c(t) ≥ 0 determine the rate of initial tumor cell damage from the treatment. the above assumptions and interpretations from [32] can also be described as follows. viewed through the lens of the glct, k0 and c(t) determine the distribution of time spent in the base state z0, which follows the first event time distribution under a non-homogeneous poisson process with rate r(t) = k0c(t) (for details, see[16]). parameters n and k1 are the shape and rate parameters, respectively, for the erlang distributed time until cell death for the cells damaged by the treatment. the treatment is assumed to have no effect on the time until cell death after the initial damage to the cell. 2.2.1. generalized tgi model. the glct-based procedure described in section 2.1 can be used to extend this model to instead assume a more general phase-type distributed time to cell death. the 2this growth rate function is an approximation of the piece-wise function that is equal to λ0z0 when w < λ0/λ1, and λ1z0/w when w ≥ λ0/λ1. see [21] for details. a procedure to derive new ode models via the glct 417 equations for zi, i = 1, . . . ,n in eqs. (2.2) can be written in matrix form, using lemma 1.1, where v =   1 0 ... 0   and m =   −k1 k1 0 · · · 0 0 −k1 k1 ... 0 ... ... ... ... ... 0 0 ... −k1 k1 0 0 · · · 0 −k1   . (2.3) this yields the more compact, and more general, set of equations below, dz0(t) dt = gf(z0(t),w(t))−k0 c(t)z0(t) (2.4a) dx(t) dt = k0 c(t)z0(t) v + m tx (2.4b) where x = [z1,z2, . . . ,zn] t, w(t) = ∑n i=0 zi(t), z0(0) = w0, zi(0) = 0, i = 1, . . . ,n. note that eqs. (2.4) generalize the tgi model in the sense that these equations accommodate any phase-type distribution assumption for the time to cell death following the initial effect of treatment, not just the erlang distribution assumed in the original tgi model and parameterized by eqs. (2.3). additionally, this matrix-vector form of the original tgi model (i.e., assuming an erlang distribution) can still be used with some benefit for both computational and mathematical analyses of the tgi model given by eqs. (2.2), where those analyses can take advantage of the matrix-vector form of these more general equations. 2.3. model 2: perscription opioid epidemic model. consider the prescription opioid epidemic model by battista et al [2], which is a system of ordinary differential equations with no explicit time delays, and (implicit) exponentially distributed dwell times in multiple states. the model assumes individuals are in one of four different states: s, p , a, and r. here s respresents the susceptible class. these individuals are not using opioids or recovering from addiction. p represents the prescribed users (those who are prescribed the drugs and using them but have no addiction). a represents addicted individuals who can be using either prescribed or ilicit opioids, and r represents the class of individuals undergoing treatment and rehabilitation to recover from addiction. the model as given in [2], where the dot over each state variable indicates a time derivative, is ṡ = −αs −βasa−βpsp + �p + δr + µ(p + r) + µ∗a (2.5a) ṗ = αs − (� + γ + µ)p (2.5b) ȧ = γp + σr + βasa + βpsp − (ζ + µ∗)a (2.5c) ṙ = ζa− (δ + σ + µ)r. (2.5d) the term αs is the number of individuals transitioning from the susceptible state to the prescribed state after being prescribed opioids per unit time, βasa is the rate of those transitioning from state s to state a after interacting with addicted individuals, and similarly βpsp represents the rate of individuals who transition from the susceptible class to the addicted class after exposure to opioids via perscription opiod users who have extra or unsecured drugs. the terms �p and δr are the rate individuals leave the prescribed users class without becoming addicted and then reenter the susceptible class at per-capita rate �, and those who leave the rehabilitation state after treatment and reenter the susceptible state at per-capita rate δ. the rates µp, µr and µ∗a are the death rates for the prescribed, rehabilitated, and addicted classes (to ensure constant population size, deaths are replaced instantaneously by new susceptible individuals). the term γp is the rate that individuals leave the prescribed class by becoming 418 p. hurtado and c. richards addicted to their prescription opioids, ζa is the rate at which addicted individuals initiate treatment, and σr is the rate at which individuals who are undergoing treatment reenter the addiction class. we may also interpret the model terms described above as follows. focusing on eq. (2.5b), for example, prescription users remain in the prescribed state for an exponentially distributed amount time (with rate � + γ + µ), and the proportion of individuals which leave the prescribed user state, and go to the susceptible state, is � �+γ+µ (see section 3.5.5 in [16]). it follows that the net rate of individual entering the susceptible state from the prescribed state is therefore � �+γ+µ (� + γ + µ)p = �p . similarly, the proportion of individuals who go on to become addicted, and who die, are given by γ �+γ+µ and µ �+γ+µ , respectively. 2.3.1. generalized opioid epidemic model. we can use the procedure described in section 2.1 to extend this model by replacing the implicit assumption of exponentially distributed dwell times in each state with arbitrary phase-type distributed dwell times. assume the dwell time distribution for the prescribed user state p is a continuous phase-type distribution parameterized by the n× 1 parameter vector vp and n×n matrix mp. then to total number of individuals in state p is given by the sum of the n sub-states pi, i = 1, . . . ,n. let x = [p1,p2, . . . ,pn] t . then by the glct (lemma 1.1), the mean field equations for our prescribed user sub-states are ẋ = vpαs + mp tx. observe that if we let vp be a one dimensional row vector with its first and only entry being a 1 and let mp = [−(� + γ + µ)] be a 1×1 matrix, we will arrive at our original equation, eq. (2.5b). recall that individuals who leave the prescribed user state either transition to the addicted state, the susceptible state, or they die. we can denote these proportions as fpa, fps, and fpd, respectively, where fij ∈ [0,1] and fpa + fps + fpd = 1. note that in the original model fpa = γ(�+γ+µ),fps = � (�+γ+µ) , and fpd = µ (�+γ+µ) . this yields the model: ṡ = −αs −βasa−βpsp + fps(−mp1)tx + δr + fpd(−mp1)tx + µr + µ∗a (2.6a) ẋ = vpαs + mp tx (2.6b) ȧ = fpa(−mp1)tx + σr + βasa + βpsp − (ζ + µ∗)a (2.6c) ṙ = ζa− (δ + σ + µ)r. (2.6d) similarly, we can generalize the addicted and rehabilitated states with phase-type dwell time distributions, assuming the respective phase-type distributions are parameterized by va, ma, vr, and mr. let y = [a1,a2, . . . ,ak] t denote the k sub-states of a, and z = [r1,r2, . . . ,rm] t the m sub-states of r. this yields the generalized model: ṡ = −αs −βasa−βpsp + (fps + fpd)(−mp1)tx + (frs + frd)(−mr1)t z + fad(−ma1)t y (2.7a) ẋ = vpαs + mp tx (2.7b) ẏ = va ( fpa(−mp1)tx + fra(−mr1)tz + βasa + βpsp ) + ma t y (2.7c) ż = vr(far(−ma1)t y) + mrtz. (2.7d) it is worth noting that the original model eqs. (2.5) from [2] are a special case of eqs. (2.7), as are any intermediate extensions of the original model obtained by applying the standard linear chain trick (lct) to impose erlang distributed dwell times on one or more of the four main states. a procedure to derive new ode models via the glct 419 2.4. model 3: within-host model of the immune-pathogen interaction. in [15], hurtado incorporated a specific (adaptive) immune response component to the innate immune response model by reynolds et al [28]. the scaled version of this within-host model, as stated in [15], is dp dt = kpgp(1−p)− kmp µp + p −k(y)np (2.8a) dn dt = n + kp p xn + n + kp p −µn p (2.8b) dy0 dt = (np)α xαy + (np) α −µy0 y0 (2.8c) dy dt = µy0 y0 −µy y (2.8d) in this model, p is the scaled pathogen (bacteria) population size, which follows a logistic growth model in the absence of an immune response. the second term in eq. (2.8a) models the effect of some baseline local immune defenses capable of neutralizing a small population of pathogen, and mathematically introduces a strong allee effect into the model. the level of innate immune activity n increases in response to the presence of pathogen, as well as from a positive feedback loop, and the interaction of this innate immune activity and pathogen stimulates progenitor cells (y0) that mature into active specific immune components (y), e.g., b-cells, which augment the pathogen-killing capacity of the innate immune components (i.e., which increase k(y)). for further details on this model, see [15] and [28]. in this model, the specific immune response delay can be thought of as an exponentially distributed maturation time (with mean 1/µy0) and the duration of the active immune response (i.e., the dwell time of mature specific immune components in state y) is also exponentially distributed (with mean 1/µy). 2.4.1. within-host model with phase-type delays in the specific immune response. both of these exponential dwell time distribution assumptions associated with the specific immune response can be replaced by phase-type distributions with respective parameters vy0, my0, vy and my, respectively. to do this, we first partition state y0 into sub-states xi, i = 1, . . . ,m, and the state y into sub-states zj, j = 1, . . . ,n, where y0 = ∑m i=1 xi and y = ∑n j=1 zj, and we let x = [x1, . . . ,xm] t and z = [z1, . . . ,zn] t. the glct (lemma 1.1) then yields the more general model dp dt = kpgp(1−p)− kmp µp + p −k(y)np (2.9a) dn dt = n + kp p xn + n + kp p −µn p (2.9b) dx dt = vy0 (np)α xαy + (np) α + my0 t x (2.9c) dz dt = −1tmytvy + mytz. (2.9d) 2.5. model 4: cell-to-cell spread of hiv. in [5], culshaw et al introduce an integro-differential equation model of the cell-to-cell spread of hiv, which incorporates a distributed time delay in the time between cells becoming infected and infectious. they then derive from this general model multiple other models which differ only in the specific assumptions on the form of this delay distribution. the consider models with no delay, with a fixed (dirac-δ) delay of τ time units, and with an exponentially distributed delay. here we will extend these models to a general phase-type distributed delay. in the most general model described in [5], state variable c(t) represents the concentration of healthy cells at time t, and i(t) is the concentration of infected cells, and 420 p. hurtado and c. richards dc dt = rcc(t) ( 1− c(t) + i(t) cm ) −kic(t)i(t) (2.10a) di dt = k′i ∫ t −∞ c(u)i(u)f(t−u)du−µii(t). (2.10b) parameter rc is the net growth rate of the healthy cell population, cm is an effective carrying capacity of the system, ki is an infection rate parameter, k ′ i/ki is the fraction of cells surviving the incubation period, and µi is the per capita death rate of infected cells (implicitly, the infected cell lifetime is exponentially distributed with mean 1/µi). initial values for c and i must be functions defined over all s ∈ (−∞,0] and are denoted φ(s) ≥ 0 and ψ(s) ≥ 0, respectively. in [5], the delay kernel f(u) is assumed to be a step function from 0 up to 1 at u = τ ≥ 0, or (in the exponential case) of the form f(u) = αn+1un n! e−αu (2.11) which is just the density function for an erlang distribution with rate α and shape n + 1 (and thus, mean (n + 1)/α and coefficient of variation 1/ √ n + 1). the weak and strong kernels referenced in [5] are just the particular cases where the shape parameter is 1 (i.e., an exponential distribution with rate α) or 2 (erlang with rate α and shape 2), respectively. in [5], culshaw et al derive the specific models described above from this more general integrodifferential equation model (eqs. (2.10)), which we have summarized below (see [5] for a comparison of the dynamics of these three models). first, assuming f(u) = δ(u − τ) is a dirac-δ function at time τ ≥ 0 yields the delay differential equation (or ode if τ = 0) below, as written in [5], with the same initial conditions as eqs. (2.10) if τ > 0 or with initial conditions c(0) = c0 ≥ 0 and i(0) = i0 ≥ 0 if τ = 0. dc(t) dt = rcc(t) ( 1− c(t) + i(t) cm ) −kic(t)i(t) (2.12a) di(t) dt = k′ic(t− τ)i(t− τ)−µii(t). (2.12b) next, the authors in [5] assumed a ‘‘weak kernel” (i.e., exponentially distributed delay with rate α) and derived the following system of odes: dc(t) dt = rcc(t) ( 1− c(t) + i(t) cm ) −kic(t)i(t) (2.13a) dx(t) dt = αc(t)i(t)−αx(t) (2.13b) di(t) dt = k′i x(t)−µii(t). (2.13c) 2.5.1. ode model of cell-to-cell hiv spread with phase-type lags in cells becoming infectious. we can extend the above models as follows. first, substituting y (t) = ki α x(t) yields a more natural (in a procedure to derive new ode models via the glct 421 the context of the lct, and in terms of the units of x and y ) set of equations dc(t) dt = rcc(t) ( 1− c(t) + i(t) cm ) −kic(t)i(t) (2.14a) dy (t) dt = ki c(t)i(t)−αy (t) (2.14b) di(t) dt = k′i ki αy (t)−µii(t). (2.14c) from these equations, it is straightforward to derive a more general ode model using our glct-based procedure. applying the lct to eqs. (2.14) yields eqs. (2.15) below, which correspond to any choice of α > 0 and non-negative integer value n ≥ 0 for the delay kernel f in eq. (2.11) (i.e., any erlang distribution with shape n + 1 and rate α). dc(t) dt = rcc(t) ( 1− c(t) + i(t) cm ) −kic(t)i(t) (2.15a) dy1(t) dt = ki c(t)i(t)−αy1(t) (2.15b) dyi(t) dt = αyi−1(t)−αyi(t), i = 2, . . . ,n + 1 (2.15c) di(t) dt = k′i ki αyn+1(t)−µii(t). (2.15d) re-writing the above equations in the particular matrix form suggested by the glct (lemma 1.1) yields the more general set of equations below, which are the desired set of model equations for which the erlang distribution assumption (with parameters n + 1 and α) have been replaced by a phase-type distribution parameterized by the length k vector v and k ×k matrix m, where y = [y1, . . . ,yk]t. dc(t) dt = rcc(t) ( 1− c(t) + i(t) cm ) −kic(t)i(t) (2.16a) dy(t) dt = ki c(t)i(t) v + m t y(t) (2.16b) di(t) dt = − k′i ki 1t mt y(t)−µii(t). (2.16c) further generalizations, e.g., to accommodate time-varying v, m, or survival fraction f = k′i/ki are also possible, as detailed in [16]. note that it is also straightforward to further extend this model by using lemma 1.1 to replace the exponential dwell time assumption (with rate µi) in state i, eq. (2.16c) by a phase-type distribution assumption. 3. discussion mean field state transition models, written as ordinary differential equations (odes), are widely used throughout the sciences, but too often they include overly simplistic assumptions regarding time delays and the duration of time individual entities spend in specific states. in this paper, we introduce a relatively simple, novel procedure for implementing the generalized linear chain trick (glct) to expedite the derivation of ode models that can incorporate a broader range of underlying assumptions. we use this procedure to derive new models, based upon existing models taken from the peer reviewed literature, illustrating the relative ease of deriving new models within the context of the glct. these examples include a mix of biological applications and different mathematical frameworks (ddes, odes, and integro-differential equations), which reflect a mix of different implicit and explicit delay and dwell time assumptions. 422 p. hurtado and c. richards these straightforward generalizations illustrate how modelers can easily incorporate phase-type distributed delays and dwell times into ode models, and how underlying stochastic model assumptions are reflected in the structure of corresponding ode models. importantly, these more general model formulations can also be used in the computational and mathematical analysis of models that only assume erlang distributions (i.e., that could otherwise be derived using the standard linear chain trick). for an example, in [17] we illustrate the potential computational benefits of using a glct formulation of models with erlang dwell time assumptions when computing numerical solutions to such models. the benefits of using ode models derived using the glct extend beyond the benefits of writing the model in matrix form. for example, concepts from markov chain theory, including reward process theory, can be employed in the analysis (and interpretation of results) of ode models derived using this glct-based procedure [18]. these generalized models also lay the groundwork for data-driven model formulations that incorporate coxian, hyperexponential, hypoexponential (i.e, generalized erlang) and other phase-type distributions into these and similar mean field ode models. statistical tools such as butools [14, 13] allow modelers to fit phase-type distributions to data, thereby allowing modelers to build approximate empirical distributions into ode models using the glct. however, it is important to note that there are limitations to approximating some delay or dwell time assumptions with phase-type distributions. for example, delay distributions with compact support (e.g., a continuous uniform distribution) may not be well approximated by phase-type distributions. there are also models with more complex state transition or dwell time assumptions that may be more appopriate to derive using the more rigorous approaches detailed in [16]. in closing, we hope this glct-based procedure for the derivation of new ode models – that approximate or generalize existing models – proves to be helpful to modelers in their efforts to build better models, to check the consequences of certain simplifying assumptions, and to gain better intuition for how underlying assumptions are reflected in the structure of ode model equations. acknowledgements the authors thank the anonymous reviewers of this manuscript for their constructive feedback, and dr. deena schmidt for conversations and comments that improved this manuscript. the authors also thank the organizers and sponsors of the second international conference on applications of mathematics to nonlinear sciences (amns-2019), held june 27-30, 2019, in pokhara, nepal, and the editors of this thematic issue. funding this work was supported by a grant awarded to pjh by the sloan scholars mentoring network of the social science research council with funds provided by the alfred p. sloan foundation; 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studies in nonlinearity, westview press, july 2014. 36. n. terranova, m. germani, f. del bene, and p. magni, a predictive pharmacokinetic–pharmacodynamic model of tumor growth kinetics in xenograft mice after administration of anticancer agents given in combination, cancer chemoth. pharm. 72 (2013), no. 2, 471--482. 37. h.j. wearing, p. rohani, and m.j. keeling, appropriate models for the management of infectious diseases, plos med. 2 (2005), no. 7, 0621--0627. corresponding author, department of mathematics and statistics, university of nevada, reno; reno, nevada, usa e-mail address: phurtado@unr.edu department of mathematics and statistics, university of nevada, reno; reno, nevada, usa e-mail address: cj.richards@nevada.unr.edu 1. introduction 1.1. generalized linear chain trick 1.2. deriving odes from delay differential equations & distributed delay equations 2. results 2.1. glct-based procedure for deriving ode models 2.2. model 1: tumor growth inhibition (tgi) model 2.3. model 2: perscription opioid epidemic model 2.4. model 3: within-host model of the immune-pathogen interaction 2.5. model 4: cell-to-cell spread of hiv 3. discussion acknowledgements funding disclosure statement references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 4, december 2020, pp.355-372 https://doi.org/10.5206/mase/10847 controlling rabies epidemics in nepal with limited resources: optimal control theory approach buddhi pantha, hem raj joshi, and naveen k. vaidya abstract. in many developing countries, including nepal, rabies epidemics constitute a serious public health concern, partly because of limited resources for proper implementation of control measures. in this study, we develop an extended model by incorporating various controls into the transmission dynamics model with both dog and jackal vectors. we apply the optimal control theory on the developed model system to identify optimal control strategy for mitigating rabies burden in nepal with limited resources. among the potential control strategies, human vaccination, dog vaccination, dog culling, dog sterilization, and jackal vaccination, considered in this study, our results show that a combination of dog vaccination and dog culling is the most effective strategy to control rabies in nepal. our optimal control solutions provide strategies for optimal implementation of these controls to suppress rabies prevalence in humans, dogs and jackals of nepal using the minimum cost associated with controls. we found that given limited resources, implementing controls in a time-dependent manner with a higher level at the beginning of the outbreaks and reducing them during later part of the epidemics can provide maximum benefits. 1. introduction rabies, a viral zoonotic disease, remains an ongoing burden in many developing countries, including nepal. because of extremely high fatality rate (almost 100%) in rabid humans or animals having symptoms such as violent movements, uncontrolled excitement, fear of water (hydrophobia), an inability to move parts of the body, confusion, and loss of consciousness [2, 6, 27, 31], this disease poses extreme threats of public health concerns. while multiple control strategies are available, including successful vaccine, limited resources and lack of proper allocation of resources often make developing countries fail to control rabies epidemics; it is also called a poor man’s disease because most of deaths (≥ 95%) occur in asian and african countries [33]. therefore, well-designed planning is necessary before implementing these control strategies to achieve optimal outcomes using limited resources. mathematical modeling can provide an important means for identifying ideal planning strategies. rabies virus is mainly transmitted through infected animal bites [8]. among the potential animals, dogs are primarily considered as vector for the transmission of rabies. there are many existing models that describe transmission dynamics of the rabies with dogs as primary vectors [1, 2, 3, 7, 8, 9, 11, 12, 17, 18, 20, 22, 27, 30, 32]. however, in the context of some countries like nepal, while dogs remain primary vectors in urban epidemiological cycle [26], jackals as secondary vectors in sylvatic epidemiological cycle received by the editors 30 june 2020; accepted 1 december 2020; published online 10 december 2020. 2010 mathematics subject classification. primary 35k55; 92d30; secondary 46e25, 20c20. key words and phrases. dog and jackal vectors; mathematical model; optimal control theory; rabies in nepal; resource limitation. authors would like to thank association of nepalese mathematicians in america (anma) for organizing a workshop on collaborative research in mathematical sciences (may 2527, 2018), during which this work was initiated. vaidya’s work was supported by nsf grants dms-1951793, dms-1616299, dms-1836647, and deb-2030479 from national science foundation of usa, and ugp award from san diego state university. 355 356 b. pantha, h. joshi, and n. vaidya also play an important role in the persistent ongoing rabies epidemics [21]. such persistent low level of rabies infection in jackals were also found in zimbabwe [24]. in a recent study, we developed a model that couple both dog and jackal along with human population to describe transmission dynamics of rabies in nepal, and identified that consideration of both vectors is essential for successful mitigation of rabies in nepal [21]. in particular, our model predicted that even though intraspecies transmission is prevented among each animal species (dogs and jackals), the rabies can still persist due to interspecies transmission [21]. nepal has pledged to end rabies by 2030, but dog-bite cases are rising and as many as 26,312 people were administered with post-exposure anti-rabies vaccines at government health facilities in 2018 [14]. despite continuous effort to control rabies, ongoing epidemic can partly be attributable to limited resources for implementing proper control and existence of secondary vectors. it is thus important to identify optimal strategy to implement control programs regarding underlying situation of two different vectors and limited resources. for such purposes, optimal control theory has been proved to be useful tool as in many previous epidemic controls, including rabies epidemics [5, 7, 13, 15, 16, 19, 20, 23, 25]. in this study, we introduce effects of various controls into the transmission dynamics model incorporating both dog and jackal populations [21]. in particular, we focus on human vaccination, dog vaccination, dog culling, dog sterilization, and jackal vaccination through bait as well as combinations of them. using the developed dynamical system model, we further formulate an optimal control problem to take resource limitation into account. implementing optimal control theory and related numerical method, we compute the optimal control strategy for successful control of rabies epidemic in nepal. 2. mathematical model with controls based on our previous model of rabies transmission dynamics in nepal [21], we develop an extended model by incorporating various control measures to describe the dynamics of rabies transmission in multiple groups of species (jackals, dogs, and humans). specifically, our model is a coupled system of differential equations that describe the rate of change of subpopulations of jackals (j), dogs (d), and humans (h) under various control programs implemented in the community. we use subscripts j, d and h in the variables and parameters to represent them corresponding to jackals, dogs, and humans, respectively. the total population of each species is divided into four subpopulations: susceptible (sj,sd,sh), exposed (ej,ed,eh), vaccinated (vj,vd,vh), and infected (ij,id,ih). the population in each group is recruited with rate λ into susceptible class and die with natural death rate µ. susceptible population get infected and enter into exposed class with rate β. also, the exposed humans or animals transit into infected class with rate γ and the infected populations die due to rabies with rate δ. as observed in the context of nepal, note that humans are infected through dogbites only, while both dogs and jackals are infected through intra-species and inter-species transmissions. there are various potential prevention and control strategies that can be applied to break the jackaldog-human transmission sequence for rabies. in this study, we consider the five most frequently used strategies: human vaccination (post-exposure), dog vaccination (both pre and post-exposure), dog culling, dog sterilization, and jackal vaccination through bait. we use u1 to denote the rate at which the exposed humans get the rabies vaccine. the vaccination program for the rabies in dogs includes both preand post-exposure vaccination. as per guidelines in [4], if vaccinated dogs are exposed, revaccination should be administered immediately. therefore, we apply the dog vaccination to both susceptible controlling rabies in nepal 357 and exposed dogs at the same rate of u2, and we assume that the immunity is not lost for the period of dynamics considered in the study. the third rabies control strategy is dog culling, which we denote using the rate u3 implemented to cull dogs from all classes, regardless of their infection status. the bait vaccination for jackals is applied using foods containing the rabies vaccine that is spread in different locations so that the jackals consume the foods, reducing the rabies contraction [6, 28]. we denote the rate of jackal vaccination for both susceptible and exposed jackals by u4. the dog sterilization strategy is used to control reproductive rate of dogs, eventually reducing the recruitment rate. we denote the net effectiveness of dog sterilization by u5 so that the dog recruitment rate changes to (1 −u5)λd. as described above, the transmission dynamics of rabies under these five control strategies can be represented using the following system of differential equations. s′j = λj −βjjsjij −βdjsjid − (µj + u4)sj e′j = βjjsjij + βdjsjid − (γj + µj + u4)ej v ′j = u4(sj + ej) −µjvj i′j = γjej − (µj + δj)ij s′d = (1 −u5)λd −βjdsdij −βddsdid − (µd + u2 + u3)sd e′d = βjdsdij + βddsdid − (γd + µd + u2 + u3)ed v ′d = u2(ed + sd) − (µd + u3)vd (2.1) i′d = γded − (µd + δd + u3)id s′h = λh −βdhidsh −µhsh e′h = βdhidsh − (u1 + µh + γh)eh v ′h = u1eh −µhvh i′h = γheh − (µh + δh)ih the model parameters related to the context of nepal [21] are given in table 1. 3. impact of controls on rabies epidemics in nepal to identify the most impactful controls, we first evaluate the effects on constant level of control on preventing rabies epidemic and/or reducing rabies prevalence. in our previous study [21], we analyzed the impact of implementing the single control at a time, and found that the use of only one control is not able to mitigate the disease unless the level of control is significantly high. for example, rabies prevalence in dog and jackal can be reduced to low level only if the annual culling for the dog is more than 40% effective for 10 years or the effectiveness of dog vaccination is more than 60%. the requirement of unusually high level of control for a longer period of time implies that the multiple control strategies need to be applied simultaneously for the successful control of rabies. on the other hand, it is unlikely for developing countries like nepal to implement many control programs at the same time due to resource limitation. therefore, we mainly focus on combinations of two control programs. here, we consider combinations of two different controls and evaluate which control combinations are more effective in reducing the basic reproduction number, r0, as well as reducing the rabies prevalence among dog and jackal populations. the most effective combination of two control measures are then considered to identify the optimal planning for implementing them in resource limited setting. 358 b. pantha, h. joshi, and n. vaidya table 1. the model variables and parameters. model variables and initial values variable description initial values sj susceptible jackals 73125 ej exposed jackals 368 vj vaccinated jackals 0 ij infected jackals 73 sd susceptible dogs 15.898 × 105 ed exposed dogs 10 4 vd vaccinated dogs 4 × 105 id infected dogs 200 sh susceptible 25.265 × 106 eh exposed 15534 ih infected 1000 vh vaccinated 14000 model parameters params. description value µj jackal mortality rate 0.125 µd dog mortality rate 0.2 µh human mortality rate 0.0142 λj jackal recruitment rate µj ×nj (0) λd dog recruitment rate µd ×nd(0) λh human recruitment rate µh ×nh (0) δj jackal rabies related mortality rate 36.5 δd dog rabies related mortality rate 36.5 δh human rabies related mortality rate 36.5 γj jackal rate of moving from exposed to infected 6.64 γd dog rate of moving from exposed to infected 2 γh human rate of moving from exposed to infected 2 βjj transmission rate from jackal to jackal 3.79 × 10−5 βdj transmission rate from dog to jackal 1.90 × 10−5 βjd transmission rate from jackal to dog 1.52 × 10−5 βdd transmission rate from dog to dog 2.74 × 10−5 βdh transmission rate from dog to human 1.71 × 10−6 ud dog vaccination rate(susceptible and exposed) 0.03 uh human vaccination rate (pep) 2.05 3.1. impact on the basic reproduction number. the basic reproduction number, r0, known as the expected number of secondary cases produced by a single (typical) infection in an entirely susceptible population [10], can be used to determine whether the outbreak occurs (r0 > 1) or infection dies out (r0 < 1) [10]. applying the next generation matrix method [10] to our model, we consider the subsystem containing all of the equations except the equations for sh, sd and sj. this subsystem is then linearized about the disease free equilibrium (dfe), given by( λj µj + u4 , 0, u4λj µj(µj + u2) , 0, (1 −u5)λd (µd + u2 + u3) , 0, u2(1 −u5)λd (µd + u3)(µd + u2 + u3) , 0, λh µh , 0, 0, 0 ) . from the resulting equations, we obtain a matrix f containing infection terms and a matrix v containing transfer terms, as follows. f =   f11 f12 0f21 f22 0 0 f32 0   , and v =   v11 0 00 v22 0 0 0 v33   , controlling rabies in nepal 359 where f11 =   0 0 βjj λj µj +u4 0 0 0 0 0 0   , f12 =   0 0 βdj λj µj +u4 0 0 0 0 0 0   , f21 =   0 0 (1−u5)βjdλd (µd+u2+u3) 0 0 0 0 0 0   , f22 =   0 0 (1−u5)βddλd (µd+u2+u3) 0 0 0 0 0 0   , f32 =   0 0 βdh λh µh 0 0 0 0 0 0   , v11 =   γj + µj + u4 0 0−u4 µj 0 −γj 0 µj + δj   , v22 =   γd + µd + u2 + u3 0 0−u2 µd + u3 0 −γd 0 µd + δd + u3   , and v33 =   u1 + µh + γh 0 0−u1 µh 0 −γh 0 µh + δh   . this implies fv−1 =   f11v−111 f12v−122 0f21v−111 f22v−122 0 0 f32v−122 0   . the basic reproduction number is then given by the spectral radius of the matrix fv−1. therefore, r0 = ρ(fv−1) = 1 2 ( rj0 + r d 0 + √ (rj0 −rd0 )2 + 4ce ) , where rj0 = βjjγjλj (µj + u4)(δj + µj)(γj + µj + u4) , rd0 = (1 −u5)βddγdλd (δd + µd + u3)(µd + u2 + u3)(γd + µd + u2 + u3) , c = βdjγdλj (µj + u4)(δd + µd + u3)(γd + µd + u2 + u3) , and (3.1) e = (1 −u5)βjdγjλd (δj + µj)(γj + µj + u4)(µd + u2 + u3) . we now use the formula for r0 derived above and parameters given in table 1 to compute the value of basic reproduction number for various control strategies with two controls taken at a time (figure 1). note that the human vaccination at its base level is always included in all strategies as it cannot be avoided in practice. we also identified the control levels for which r0 is less than 1, leading to eradication of rabies in nepal. while each combination strategy has certain levels that can bring r0 below 1, the level required is quite high for some strategies compared to others. as presented in figure 1 we observe that jackal vaccination is one of the least effective strategies on lowering the basic reproduction number; the increase in the coverage of jackal bait vaccination has negligible impact on the basic reproduction number (figure 1a, 1c, 1e). similarly, the dog sterilization does not show significant effectiveness either to reduce r0 (figure 1a, 1b). the dog culling and dog vaccination on the other hand are highly impactful on bringing the value of r0 below 1 (figure 1d). in particular, a combined 360 b. pantha, h. joshi, and n. vaidya figure 1. heatmap of r0 for various combination of pair of controls. the dotted curve represents r0 = 1. strategy, including the dog vaccination and dog culling control measures, is the best approach to reduce r0, thereby preventing or eradicating the rabies in nepal (figure 1d). 3.2. impact on long-term rabies prevalence. in this section, we present the effects of combined strategies with two control measures at a time on the long term prevalence of rabies among dogs and jackals. the parameter values and the initial values are used as given in table 1. the runge-kutta method of fourth order is applied for the model simulations. since there is no transmission of rabies from humans to animals, the human vaccination does not have any effect on the rabies prevalence in dog and jackal populations. therefore, we keep the human vaccination at a constant level estimated using the data from nepal [21] and focus on other four control measures. in the following simulations (figure 2-7), we compute the prevalence of rabies in dog and jackal populations with the application of combined strategies for the period of 10 years. in our simulations, rate of each control strategy is estimated from the target percentage of the population that are to be covered by that strategy. for example, with the vaccination rate of u2, the target dog population, xd, can be approximated using the solution of dxd dt = −u2xd, i.e, xd(t) = xd(0)e−u2t. then for a program that aims to vaccinate η2% of the dog population in a period of t years, the vaccination rate u2 needs to be in such a way that xd(t) = (1 −η2/100)xd(0) = xd(0)e−u2t, which gives u2 = − ln(1−η2/100) t . similarly, the dog culling rate can be written as u3 = − ln(1−η3/100) t , where η3 is the % coverage for dog culling in a period of t years. first, we consider combined strategy with dog vaccination and dog culling control measures and compute the prevalence of rabies in dog and jackal populations. as expected, our model prediction shows that applying one control strategy at a lower level requires another control strategy at higher level to achieve sufficient decrease in the long-term prevalence of rabies among dogs and jackals. for example, annual dog vaccination coverage at a level of 5% for a period of 10 years requires more than 5% of controlling rabies in nepal 361 figure 2. the rabies prevalence in (a) dog, and (b) jackal populations under various levels of annual coverage of dog vaccination and dog culling. annual dog culling for that period to lower the rabies prevalence among dogs below 0.006%, and similar dog vaccination and culling coverage is needed to achieve the same low level of rabies prevalence in jackal. next, we consider dog vaccination and jackal bait vaccination together to compute the prevalence of rabies in dog and jackal populations. as shown in figure 3, we observe that jackal vaccination does not figure 3. the rabies prevalence in (a) dog and (b) jackal populations under various level of annual coverage by dog vaccination and jackal vaccination. have significant impact in lowering the rabies prevalence in the dog population, i.e., the rabies prevalence among dogs remains almost the same for any level of coverage by jackal vaccination. however, the rabies prevalence among jackals, is impacted by this combination of dog vaccination and jackal vaccination. for example, a combined program with 5% coverage of each of dog vaccination and jackal bait vaccination results in the rabies prevalence in jackal of about 0.033% while an increase in the jackal vaccination to 15% keeping the same 5% dog vaccination coverage decreases the rabies prevalence in jackal to 0.018%. the third strategy we consider is the combination of the dog vaccination and the dog sterilization. note that sterilization does not control the rabies incidence directly, it rather controls the disease in a 362 b. pantha, h. joshi, and n. vaidya figure 4. the rabies prevalence among (a) dog and (b) jackal populations under various level of annual coverage by dog vaccination and dog sterilization. long run by decreasing the reproduction of the dog and eventually reducing the susceptible populations of dog. therefore, for fare comparison as in our previous work [21], we measure the sterilization strength in a 5 year time frame as opposed to other strategies which are measured in 1 year time frame. in this strategy, with dog sterilization strength of 15% coverage over 5 years, at least 10% of annual coverage of dog vaccination is required for 10 years to keep the prevalence in dog population below 0.003% (figure 4), while at least 11% dog vaccination coverage is required for 10 years to keep the rabies prevalence in jackal population below 0.003%. we observe that dog sterilization does not have significant impact in lowering the rabies prevalence in dog and jackal populations. the fourth strategy is the combination of the dog sterilization and the jackal vaccination. this stratfigure 5. the rabies prevalence among (a) dog and (b) jackal populations under various level of annual coverage by jackal vaccination and dog sterilization. egy does not have significant impact on the rabies prevalence in dog population (figure 5). however, it has considerable impact on the rabies prevalence in jackal population. for instance, dog sterilization level of 10% and jackal vaccination coverage of 15% for 10 years can bring the rabies prevalence below 0.050% in jackal population. for a higher dog sterilization coverage (for example, 17% for 10 years), the same level of rabies prevalence can be achieved with lower jackal vaccination level of 4%. controlling rabies in nepal 363 the fifth strategy considered in this study is the combination of dog sterilization and culling. in this figure 6. the rabies prevalence among (a) dog and (b) jackal populations under various level of annual coverage by jackal vaccination and dog culling. combination of strategies, the dog culling has significant impact in reducing rabies prevalence in both dog and jackal populations (figure 6a), but the dog sterilization has only a little impact in reducing the rabies prevalence in both populations (figure 6b). for example, at 5% of dog culling for 10 years, a change of the level of dog sterilization from 5 to 15% causes the rabies prevalence in dog population to change from 0.0351% to 0.0122%. in this change, the prevalence in jackal populations also changes from 0.0289% to 0.0095%. the last strategy considered in this study is the combination of the jackal vaccination and the dog culling. as in the previous combination of strategies, dog culling has significant impact on reducing figure 7. the rabies prevalence among (a) dog and (b) jackal populations under various level of annual coverage by jackal vaccination and dog culling. rabies prevalence in both populations but the jackal vaccination plays a noticeable role in reducing rabies prevalence in jackal population only. for example, at 5% of dog culling for 10 years, the rabies prevalence in jackal population changes from 0.45% to 0.011% on changing the level of jackal vaccination from 0 to 25%, but the prevalence in dog populations does not change significantly (0.055% for 0% 364 b. pantha, h. joshi, and n. vaidya coverage and 0.047% for 25% coverage of jackal vaccination). for comparison purpose, we consider the rabies prevalence among dogs and jackal populations under each combined strategy with 10% coverage level of each of two strategies included in the combination. for this level of (10%, 10%) coverage, our results show that the rabies prevalence for dog population remains 0.0001%, 0.0040%, 0.0036%, 0.1362%, 0.0023% and 0.0056% for combined strategies with dog vaccination and dog culling, jackal vaccination and dog vaccination, dog vaccination and dog sterilization, dog sterilization and jackal vaccination, dog culling and dog sterilization, and jackal vaccination and dog culling, respectively. in the jackal population the corresponding prevalence is 0.0001%, 0.0026%, 0.0041%, 0.0652%, 0.0027% and 0.0025%, respectively. therefore, dog vaccination and dog culling are the most effective measures to prevent and control rabies epidemics in nepal. in the following section, we focus on the optimal strategy to implement the combination of dog vaccination and dog sterilization under resource limited settings. 4. optimal control of rabies in nepal in this section, we use our model to formulate optimal control problem, which allows us to identify optimal time-dependent strategies under limited resources to achieve maximum benefit from the control strategy implementation. while we formulate general optimal control problem including all possible control strategies, we particularly emphasize on two most effective strategies, namely the dog vaccination and the dog culling, identified in section 3. 4.1. formulation of optimal control problem. we consider controls as time-dependent functions, i.e, ui = ui(t) for i = 1, 2, ..., 5. to incorporate resource limitation while controlling the rabies burden, we set a goal of minimizing the number of exposed and infected humans and animals as well as minimizing the anticipated cost for control implementation for a fixed period of time, say tf . to achieve this goal, we formulate an objective functional as follows. j(u1,u2,u3,u4,u5) = min (u1,u2,u3,u4,u5)∈u ∫ tf 0 [a1(eh + ih) + a2(ed + id) + a3(ej + ij) + b1u1eh + b2u2(sd + ed) + b3u3nd + b4u4nj + b5u5(nd − id) +c1u 2 1 + c2u 2 2 + c3u 2 3 + c4u 2 4 + c5u 2 5 ] dt, (4.1) where ai’s, bi’s, and ci’s are the positive constants, associated with weights corresponding to disease outcome and costs. here, we assume that the admissible control set u is given by u ={(u1,u2,u3,u4,u5) ∈ r5 : 0 ≤ ui(t) ≤ bi for i = 1 . . . 5 and ui are lebesgue measurable}, where bi’s are positive constants related to the availability of resources. we consider the formulated optimal control under dynamical system given by eqs. (2.1). for the control problem formulated above, we can apply a result from lukes [29] to prove the existence and uniqueness of solutions for the state system (2.1) with the given controls. the existence and uniqueness results for our optimal control problem can be summarized as in theorem 4.1. theorem 4.1. given controls u = (u1,u2,u3,u4,u5) ∈ u, there exist non-negative bounded solutions (sj,ej,vj,ij,sd,ed,vd,id,sh,eh,vh,ih) to the state system (2.1) in the finite interval [0,t] with given initial conditions. controlling rabies in nepal 365 the structure of system (2.1) gives the non-negativity and uniform boundedness of the state solutions. as stated in theorem 4.2, we can also assert the existence of the optimal controls based on the results from [16]. theorem 4.2. there exists an optimal control tuple u∗ = (u∗1,u ∗ 2,u ∗ 3,u ∗ 4,u ∗ 5) ∈ u with corresponding states (s∗j,e ∗ j,v ∗ j ,i ∗ j,s ∗ d,e ∗ d,v ∗ d,i ∗ d,s ∗ h,e ∗ h,v ∗ h,i ∗ h) that minimizes the objective functional j(u1,u2,u3,u4,u5). by using pontryagin’s maximum principle as stated in [16, 23], we are able to derive necessary conditions for our optimal control and corresponding states. the hamiltonian of the system is: h = a1(eh + ih) + a2(ed + id) + a3(ej + ij) + b1u1eh + b2u2(sd + ed) + b3u3nd + b4u4nj + b5u5(sd + ed + vd) + c1u 2 1 + c2u 2 2 + c3u 2 3 + c4u 2 4 + c5u 2 5 + λ1(λj −βjjsjij −βdjsjid − (µj + u4)sj) + λ2(βjjsjij + βdjsjid − (γj + µj + u4)ej) + λ3(u4(sj + ej) −µjvj) + λ4(γjej − (µj + δj)ij) + λ5((1 −u5)λd −βjdsdij −βddsdid − (µd + u2 + u3)sd) + λ6(βjdsdij + βddsdid − (γd + µd + u2 + u3)ed) + λ7(u2(ed + sd) − (µd + u3)vd) + λ8(γded − (µd + δd + u3)id) + λ9(λh −βdhidsh −µhsh) + λ10(βdhidsh − (u1 + µh + γh)eh) + λ11(u1eh −µhvh) + λ12(γheh − (µh + δh)ih). for given optimal controls u1, ...,u5, there exist λi, i = 1, . . . , 12, with derivative λ ′ i(t) given by λ′i(t) = − ∂h ∂ (ith state variable) . therefore, we obtain λ′1(t) = −(b4u4 −λ1(βjjij + βdjid + (µj + u4)) + λ2(βjjij + βdjid) + λ3u4) , λ′2(t) = −(a3 + b4u4 −λ2(γj + µj + u4) + λ3u4 + λ4γj) , λ′3(t) = −(b4u4 −λ3µj) , λ′4(t) = −(a3 + b4u4 −λ1βjjsj + λ2βjjsj −λ4(µj + δj) −λ5βjdsd + λ6βjdsd) , λ′5(t) = −(b2u2 + b3u3 + b5u5 −λ5(βjdij + βddid + µd + u2 + u3) , +λ6(βjdij + βddid) + λ7u2) , λ′6(t) = −(a2 + b3u3 + b5u5 −λ6(γd + µd + u2 + u3) + λ7u2 + λ8γd) , λ′7(t) = −(b3u3 + b5u5 −λ7(µd + u3), ) λ′8(t) = −(a2 + b3u3 −λ1βdjsj + λ2βdjsj −λ5βddsd , (4.2) +λ6βddsd −λ8(µd + δd + u3) −λ9βdhsh + λ10βdhsh) , λ′9(t) = −(−λ9(βdhid + µh) + λ10βdhid) , λ′10(t) = −(a1 + b1u1 −λ10(u1 + µh + γh) + λ11u1 + λ12γh) , λ′11(t) = −(−λ11µh) , λ′12(t) = −(a1 −λ12(µh + δh)) , and the transversality conditions are λi(tf ) = 0, i = 1, . . . 12. 366 b. pantha, h. joshi, and n. vaidya the optimal control solutions u∗i , i = 1, 2, ..., 5, can then be obtained by setting the derivative of hamiltonian system with respect to each control to zero, i.e., ∂h/∂ui = 0, i = 1, 2, ..., 5, where ∂h ∂u1 = b1eh + 2c1u1 −λ10eh + λ11eh, ∂h ∂u2 = b2(sd + ed) + 2c2u2 −λ5sd −λ6ed + λ7(ed + sd), ∂h ∂u3 = b3(sd + ed + vd + id) + 2c3u3 −λ5sd −λ6ed −λ7vd −λ8id, ∂h ∂u4 = b4(sj + ej + vj + ij) + 2c4u4 −λ1sj −λ2ej + λ3(sj + ej), ∂h ∂u5 = b5(sd + ed + vd) + 2c5u5 −λ5λd, we obtain the optimal control solution as follows. u∗1 = min [ b1, max [ a1,− 1 2c1 (b1 −λ10 + λ11) eh ]] , u∗2 = min [ b2, max [ a2,− 1 2c2 (b2(sd + ed) −λ5sd −λ6ed + λ7(ed + sd)) ]] , u∗3 = min [ b3, max [ a3,− 1 2c3 (b3(sd + ed + vd + id) −λ5sd −λ6ed −λ7vd −λ8id) ]] , u∗4 = min [ b4, max [ a4,− 1 2c4 (b4(sj + ej + vj + ij) −λ1sj −λ2ej + λ3(sj + ej)) ]] , u∗5 = min [ b5, max [ a5,− 1 2c5 (b5(sd + ed + vd) −λ5λd) ]] . 4.2. estimation of weight parameters and bounds for controls. it is important to determine the reasonable weight parameters ai,bi,ci introduced into the objective functional, since the outcome of the minimization procedure may highly be impacted by the choice of these weights. here, we follow a similar technique used in mallela et al. [19] to make proper choice of these weight constants. to estimate the reasonable proportion of weights, we take a1 = 1, and compute other weight constants in such a way that the term corresponding to each weight is approximately the same as the term corresponding to a1. for example, we estimate a2 using∫ tf 0 a1(eh + ih)dt = ∫ tf 0 a2(ed + id)dt, and obtain a2 = 1.0702. the similar technique allows us to obtain a3 = 95.8453, b1 = 0.5144, b2 = 0.0614, and b3 = 0.1259. to estimate ci’s we use the average value, u av i , of minimum and maximum values of controls. the minimum value for both dog vaccination and culling rates are assumed to be zero for no vaccination and culling. for the upper bound, we assume that the available resource for dog vaccination corresponds to the maximum capacity of covering 40% of dogs in nepal in a year. using dxd dt = −u2xd, where xd is dog population remained to be vaccinated, we can compute that the maximum resource (i.e., 40% coverage) is equivalent to the dog vaccination rate of u2 = 0.5. thus, we take the bounds for dog vaccination rate as 0 ≤ u2 ≤ 0.5. next, as capturing the dogs, culling them and disposing them need more resources, culling process needs more manpower than vaccination process, and thus lower coverage for culling can be achieved with limited resources. let us assume that the maximum available controlling rabies in nepal 367 resources for dog culling can cover 20% of dogs per year. with this assumption, the bounds for the dog culling rate is 0 ≤ u3 ≤ 0.22. then we use a1 ∫ tf 0 (eh + ih)dt = ci ∫ tf 0 (uavi ) 2dt, i = 1, 2, 3. to estimate c′is. the computation from our model solution provides c1 = 4.7562×10 4, c2 = 1.5991× 106, c3 = 8.2596 × 105. since we intend to obtain optimal time-varying strategy with combined two most effective controls, u2(t) and u3(t), with underlying constant human post-infection vaccination (u1), the remaining two controls u4 and u5 are taken to be 0. as a result, the corresponding weight parameters vanish, i.e., b4 = b5 = c4 = c5 = 0. 4.3. method for numerical computation. in this section, we briefly summarize the computational method used to obtain the optimal control solutions. our technique is similar to the iterative algorithm introduced by lenhart and workman [16]. in particular, we use a backward-forward sweep iterative method with a fourth order runge-kutta scheme. starting with initial guesses for the controls, the state equations (2.1) are solved forward in time. then, the resulting state values are used to solve the adjoint equations (4.2) backward in time. the controls are then updated. this iteration process is repeated until the convergence is achieved. the convergence of iteration is defined as a condition, at which the value of variables in two successive iterations are negligibly close (i.e., their difference is smaller than a desired small number). the algorithm implemented in our study can be summarized as follows: step 1: input initial guess for controls over the interval [0, tf ]. step 2: using the initial values of state variables and the values of the controls, solve the system (2.1) forward in time, i.e., from t = 0 to t = tf . step 3: using the transversality conditions λi(tf ) = 0, i = 1, 2, ..., 12 as well as the values of the state variables from step 2 and the values of controls, solve the adjoint system (4.2) backward in time, i.e., from t = tf to t = 0. step 4: update the controls with the characterization u∗i , i = 1, 2, ...5 using the values of state and adjoint variables. step 5: if the convergence is achieved, output the current values of solution, otherwise return to step 2. 4.4. solutions for the optimal control of rabies in nepal. as discussed in section (3), as well as in our previous work [21], the dog vaccination and the dog culling are two most effective intervention strategies to control the rabies epidemics in nepal. here, we present the numerical solutions of how these strategies can be implemented optimally given limited resources in the context of nepal. first, we consider programs with only one of these two strategies and then consider a program with these two strategies combined. as mentioned earlier, note that there is always underlying post-infection human vaccination as this can not be avoided in practice. 4.4.1. control program with dog vaccination only. we used the bounds for the vaccination rate 0 ≤ u2 ≤ 0.5 and the weight parameters from section 4.2. all other model parameters are taken from table 1. our model simulations at the boundary levels of the dog vaccination control (figure 8) show that without dog vaccination (u2 = 0) the prevalence of rabies in dog population increases continuously and reaches at a high level of 0.2543% as early as the ninth year, and in jackal population, the rabies prevalence increases and reaches 0.2162% in the eighth year. in human population, the rabies case increases continuously and reaches about 169,000 as soon as eighth year. on the other hand, with the 368 b. pantha, h. joshi, and n. vaidya figure 8. the rabies prevalence among (a) dog population, (b) jackal population, (c) number of human rabies cases and (d) optimal dog vaccination profile, u2, in the control program with dog vaccination only. highest level of dog vaccination (u2 = 0.5), i.e., use of maximum resources, the prevalence of rabies in both dog and jackal populations reach approximately zero as early as fourth year and the number of rabies in humans is about one. our optimal control solution indicates that the optimal strategy should be with the dog vaccination control profile u2(t) starting at rate 0.18 and tapering down to zero as shown in figure 8d. with this profile, the value of objective functional can be brought to 36% less than without control and 64% less than with highest control (i.e., the value of j is 1.5310×106 without control, 6.2895 × 106 with highest control, and 2.2690 × 106 with optimal control), thereby utilizing the resources optimally. with this optimal vaccination strategy, the rabies prevalence in dog and jackal populations decreases slowly and maintains at about 0.0026%, as opposed to 0.2458% and 0.2027%, respectively, without vaccination. in this case, there are only about 2,510 human rabies infections at the end of tenth year. 4.4.2. control program with dog culling only. the bounds for the dog culling rate, 0 ≤ u3 ≤ 0.22 and the weight parameters are taken from section 4.2. all other model parameters are taken from table 1. from the model simulations (figure 9), we observe that without the dog culling program (u3 = 0), the figure 9. the rabies prevalence among (a) dog population, (b) jackal population (c) number of human rabies cases and (d) optimal dog culling profile u3 in the control program with dog culling only. prevalence of rabies in dog population increases immediately after the outbreak begins. the prevalence then reaches as high as 0.2543% in the ninth year. similarly, the rabies prevalence in jackal population peaks reaching to 0.2162% in eighth year and the number of human rabies reaches about 169,000. if the highest level of resource for culling is implemented (u3 = 0.22), the rabies prevalence in dog and jackal populations reaches negligible level (close to zero) at about 6 years. also, the number of human rabies cases decreases rapidly and reaches about 6 at the end of tenth year. controlling rabies in nepal 369 in both cases, without dog culling (u3 = 0) and program with highest level of culling (u3 = 0.5), the objective functional remains higher at j = 3.5310 × 106 and j = 2.8437 × 106, respectively, asserting that neither of them is an optimal strategy. our optimal control solution implies that with the dog culling strategy, u3(t), starting at the rate 0.184 and tapering down to zero, as shown in figure 9d, can bring down the values of objective functional to 2.3017 × 106, which is 35% lower than without dog culling and 20% lower than the value in the highest culling rate. under this optimal culling strategy, the peak rabies prevalence in dog and jackal populations are significantly lower than no culling strategy, while utilizing minimum resources. with this strategy, at the end of the study period of the tenth year, the rabies prevalence among dog and jackal populations remain 0.004% and 0.003%, respectively and the number of human rabies infections are about 2,608. 4.4.3. control program with dog vaccination and dog culling combined. we now consider a combination of dog vaccination and dog culling strategies, and identify the optimal way of implementing them under resource limitation. as above, we take the range of u2 and u3 as 0 ≤ u2 ≤ 0.5, 0 ≤ u3 ≤ 0.22, respectively. as presented in figure 10, under the combined program with the highest level of both figure 10. the rabies prevalence among (a) dog population, (b) jackal population, (c) number of human rabies cases (d) optimal dog vaccination profile, u2, and (e) optimal dog culling profile, u3, in the control program with dog vaccination and dog culling combined. dog vaccination and dog culling (u2 = 0.5,u3 = 0.22), the prevalence of rabies in both dog and jackal populations as well as the number of human rabies cases approach to zero as early as in 3 years, while the long term prevalence remains 0.2543% and 0.2162%, respectively, in the absence of the program (u2 = 0,u3 = 0). in this combined approach, the optimal benefit can be achieved by implementing the dog vaccination and dog culling at the level of 0.11 and 0.09, respectively, at the beginning of the outbreak, and tapering down both to zero as shown in figures 10(d,e). in this combined optimal strategy, the value of objective functional is j = 2.2532 × 106, which is about 37% lower than without control program (j = 3.5310×106) and about 67% lower than the highest level of both dog vaccination 370 b. pantha, h. joshi, and n. vaidya and dog culling (j = 6.9613 × 106). with this strategy, the rabies prevalence stays approximately at the level of 0.00001% in both dog and jackal populations with only about 94 human rabies cases at the end of tenth year. as the more resources become available, we can utilize the weight constants of our optimal control model to represent the high resource scenario by assigning lower weight constants corresponding to the cost for controls (i.e., smaller bi’s and ci’s). for example, for 10-fold and 1000-fold higher resource availability, we take 10-fold and 1000-fold lower values of b2,c2,b3, and c3 than in the base case. for a higher resource availability (or lower b2,c2,b3,c3), the optimal control solution results in the optimal dog vaccination and dog culling strategies with a level higher than a case of limited resource (allowing the use of more resources) and tapering down (figure 11). in this case, the value of the objective functional also comes out to be smaller (j = 2.2532×106 for the base case, j = 2.1062×106 for 10-fold higher resource, and j = 2.0724×106 for 1000-fold higher resource). for a higher resource availability, the number of human rabies cases as well as the prevalence of rabies in dog and jackal populations can be maintained at extremely low level with faster pace, particularly in the case of 1000-fold higher resource, compared to the case when resources are more limited (figure 11). figure 11. the rabies prevalence among (a) dog population, (b) jackal population, (c) number of human rabies cases (d) optimal dog vaccination profile, u2, and (e) optimal dog culling profile, u3, for 10-fold higher (red dashed curve) and 1000-fold higher(black dotted curve) resource availability (i.e., 10 times and 1000 times smaller values of b2,c2,b3,c3 than in the base case). 5. conclusion since the rabies is mostly problematic in developing counties of asia and africa, the control of rabies epidemics poses challenges due to limited resources available in these countries. the proper evaluation of control strategies and identifying optimal way of implementing such strategies are critical for controlling rabies in nepal 371 reducing rabies burden. in this study, we extended our basic rabies transmission dynamics model by adding the effects of commonly practiced control strategies, such as human post exposure vaccination, dog vaccination, dog culling, dog sterilization, and jackal bait vaccination. furthermore, we formulated the optimal control model, which was then used to obtain the optimal time-varying strategy of implementing controls to mitigate rabies in nepal under limited resources. our model predicts that the dog vaccination and the dog culling are the most effective two control strategies to bring the basic reproduction number to low value, and also to reduce the human rabies cases and prevalence of rabies among dogs and jackals in nepal. the optimal control formulation allowed us to identify time-dependent implementation of these two control strategies to achieve maximum benefit under limited resources. in general, applying higher level of controls at the beginning of outbreak and reducing during later part of the epidemic provide a maximum benefit, in both programs with single control strategy and two strategies combined. as revealed in our optimal control results, availability of more resources allows us to apply higher level of controls for longer period, resulting in lower level of rabies prevalence. we acknowledge that parameters used in our model are estimated from the literature or from the limited data set. therefore, there may be some discrepancy between the model predictions and the actual prevalence of rabies in nepal; more data sets from nepal may help achieve better predictions of the model. the resource related parameters are particularly difficult to estimate, implying some uncertainty in quantitative optimal control results. however, the qualitative conclusion of results remain the same for a wider range of parameters, and therefore can be useful for 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[33] world health organization/nepal: http://www.searo.who.int/nepal/en/, accessed in october 2018. corresponding author, department of science and mathematics, abraham baldwin agricultural college, tifton, ga, 31793, usa e-mail address: bpantha@abac.edu department of mathematics, xavier university, cincinnati, oh, usa e-mail address: joshi@xavier.edu department of mathematics and statistics, computational science research center, & viral information institute, san diego state university, san diego, ca, usa e-mail address: nvaidya@sdsu.edu mathematics in applied sciences and engineering https://doi.org/10.5206/mase/10508 volume 1, number 1, march 2020, pp.65-84 https://ojs.lib.uwo.ca/mase modeling the impact of host resistance on structured tick population dynamics mahnaz alavinejad, jemisa sadiku, and jianhong wu abstract. for a variety of tick species, the resistance, behavioural and immunological response of hosts has been reported in the biological literature but its impact on tick population dynamics has not been mathematically formulated and analyzed using dynamical models reflecting the full biological stages of ticks. here we develop and simulate a delay differential equation model, with a particular focus on resistance resulting in grooming behaviour. we calculate the basic reproduction number using the spectral analysis of delay differential equations with positive feedback, and establish the existence and uniqueness of a positive equilibrium when the basic reproduction number exceeds unit. we also conduct numerical and sensitivity analysis about the dependence of this positive equilibrium on the the parameter relevant to grooming behaviour. we numerically obtain the relationship between grooming behaviour and equilibrium value at different stages. 1. introduction lyme disease is the most reported athropod-borne illness and it was first recognized in 1976 in lyme, connecticut usa [22]. borrelia burgdorferi is a tick-borne spirochete responsible for lyme disease which is found in nymphal ixodes dammini and has the highest chance to be transmitted to the host if the infected tick feeds for a duration of 72 hours or more [16, 23, 10]. once an infected tick bites the host, a skin lesion called erythema migrans (em) starts emerging and more than 95% of those patients diagnosed with lyme disease have em on the tick biting site [5, 22]. once the bacterium enters the body it starts spreading in many organs and tissues through the lymph system and blood [5]. as time progresses the patient will experience headache, neck pain, fever, fatigue, and migratory musculoskeletal pain [23, 10, 5]. the government of canada has data representing an increase of 144 cases in 2009 to 2025 cases in 2017 [17]. the i.scopularis also known as a black-legged tick is the main carrier of b. burgdorferi and has a life cycle of nearly two years [22]. the tick population undergoes three main stages: l-larvae, n-nymph, a-adult and to move from one stage to the other ticks will quest feed and molt [13, 26, 18, 19]. larvae and nymph feed on small rodents such as mice while adult ticks are more selective when it comes to their host since their body is larger compared to larvae and nymph and therefore the host must be a large mammal such as a deer. for ticks to move from one stage to the next it requires three hosts per stage and often the tick may use the same host for all three blood meals [13, 26, 18, 19]. female ticks lay eggs in the spring and larvae hatch during late summer. the larvae that feeds during the late summer starts molting to nymph during winter. the nymph then received by the editors 12 march 2020; revised 29 march 2020; accepted 29 march 2020; published online 31 march 2020. 2000 mathematics subject classification. primary 92d25; secondary 34k20. key words and phrases. ticks, host resistance, population dynamics. this research was supported by natural sciences and engineering research council of canada and the canada research chair program. 65 66 m. alavinejad, j. sadiku, and j. wu starts feeding in the spring of the following year and molts into adult on the same year. adult ticks die shortly right after they lay their eggs in the early spring [26, 18]. when a tick bites a host the expression of immunity varies depending on different hosts and tick species.the effects on ticks can vary from a simple rejection of the tick to interfering with the duration of feeding, inhibition of egg laying, also decreasing their viability to death of the tick while feeding. in addition, studies reveal that when female ticks feed on immune cattle their body of fully engorged tick was reduced by 30% [12, 24, 2]. according to brown [3] hosts with resistance respond to tick bites with an intensified grooming behaviour and the attachment site is marked by serous exudes which could engulf the tick. in an experiment conducted on resistant guinea pigs bitten by dermacentor andersoni, basophilia is present on the biting site. the attachment of a tick on a tick-sensitizes host is characterized by packs of basophils located in the intraepidermal vesicles. when ticks’ extracts are injected into ticksensitized host it causes a skin reaction and the plasma of the host expresses anti-tick antibodies which suggests a present mediated immune response. in case of unbitten animals, the reaction starts with neutrophils and the feeding site is characterized by an hemorrhagic as feeding progresses. basophils start to also accumulate to the feeding site, however little degranulation occurs. in an experiment to study the effect of resistance of guinea pigs to ticks, basophil degranulation at tick feeding sites, resulted in tick rejection after tick-attachment: 29% after 6 hours, 18% after 12 hours, 22% after 24 hours, 37% after 48 hours and 7.3% after 72 and 96 hours. this shows that ticks are most susceptible to the resistance at 6, 12, 24 and 48 hours after attachment which are corresponding to the attachment time and fast feeding period [4]. there have been intensive studies modelling the dynamics of tick-host interaction and the transmission of various pathogens. different aspects have also been included such as: seasonality , environmental changes, geographical heterogeneity and so on. on the other hand, few models incorporate delays in the development of tick from each life stage to the next [6, 25, 27]. jennings et al. [9] studied the effect of host resistance on tick population dynamics. they developed a mathematical model, described by a system of ordinary differential equations, focusing on tick-host interaction where the tick’s life cycle was divided into two main stages, adult and juvenile, and the host was subdivided into host with no immunity and host with immunity. their focus is to show how immunity affects the extinction or persistence of tick dynamics. however, their model does not include all biological stages (and sub-stages) of ticks and the possibility of different immunological response for each stage is ignored. here, we consider a stage-structured model that involves the full biological dynamics of tick and the emphasis is on the grooming behaviour of the host and its impact on tick population dynamics. we analyze the grooming behaviour in the mathematical model as a reduction in the successful attachment rates of ticks on the host i.e., the host-finding rates are reduced by a fraction for the host that shows resistance to tick bites. the model studies three main stages of tick’s life cycle in which the ticks interact with hosts during questing-feeding-molting process. there is one more stage that we consider between adult and egg which is egg laying female. the host is divided into two compartments: host with resistance (host has been bitten by ticks before) and host with no resistance (host that has not been exposed to ticks). we observe that the basic reproduction number does not change with the resistance factor, however, numerical simulations show that the value of the positive equilibrium for different stages of tick population, and the dynamical behaviour of the solutions change with varying the resistance factor. also, the sensitivity analysis demonstrates the dependence of the solutions on different parameters. modeling the impact of host resistance on structured tick population dynamics 67 lq lf lm nq nf nm aqafaelfe βl(αlhr+ + hr−)lq δlf δe−d lmτ(l,n)lf (t− τ(l,n)) βn (αnhr+ + hr−)nq δlf δe−d nmτ(n,a)nf (t− τ(n,a)) βa(αahr+ + hr−)aqεδaf γ(aelf ) γ (a e lf (t − τ ( e ,l ) )) d l q l q d l m l m d n q n q d n m n m d a q a q d a e l f a e lfd e e figure 1. flow diagram for the ticks’ life cycle and their interaction with hosts 2. the model formulation we start the model aiming to focus on the grooming behaviour. we model the three stages of larvae, nymph, and adult. the larvae and nymph populations are subdivided into questing, feeding and molting. on the other hand, for the adult population we consider adult egg laying female aelf , adult questing (aq) and feeding (af ). since a single female tick lays several thousands eggs the birth rate is the entry into population which is represented by ricker function, γ(a) = pae−qa [19, 15]. tick dynamics are regulated by insufficient resources for blood meal and this is illustrated in parameter q. the delay functions, demonstrating the time delays of ticks molting from one stage to another, are constants. in the model, τ(e,l), τ(l,n), τ(n,a) represent the time it takes for ticks to molt from egg to larvae, larvae to nymph and nymph to adult, respectively. the host population is divided into two compartments: hr+ represents the bitten host compartment that have developed with immunity; hr− represents the compartment of hosts that have never been bitten before and therefore without immunity. h is the total host population with a birth rate b and a mortality rate µ. the densitydependent regulations of the host population is described by k, c, and b − µ. the variables and parameters and their meaning are given in tables 3 and 1. the life cycle of ticks and their interaction with hosts is illustrated in figure 1. we suppose the successful attachment rates are reduced by a fraction αl for larvea, αn for nymph and αa for adult ticks. based on the results from [4] we can assume that α is in the range [0.6, 0.95], however we will study the effect of α values in [0, 1]. we also assume the hosts with no resistance develop resistance to ticks at a rate, denoted by κ, that depends on the tick densities, tick attachment rates and the immune system response. 68 m. alavinejad, j. sadiku, and j. wu table 1. definition of parameters and their values symbol meaning value reference dlq per capita mortality rate of lq 0.6 × 10−2 per day [15] dlm per capita mortality rate of lm 0.3 × 10−2 per day [15] dnq per capita mortality rate of nq 0.6 × 10−2 per day [15] dnm per capita mortality rate of nm 0.2 × 10−2 per day [15] daq per capita mortality rate of aq 0.6 × 10−2 per day [15] daelf per capita mortality rate of aelf 1 per day [28] de per capita mortality rate of e 0.2 × 10−2 per day [15] βl successful attachment rate of 0.6 × 10−3 per day per host [11] questing larva to host βn successful attachment rate of 0.6 × 10−3 per day per host [11] questing nymph to host βa successful attachment rate of 0.2 × 10−2 per day per host [11] questing adult to host β∗l rate of developing resistance to larva κ×βl per day per tick calculated β∗n rate of developing resistance to nymph κ×βn per day per tick calculated β∗a rate of developing resistance to adult κ×βa per day per tick calculated δ detachment rate 0.01 per day [20] αl host grooming effect for larva 0.4, [0, 1] unitless assumed αn host grooming effect for nymph 0.6, [0, 1] unitless assumed αa host grooming effect for adult 0.5, [0, 1] unitless assumed � female proportion 0.5 unitless [7] τ(e,l) the delay of development 21 days [15] form egg to larvae τ(l,n) the delay of development 101.18 ×temp−2.25, 200 days [15] form larvae to nymph τ(n,a) the delay of development 1596 ×temp−1.21, 61 days [15] form nymph to adult b birth rate of the host 0.66 × 10−3 per day [25] µ death rate of the host 0.33 × 10−3 per day [25] c crowding 3.5 × 10−4 per day calculated k carrying capacity of deers 20 [15] p maximum number of eggs 3000 [15] per female adult tick q the strength of density dependence 0.001 unitless [19] κ constant factor for resistance development 0.0001 unitless assumed modeling the impact of host resistance on structured tick population dynamics 69 table 2. modified parameter values to get different values for r0 symbol modified value comments de 1.2 × 0.2 × 10−2 p+20%p dlq 1.2 × 0.6 × 10−2 p+20%p dlm 1.2 × 0.3 × 10−2 p+20%p dnq 1.2 × 0.6 × 10−2 p+20%p dnm 1.2 × 0.2 × 10−2 p+20%p daq 1.2 × 0.6 × 10−2 p+20%p βl 0.1 × 0.6 × 10−3, 0.2 × 0.6 × 10−3 10%p, 20%p βn 0.3 × 0.6 × 10−3, 0.5 × 0.6 × 10−3 30%p, 50%p βa 0.5 × 0.2 × 10−2 50%p fixed β∗l κ×βl changed by changing βl β∗n κ×βn changed by changing βn β∗a κ×βa changed by changing βa αl 0.4 varied in [0, 1] αn 0.6 varied in [0, 1] αa 0.5 varied in [0, 1] c 1.2 × 3.5 × 10−4 p + 20%p fixed q 0.001 not changed table 3. definition of variables and their initial values symbol meaning initial value lq number of questing larvae lq0 = 1 × 106 lf number of feeding larvae lf0 (θ) = 0, −τ(e,l) ≤ θ ≤ 0 lm number of molting larvae nq number of questing nymph nq0 = 0 nf number of feeding nymph nf0 (θ) = 0, −τ(l,n) ≤ θ ≤ 0 nm number of molting nymph aq number of questing adult aq0 = 0 af number of feeding adult af0 = 0 aelf number of egg laying female adult aelf0 (θ) = 0, −τ(n,a) ≤ θ ≤ 0 e number of eggs h number of hosts hr+ number of hosts with resistance hr+ = 0 hr− number of host with no resistance 70 m. alavinejad, j. sadiku, and j. wu in order to make the model comprehensible we neglect few biological factors of tick dynamics. there are multiple blood meals that take place during molting procedures however in our model we consider only a homogeneous molting process, that is, ticks feed once, drop and molt with a constant time delay. the death rate depends on the stage of the tick (egg, larvae, nymph, adult) and also on whether the tick is questing or feeding. however, we consider a constant mortality rate. impact of climate change on development of ticks having a non linear relationship with increasing ambient temperature has not also been modelled. in addition, the questing rate is considered constant, even though it decreases as the temperatures and the day light decreases. the model is described by the following system of delay differential equations:   dlq dt = e−d eτ(e,l)γ(aelf (t− τ(e,l))) −βllq(t)(αlhr+(t) + hr−(t)) −dlqlq(t) dlf dt = βllq(t)(αlhr+(t) + hr−(t)) − δlf (t) dlm dt = δlf (t) −dlmlm(t) − δψe−d lmτ(l,n)lf (t− τ(l,n)) dnq dt = δψe−d lmτ(l,n)lf (t− τ(l,n)) −βnnq(t)(αnhr+(t) + hr−(t)) −dnqnq(t) dnf dt = βnnq(t)(αnhr+(t) + hr−(t)) − δnf (t) dnm dt = δnf (t) −dnmnm(t) − δψe−d nmτ(n,a)nf (t− τ(n,a)) daq dt = δψe−d nmτ(n,a)nf (t− τ(n,a)) −βaaq(t)(αahr+(t) + hr−(t)) −daqaq(t) daf dt = βaaq(t)(αahr+(t) + hr−(t)) −δaf (t) daelf dt = εδaf (t) −daelf aelf (t) de dt = γ(aelf (t)) −dee(t) −e−d eτ(e,l)γ(aelf (t− τ(e,l))) dhr− dt = bh(t) −µhr−(t) − c k h(t)hr−(t) − (β∗llq(t) + β ∗ nnq(t) + β ∗ aaq(t))hr−(t) dhr+ dt = −µhr+(t) − c k h(t)hr+(t) + (β ∗ llq(t) + β ∗ nnq(t) + β ∗ aaq(t))hr−(t) (2.1) where γ(a) = pae−qa is the birth function. here, we use the following equation for the host population dynamics dh(t) dt = (b−µ)h(t) − c k (h(t))2 (2.2) where h(t) = hr−(t) + hr+(t). note that the positive equilibrium of this equation is given by h̄ = (b−µ) c k. interpreting k as an environmental constraint, and in order to have h̄ ≤ k we assume c ≥ (b−µ), with h̄ = k when the equality holds. modeling the impact of host resistance on structured tick population dynamics 71 from system (2.1) we can get the following integral equations for lm(t), nm(t) and e(t) lm(t) = lm(0) − ∫ 0 −τ(l,n) e−d lm (−s)δlf (s)ds + ∫ t t−τ(l,n) e−d lm (t−s)δlf (s)ds nm(t) = nm(0) − ∫ 0 −τ(n,a) e−d nm (−s)δnf (s)ds + ∫ t t−τ(n,a) e−d nm (t−s)δnf (s)ds e(t) = e(0) − ∫ 0 −τ(e,l) e−d e (−s)γ(aelf (s))ds + ∫ t t−τ(e,l) e−d e (t−s)γ(aelf (s))ds (2.3) therefore system (2.1) is equivalent to the following  dlq dt = e−d eτ(e,l)γ(aelf (t− τ(e,l))) −βllq(t)(αlhr+(t) + hr−(t)) −dlqlq(t) dlf dt = βllq(t)(αlhr+(t) + hr−(t)) − δlf (t) dnq dt = δψe−d lmτ(l,n)lf (t− τ(l,n)) −βnnq(t)(αnhr+(t) + hr−(t)) −dnqnq(t) dnf dt = βnnq(t)(αnhr+(t) + hr−(t)) − δnf (t) daq dt = δψe−d nmτ(n,a)nf (t− τ(n,a)) −βaaq(t)(αahr+(t) + hr−(t)) −daqaq(t) daf dt = βaaq(t)(αahr+(t) + hr−(t)) −δaf (t) daelf dt = εδaf (t) −daelf aelf (t) dhr− dt = bh(t) −µhr−(t) − c k h(t)hr−(t) − (β∗llq(t) + β ∗ nnq(t) + β ∗ aaq(t))hr−(t) dhr+ dt = −µhr+(t) − c k h(t)hr+(t) + (β ∗ llq(t) + β ∗ nnq(t) + β ∗ aaq(t))hr−(t) (2.4) together with (2.3). for further analyses of this model we use the theory of monotone dynamical systems [21]. let τ = (τ1, · · · ,τ12) where τi ≥ 0, τ2 = τ(l,n), τ5 = τ(n,a)), τ9 = τ(e,l) are non zero and τi = 0 for i 6= 2, 5, 9. assume |τ| = max{τi}. let cτ be the product of banach spaces cτi = c([−τi, 0],r), i.e., cτ = 12∏ i=1 c([−τi, 0],r). let xt = (x 1 t , · · · ,x12t ) ∈ cτ be given by xit(θ) = x i(t + θ), i = 1, · · · , 12. where x(t) = (x1(t), · · · ,x12(t)) = (lq,lf,lm,nq,nf,nm,aq,af,aelf,e,hr−,hr+). then the right hand side of the equation (2.1) is given by x′(t) = f(xt). (2.5) we assume the initial data is non-negative. so we will assume the initial data x0 is in the banach space c+τ defined below c+τ = {φ ∈ cτ : φi(θ) ≥ 0,−τi ≤ θ ≤ 0}. 72 m. alavinejad, j. sadiku, and j. wu also, for the initial data to be continuous and positive we assume: lm(0) ≥ ∫ 0 −τ(l,n) e−d lm (−s)δlf (s)ds nm(0) ≥ ∫ 0 −τ(n,a) e−d nm (−s)δnf (s)ds e(0) ≥ ∫ 0 −τ(e,l) e−d e (−s)γ(aelf (s))ds. (2.6) the fundamental theory of functional differential equations implies that the solutions exist and are unique for all t ≥ 0. we now show that the solutions will be positive and remain bounded. theorem 2.1. let xi(0) > 0 and xi(θ) ≥ 0 for −τi ≤ θ < 0, for i = 1, · · · , 12. then the solutions to the system (2.4) are positive and bounded for all t ≥ 0. proof. consider the first equation in (2.4). first we look at the solution on [0,τ]: if there exists t1 ∈ (0,τ) such that lq(t1) = 0, then the derivative dlq(t)/dt at t1 is dlq(t) dt ∣∣∣∣ t1 = e−d eτ(e,l)γ(aelf (t1 − τ(e,l))). (2.7) since initial data for aelf on [−τ, 0] is positive, the derivative of lq at t1 is positive and therefore lq(t) is increasing, so it can not be negative. the same argument can be applied for [τ, 2τ]. this proves that lq(t) ≥ 0 for all t ≥ 0. if there exists t2 such that lf (t2) = 0, then the derivative of lf at t2 is given by dlf (t) dt ∣∣∣∣ t2 = βllq(t)(αlhr+(t) + hr−(t)) (2.8) which is positive since lq(t2), hr+(t2) and hr−(t) are positive. thus lf is increasing at t2 so it can not be negative. the same argument applies for other equations. therefore the solutions are positive. from equation (2.2) it is clear that h(t) is positive and bounded by the carrying capacity k. also the above discussion shows that hr− and hr+ are positive for all t ≥ 0. we show the boundedness of the tick population as follows. let t > 0 and τ = max{τ(e,l),τ(l,n),τ(n,a)}. we integrate the first equation in the original system (2.1) lq(t) = e −dlq t−βl ∫ t 0 (αlhr+(u)+hr−(u))du ∫ t 0 ed lq s+βl ∫ s 0 (αlhr+(u)+hr−(u))du ( e−d eτ(e,l)γ(aelf (s− τ(e,l))) ) ds + e−d lq t−βl ∫ t 0 (αlhr+(u)+hr−(u))dulq(0) therefore sup −τ≤t≤t lq(t) ≤ lq(0) + e−d eτ(e,l) dlq sup −τ≤t≤t γ(aelf (t)) using sup −τ≤t≤t e−βl ∫ t s (αlhr+(u)+hr−(u))du = 1 and ∫ t 0 e−d lq (t−s)ds < 1/dlq. using the fact that γ(aelf (t)) ≤ p/qe for all t ≥ 0, we see that sup −τ≤t≤t lq(t) ≤ c where c = lq(0) + pe −deτ(e,l)/qedlq is independent of t . therefore lq(t) ≤ c for all −τ ≤ t < ∞. modeling the impact of host resistance on structured tick population dynamics 73 integrating the next equations and taking the supermom we have: sup −τ≤t≤t lf (t) ≤ sup −τ≤t≤0 lf0 (t) + βlk δ sup −τ≤t≤t lq(t) sup −τ≤t≤t nq(t) ≤ nq(0) + e−d lmτ(l,n) dnq sup −τ≤t≤t lf (t) sup −τ≤t≤t nf (t) ≤ sup −τ≤t≤0 nf0 (t) + βnk δ sup −τ≤t≤t nq(t) sup −τ≤t≤t aq(t) ≤ aq(0) + e−d nmτ(n,a) daq sup −τ≤t≤t nf (t) sup −τ≤t≤t af (t) ≤ af (0) + βak δ sup −τ≤t≤t aq(t) sup −τ≤t≤t aelf (t) ≤ sup −τ≤t≤0 aelf0 (t) + �δ daelf sup −τ≤t≤t af (t). combining the above inequalities and assuming that the initial data are bounded we can see that these tick stages are bounded on −τ ≤ t < ∞. we can get similar inequalities from system (2.3). this proves that all tick stages are bounded. � since the host population stabilizes quickly at h̄ = (b−µ)k/c, the limiting system is as follows   dlq dt = e−d eτ(e,l)γ(aelf (t− τ(e,l))) + βl(1 −αl)lq(t)hr+(t) − (βlh̄ + dlq )lq(t) dlf dt = −βl(1 −αl)lq(t)hr+(t) + βlh̄lq(t) −δlf (t) dnq dt = δψe−d lmτ(l,n)lf (t− τ(l,n)) + βn (1 −αn )nq(t)hr+(t) − (βnh̄ + dnq )nq(t) dnf dt = −βn (1 −αn )nq(t)hr+(t) + βnh̄nq(t) −δnf (t) daq dt = δψe−d nmτ(n,a)nf (t− τ(n,a)) + βa(1 −αa)aq(t)hr+(t) − (βah̄ + daq )aq(t) daf dt = −βaaq(t)(1 −αa)hr+(t) + βah̄aq(t) − δaf (t) daelf dt = εδaf (t) −daelf aelf (t) dhr+ dt = −µhr+(t) − c k h̄hr+(t) + (β ∗ llq(t) + β ∗ nnq(t) + β ∗ aaq(t))(h̄ −hr+(t)) (2.9) from now on we refer to this system as the main system of our model unless otherwise stated. 3. analyses in this section we give the necessary condition for existence and uniqueness of the positive equilibrium point and the conditions for local stability of the tick free equilibrium. 74 m. alavinejad, j. sadiku, and j. wu 3.1. equilibria. let x∗ denote the vector (lq,lf,nq,nf,aq,af,aelf,hr+) in r8, and let f(x) be the right hand side of (2.9). in order to find all equilibria we need to solve the system f(x) = 0:  0 = e−d eτ(e,l)γ(aelf (t− τ(e,l))) + βl(1 −αl)lq(t)hr+(t) − (βlh̄ + dlq )lq(t) 0 = −βl(1 −αl)lq(t)hr+(t) + βlh̄lq(t) − δlf (t) 0 = δψe−d lmτ(l,n)lf (t− τ(l,n)) + βn (1 −αn )nq(t)hr+(t) − (βnh̄ + dnq )nq(t) 0 = −βn (1 −αn )nq(t)hr+(t) + βnh̄nq(t) − δnf (t) 0 = δψe−d nmτ(n,a)nf (t− τ(n,a)) + βa(1 −αa)aq(t)hr+(t) − (βah̄ + daq )aq(t) 0 = −βaaq(t)(1 −αa)hr+(t) + βah̄aq(t) − δaf (t) 0 = εδaf (t) −daelf aelf (t) 0 = −µhr+(t) − c k h̄hr+(t) + (β ∗ llq(t) + β ∗ nnq(t) + β ∗ aaq(t))(h̄ −hr+(t)) (3.1) at the tick-free equilibrium, where all tick stages are equal to zero, we have hr+ = 0. let hr+ 6= h̄ so that (h̄−(1−αl)hr+), (h̄−(1−αn )hr+), (h̄−(1−αa)hr+) > 0. we want to derive conditions for existence and uniqueness of a (strongly) positive equilibrium point (xi > 0 for all i = 1, · · · ,n). from the equations in (3.1) we get the following lq = daelf (βn (h̄ − (1 −αn )hr+) + dnq )(βa(h̄ − (1 −αa)hr+) + daq ) s2s3�βlβnβl(h̄ − (1 −αl)hr+)(h̄ − (1 −αn )hr+)(h̄ − (1 −αa)hr+) aelf lf = daelf (βn (h̄ − (1 −αn )hr+) + dnq )(βa(h̄ − (1 −αa)hr+) + daq ) s2s3δ�βnβa(h̄ − (1 −αn )hr+)(h̄ − (1 −αa)hr+) aelf nq = daelf (βa(h̄ − (1 −αa)hr+) + daq ) s3�βnβa(h̄ − (1 −αn )hr+)(h̄ − (1 −αa)hr+) aelf nf = daelf (βa(h̄ − (1 −αa)hr+) + daq ) s3δ�βa(h̄ − (1 −αa)hr+) aelf aq = daelf �βa(h̄ − (1 −αa)hr+) aelf af = daelf �δ aelf (3.2) where s1 = e −deτ(e,l) , s2 = ψe −dlmτ(l,n) and s3 = ψe −dnmτ(n,a) . from the first equation in the system (3.1) we get lq = s1γ(aelf ) (βl(h̄ − (1 −αl)hr+) + dlq ) (3.3) and therefore γ(aelf ) = d aelf (βl(h̄−(1−αl)hr+)+dlq )(βn (h̄−(1−αn )hr+)+dnq )(βa(h̄−(1−αa)hr+)+daq ) s1s2s3�βlβnβa(h̄−(1−αl)hr+)(h̄−(1−αn )hr+)(h̄−(1−αa)hr+) aelf . (3.4) modeling the impact of host resistance on structured tick population dynamics 75 since γ(aelf ) = paelfe −qaelf we have the following cases: aelf = 0 or pe−qaelf = d aelf (βl(h̄−(1−αl)hr+)+dlq )(βn (h̄−(1−αn )hr+)+dnq )(βa(h̄−(1−αa)hr+)+daq ) s1s2s3�βlβnβa(h̄−(1−αl)hr+)(h̄−(1−αn )hr+)(h̄−(1−αa)hr+) (3.5) finally, we reduce the system (3.1) to the following system 0 = γ(hr+) −pe−qaelf (3.6a) 0 = −bhr+ + ω(hr+)(h̄ −hr+)aelf (3.6b) where γ(hr+) = daelf (βl(h̄ − (1 −αl)hr+) + dlq )(βn (h̄ − (1 −αn )hr+) + dnq )(βa(h̄ − (1 −αa)hr+) + daq ) s1s2s3�βlβnβa(h̄ − (1 −αl)hr+)(h̄ − (1 −αn )hr+)(h̄ − (1 −αa)hr+) ω(hr+) = β ∗ l daelf (βn (h̄ − (1 −αn )hr+) + dnq )(βa(h̄ − (1 −αa)hr+) + daq ) s2s3�βlβnβa(h̄ − (1 −αl)hr+)(h̄ − (1 −αn )hr+)(h̄ − (1 −αa)hr+) + β∗n daelf (βa(h̄ − (1 −αl)hr+) + daq ) s3�βnβa(h̄ − (1 −αn )hr+)(h̄ − (1 −αa)hr+) + β∗a daelf �βa(h̄ − (1 −αa)hr+) from (3.6b) we have aelf = bhr+ ω(hr+)(h̄ −hr+) given that hr+ 6= h̄ and ω(hr+) 6= 0 (it can be proved that this holds). substituting this in the equation (3.6a) we get the following γ(hr+) = pe −q bhr+ ω(hr+)(h̄−hr+). (3.7) this is a nonlinear equation for hr+and we need to determine under what conditions this equation has a unique positive solution. let g(hr+) be the right hand side of equation (3.7). the functions γ and g have the following properties: (i) γ is a rational function and is strictly increasing for 0 < hr+ < h̄; (ii) γ(0) > 0 is given by daelf (βlh̄ + d lq )(βah̄ + d aq )(βnh̄ + d nq ) s1s2s3�βlβnβah̄3 ; (iii) g is a negative exponential function and it approaches zero (exponentially) as hr+ approaches h̄; (iv) g(0) = p. from these properties we can see that the equation (3.7) has at least one solution 0 < hr+ < h̄, if and only if g(0) > γ(0), i.e., p > daelf (βlh̄ + d lq )(βah̄ + d aq )(βnh̄ + d nq ) s1s2s3�βlβnβah̄3 . this solution is unique if g(hr+) is monotonically decreasing, and this holds if and only if d dhr+ ( hr+ ω(hr+)(h̄ −hr+) ) > 0 for all hr+ ∈ (0,h̄). 76 m. alavinejad, j. sadiku, and j. wu theorem 3.1. let rv0 = ps1s2s3�βlβnβah̄ 3 daelf (βlh̄ + d lq )(βah̄ + d aq )(βnh̄ + d nq ) . if rv0 > 1, then system (2.9) has a positive equilibrium point. if additionally d dhr+ ( hr+ ω(hr+)(h̄ −hr+) ) > 0 holds, then the positive equilibrium is unique. 3.2. stability of the tick-free equilibrium. first we linearize system (2.9) about a given equilibrium point using the fréchet derivative of the function f(x), given by the right hand side of the system (2.9): df(x∗)x = lim h→0 (f(x∗ + hx) −f(x∗) h ) the linearized system is given by df1(x ∗)x = pe−d eτ(e,l)aelf (t− τ(e,l))e−qa ∗ elf (t−τ(e,l)) −pqe−d eτ(e,l)aelf (t− τ(e,l))a∗elf (t− τ(e,l))e −qa∗elf (t−τ(e,l)) + (1 −αl)βl(l∗q(t)hr+(t) + lq(t)h ∗ r+(t)) − (βlh̄ + d lq )lq(t) df2(x ∗)x = −(1 −αl)βl(l∗q(t)hr+(t) + lq(t)h ∗ r+(t)) + βlh̄lq(t) −δlf (t) df4(x ∗)x = δψe−d lmτ(l,n)lf (t− τ(l,n)) + (1 −αn )βn (n∗q (t)hr+(t) + nq(t)h ∗ r+(t)) − (βnh̄ + d nq )nq(t) df5(x ∗)x = −(1 −αn )βn (n∗q (t)hr+(t) + nq(t)h ∗ r+(t)) + βnh̄nq(t) − δnf (t) df7(x ∗)x = δψe−d nmτ(n,a)nf (t− τ(n,a)) + (1 −αa)βa(a∗q(t)hr+(t) + aq(t)h ∗ r+(t)) − (βah̄ + d aq )aq(t) df8(x ∗)x = −(1 −αa)βa(a∗q(t)hr+(t) + aq(t)h ∗ r+(t)) + βah̄aq(t) −δaf (t) df9(x ∗)x = εδaf (t) −daelf aelf (t) df12(x ∗)x = −(µ + c k h̄)hr+(t) + h̄(β ∗ ll ∗ q(t) + β ∗ nn ∗ q (t) + β ∗ aa ∗ q(t)) − ( (β∗llq(t) + β ∗ nnq(t) + β ∗ aaq(t))h ∗ r+(t) + (β ∗ ll ∗ q(t) + β ∗ nn ∗ q (t) + β ∗ aa ∗ q(t))hr+(t) ) (3.8) the linearized system about the tick-free equilibrium point is as follows:  df1(x ∗)x = ps1aelf (t− τ(e,l)) − (βlh̄ + dlq )lq(t) df2(x ∗)x = βlh̄lq(t) − δlf (t) df4(x ∗)x = δs2lf (t− τ(l,n)) − (βnh̄ + dnq )nq(t) df5(x ∗)x = βnh̄nq(t) − δnf (t) df7(x ∗)x = δs3nf (t− τ(n,a)) − (βah̄ + daq )aq(t) df8(x ∗)x = βah̄aq(t) −δaf (t) df9(x ∗)x = εδaf (t) −daelf aelf (t) df12(x ∗)x = −(µ + c k h̄)hr+(t) (3.9) using the theory of monotone dynamical systems we can see that system (2.9) is cooperative ([21] corollary 3.2) and therefore stability of the zero equilibrium of system (3.9) is given by the stability of the corresponding ode system. modeling the impact of host resistance on structured tick population dynamics 77 theorem 3.2. if rv0 < 1, then x = 0 is the only equilibrium point of the system (2.9) and is locally asymptotically stable. when rv0 > 1, there exists a positive equilibrium point and x = 0 is unstable. proof. we use the method of next generation matrix for the ode system given by x′(t) = jx(t) where the matrix j is obtained from system (3.9): j =   −βlh̄ −dlq 0 0 0 0 0 ps1 0 βlh̄ −δ 0 0 0 0 0 0 0 δs2 −βnh̄ −dnq 0 0 0 0 0 0 0 βnh̄ −δ 0 0 0 0 0 0 0 δs3 −βah̄ −daq 0 0 0 0 0 0 0 βah̄ −δ 0 0 0 0 0 0 0 �δ −daelf 0 0 0 0 0 0 0 0 −b   the matrix j can be written as j = f − v . the zero equilibrium is locally asymptotically stable if ρ(fv −1) < 1 (ρ is the spectral radius of fv −1) and it is unstable if ρ(fv −1) > 1. we can see that ρ(fv −1) = rv0 = ps1s2s3�βlβnβah̄ 3 daelf (βlh̄ + d lq )(βah̄ + d aq )(βnh̄ + d nq ) . � 4. numerical simulations in this section we study the long-term dynamical behaviour of the system using numerical simulations and perform a sensitivity analysis for different parameters. 4.1. model parametrization and validation. the observation of the dynamical behaviour of each stage of the tick population is demonstrated by applying dde23 packages in matlab to system (2.9). the model is parameterized using parameter values available in mathematical and ecological literature ([7, 11, 15, 20, 19, 28]). parameter values and initial conditions are given in tables 1-3. we note that the grooming behaviour does not impact the initial growth of the tick population, since parameters reflecting the grooming factor do not change the value of the basic reproduction number. we consider three cases to illustrate the dynamics of tick population in the presence of grooming factor. however, in these cases we fix the values for parameters related to the grooming behaviour. in the first case (figure 2) the basic reproduction number is below the threshold value i.e., rv0 < 1, the tick-free equilibrium is locally asymptotically stable and therefore all stages of ticks go extinct. in case 2 (figure 3) the basic reproduction number is slightly greater than one, the tick-free equilibrium point becomes unstable and the solutions approach the positive equilibrium without any initial oscillatory behaviour. in case 3 (figure 5) the solutions oscillate initially and then approach the positive equilibrium. when the resistance related parameter values are fixed and the rest of the parameters vary, the positive equilibrium becomes unstable and a limit cycle appears. therefore, the solutions oscillate about the equilibrium point. figure 4 shows how a limit cycle appears as the value of αa increases from 0 to 1. to study the population behaviour without grooming factor we set αl = αn = αa = 1 and κ = 0 and for intense grooming behaviour the αl = αn = αa = 0. in addition, we observe the dynamics for a mild grooming behaviour where αl = 0.4,αn = 0.6,αa = 0.5 and κ = 0.1 × 10−5. the equilibrium value for all stages are higher when there is no grooming behaviour. in particular, the value of the adult egg laying females at the equilibrium is 693 for a mild resistance behaviour and 1.9×103, when there is no resistance (figure 3 and the left side of figure 6). we also see that by decreasing the resistance solutions 78 m. alavinejad, j. sadiku, and j. wu (a) (b) (c) (d) figure 2. case 1, rv0 < 1 where βl = 0.6×10−4, βn = 1.8×10−4 and p = 200 yields rv0 = 0.89. with non-oscillatory behaviour show damped oscillation. in a maximum intensified grooming behaviour the tick attachment rates to hosts with resistance are reduced to 0, therefore high resistance of hosts affects the tick equilibrium values significantly. for instance, in figure 6 the equilibrium value for aelf reduces from 1.9 × 103, when there is no resistance, to 78 when the resistance is very high. comparing the right side of figure 5 with 7, demonstrates the effect of resistance factors on the dynamical behaviour of the solutions. reducing the resistance from high to a mild resistance results in an increase in the value of the equilibrium of aelf from 78 to 1600. however, in the absence of host resistance, the tick population at different stages oscillate about a positive equilibrium (aelf ≈ 2.7×103). in other words, by decreasing the grooming behaviour (increasing the value of αl, αn and αa from 0 to 1), there is more available resources for ticks to feed on. therefore, the dynamical behaviour of tick population at different stages changes from solutions converging to the positive equilibrium to oscillatory solutions. the dynamics of the feeding ticks are similar to those of questing ticks and therefore we exclude the pictures on this paper. when we ignore the resistance behaviour in case 2 and 3, the host population with resistance hr+ is equal to 0 and it reaches a positive equilibrium point when αl = αn = αa = 0. 4.2. lhs and prcc. we perform latin hypercube sampling to further analyze the effects of each parameter on the dynamics of each life stage of the ticks [1, 8] before we proceed to performing prcc a verification of monotonicity is necessary to ensure the correct range of the parameters for prcc analysis. next, we calculate the prcc, which determines the contribution of each parameter to the output variable such the population of larvae questing. a prcc value significantly greater than 0 indicates a positive correlation and for prcc significantly less than 0, a negative correlation between the parameter and the output [14]. in figure 8, the prcc for the larvae questing population demonstrates modeling the impact of host resistance on structured tick population dynamics 79 (a) (b) (c) (d) figure 3. in case 2 the values of p and κ have changed to p = 1500 and κ = 0.1×10−5 and the reproduction number increased to rv0 = 6.71. the simulations run for a time span of 10000 days. the equilibrium points for each stage of questing, feeding and adult egg laying female tick are as follows: lq = 6.5 × 107,nq = 1.6 × 106,aq = 1.6 × 105 lf = 2.9×106,nf = 2.9×105,af = 1.4×105,aelf = 693. in addition, the equilibrium point of the host with resistance is 13. figure 4. the solutions oscillate about the equilibrium point as we change the value of αa in the interval [0, 1] for parameter values in case 3. the values for αl and αn are 0.6 and 0.8. 80 m. alavinejad, j. sadiku, and j. wu (a) (b) (c) (d) figure 5. in case 3 the values of βl and βn have changed to βl = 1.2×10−4,βn = 3 × 10−4 producing a higher reproduction number, rv0 = 16.9. the simulation are again running for a time span 10000 days. the equilibrium points for each stage of questing, feeding and adult egg laying female tick are as follows: lq = 5.7 × 107,nq = 2.3 × 106,aq = 3.8 × 105,lf = 4.8 × 106,nf = 6.9 × 105,af = 3.2 × 105,aelf = 1600. in addition, the equilibrium point of the host with resistance is 14. the negative correlation with the death rates daelf , dnm, dlm, dnq, daq, dlq, de and dlq having the highest effect on this stage. the detachment rate δ does not have a an impact, however the parameters related ticks’ biological characteristics, p, q, �, have a significant effect. we also observe that the host finding rates βa, βn ,βl, have positive correlation with larvae questing dynamics. for the values of most parameters that are taken from the literature, we would expect to see a reasonable correlation between the parameter and the output (in a range where the output is monotonically increasing or decreasing with parameter). for instance the output value of lq (and therefore lf ) at the equilibrium is supposed to decrease with an increase of the larvae questing death rate (negative correlation). 5. conclusion in this paper we formulated a delay differential model for black leg ticks, stratified based on stage and activity, with a particular focus on the host grooming behaviour. the basic reproduction number was calculated and the condition for local stability of tick-free equilibrium, for which the tick population go extinct, and also for existence and uniqueness of a positive equilibrium was given. model parameterization and numerical simulations were carried out to demonstrate the dynamics of tick and host population with and without the grooming behaviour and the effect of the resistance factor on the value of equilibrium points are studied. parameters related to the grooming and resistance factors, αl, αn , αa, and κ have no effect on the initial growth rate of ticks since these parameters do not change the modeling the impact of host resistance on structured tick population dynamics 81 (a) (b) (c) (d) figure 6. the parameter values are the same as in case 2 except the αl = αn = αa = 1 (on the left). the equilibrium points are as follows: lq = 5.0 × 107, nq = 2.3×106, aq = 2.4×105, aelf = 1.9×103. there is no resistance and hence hr+ = 0. in case of αl = αn = αa = 0 (on the right) the equilibrium points are lq = 1.3×107, nq = 3.2 × 105, aq = 2.4 × 104, aelf = 78. since now we introduce resistance, hr+ = 10. value of rv0 . however, with an increase of the intensity of the grooming behaviour from no resistance to a high level of resistance, where either the hosts show intensified grooming behaviour or ticks are withdrawn from feeding or dead, the values of equilibrium points of all tick stages decrease. from the numerical simulations we observed structural changes of the dynamical behaviour of the tick population by changing the parameter values reflecting the effect of the host resistance. also, the intensified resistance results in higher equilibrium values for hr+. a sensitivity analysis of the positive equilibrium value to the parameters was carried out by performing lhs and prcc. from prcc we observed high positive correlation between the maximum number of eggs per female adult tick (p) and larvae questing; as more eggs are produced the higher the number of larvae questing. the female proportion parameter (�) is also positively correlated to larvae questing. as the female rate proportion increases the higher number of egg production and therefore increasing the value of larvae questing. in contrast, the value of the strength of density dependence (q) and death rate of larvae questing (dlq) are negatively correlated with the population of larvae questing. as the death rate increases there will be a lower population size of larvae questing. lastly, as the number of larvae questing increases there will be harder to find resources to survive, hence as q increases the number the lq decreases. this study has some limitations. the death rates are assumed to be constants for each stage of the tick and we have ignored the possibility of death during the feeding process resulting from serous exudes 82 m. alavinejad, j. sadiku, and j. wu (a) (b) (c) (d) figure 7. the parameter values are the same as in case 3 except αl = αn = αa = 1 (the left). the equilibrium points are as follows: lq ≈ 2.8 × 107, nq ≈ 2.1 × 106, aq ≈ 3.5×105, aelf ≈ 2.7×103. since resistance factor is not introduced the hr+ = 0. on the right side the αl = αn = αa = 0 and the equilibrium points are as follows: lq = 1.4 × 107, nq = 4.0 × 105, aq = 3.8 × 104, aelf = 80. the resistance factor increase the population size from zero to hr+ = 11. figure 8. prcc for most of the parameters used in the model at the equilibrium point of lq. the value of each parameter is taken from 1and case 2 for a range of (+/−)20% modeling the impact of host resistance on structured tick population dynamics 83 which could engulf the tick. also, interpreting the host resistance as a kind of immunity to ticks we can consider the situation where the host resistance decreases in time the hosts lose immunity to ticks. the molting process is demonstrated by constant delay functions. future work could incorporate the temperature and humidity on molting process and explore 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[28] x. wu, v. r. s. k. duvvuri and j. wu, modelling dynamical temperature influence on tick ixodes scapularis population, international congress on environmental modelling and software 529, 2010. corresponding author. department of mathematics and statistics, york university e-mail address: mahnazal@yorku.ca department of mathematics and statistics, york university e-mail address: jemisa18@yorku.ca department of mathematics and statistics, york university e-mail address: wujh@yorku.ca mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 2, number 4, december 2021, pp.273-289 https://doi.org/10.5206/mase/14233 an extended seiard model for covid-19 vaccination in mexico: analysis and forecast ángel g. c. pérez and david a. oluyori abstract. in this study, we propose and analyse an extended seiard model with vaccination. we compute the control reproduction number rc of our model and study the stability of equilibria. we show that the set of disease-free equilibria is locally asymptotically stable when rc < 1 and unstable when rc > 1, and we provide a sufficient condition for its global stability. furthermore, we perform numerical simulations using the reported data of covid-19 infections and vaccination in mexico to study the impact of different vaccination, transmission and efficacy rates on the dynamics of the disease. 1. introduction the coronavirus disease 2019 (covid-19) pandemic, caused by the severe acute respiratory syndrome coronavirus 2 (sars-cov-2) has caused a worldwide crisis due to its effects on society and global economy. due to the absence of specific anti-covid-19 medical treatments, most countries had been relying on non-pharmaceutical interventions, such as wearing of face masks, social/physical distancing, partial/total lockdown, travel restrictions, and closure of schools and work centres, in order to curtail the spread of the disease before december 2020. however, these measures have been insufficient to mitigate the pandemic globally as medical facilities were overstretched and death toll heightened. vaccination has been an effective strategy in combatting the spread of infectious diseases, e.g., pertussis, measles, and influenza. historically, the eradication of smallpox has been considered as the most remarkable success of vaccination ever recorded [40]. so far, the development and testing of vaccines against sars-cov-2 has occurred at an unprecedented speed and, in the last months, several vaccines have been approved for use in many countries, and their deployment is already underway. in a pandemic situation such as this, current preventive vaccines consisting of inactivated viruses do not protect all vaccine recipients equally as the protection conferred by the vaccine is dependent on the immune status of the recipient [3]. over the past few decades, a large number of simple compartmental models with vaccination have been proposed in the literature to assess the effectiveness of vaccines in combatting the infectious diseases [19, 28, 4, 1, 16, 21, 9, 8, 37, 30]. with the recent development of anti-covid vaccines, several models have been proposed to provide insight into the effect that inoculation of a certain portion of the population will have on the dynamics of the covid-19 pandemic. the authors in [10] studied an seihrdv model and an smeihrdv model (the latter including a semi-susceptible class) and fitted the parameters using data from several countries to evaluate the effect of social distancing and vaccination on controlling covid-19. another model was studied in [25] to compare the outcomes of single-dose and two-dose anti-covid vaccination regimes. the global stability of a two-strain covid-19 model received by the editors 13 september 2021; accepted 8 november 2021; published online 10 november 2021. 2010 mathematics subject classification. 92d30, 34d20, 37m05. key words and phrases. coronavirus, vaccine efficacy, asymptomatic infection, stability. 273 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14233 274 á. g. c. pérez and d. a. oluyori with vaccination against one strain was studied in [32]. the stability analysis of an seir model with prophylactic and therapeutic vaccines was performed in [38]. in [20], the effect of immunity, vaccination, and reinfection with changing parameters was analysed using an sevis model, while in [13] an siqrd model was used to simulate several scenarios of vaccine delivery in indonesia. since it is known that individuals infected with sars-cov-2 can transmit the virus to other people without presenting symptoms of the disease, some authors have proposed models that distinguish between symptomatic and asymptomatic infections. for instance, an ageand region-structured model was proposed in [22] to simulate the rollout of a two-dose vaccination programme in the uk using the pfizer–biontech and oxford–astrazeneca vaccines. an agent-based transmission model was used in [34] to project the impact of a two-dose vaccination campaign with the pfizer–biontech and moderna vaccines in ontario. on the other hand, a sensitivity analysis and uncertainty quantification of an seisiaqr model with vaccination strategy was conducted in [23]. a three-patch metapopulation epidemic model structured by risk was used to investigate several vaccination scenarios in mexico city in [27]. the threshold dynamics of a covid-19 model combining vaccination and treatment was established in [12], while [11] considered an se(is)(ih)ar model with vaccination and antiviral controls. some other covid-19 models with vaccination can be found in [14, 31, 5, 2, 26]. of the works mentioned above, only [11, 12, 32, 38] studied the global stability of the equilibrium states; moreover, none of these studies considered that people who become infected after being vaccinated may have reduced infectivity and a lower chance of showing severe symptoms. hence, in the present work, we aim to study a model that takes into account all of these factors and perform a local and global stability analysis of the disease-free equilibria, as well as present an application of this model to the covid-19 epidemic in mexico. in mid-november 2020, mexico passed the mark of 1,000,000 confirmed cases and 100,000 deaths due to covid-19. on 11 december 2020, the mexican government’s medical safety commission approved the emergency use of the pfizer–biontech coronavirus vaccine, with the first 250,000 doses intended for health workers. the inoculation of frontline health personnel started that year on 24 december. four other covid-19 vaccines were approved between january and february 2021, and vaccination of people over 60-years old with the astrazeneca vaccine began in february. the first vaccines to be applied in mexico followed a two-dose regime, until the single-dose cansino vaccine began to be applied to a portion of the population in april 2021. the johnson & johnson vaccine, also single-dose, was deployed in june 2021 to inoculate the population over age 18 in the municipalities of the northern border. the government expected to have vaccinated all adult population with at least one dose by october 2021, after which, the application of vaccines would continue for people aged 12 to 17 with comorbidities. in this paper, we propose a differential equation model to simulate the application of a two-dose vaccine against covid-19, considering the possibility of vaccine leakiness and asymptomatic infections. the motivation of this study is derived from the work of the authors in [7, 6], who considered an seiard mathematical model to investigate the outbreak of covid-19 in mexico. therefore, in the present work, we incorporate the vaccination component to the model in [6] to derive an extended seiard model to examine the effectiveness of the covid-19 jabs which are currently being deployed to many countries to help combat the raging pandemic situation. the rest of this paper is organized as follows. in section 2, we present the equations and assumptions of the extended seiard model with vaccination. in section 3, we perform a theoretical analysis of the model, compute its control reproduction number and study the stability of the disease-free equilibria. in section 4, we carry out numerical simulations using reported data on covid-19 infections and vaccination in mexico. lastly, we provide some discussions and concluding remarks in section 5. an extended seiard model for covid-19 vaccination 275 figure 1. flow diagram of the model with vaccination. 2. model formulation to derive the mathematical model, we divide the unvaccinated population into susceptible (s), exposed (e), symptomatic infectious (i), asymptomatic infectious (a), and recovered (r). the number of individuals in each subpopulation at time t is denoted by s(t), e(t), etc. since the time scale we consider in our analysis is considerably shorter than the mean lifespan of individuals in the population, our model does not consider the natural death and birth rates. susceptible individuals become exposed by contact with symptomatic infectious individuals at a rate β1 and by contact with asymptomatic infectious individuals at a rate β2. the exposed individuals become infectious at a rate w: a proportion p1 of them will show symptoms, while the rest remains asymptomatic. we assume that the symptomatic class has a disease-induced death rate, denoted by δ1. both symptomatic and asymptomatic infectious people recover at a rate γ. we also assume that the susceptible population s is vaccinated at a rate v ≥ 0 (the number of first doses administered per day at time t is given by vs(t)). individuals who have received only the first dose of the vaccine are included in the class v1, and they move to the class v2 upon receiving the second dose, which occurs at a rate θ. due to vaccine leakiness, vaccination of an individual does not completely remove the risk of infection. hence, we also assume that the vaccinated population can become exposed (ev ), symptomatic infectious (iv ) and asymptomatic infectious (av ). individuals in the class v1 (respectively, v2) move to the class ev due to contact with symptomatic infectious people at a rate (1 − η1)β1 (respectively, (1 − η2)β1) and by contact with the asymptomatic infectious at a rate (1 − η1)β2 (respectively, (1 − η2)β2), where η1 is the efficacy of the vaccine after one dose (η2 is the efficacy of the vaccine after two doses). the infectivity of individuals in the iv and av classes is reduced by a factor 1 −q with respect to that of individuals in the i and a classes. the population in the class ev becomes infectious at a rate w; we assume that the proportion of people from this class who become symptomatic infectious is p2, which may be different from that of unvaccinated people due to the effect of the vaccine in reducing the severity of the infection. likewise, the disease-induced death rate δ2 is lower for the vaccinated population. individuals in the iv and av classes also move to the r class upon recovery at a rate γ. we will denote by n(t) the total population at time t, which is given by n(t) = s(t) + e(t) + i(t) + a(t) + v1(t) + v2(t) + ev (t) + iv (t) + av (t) + r(t). 276 á. g. c. pérez and d. a. oluyori hence, our model is described by the following system of differential equations: ṡ = − s n β1(i + qiv ) − s n β2(a + qav ) −vs, ė = s n β1(i + qiv ) + s n β2(a + qav ) −we, i̇ = p1we − (δ1 + γ)i, ȧ = (1 −p1)we −γa, v̇1 = vs − (1 −η1) v1 n β1(i + qiv ) − (1 −η1) v1 n β2(a + qav ) −θv1, v̇2 = θv1 − (1 −η2) v2 n β1(i + qiv ) − (1 −η2) v2 n β2(a + qav ), ėv = (1 −η1) v1 n β1(i + qiv ) + (1 −η1) v1 n β2(a + qav ) + (1 −η2) v2 n β1(i + qiv ) + (1 −η2) v2 n β2(a + qav ) −wev , i̇v = p2wev − (δ2 + γ)iv ȧv = (1 −p2)wev −γav , ṙ = γ(i + a + iv + av ). (2.1) we define an additional variable d(t) that denotes the number of people deceased due to covid-19, which is governed by the equation ḋ = δ1i + δ2iv . (2.2) we assume that β1, β2, θ, w and γ are positive, v,δ1,δ2 ≥ 0 and q,η1,η2,p1,p2 ∈ [0, 1]. the interpretation of parameters is summarized in table 1. the flow diagram of the model can be seen in figure 1. 3. theoretical analysis in this section, we will derive some theoretical results for model (2.1). our aim is to investigate the behaviour of the epidemic in the next few years after the vaccination campaign is over. thus, we will mainly focus on the case when the vaccination rate v is zero, but the vaccinated subpopulations (v1, v2, ev , iv , av ) may have positive initial values. first, we will determine the disease-free equilibria of the model, i.e., the equilibria with e = i = a = ev = iv = av = 0. in the case when v is positive, the disease-free equilibria (dfe) of system (2.1) are given by the set s = { (s,e,i,a,v1,v2,ev ,iv ,av ,r) = (0, 0, 0, 0, 0,v ∗ 2 , 0, 0, 0,r ∗) ∈ r10 : v ∗2 > 0, r ∗ > 0 } . the equilibria in s correspond to the case when the whole population has been fully vaccinated or recovered from covid-19, and there are no susceptible individuals. when v = 0, model (2.1) has a continuum of disease-free equilibria, given by the set s0 = { (s,e,i,a,v1,v2,ev ,iv ,av ,r) = (s ∗, 0, 0, 0, 0,v ∗2 , 0, 0, 0,r ∗) ∈ r10 : s∗ ≥ 0, v ∗2 > 0, r ∗ > 0 } . each equilibrium p0 ∈ s0 represents a scenario when the anti-covid vaccination programme has ended, and a certain number of individuals v ∗2 has been vaccinated to achieve herd immunity in the population. we will compute the control reproduction number rc of the model based on this expression for the dfe. an extended seiard model for covid-19 vaccination 277 using the notation in [33], we determine the matrix of new infections f and the transition matrix v , considering only the infected compartments (e, i, a, ev , iv and av ). first, we define f =   s n β1(i + qiv ) + s n β2(a + qav ) 0 0 (1 −η1)v1n [ β1(i + qiv ) + β2(a + qav ) ] + (1 −η2)v2n [ β1(i + qiv ) + β2(a + qav ) ] 0 0   and v =   we −p1we + (δ1 + γ)i −(1 −p1)we + γa wev −p2wev + (δ2 + γ)iv −(1 −p2)wev + γav   . then, the derivative of f at a disease-free equilibrium p0 ∈s0 is f =   0 s ∗ n∗ β1 s∗ n∗ β2 0 s∗ n∗ qβ1 s∗ n∗ qβ2 0 0 0 0 0 0 0 0 0 0 0 0 0 (1 −η2) v ∗2 n∗ β1 (1 −η2) v ∗2 n∗ β2 0 (1 −η2) v ∗2 n∗ qβ1 (1 −η2) v ∗2 n∗ qβ2 0 0 0 0 0 0 0 0 0 0 0 0   where n∗ = s∗ +v ∗2 +r ∗ denotes the total population at the equilibrium. the derivative of v evaluated at p0 is v =   w 0 0 0 0 0 −p1w δ1 + γ 0 0 0 0 −(1 −p1)w 0 γ 0 0 0 0 0 0 w 0 0 0 0 0 −p2w δ2 + γ 0 0 0 0 −(1 −p2)w 0 γ   . it follows that fv −1 =   a11 s∗β1 n∗(δ1+γ) s∗β2 n∗γ a14 s∗qβ1 n∗(δ2+γ) s∗qβ2 n∗γ 0 0 0 0 0 0 0 0 0 0 0 0 a41 (1−η2)v ∗2 β1 n∗(δ1+γ) (1−η2)v ∗2 β2 n∗γ a44 (1−η2)v ∗2 qβ1 n∗(δ2+γ) (1−η2)v ∗2 qβ2 n∗γ 0 0 0 0 0 0 0 0 0 0 0 0   , 278 á. g. c. pérez and d. a. oluyori where a11 = s∗ n∗ [ β1p1 δ1 + γ + β2(1 −p1) γ ] , a14 = s∗ n∗ q [ β1p2 δ2 + γ + β2(1 −p2) γ ] , a41 = v ∗2 n∗ (1 −η2) [ β1p1 δ1 + γ + β2(1 −p1) γ ] , a44 = v ∗2 n∗ q(1 −η2) [ β1p2 δ2 + γ + β2(1 −p2) γ ] . the control reproduction number rc of model (2.1) is given by rc = ρ ( fv −1 ) , where ρ denotes the spectral radius. hence, rc = s∗ n∗ [ β1p1 δ1 + γ + β2(1 −p1) γ ] + v ∗2 n∗ q(1 −η2) [ β1p2 δ2 + γ + β2(1 −p2) γ ] . (3.1) the quantity rc measures the average number of new covid-19 cases generated by a typical infectious individual introduced into a population where a fraction v ∗2 /n ∗ has been fully vaccinated using a vaccine with efficacy η2. according to [33, theorem 2], we can obtain the following result about the control reproduction number. theorem 3.1. a disease-free equilibrium p0 = (s ∗, 0, 0, 0, 0,v ∗2 , 0, 0, 0,r ∗) of system (2.1) with v = 0 is locally asymptotically stable if rc < 1, and it is unstable if rc > 1. the epidemiological interpretation of theorem 3.1 is that, if rc < 1, then a small perturbation from a disease-free equilibrium will not generate an epidemic outbreak. on the other hand, if rc > 1, the epidemic curve will initially show an exponential growth, then reach a peak and start to decrease until becoming extinct. the following theorem gives a sufficient condition for the global stability of the disease-free equilibria. theorem 3.2. suppose that β1p1 δ1 + γ + β2(1 −p1) γ < 1 and q(1 −η2) [ β1p2 δ2 + γ + β2(1 −p2) γ ] < 1. (3.2) then, the disease-free equilibrium p0 = (s ∗, 0, 0, 0, 0,v ∗2 , 0, 0, 0,r ∗) of system (2.1) with v = 0 is globally asymptotically stable. proof. consider the following lyapunov function: l = g1e + g2i + g3a + g4ev + g5iv + g6av + g7v1, where g1 = γ(δ1 + γ), g2 = γβ1, g3 = (δ1 + γ)β2, g4 = γ(δ1 + γ) 1 −η2 , g5 = qγβ1(δ1 + γ) δ2 + γ , g6 = qβ2(δ1 + γ), g7 = γ(δ1 + γ) 1 −η2 . an extended seiard model for covid-19 vaccination 279 the time derivative of l evaluated at the solutions of system (2.1) with v = 0 is given by l̇ = g1 [ s n β1(i + qiv ) + s n β2(a + qav ) −we ] + g2 [ p1we − (δ1 + γ)i ] + g3 [ (1 −p1)we −γa ] + g4 [ (1 −η1) v1 n β1(i + qiv ) + (1 −η1) v1 n β2(a + qav ) + (1 −η2) v2 n β1(i + qiv ) + (1 −η2) v2 n β2(a + qav ) −wev ] + g5 [ p2wev − (δ2 + γ)iv ] + g6 [ (1 −p2)wev −γav ] + g7 [ −(1 −η1) v1 n β1(i + qiv ) − (1 −η1) v1 n β2(a + qav ) −θv1 ] . after cancelling terms and simplifying, we obtain l̇ = γβ1(δ1 + γ)(i + qiv ) ( s n + v2 n − 1 ) + γβ2(δ1 + γ)(a + qav ) ( s n + v2 n − 1 ) + wγ(δ1 + γ) [ β1p1 δ1 + γ + β2(1 −p1) γ − 1 ] e + wγ(δ1 + γ) 1 −η2 [ qβ1p2(1 −η2) δ2 + γ + qβ2(1 −p2)(1 −η2) γ − 1 ] ev − γθ(δ1 + γ) 1 −η2 v1. since s(t) + v2(t) ≤ n(t) for all t, we have sn + v2 n ≤ 1. combining this with the hypothesis (3.2), we can see that l̇ ≤ 0, and l̇ = 0 if and only if e(t) = 0 and ev (t) = 0. substituting e(t) = 0 and ev (t) = 0 in system (2.1) with v = 0 shows that (s,e,i,a,v1,v2,ev ,iv ,av ,r) → (s∗, 0, 0, 0, 0,v ∗2 , 0, 0, 0,r ∗) as t → ∞. hence, the largest positively invariant set where l̇ = 0 is the continuum of disease-free equilibria. therefore, by lasalle’s invariance principle, we conclude that the dfe is globally asymptotically stable. � 3.1. impact of vaccination coverage on the control reproduction number. next, we will study how the control reproduction number rc is affected by some of the model parameters. by equation (3.1), we know that rc does not only depend on the parameters of system (2.1), but also on the final proportions of unvaccinated susceptible people (s∗/n∗) and fully vaccinated people (v ∗2 /n ∗) at the time when vaccines are no longer being deployed to the population. we recall that a disease-free equilibrium takes the form p0 = (s ∗, 0, 0, 0, 0,v ∗2 , 0, 0, 0,r ∗), where the total population is n∗ = s∗ + v ∗2 + r ∗. if we define x = v ∗2 n∗ as the proportion of fully vaccinated people and y = r∗ n∗ as the proportion of people recovered from covid-19, we can rewrite the expression for the control reproduction number as rc(x,y) = [ β1p1 δ1 + γ + β2(1 −p1) γ ] (1 −x−y) + q(1 −η2) [ β1p2 δ2 + γ + β2(1 −p2) γ ] x. figure 2 depicts the value of rc as function of the proportions x and y, using several values for the transmission rates and efficacy after the second vaccine dose. other parameter values were taken as in table 1. we can see that an increase in either x or y contributes to reducing the reproduction number, and therefore, is helpful towards achieving herd immunity. 280 á. g. c. pérez and d. a. oluyori figure 2. value of the control reproduction number as function of the proportion of fully vaccinated individuals (horizontal axis) and recovered individuals (vertical axis). herd immunity occurs when a large portion of the population has become immune to the disease due to vaccination or natural recovery, which makes spread of the disease difficult. thus, the minimal level of vaccination coverage that is required to achieve herd immunity (that is, making rc < 1) will also depend on the percentage of the population that has been infected and then successfully recovered. comparing the different panels of figure 2, we can see that increasing the vaccine efficacy η2 reduces the vaccination coverage needed to make rc < 1 for a fixed proportion of recovered people. however, this reduction is small compared to the effect gained by decreasing the transmission rate. for example, when η2 = 0.65 and the recovered population is close to zero, it is necessary to vaccinate 46% of population to obtain rc = 1 in the case of 120% transmission rate, 33% in the case of baseline transmission rate, and only 11% of population in the case of 80% transmission rate (bottom row of figure 2). 4. numerical simulations in this section, we perform some numerical simulations for model (2.1) to provide estimates for the evolution of the covid-19 outbreak in mexico. 4.1. data fitting and estimation of parameters. we used cumulative data provided by the johns hopkins university repository [18] to fit the parameters of model (2.1) in the absence of vaccination. we considered the data for reported covid-19 infections, deaths and recovered cases during the period from 12 november 2020 to 24 december 2020, which is before the vaccination programme in mexico began. for this part, we considered system (2.1) with v = 0 and the vaccinated subpopulations v1, v2, ev , av and iv equal to zero. for the other variables, we used the initial conditions s(0) = 1.1938 × 108, e(0) = 1.58582 × 105, i(0) = 1.58582 × 105, a(0) = 1.1629 × 106, r(0) = 6.134975 × 106, d(0) = 97056. an extended seiard model for covid-19 vaccination 281 table 1. baseline values for the parameters used in the simulations. parameter description value source β1 transmission rate by contact with i and iv classes 0.2 day −1 fitted to data β2 transmission rate by contact with a and av classes 0.0330 day −1 fitted to data q relative infectivity of individuals in iv and av classes 0.52 [17] v vaccination rate variable assumed θ application rate of second doses 1/70 day−1 assumed η1 vaccine efficacy rate after one dose 0.463 [35] η2 vaccine efficacy rate after two doses 0.557 [35] w transfer rate from exposed to infectious 0.25 day−1 [29] p1 portion of people in e class that develop symptoms 0.12 [6] p2 portion of people in ev class that develop symptoms 0.089 estimated δ1 disease-induced death rate of i class 3.2135 × 10−3 day−1 fitted to data δ2 disease-induced death rate of iv class 0 assumed γ recovery rate of infectious individuals 3.6987 × 10−2 day−1 fitted to data the values for i(0), r(0) and d(0) were chosen based on the reported data for 12 november 2020. the value of a(0) was chosen so the symptomatic infections represent 0.12 times the total infections, i.e., i(0) = 0.12 [a(0) + i(0)]. the value of e(0) was assumed equal to i(0), and s(0) was estimated as s(0) = n −e(0) − i(0) −a(0) −r(0) −d(0), where n is the population of mexico. we regarded as fixed parameters w = 0.25, which corresponds to a latent period of 4 days [29], and a proportion p1 = 0.12 of symptomatic infections [6]. the set of differential equations was solved using matlab 2016b with the ode45 solver. the values for β1, β2, δ1 and γ were estimated by minimizing the sum of squared errors (sse), which is calculated as follows. for a given vector of parameters x, we compute numerically the i(t) and d(t) components of the solutions for our model, as well as the estimated number of people recovered from symptomatic infections ri(t) and the cumulative number of symptomatic infected cases c(t), defined by ṙi(t) = γi(t), c(t) = i(t) + ri(t) + d(t). then the sum of squared errors is given by sse(x) = n∑ i=1 [ k1 (c(ti) −c exp i ) 2 + k2 (d(ti) −d exp i ) 2 + k3 (ri(ti) −r exp i ) 2 ] , where c exp i , d exp i and r exp i denote the experimental data for cumulative infections, deaths and recoveries, respectively, reported for day ti (i = 1, . . . ,n), while k1, k2 and k3 are coefficients used to compensate the order of magnitude for the data. in our simulations, we used k1 = 20, k2 = 10 and k3 = 1. the global minimum of the sse function was obtained by applying three consecutive searches: a gradient-based method, a gradient-free algorithm and, again, a gradient-based method. the best fit values for β1, β2, δ1 and γ are shown in table 1. figure 3 depicts a comparison between the model solutions and the observed cumulative covid-19 data before the vaccination period. 4.2. simulations for the model with vaccination. we will now simulate the solutions to model (2.1) to assess the impact of the vaccination campaign that started in mexico in december 2020 to combat the covid-19 pandemic. as of august 2021, seven covid-19 vaccines have received emergency use authorization for their deployment in mexico: • bnt162b2 (pfizer–biontech), 282 á. g. c. pérez and d. a. oluyori nov 12 nov 19 nov 26 dec 03 dec 10 dec 17 dec 24 2020 0.8 1 1.2 1.4 p e o p le 10 6 cumulative number of symptomatic cases model solutions reported data nov 12 nov 19 nov 26 dec 03 dec 10 dec 17 dec 24 2020 0.8 1 1.2 1.4 p e o p le 10 5 death toll model solutions reported data nov 12 nov 19 nov 26 dec 03 dec 10 dec 17 dec 24 2020 6 8 10 12 p e o p le 10 5 recovered from symptomatic infection model solutions reported data figure 3. reported cumulative number of symptomatic cases, covid-19 deaths and recovered cases in mexico for the pre-vaccination period, and simulations using model (2.1) with the parameters in table 1 and v = 0. • azd1222 (oxford–astrazeneca), • sputnik v (gamaleya institute), • coronavac (sinovac), • bbv152 (covaxin), • ad5-ncov (cansino), and • ad26.cov2-s (johnson & johnson). the first five of these vaccines require two doses, while cansino and johnson & johnson are single-dose vaccines [15]. efficacy estimates for each vaccine based on data from clinical trials are subject to change with the emergence of new analyses. an interim analysis for the oxford–astrazeneca vaccine [35] estimated an efficacy against infection (symptomatic or asymptomatic) of 46.3% (31.8%–57.8%), considering people who had a nucleic acid amplification test (naat)-positive swab more than 21 days after a single dose, and 55.7% (41.1%–66.7%) for people who tested positive more than 14 days after a second dose of the vaccine. however, a more recent study [36] estimated an efficacy of 63.9% (46.0%–76.9%) after one dose and 59.9% (35.8%–75.0%) after two standard doses given 12 or more weeks apart. due to longer dose intervals being associated with greater efficacy against symptomatic infection, the who has recommended to administer the oxford–astrazeneca vaccine with an interval of 8 to 12 weeks between first and second doses [39]. based on the above, we will assume in our simulations an average length of 1/θ = 70 days for the inter-dose period, and we will use η1 = 0.463 and η2 = 0.557 as baseline values for the efficacy parameters. furthermore, we assume a reduction of 48% in the infectivity of individuals becoming infected after being vaccinated (i.e., q = 0.52), following the estimations in [17]. an extended seiard model for covid-19 vaccination 283 table 2. estimated values for the vaccination rate v. date value (day−1) 24 dec 2020 – 11 jan 2021 4.0 × 10−5 12 jan 2021 – 15 jan 2021 7.9 × 10−4 16 jan 2021 – 14 feb 2021 6.0 × 10−5 15 feb 2021 – 7 mar 2021 7.3 × 10−4 8 mar 2021 – 14 mar 2021 0.0021 15 mar 2021 – 31 mar 2021 0.0017 1 apr 2021 – 15 apr 2021 0.0035 16 apr 2021 – 15 may 2021 0.0017 16 may 2021 – 27 jul 2021 0.0060 jan 2021 mar 2021 may 2021 jul 2021 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p e o p le 10 7 0% 5% 10% 15% 20% 25% 30% 35% % o f m e x ic a n p o p u la ti o n vaccinated with at least one dose model solutions reported data jan 2021 mar 2021 may 2021 jul 2021 0 0.5 1 1.5 2 2.5 p e o p le 10 7 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% % o f m e x ic a n p o p u la ti o n fully vaccinated figure 4. covid-19 vaccination coverage in mexico from 24 december 2020 to 27 july 2021. x represents real data, continuous lines represent model simulations using the vaccination rate in table 2. for computing the proportion of infectious vaccinated individuals that show symptoms of the disease (p2), we follow [35], who reported 37 cases of symptomatic covid-19 disease out of a total of 68 naat-positive swabs in the group of people vaccinated with azd1222, and 112 symptomatic cases out of 153 naat-positive cases in the control group. this yields a reduction from 0.732 to 0.544 in the symptomatic proportion after vaccination. since we have chosen p1 = 0.12, we will take p2 = 0.089 so that p1 : p2 = 0.732 : 0.544. furthermore, we assume that the death rate δ2 of infectious vaccinated people is zero since it is widely accepted that current anti-covid vaccines provide full protection against severe infections. we used the daily data on covid-19 vaccinations in mexico obtained from [24] to estimate the value of the vaccination rate v over nine different date ranges, as shown in table 2. we plot in figure 4 a comparison of the reported number of vaccinated people and the simulations obtained with model (2.1) for the period 24 december 2020 – 27 july 2021. in these graphs, we considered the total population of mexico as 127 090 000 people. in order to obtain long-term projections for the vaccination coverage in mexico, we simulated two different scenarios. first, we assumed that the vaccination rate is kept constant at its baseline value on 27 july 2021 (0.6% of susceptible population per day, which equals roughly 294 000 first doses per day). second, we assumed that the vaccination rate increases to twice its baseline value starting on september 284 á. g. c. pérez and d. a. oluyori jan 2021 jul 2021 jan 2022 jul 2022 jan 2023 0 10 20 30 40 50 60 70 80 90 m il li o n s o f p e o p le 0% 10% 20% 30% 40% 50% 60% 70% % o f m e x ic a n p o p u la ti o n vaccinated with at least one dose reported data baseline vaccination rate 200% vaccination rate jan 2021 jul 2021 jan 2022 jul 2022 jan 2023 0 10 20 30 40 50 60 70 80 90 m il li o n s o f p e o p le 0% 10% 20% 30% 40% 50% 60% 70% % o f m e x ic a n p o p u la ti o n fully vaccinated figure 5. long-term projections of covid-19 vaccination coverage in mexico. x represents real data, continuous lines represent simulations using the baseline vaccination rate, and dashed lines represent simulations using 200% vaccination rate from september 2021 onwards. 2021. figure 5 shows that, if vaccines continue to be delivered at their baseline rate, the vaccination coverage with at least one dose will have reached 70% of mexican population by january 2023. on the other hand, if the vaccination rate is doubled, the same coverage with at least one dose could be achieved by may 2022, and 70% of the mexican population could be fully vaccinated by september 2022. 4.2.1. assessing the effect of vaccination and different transmission rates. we will next compute the solutions of model (2.1) to simulate the evolution of the pandemic in mexico as the vaccination campaign takes place. we consider the initial date for simulations as 24 december 2020. based on the results obtained in subsection 4.1, we use the initial conditions s(0) = 1.1622 × 108, e(0) = 3.4415 × 105, i(0) = 2.1247 × 105, a(0) = 1.6521 × 106, r(0) = 8.5421 × 106, d(0) = 1.2128 × 105, and v1(0) = v2(0) = ev (0) = iv (0) = av (0) = 0. in these subsection, we will consider different values for the transmission rates β1 and β2 to account for the possibility that the number of infectious contacts between people may increase or decrease due to resumption of economical activities, compliance with social/physical distancing, wearing of face masks, etc. hence, we consider three cases: when β1 and β2 are kept with the values in table 1, when both of them decrease to an 80% of these values, and when they increase to a 120%. the values for other parameters are fixed as in tables 1 and 2. figure 6 depicts the time evolution of the number of infectious covid-19 cases with symptoms (i(t)+ iv (t)) and the death toll (d(t)) for each of the above cases. in each graph, we have plotted the solutions assuming the baseline vaccination rate and the 200% vaccination rate, as well as a counterfactual case with no vaccination. figure 6(a) shows that, in the case of low transmission rate, the number of active cases would start to decrease in the early months of 2021, and the epidemic would be almost extinguished by january 2022. in the cases with higher transmission rate (figures 6(b) and (c)), the epidemic curve would reach its peak around may 2021, and the number of active symptomatic cases would be less than 1000 by may 2022. figures 6(d)–(f) show that the cumulative number of deaths would be around 250 000 for low transmission, 390 000 for baseline transmission, and 580 000 for high transmission rates. an extended seiard model for covid-19 vaccination 285 jan 2021 jan 2022 jan 2023 0 2 4 6 8 10 i( t) + i v (t ) 10 5 80% transmission rate no vaccination baseline vaccination rate 200% vaccination rate jan 2021 jan 2022 jan 2023 2 4 6 8 d (t ) 10 5 jan 2021 jan 2022 jan 2023 0 2 4 6 8 10 i( t) + i v (t ) 10 5 baseline transmission jan 2021 jan 2022 jan 2023 2 4 6 8 d (t ) 10 5 jan 2021 jan 2022 jan 2023 0 2 4 6 8 10 i( t) + i v (t ) 10 5 120% transmission rate jan 2021 jan 2022 jan 2023 2 4 6 8 d (t ) 10 5 (f) (c)(b)(a) (e)(d) figure 6. simulations of model (2.1) using different values for the transmission rates. (a) and (d): 80% transmission rate. (b) and (e): baseline transmission rate. (c) and (f): 120% transmission rate. top row: number of active symptomatic infectious cases. bottom row: cumulative number of deaths. we can also see that an increase in the vaccination rate to double its baseline value does not result in a considerable change in the number of infections or deaths, although the vaccination scenarios result in around 250 000 less deaths compared with the case with no vaccination. on the other hand, comparing figures 6(d) and (e) shows that more than 130 000 deaths can be avoided by reducing the transmission rate to 80%, while a 20% increase in the transmission rate would result in almost 200 000 additional deaths (figure 6(f)). this suggests that decreasing the number of infectious contacts by complying with preventive measures is more effective than simply accelerating the deployment of vaccines. 4.2.2. assessing the effect of different vaccine efficacy rates. given that there is still uncertainty regarding the efficacy of anti-covid vaccines against infection, including asymptomatic cases, we will also simulate the solutions of model (2.1) using different values for the parameters η1 and η2. figure 7 shows the number of active infectious cases with symptoms (i(t) + iv (t)) and without symptoms (a(t) + av (t)), as well as the death toll (d(t)), using different efficacy rates: in addition to the baseline case (η1 = 0.463, η2 = 0.557), we include a case with lower efficacy (η1 = 0.4, η2 = 0.45) and a case with higher efficacy (η1 = 0.6, η2 = 0.65). here, we have plotted all solutions using the baseline vaccination rate. the simulations show that lower efficacy results in an additional 7 429 symptomatic cases and 79 624 asymptomatic cases at the peak of the infection curve, compared with the case with higher efficacy. however, this does not significantly affect the time when the peak occurs. moreover, lower efficacy also results in 4 953 additional deaths. 5. conclusion in this work, we studied a model for covid-19 with vaccination. our work was based on the seiard model proposed in [6], which included an exposed (latent) compartment and different transmission rates for the symptomatic and asymptomatic infectious individuals; we extended this model by incorporating 286 á. g. c. pérez and d. a. oluyori jan 2021 jan 2022 jan 2023 0 1 2 3 4 5 i( t) + i v (t ) 10 5 symptomatic infectious 1 =0.4, 2 =0.45 1 =0.463, 2 =0.557 1 =0.6, 2 =0.65 jan 2021 jan 2022 jan 2023 0 0.5 1 1.5 2 2.5 3 3.5 a (t )+ a v (t ) 10 6 asymptomatic infectious jan 2021 jan 2022 jan 2023 1 1.5 2 2.5 3 3.5 4 d (t ) 10 5 death toll figure 7. simulations of model (2.1) using different values for the vaccine efficacy rates. left panel: number of active symptomatic infectious cases. central panel: number of active asymptomatic infectious cases. right panel: cumulative death toll. vaccinated compartments and considering a two-dose vaccination regime. although several covid-19 models with vaccination have been proposed in the literature, most studies have focused only on carrying out numerical simulations, while our work shed some light on the theoretical properties of model (2.1). when compared to other models analysed in the literature, we can see that the novelty of our model lies in the inclusion of a parameter q representing the reduction in infectivity due to vaccination, as well as the different probabilities (p1 and p2) of developing covid-19 symptoms depending on whether the infected person has been vaccinated. we showed that our model has multiple disease-free equilibria and computed the control reproduction number rc using the next-generation matrix method. we established that the set of disease-free equilibria is locally asymptotically stable when rc < 1 and unstable when rc > 1. furthermore, we determined a condition that guarantees the global asymptotic stability of the dfe. we performed a numerical simulation on our model using repository data on the outbreak of covid19 in mexico and the daily data on covid-19 vaccinations to estimate the value of the vaccination rate over nine different date ranges. we used the efficacy estimates based on data from clinical trials of the oxford–astrazeneca vaccine, which is the one that is being more widely distributed in mexico at the time of this writing. we remark that, in this article, we considered vaccine efficacy in the sense of protection against covid-19 infection (symptomatic or asymptomatic), while other works consider efficacy as protection against symptomatic infection only. we simulated two different scenarios to obtain projections for the vaccination coverage in the next few years. first, we assumed that the vaccination rate is kept constant by vaccinating the same proportion of susceptible individuals per day, and secondly, we assumed that the vaccination rate increases to twice its baseline value from september 2021 onwards. our study showed that if vaccines continue to be delivered at their baseline rate, by january 2023 the first dose will be applied to 90 million people, which represents roughly the total mexican population over age 18. on the other hand, if the vaccination rate is doubled, the total adult population could be partially vaccinated by may 2022 and fully vaccinated by september 2022. in the case of low transmission rate, the number of active cases would start to decrease in the early months of 2021, and the epidemic would be almost eradicated in early 2022, while in the cases with medium to high transmission rate the epidemic curve would reach its peak around may 2021 and would be close to zero by mid-2022. our simulations show that keeping a low transmission rate (by wearing face masks, complying with social/physical distancing, etc.) is the most effective method to reduce the death toll. for example, reducing the transmission rate to 80% its baseline value results in 130 000 less deaths, while doubling the an extended seiard model for covid-19 vaccination 287 vaccination rate does not yield a significant reduction in death toll. also, decreasing the transmission rate is more effective to reduce the control reproduction number and achieve herd immunity than deploying vaccines with higher efficacy rates. our model has certain limitations that could affect the results presented in this study. firstly, our model assumes that the time between first and second doses and the efficacy rate are the same for all vaccines applied in the population; 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[39] world health organization, the oxford/astrazeneca covid-19 vaccine: what you need to know, https://www.who.int/en/news-room/feature-stories/detail/the-oxford-astrazeneca-covid-19-vaccine-whatyou-need-to-know, accessed 25 march 2021. [40] world health organization, who advisory committee on variola virus research: report of the thirteenth meeting, tech. report, world health organization, 2011. https://github.com/cssegisanddata/covid-19/ https://github.com/cssegisanddata/covid-19/ https://www.who.int/en/news-room/feature-stories/detail/the-oxford-astrazeneca-covid-19-vaccine-whatyou-need-to-know an extended seiard model for covid-19 vaccination 289 corresponding author, facultad de matemáticas, universidad autónoma de yucatán, mérida, yucatán, mexico e-mail address: agcp26@hotmail.com department of mathematics, school of physical science, ahmadu bello university, zaria, kaduna state, nigeria e-mail address: oluyoridavid@gmail.com 1. introduction 2. model formulation 3. theoretical analysis 3.1. impact of vaccination coverage on the control reproduction number 4. numerical simulations 4.1. data fitting and estimation of parameters 4.2. simulations for the model with vaccination 5. conclusion references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 2, number 2, june 2021, pp.72-86 https://doi.org/10.5206/mase/11137 the replicator dynamics of generalized nash games jason lequyer and monica-gabriela cojocaru abstract. generalized nash games are a powerful modelling tool, first introduced in the 1950’s. they have seen some important developments in the past two decades. separately, evolutionary games were introduced in the 1960’s and seek to describe how natural selection can drive phenotypic changes in interacting populations. the dynamics of evolutionary games are frequently studied using the replicator equation, however there is no general theory about how to derive these kinds of dynamics for more complex games, such as generalized nash games. in this paper we extend and generalize the replicator equation by using an analogy with the projected dynamical system, and show how this extension can be used to derive a replicator equation for a wide range of problems. we also show how this extension enables us to to consider the dynamics of a new type of evolutionary games where we add shared inter-species contraints. 1. introduction nash games as we know them were first introduced in 1950 by nash [17] and have a wide array of applications in applied sciences, most notably economics and engineering. the generalized nash game, the subject of this paper, was described only four years later by arrow and debreu [1], but took much longer to unravel and has not yet gained the currency of its precursor. a generalized nash game (gn) seeks to describe a situation where each player’s choice of strategy somehow affects the choices available to his/her opponents. however, since everyone takes their turn all at the same time, this leads to games that cannot be played by normal people, at least not in the traditional sense. to illustrate, a classic example of a nash game is rock-paper-scissors. a gn version of this is rock-paper-scissors where ties are prohibited, i.e. if one player picks rock, another cannot also pick rock. but this game is impossible to play with another individual because a player cannot possibly know in advance what their opponent is going to pick, thus one cannot knowingly adhere to the mentioned restriction. for this reason, ichiishi calls them pseudo-games [13]. despite this artificiality, there are settings where outside forces ensure the satisfaction of constraints and, moreover, the model has explanatory value even in circumstances where this is not the case [9]. in general, it is difficult to find the equilibrium points of a gn. however the issue becomes easier if each player is subject to identical constraints (as far as variables under that player’s control are concerned). this is known as a gnsc (generalized nash game with shared constraints). there received by the editors november 10 2020; revised 3 march and 8 may 2021 respectively, accepted 9 may 2021; published online 18 may 2021. 2010 mathematics subject classification. primary 91a22. key words and phrases. evolutionary games; generalized nash games; variational inequalities; projected dynamical systems; replicator equation. monica-gabriela cojocaru was supported in part by national science and engineering research council (nserc)discovery grant #400684. 72 the replicator dynamics of generalized nash games 73 are many computational methods for finding some equilibria of the gnsc [9] and recent work gives a method for extracting all such points [15, 6]. another type of game we consider is the evolutionary game, first described by maynard smith [19]. evolutionary games seek to model the evolution of phenotypes as a function of natural selection, such as in the well-known hawk-dove game [12]. the dynamics of these games can often be described by the replicator equation of taylor and jonker [20]. in this paper we build a bridge between the most fundamental type of evolutionary game and the gnsc, by establishing a connection between their nash equilibria via the replicator equation. this bridge allows us to extend the existing model to accommodate new types of problems, as gnsc’s are richer and more diverse than population games. we build this bridge by extending the replicator equation so that it may be applied to gnscs and we derive this extension using an analogy between the replicator equation and what is known as the projected dynamical system. the projected dynamical system (pds) is a type of discontinuous dynamical system that was first introduced in the 70’s by henry [11], studied further in the 90’s by dupuis and nagurney [8] and extended in the early 2000’s by cojocaru and jonker [5] to hilbert spaces of any dimension. pds are intimately linked to gnscs in that the steady state set of a pds is a subset of the nash equilibria of its associated gnsc. the pds is useful to us in this paper because it gives a known, distinct, game dynamic that’s already applicable to evolutionary games and gnscs alike. the relationship between the projection dynamic and the replicator dynamic was first studied by lahkar and sandholm [18, 14]. in their papers, the authors elucidate similarities between the revision protocols implied by the two dynamics, and establish some properties of the solution trajectories of these systems for population games. in this paper we aim to expand this analogy beyond just population games, by extending the replicator equation and showing that a key theorem still holds that relates the rest points of our extended replicator equation to those of the projection dynamical system. in doing so, we allow for these two dynamics to be considered for gnscs, which are varied and more general than population games. to show that our extension of the replicator equation is useful, we prove a part of the folk theorem of evolutionary game theory, namely that every stable rest point of the replicator equation is a nash equilibrium of the corresponding population game [7, 12]. as such, our theorem 4.2 generalizes this aspect of the folk theorem so that it may be applied to any gnsc defined on a polytope. then, after associating these two concepts, we show how we can extend evolutionary games under our new framework in section 5. but to accomplish this, we first need a way to frame population dynamics problems as nash games, which we illustrate in section 3 for a standard nash game, and in section 5 for a gnsc. 1.1. contribution and significance. in this paper we extend the replicator equation for simple evolutionary games, so that it can be applied to gnscs. we show how with this extension, we can introduce shared inter-species constraints into evolutionary games and recover a corresponding replicator equation to study its dynamics. specifically, our contributions are as follows: (1) we show how the existing replicator equation can be written as a sum of projections for simple evolutionary games, in equation (4.1) . (2) we use this new way of writing the replicator equation to extend it from simple evolutionary games to gnscs in (4.3), and we show that a part of the folk theorem holds for this representation (theorem 4.2), which allows us to use this extended replicator equation to find nash equilibria of the associated gnsc 74 jason lequyer and monica-gabriela cojocaru (3) we show how we can add shared inter-species constraints to population games and find the corresponding replicator equation using this extension (example 5.2). 2. brief mathematical background 2.1. convex analysis. we will recall some basic definitions used in convex analysis, for ease of reading. most of these definitions and results are drawn from or based on those found in [3]. given a finite set of vectors β = {x1, . . . ,xm} where xi ∈ rn we say that a vector y is an affine combination of β if we can find λ1, . . . ,λm ∈ r such that y = λ1x1 + . . . + λmxm, λ1 + . . . + λm = 1. if each λi ≥ 0, then we say that y is a convex combination of β. a set k ⊆ rn is said to be convex if k contains every convex combination of vectors in k. given a set k ⊆ rn, we can construct a convex set by taking all convex combinations of vectors in k or construct an affine set by taking all affine combinations. we call these the convex hull and affine hull of k respectively and we formally define them as conv(k) = {y ∈ rn : y = λ1x1 + . . . + λmxm, ∑ λi = 1, λi ∈ r+, xi ∈ k}, aff(k) = {y ∈ rn : y = λ1x1 + . . . + λmxm, ∑ λi = 1, λi ∈ r, xi ∈ k}. the affine hull is important for defining the relative interior of a set. in optimization we are often working with low dimensional sets embedded in higher dimensional spaces, so we need a more general notion for the interior of a set: the relative interior fulfills this role. given some set k, the relative interior of k (ri(k)) is defined as ri(k) = {x ∈ k : b�(x) ∩aff(k) ⊆ k for some � > 0}, where b�(x) is the open ball of radius �, centered at x. we also often consider the normal cone of a convex set. given some convex set k ⊆ rn, we define the normal cone of k at some point x ∈ k to be nk(x) = {y ∈ rn : 〈y,x∗ −x〉≤ 0 ∀x∗ ∈ k}. in addition to convex sets, we also consider convex functions. given some convex set k ⊆ rn, a function θ : k → r is said to be convex if for every t ∈ [0, 1] and x1,x2 ∈ k, we have f(tx1 + (1 − t)x2) ≤ tf(x1) + (1 − t)f(x2) 2.2. polytopes. now we will give a brief review of polytopes. a bounded, convex polytope is defined as the convex hull of a non-empty finite set of real points. for simplicity, we will just call this a polytope for the remainder of our paper. let p be a polytope. the set s = {x1, . . . ,xn} such that p = conv(s) is not unique; however, there does exist a unique minimal spanning set. we call this set ext(p) and its elements are called vertices of p [14]. a convex subset f of p is called a face when for every distinct x,y ∈ p , if the line segment connecting x and y intersects f at some point other than x or y, then f contains the entire line segment. a face is itself a polytope, the convex hull of some subset of the vertices of p , therefore we can denote a face fr where r ⊆ ext(p). note that p = fext(p) is a face of itself. if the line segment connecting two vertices vi,vj is a face of k, we call them adjacent vertices and denote this relationship vi vj. theorem 2.1. suppose p is a polytope and fr is a face of p . let x ∗ ∈ fr and r = {v1, . . . ,vn}. then for every vi ∈ r, we have that span(fr −x∗) = span{vi −vj : vj ∈ r, vi vj}. proof. this follows from corollary 11.7 in [10]. � the replicator dynamics of generalized nash games 75 for any face fr, we define the dim(fr) as the dimension of the vector space associated with the affine hull of fr. equivalently, for every x ∗ ∈ fr, we have that dim(fr) = dim(span(fr −x∗)). theorem 2.2. suppose fr is a face of some polytope k and x ∈ fr. then fr is the lowest dimensional face containing x iff x ∈ ri(fr). proof. see [10], theorem 5.6. � if fr is a face of fr′ and dim(fr) < dim(fr′) we say that fr is a proper face of fr′. if dim(fr) = dim(fr′) − 1 we call fr a facet of fr′. theorem 2.3. suppose fr is a face of some polytope k and dim(fr) ≤ dim(k) − 2. then fr is the intersection of at least two facets of k with k. if dim(fr) = dim(k)−1 then fr is the intersection of exactly one facet of k with k. proof. see [10], theorem 10.4. � we say that two distinct faces fr and fr′ are adjacent if fr∩fr′ is nonempty and dim(fr) = dim(fr′). lemma 2.4. if fr is a facet of k, x ∈ fr and x /∈ ri(fr), then x ∈ fr′ for some facet fr′ adjacent to fr. proof. since x /∈ ri(fr), then by theorem 2.2 we can find a face fr∗ with dim(fr∗) < dim(fr) such that x ∈ fr∗. then dim(fr∗) ≤ dim(k) − 2, hence by theorem 2.3 fr∗ is the intersection of at least two facets of k. any such facet is adjacent to fr and contains x. � we can equivalently describe a convex bounded polytope k in terms of a set of affine functions h = {g1, . . . ,gn}, where k = {x : g1(x) ≥ 0, . . . ,gn(x) ≥ 0}. we call this the halfspace representation, and just like with the vertex representation, the choice of h is not unique. the fact that we can express bounded polytopes in these two different ways was first proved by krein and milman [16]. any face of k is just k intersected with some combination of hyperplanes gi(x) = 0 where gi ∈ h. given such a representation of k, we can define the faces based on which functions gi are nonzero somewhere on that face. specifically, we can denote the faces of k as fh where h = {g ∈ h : g(x) > 0 for some x ∈ fh} and it will hold that if fh is a face of fh ′ , then h ⊆ h′ with h = h′ iff fh = fh ′ . we will refer to the set h as the halfspace identifier of the face fh with respect to h. we now have both a top-down and bottom-up representation for a face. lemma 2.5. suppose fh is a face of some polytope k, where k has halfspace representation h ⊇ h = {g1, . . . ,gn}. then gi(x) > 0 for all x ∈ ri(fh), gi ∈ h. proof. let gi ∈ h. suppose we can find x∗ ∈ ri(fh) such that gi(x∗) = 0. since gi ∈ h, we can also find x ∈ fh such that gi(x) > 0. consider the line segment connecting x∗ and x: γ(λ) = (1 −λ)x + λx∗. for λ ∈ [0, 1]. since x∗ ∈ ri(fh) we can extend this segment beyond x∗ slightly so that γ(λ) ∈ fh for all λ ∈ [0, 1 + �] for some � > 0. since gi(x) is affine, this gives us: gi(γ(1 + �)) = gi((1 − (1 + �))x + (1 + �)x∗) = gi(x∗) + �(gi(x∗) −gi(x)) < 0, a contradiction. � 76 jason lequyer and monica-gabriela cojocaru 2.3. variational inequalities and projected dynamical systems. given a subset k of rn and a mapping f : rn → rn, the variational inequality problem (vi) is to find a solution x to the inequality (see [5]) 〈f(x),y −x〉≥ 0, ∀y ∈ k. (2.1) a projected differential equation is defined as follows: given some closed convex set k ⊆ rn, we define the projection operator pk at a point x, as the unique element pk(x) ∈ k such that ‖x−pk(x)‖≤‖x−y‖, ∀y ∈ k. we then define the vector projection operator from a vector v ∈ rn to a vector x ∈ k as πk(x,v) = lim δ→0+ pk(x + δv) −x δ . note that this is equivalent to taking the gateaux derivative of the projection operator onto k, in the direction of v. now, for some vector field −f : rn → rn, the class of differential equations known as projected differential equations takes the form ẋ = πk(x,−f(x)), x(0) ∈ k. (2.2) last but not least, it is known from [5], that if k is closed and convex then for any x∗ such that 0 = πk(x ∗,−f(x∗)) we have that 〈f(x∗),y −x∗〉≥ 0, ∀y ∈ k, and the converse is also true. in general the projected system we introduced here has a discontinuous righthand side, though existence and qualitative studies have been known since the first work by henry in 1970’s [11], aubin and cellina in the 80’s [2] and followed up by many others (see [8, 5], see [4] and the extensive references therein). there is a similar notion of a projected equation (see [2, 21]) defined by ẋ = pk(x−αf(x)), x(0) ∈ k for a given value of α > 0 real number. (2.3) the righthand side of this equation is continuous and, with good values of α, solutions of this projected equation amount to “smoothed out” (differential) approximations of the trajectories of (2.2). in general it is up to practitioners to choose one of the two versions of the projected system. if continuity in the equation’s vector field is desired, then equation (2.3) can be considered. we take here the point of view of the pds (2.2). 3. games and the replicator equation a generalized nash game (gn) is characterized by a finite set of players {p1, . . . ,pn}, where player pi controls the variable xi and has an objective function θi(x1, . . . ,xn). the goal of each player is to minimize their objective function subject to some constraint set xi ∈ ki(x−i), where x−i denotes (x1, . . . ,xi−1,xi+1, . . . ,xn). the key feature here is that each ki depends on variables beyond player i’s control. a nash equilibrium is any strategy (x1, . . . ,xn) where no player can lower their objective function by unilaterally altering their strategy, i.e. for every i ∈{1, . . . ,n}, for every x∗i ∈ ki(x−i), we have θi(xi,x−i) ≤ θi(x∗i ,x−i). here is the basic form of a generalized nash game: player 1(x1) : min θ1(x1,x−1) s. t. { x1 ∈ k1(x−1) , · · · , player n(xn) : min θn(xn,x−n) s. t. { xn ∈ kn(x−n) the replicator dynamics of generalized nash games 77 for the remainder of this paper we will assume that ki(x−i) is closed and convex and ∇θi is lipschitz continuous for each i. if there additionally exists a convex set k such that for each i we have that ki(x−i) = {xi : (xi,x−i) ∈ k}, then we call the game a gnsc, or a generalized nash game with shared constraints, named because in the case of sets defined by inequalities, it is easy to see that this restriction amounts to saying that each player’s strategy set ki(x−i) can be defined by the exact same set of inequality constraints. now, let us turn to a more specific kind of problem, evolutionary games. an evolutionary game is a game where there is a population of agents, whose strategies evolve according to some rule, that may model various adaptation processes. in the simple two-player symmetric case, evolutionary games are matrix games where each member of a population has a choice among n pure strategies {e1, . . . ,en}. in the associated matrix a, we have that aij = π(ei,ej) is just the payoff of playing pure strategy ei in a population that exclusively plays strategy ej [7]. all strategies must belong to the simplex ∆n = {(p1, . . . ,pn) : ∑n i=1 pi = 1, pi ≥ 0}. a nash equilibrium is any strategy x ∈ ∆ n such that [12] π(p,x) ≤ π(x,x) for every p ∈ ∆n. although these games are usually described in the literature as we did above, it is easy to see that they are essentially solving the following nash game: player 1(x) : max xtay s. t. { x ∈ k , player 1’(y) : min 1 2 (y −x)t (y −x) s. t. { y ∈ rn, where k = ∆n. in this system y is just a shadow variable that tests strategy x against all other strategies, and its objective function ensures x = y at all solutions. this system is known from [9] to share its nash equilibria with the solutions to the variational inequality( x∗ y∗ ) : 〈( −ay∗ y∗ −x∗ ) , ( x y ) − ( x∗ y∗ )〉 ≥ 0, ∀ ( x y ) ∈ k ×rn. (3.1) note that at any solution to (3.1) we must have that y∗−x∗ = 0, otherwise we could always find some y ∈ rn to make the inner-product negative. therefore this variational inequality shares its solutions (in x) with the problem find x∗ s.t. : 〈−ax∗,x−x∗〉≥ 0, ∀x ∈ k, (3.2) which is known from [8] to share its solutions with the rest points of the projected dynamical system ẋ∗ = πk(x ∗,ax∗). for simplicity of notation, let us denote x := x∗ going forward, hence we write ẋ = πk(x,ax). (3.3) it is also known [7] that the replicator equation associated to our game is ẋi = xi(e t i ax−x tax), i ∈{1, . . . ,n} (3.4) we therefore have two different dynamics we can use to study evolutionary games, (3.3) and (3.4). in [18, 14], the authors throughly study the relationship between these two dynamics and the revision protocol implied by each. we would like to extend these dynamics to the much more general problem of gnscs. the projection dynamic of course, is already used extensively in the study of gnscs. however the replicator equation has not to our knowledge ever been applied to gnscs; equation (3.4) only gives 78 jason lequyer and monica-gabriela cojocaru us the dynamics of very elementary evolutionary games. versions of (3.4) have been adapted for more sophisticated kinds of population games (see [7] for an overview), however so far there is no analogue for anything as broad as gnscs. in the next section we build this analogue, by devising a method to derive a replicator equation from a given projected dynamical system. 4. results 4.1. extending the replicator equation. it is known by the folk theorem of evolutionary game theory [7], that any stable rest point of (3.4) in k = ∆n must be a nash equilibrium. this implies that such a point is also a rest point of (3.3), which raises the question: what is the relationship between the system in (3.4) and the system in (3.3)? notice that we can rewrite (3.4) in the following way ([·] denotes the iverson bracket) ẋ = n∑ i=1 n∑ j=1 xixj(ax · (ei −ej))(ei −ej)[i < j], (4.1) where ei and ej are just the coordinates of two adjacent vertices on the simplex ∆ n, ax·(ei−ej))(ei−ej) is the un-normalized projection of ax onto the line connecting these two vertices and xi and xj are just the constraints which aren’t identically zero on that line. this system is similar to (3.3), however we exclusively project onto the edges of the polytope k instead of the tangent cone of the entire set. this system is continuous, but the cost of this continuity is that we generate new rest points which are not necessarily equilibria of the original game. however we can find some, but not all, of these equilibria via stability tests [7]. in the spirit of this process, let us now try and apply this technique to a more general type of problem. suppose k is a bounded convex polytope in rn (see chapter 2 for background on polytopes). then k has some half-space representation h = {g1, . . . ,gk}, where each gi is affine and k = {x : g1(x) ≥ 0, . . . ,gk(x) ≥ 0}. let −f : rn → rn be lipschitz continuous. consider the projected differential equation ẋ = πk(x,−f(x)) (4.2) number the vertices of k as {v1, . . .vm}. for each vi vj, we can find a halfspace identifier in h for the edge connecting these two vertices hij ⊆ h. let gij = ∏ g∈hij g. then, mirroring the procedure we used with the replicator equation, we can consider the following classical dynamical system ẋ = m∑ i=1 m∑ j=1 gij(x)((vi −vj) · (−f(x))(vi −vj)[i < j and vi vj]. (4.3) note that we are essentially performing vector projection onto each edge, without the usual normalizing term. (4.3) is our proposed way of extending the replicator equation to gnscs. it is continuous, just like the original replicator equation, however there is no guarantee that our extension has any relation at all to the associated gnsc, since we have no idea whether the folk theorem holds for this system. while proving the folk theorem was trivial for the case of elementary evolutionary games (see [12] for a short and simple proof), there is no obvious way to extend this proof to gnscs. therefore the next subsection is dedicated to proving that a result analogous to a part of the folk theorem holds for our system in (4.3) (theorem 4.2). equipped with this theorem we can then relate the rest points of our system in (4.3) to the nash equilibria of our original gnsc, which we do in section 5. the replicator dynamics of generalized nash games 79 4.2. connecting the extended replicator equation to gnscs. for absolute clarity, in the theorems that follow when we say stable rest point we mean the usual definition: a rest point x∗ is stable if and only if, for every � > 0, there exists a δ > 0 such that for every solution x(t), if t0 is such that ‖x(t0)−x∗‖ < δ, then ‖x(t)−x∗‖ < � for every t ≥ t0. also, when we say face invariant, we mean that if any solution x(t) lies on some face at time t0, then it will remain on that face for all future t ≥ t0 (usually called forward invariance). theorem 4.1. the system in (4.3) is face invariant. theorem 4.2. a stable rest point of (4.3) is a rest point of (4.2). before we can prove these theorems, we need four more lemmas about polytopes. lemma 4.3. let fr be a face of some polytope k and x ∗ ∈ fr. we have that span(fr−x∗)∩(k−x∗) = fr −x∗. proof. (⇒) assume x ∈ span(fr −x∗) ∩ (k −x∗) and x /∈ fr −x∗. now consider any y ∈ ri(fr −x∗). by convexity we have that λx + (1 − λ)y ∈ span(fr − x∗) ∩ (k − x∗) for every λ ∈ [0, 1]. since y ∈ ri(fr −x∗), then for some λ ∈ (0, 1) we must have that λx + (1 −λ)y ∈ fr −x∗. therefore by the definition of a face, fr contains x, a contradiction. the (⇐) direction is obvious. � lemma 4.4. suppose fr1 and fr2 are two adjacent facets of fr′. then for every x ∗ ∈ fr1 ∩fr2 , we have that span(fr′ −x∗) = span((fr1 −x ∗) ∪ (fr2 −x ∗)). proof. clearly span(fr1 −x∗) and span(fr2 −x∗) are subspaces of span(fr′ −x∗). since fr is a facet of fr′, then dim(fr1 −x∗) + 1 = dim(fr′ −x∗) = dim(fr2 −x∗) + 1. therefore it suffices to show that span(fr1 −x∗) 6= span(fr2 −x∗) which follows easily from lemma 4.3. � lemma 4.5. suppose fr is a facet of fr′. then for every x ∗ ∈ fr there is a unit vector nr(x∗) ∈ (fr′ −x∗) called the inner-normal of fr at x∗ on fr′, such that for every v ∈ span(fr′ −x∗) nr(x ∗) ·u = 0, for every u ∈ span(fr −x∗) (4.4) v = u + knr(x ∗), for some u ∈ span(fr −x∗), k ∈ r (4.5) if x∗ ∈ ri(fr), then nr(x∗) is a feasible direction, (4.6) with k > 0 for v ∈ (fr′ −x∗) \ (fr −x∗). proof. clearly span(fr − x∗) is a subspace of span(fr′ − x∗). since fr is a facet of fr′, then dim(fr − x∗) = dim(fr′ − x∗) + 1. then we can take any orthonormal basis of span(fr − x∗) and extend it to an orthonormal basis of span(fr′ −x∗) by the addition of a single vector, call it w. suppose that v1, v2 ∈ (fr′ − x∗) \ (fr − x∗). then from the last paragraph, we know we can write v1 = u1 + k1w and v2 = u2 + k2w, with u1, u2 ∈ span(fr −x∗) and k ∈ r. if k1 = 0 or k2 = 0, this would contradict lemma 4.3. assume that k1 < 0 < k2. then the line connecting v1 and v2 intersects fr, but contains points that aren’t in fr, contradicting the fact that fr is a face. therefore k1 and k2 must have the same sign, and so by choosing nr(x ∗) = w or nr(x ∗) = −w as appropriate, we get that (4.4) and (4.5) hold. now suppose x∗ ∈ ri(fr). then for p > 0 sufficiently small, we will have that −pu1 ∈ (fr − x∗). hence by convexity, λv1 + (1 − λ)(−pu1) ∈ (fr′ − x∗) for all λ ∈ [0, 1]. and we can choose our λ so that λv1 + (1 −λ)(−pu1) = λknr, proving (4.6). � 80 jason lequyer and monica-gabriela cojocaru lemma 4.6. suppose fr is a polytope, and fr′ is a proper face of fr. then for every vi ∈ r′ there exists vj ∈ r \r′ such that vi vj. proof. let x∗ ∈ fr, vi ∈ r′. from theorem 2.1 we know that span(fr − x∗) = span{vi − vj : vj ∈ r, vi vj}. if there doesn’t exist vj ∈ r \r′ with the desired property then span{vi −vj : vj ∈ r, vi vj} = span{vi −vj : vj ∈ r′, vi vj}, and therefore span(fr −x∗) = span(fr′ −x∗), a contradiction. � equipped with the above lemmas, we can now prove theorem 4 and 5. proof of theorem 4.1. let fh = fr be the lowest dimensional face of k such that x ∈ fh. suppose fh ′ = fr′ = conv(vi,vj) 6⊆ fr = fh. then h′ 6⊆ h. therefore gij(x) = 0. taking this together with theorem 2.2 ensures that we remain on fh and therefore every face to which x belongs. � proof of theorem 4.2. if k is a singleton, then the result is trivial, therefore assume dim(k) ≥ 1. suppose x∗ is not a rest point of (4.2), but is a rest point of (4.3). then −f(x∗) /∈ nk(x∗). let fr be the lowest dimensional face containing x∗ such that −f(x∗) /∈ nfr (x∗). we have that ẋ∗ = m∑ i=1 m∑ j=1 gij(x ∗)((vi −vj) · (−f(x∗))(vi −vj). (4.7) where the sum is over all i and j such that i < j, vi vj and vi,vj ∈ r. since x∗ is a rest point of (3.4) we must have that ẋ∗ = 0. if x ∈ ri(fr), this means that gij(x∗) > 0 for every vi,vj ∈ r (lemma 2.5). thus (vi −vj) · (−f(x∗)) = 0 for every vi,vj ∈ r, vi vj. however since for any fixed vi we have that fr − x∗ ⊆ span{vi − vj : vj ∈ r , vi vj} (theorem 2.1) then this contradicts the fact that −f(x∗) /∈ nfr (x∗). therefore x∗ /∈ ri(fr), hence there must exist some facet fr∗ of fr such that x ∈ fr∗ (theorem 2.2). now suppose x∗ /∈ ri(fr∗). then we can find another facet, fr′ adjacent to fr∗ such that x∗ ∈ fr′ (lemma 2.4). we must have that −f(x∗) ∈ nfr∗ (x ∗) and −f(x∗) ∈ nfr′ (x ∗) (otherwise we’ve found a lower dimensional face that doesn’t have this property). however by lemma 4.4, this implies that −f(x∗) ∈ nfr (x∗), a contradiction. therefore x∗ ∈ ri(fr∗) and x∗ belongs to only one facet of fr. let nr∗(x ∗) be the unit inner-normal of fr∗ on fr at x ∗ (obtained from lemma 4.5). since −f(x∗) /∈ nfr (x ∗), then for some v ∈ fr −x∗, we have that −f(x∗) ·v > 0. we can write v = u + knr∗(x∗) for some u ∈ fr∗ − x∗, k ∈ r (lemma 4.5). since −f(x∗) ∈ nfr∗ (x ∗), then v ∈ (fr − x∗) \ (fr∗ − x∗). therefore k > 0 (lemma 4.5). we have −f(x∗) ·v = −f(x∗) ·u + knr∗(x∗) = k(−f(x∗) ·nr∗(x∗)) > 0. thus −f(x∗) ·nr∗(x∗) > 0. for all vi ∈ r, vj ∈ r, v1 vj we have that vi −vj = u + knr∗(x∗), for some k ∈ r (theorem 2.1/lemma 4.5). hence −f(x∗) · (vi −vj) = (−f(x∗) ·nr∗(x∗))(nr∗(x∗) · (vi −vj)). thus (−f(x∗) · (vi −vj))(nr∗(x∗) · (vi −vj)) = (−f(x∗) ·nr∗(x∗))(nr∗(x∗) · (vi −vj))2, which is ≥ 0. now fix vp ∈ r, vq ∈ r∗ \r such that vp vq (exists by lemma 4.6). by lemma 4.3 we have that nr∗(x ∗) · (vp −vq) 6= 0. this implies that (−f(x∗) ·nr∗(x∗))(nr∗(x∗) · (vp −vq))2 > 0. then by continuity of f, we can find an open ball b�0 (x ∗) in fr such that for some γ > 0 γ < (−f(x) ·nr∗(x∗))(nr∗(x∗) · (vp −vq))2. the replicator dynamics of generalized nash games 81 we can further constrain �0 so that b�0 (x ∗) ∩ fr∗ ⊆ ri(fr∗) and b�0 (x∗) contains no other facets of fr aside from fr∗ (we can do this second part because x ∗ belongs to only one facet). therefore within this neighbourhood we have d((x−x∗) ·nr∗(x∗)) dt = m∑ i=1 m∑ j=1 gij(x)((vi −vj) ·−f(x))((vi −vj) ·nr∗(x∗)) = m∑ i=1 m∑ j=1 gij(x)(−f(x) ·nr∗(x∗))(nr∗(x∗) · (vi −vj))2 ≥ gpq(x)(−f(x) ·nr∗(x∗))(nr∗(x∗) · (vp −vq))2 ≥ gpq(x)γ. where the sums are over all i and j such that i < j, vi vj and vi,vj ∈ r. consider any �, δ such that 0 < δ ≤ � < �0. we know that nr(x∗) is a feasible direction by lemma 4.5, therefore let y(0) = k0nr(x∗) be a solution to the ivp, where k0 > 0 is sufficiently small so that y(0) ∈ (b�(x∗)∩fr) ⊆ (b�0 (x∗)∩fr). consider the set u = {x ∈ (b�0 ∩ fr) : (x − x∗) · nr ≥ k0}. clearly u ∩ fr∗ = {}; hence u contains no facets of fr. thereore u ⊆ ri(fr) (theorem 2.2). thus we can find λ > 0 such that gpq(x) > λ for all x ∈ u (continuity and lemma 2.5). since d((x−x ∗)·nr) dt > 0 on ri(fr) and (4.2) is face invariant (theorem 4.1) we know that y(t) is also u invariant. therefore d((y −x∗) ·nr) dt ≥ gpq(x)γ ≥ λγ, contradicting lyapunov stability. � note that the converse to theorem 4.2 is not true (see [12], exercise 7.2.2). with this theorem we show that stable rest points of our extended replicator equation are rest points of the projection dynamic. since it is known that the rest points of pds are nash equilibria, then this taken together with theorem 4.2 allows us to ultimately say that stable rest points are nash equilibria, and hence a key part of the folk theorem applies to our extension of the replicator equation. while lahkar and sandholm [17] could rely on an already established link between their games and the replicator equation, we needed to reprove that this link is still there for our extension. our hope is that this result potentially paves the way for the type of analyses conducted by lahkar and sandholm [18] to be extended into this more general setting. 5. examples and extensions in this section we will consider examples that illustrate how (4.3) achieves three basic purposes. first, it recovers the standard replicator equation, showing that what we have is in fact a generalization of that concept. second, it allows us to incorporate the shared constraints of generalized nash games into elementary evolutionary games. finally, it enables us to express a given gnsc as a classical dynamical system, regardless of whether that gnsc corresponds to any particular evolutionary game. we should recall that our method for finding the extended replicator equation associated to a game and vice versa, consists of several steps. for clarity, and since they are used to solve the examples that follow, we will enumerate these steps. first, to find the extended replicator equation associated to a generalized nash game we must (call this method 1): (1) find the variational inequality associated to the generalized nash game (2) find the projected dynamical system associated to this variational inequality 82 jason lequyer and monica-gabriela cojocaru (3) find the replicator equation associated to the projected dynamical system (as per (4.3)) to take an evolutionary game and find the extended replicator equation we must (call this method 2): (1) express the evolutionary game as a generalized nash game (as at start of section 4) (2) find the variational inequality associated to the generalized nash game (3) drop the shadow variables (as in the paragraph after (3.1)) (4) find the projected dynamical system associated to the variational inequality (5) find the replicator equation associated to the projected dynamical system (as per (4.3)) we can of course skip variational inequalities, and just move straight to a projected dynamical system, however we include this procedure for clarity of analysis and to better mirror our exposition in the previous sections. we will now apply these steps to three examples. example 5.1. the 1-species evolutionary game in (2.1) can be extended to an arbitrary number of species [7]. in the simplest case, we have two species, call them species a and species b, each of whose members can choose between n and m possible pure strategies {e1, . . . ,en} and {f1, . . . ,fm} respectively. in this case we have two associated matrices a ∈ rn×(m+n) and b ∈ rm×(m+n), where eti a(p,q) represents π1(ei, (p,q)), the payoff of playing pure strategy ei in a population where species a adopts mixed strategy p and species b adopts mixed strategy q. π2(fi, (p,q)) has a similar meaning for species b. the strategies of species a and species b respectively must belong to the simplices ∆n = {(p1, . . . ,pn) : ∑n i=1 pi = 1, pi ≥ 0} and ∆ m = {(q1, . . . ,qm) : ∑m i=1 qi = 1, qi ≥ 0}. a nash equilibrium is defined as any strategy x ∈ ∆n and y ∈ ∆m such that [6] π1(p, (x,y)) ≤ π1(x, (x,y)) for every p ∈ ∆n, π2(q, (x,y)) ≤ π2(y, (x,y)) for every q ∈ ∆m. it is known that the replicator equations associated to this problem are [7] ẋi = xi(π1(ei, (x,y)) −π1(x, (x,y))) for i = 1, . . . ,n, ẏj = yj(π2(fj, (x,y)) −π2(y, (x,y))) for j = 1, . . . ,m. if we assume each player can choose between two possible strategies, this reduces to ẋ1 = x1x2(e1a(x,y) −e2a(x,y)), ẋ2 = x1x2(e2a(x,y) −e1a(x,y)), ẏ1 = y1y2(e1b(x,y) −e2b(x,y)), ẏ2 = y1y2(e2b(x,y) −e1b(x,y)), (5.1) for some a,b ∈ r2×2. first we will attempt to derive these dynamics using our extension. we will do this by using method 2, as described at the start of this section. first we express it as a gn: player 1(x) : max xta(x′,y′), s. t. { x ∈ k1 player 1’(x′) : min 1 2 (x′ −x)t (x′ −x), s. t. { y ∈ r2 player 2(y) : max ytb(x′,y′), s. t. { y ∈ k2 player 2’(y′) : min 1 2 (y′ −y)t (y′ −y), s. t. { y ∈ r2, (5.2) where k1 = ∆ 2 = k2. let k = k1 × k2. then this problem shares its nash equilibria with the solutions (x∗,x ′∗,y∗,y ′∗) of the variational inequality  x∗ x ′∗ y∗ y ′∗   : 〈 −a(x ′∗,y ′∗) x ′∗ −x∗ −b(x ′∗,y ′∗) y ′∗ −y∗   ,   x x′ y y′  −   x∗ x ′∗ y∗ y ′∗   〉 ≥ 0, ∀   x x′ y y′   ∈ k1 ×r2 ×k2 ×r2. (5.3) the replicator dynamics of generalized nash games 83 note that at any solution of (5.3) we must have that x ′∗−x∗ = 0 = y ′∗−y∗, otherwise we could always find some x′,y′ ∈ r2 to make the inner-product negative. therefore this variational inequality shares its solutions (in (x∗,y∗)) with (x∗,y∗) : 〈(−a(x∗,y∗),−b(x∗,y∗)), (x,y) − (x∗,y∗)〉≥ 0, ∀(x,y) ∈ k, (5.4) and hence these solutions correspond to the rest points of the projected dynamical system ˙(x∗,y∗) = πk((x ∗,y∗), (a(x∗,y∗),b(x∗,y∗))). for simplicity of notation, we will denote x := x∗ and y := y∗ going forward. now k is a polytope in r4 with vertices v1 = (1, 0, 1, 0), v2 = (1, 0, 0, 1), v3 = (0, 1, 1, 0), v4 = (0, 1, 0, 1). using the halfspace representation h = {x1,x2,−x1 −x2 + 1,x1 + x2 −1,y1,y2,−y1 −y2 + 1,y1 + y2 −1}, we can then find the corresponding extended replicator equation ˙(x,y) = x1y1y2((v1 −v2) · (a(x,y),b(x,y)))(v1 −v2) +y1x1x2((v1 −v3) · (a(x,y),b(x,y)))(v1 −v3) +y2x1x2((v2 −v4) · (a(x,y),b(x,y)))(v2 −v4) +x2y1y2((v3 −v4) · (a(x,y),b(x,y)))(v3 −v4) notice that v1 −v2 = v3 −v4 and v1 −v3 = v2 −v4, hence: ˙(x,y) = (x1 + x2)y1y2((v1 −v2) · (a(x,y),b(x,y)))(v1 −v2) +(y1 + y2)x1x2((v1 −v3) · (a(x,y),b(x,y)))(v1 −v3), but x1 + x2 = 1 = y1 + y2 everywhere on k, therefore ˙(x,y) = y1y2((v1 −v2) · (a(x,y),b(x,y)))(v1 −v2) +x1x2((v1 −v3) · (a(x,y),b(x,y)))(v1 −v3). simplifying this, we get ẋ1 = x1x2(e1a(x,y) −e2a(x,y)), ẋ2 = x1x2(e2a(x,y) −e1a(x,y)) ẏ1 = y1y2(e1b(x,y) −e2b(x,y)), ẏ2 = y1y2(e2b(x,y) −e1b(x,y)), which is the known replicator equation (5.1). example 5.2. now let us try to use our method to extend the model. suppose we want to implement the shared constraint x2 + y2 ≤ 1 (perhaps being a “2-strategist” consumes some finite resource). then proceeding again by method 2, our generalized nash game then becomes: player 1(x) : max xta(x′,y′) s. t. { x ∈ k(y) player 1’(x′) : min 1 2 (x′ −x)t (x′ −x) s. t. { y ∈ r2 player 2(y) : max ytb(x′,y′) s. t. { y ∈ k(x) player 2’(y′) : min 1 2 (y′ −y)t (y′ −y) s. t. { y ∈ r2, (5.5) where k(y) = ∆2 ∩{x : x2 + y2 ≤ 1} and k(x) = ∆2 ∩{y : x2 + y2 ≤ 1}. therefore (x,y) ∈ k where k = ∆2∩{(x,y) : x2 +y2 ≤ 1}. then some of the nash equilibria for this game (in (x,y)) are solutions (x∗,y∗) to the variational inequality (x∗,y∗) : 〈(−a(x∗,y∗),−b(x∗y∗)), (x,y) − (x∗,y∗)〉≥ 0, ∀(x,y) ∈ k. (5.6) now, unfortunately our shared constraint means that although every solution to (5.6) is a solution to (5.5), the set of solutions to (5.5) may be much larger than this. we can however extend this variational inequality to a family of variational inequalities using the parametric method in [6, 15] to recover many 84 jason lequyer and monica-gabriela cojocaru solutions of (5.5). for simplicity we will only implement our shared constraint x2 + y2 ≤ 1 on (5.3) however this method can be extended to each member of the family of variational inequalities in [6]. now, (5.6) shares its solutions with the rest points of the projected dynamical system ˙(x∗,y∗) = πk((x ∗,y∗), (a(x∗,y∗),b(x∗,y∗)). notice that k is a bounded polytope with vertices: v1 = (1, 0, 1, 0), v2 = (1, 0, 0, 1), v3 = (0, 1, 1, 0). therefore, again denoting x := x∗ and y := y∗, and using the halfspace representation h = {x2,y2, 1− x2 −y2,−x1 −x2 + 1,x1 + x2 − 1,−y1 −y2 + 1,y1 + y2 − 1} we find the extended replicator equation ˙(x,y) = y2(1 −x2 −y2)((v1 −v2) · (a(x,y),b(x,y)))(v1 −v2) + x2(1 −x2 −y2)((v1 −v3) · (a(x,y),b(x,y)))(v1 −v3) + x2y2((v2 −v3) · (a(x,y),b(x,y)))(v2 −v3), if we set u = e1a(x,y)−e2a(x,y) and v = e1b(x,y)−e2b(x,y), then with a bit of algebra and invoking the fact that x1 + x2 = 1 = y1 + y2, we can arrive at the differential equation ẋ1 = x2(x1u−y2v), ẋ2 = −x2(x1u−y2v) ẏ1 = −y2(x2u−y1v), ẏ2 = y2(x2u−y1v), and by theorem 4.2, we know that the stable rest points of this classical dynamical system will be nash equilibria of our original game. we can therefore call this the replicator equation associated to the game. example 5.3. this extension does not just help us with evolutionary games, it can be used as an alternative way to consider the dynamics of any gnsc. for example, consider the following two player game: player 1(x) : min 1 2 (x− 2)2 s. t. { −x ≤ 0 x + y − 1 ≤ 0 , player 2(y) : min 1 2 (y − 3)2 s. t. { −y ≤ 0 x + y − 1 ≤ 0 (5.7) the set of nash equilibria for this game is easily calculated to be {(t, 1 − t) : 0 ≤ t ≤ 1}. notice that this problem is a gnsc, with global constraint set k = {(x,y) ∈ r2≥0 : x+y−1 ≤ 0}. then proceeding by method 1, we find the corresponding variational inequality: (x∗,y∗) : 〈(x∗ − 2,y∗ − 3), (x,y) − (x∗,y∗)〉≥ 0, ∀(x,y) ∈ k, (5.8) and then find the corresponding pds: ˙(x∗,y∗) = πk((x ∗,y∗), (−x∗ + 2,−y∗ + 3)). now the vertex representation of k is {(0, 0), (0, 1), (1, 0)} and a halfspace representation is {x ≥ 0,y ≥ 0, 1 −x−y ≥ 0}. thus we can find the corresponding extended replicator equation ẋ = x((x− 2)(x− 1) + y(y − 3)), ẏ = y(x(x− 2) + (y − 3)(y − 1)), whose only stable rest point on k is (0, 1), which does correspond to a rest point of our pds and to the original game (in fact this is the only rest point of this pds). the replicator dynamics of generalized nash games 85 6. conclusions and future work in this paper we generalize the replicator equation so that it may applied to any gnsc defined on a polytope. theorem 4.2 relates the stable rest points of this extended replicator equation with the rest points of a projected differential equation. this connection allows us to expand certain evolutionary games by introducing shared inter-species constraints via the gnsc. currently there are many different variations of the standard two player population game, for example games where the payoff is not a matrix or multiplayer games (see [7] for an overview), each of which has its own version of the folk theorem of evolutionary game theory. if further work is done to adapt parts 2 and 3 of the folk theorem to our model, then we would have a complete general version of the folk theorem for which these all could be considered special cases. in the literature, the replicator equation has already been unified with other models such as the price equation and the generalized lotka-volterra equation [14]. with this result we make yet another such connection, however it should be noted that generalized nash games are extremely broad and are a superset of all classical nash games. we also point out that our connection is reciprocal, we don’t just place the replicator equation under the umbrella of the projected dynamical system, it actually gives us a new way of looking at gnscs. this new perspective is an alternative to the projected dynamical system, in that the replicator equation frames these problems as classical (with righthand sides of class c1) dynamical systems. further work could investigate whether our results hold on an arbitrary convex set, not just a convex bounded polytope. we are optimistic that such a generalization is achievable, and it should be noted that the projected dynamical system itself was originally only shown to work on polyhedral constraints in [8] before it was shown to apply to arbitrary convex sets 11 years later in [5]. another possible direction would be adapting our method to the much broader class of generalized nash games without shared constraints. this would perhaps be the most ambitious way to continue since there are still large theoretical obstacles that need to be overcome before we can solve these types of games in general. more specifically, we would need to find a way to determine the replicator dynamics of a quasivariational inequality, which is a much less well understood mathematical object. references 1. k. j. arrow and g. debreu, existence of an equilibrium for a competitive economy, econometrica 22 (1954), no. 3, 265. 2. j.-p. aubin and a. cellina, differential inclusions, springer berlin heidelberg, 1984. 3. d. p. bertsekas, nonlinear programming, journal of the operational research society 48 (1997), no. 3, 334–334. 4. m. g. cojocaru, p. daniele, and a. nagurney, projected dynamical systems and evolutionary variational inequalities via hilbert spaces with applications1, journal of optimization theory and applications 127 (2005), no. 3, 549–563. 5. m.-g. cojocaru and leo b. jonker, existence of solutions to 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games and economic behavior 64 (2008), no. 2, 666–683. 19. j. m. smith, evolution and the theory of games, cambridge university press, cambridge ; new york, 1982. 20. p. d. taylor and l. b. jonker, evolutionary stable strategies and game dynamics, mathematical biosciences 40 (1978), no. 1-2, 145–156. 21. y. s. xia and j. wang, on the stability of globally projected dynamical systems, journal of optimization theory and applications 106 (2000), no. 1, 129–150. corresponding author, lunenfeld-tanenbaum research institute, mount sinai hospital, toronto, on, canada. room 1070 e-mail address: jlequyer@lunenfeld.ca university of guelph, 50 stone rd e, guelph, on n1g 2w1. macnaughton 549. e-mail address: mcojocar@uoguelph.ca mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 2, number 3, september 2021, pp.194-218 https://doi.org/10.5206/mase/14147 cluster solutions in networks of weakly coupled oscillators on a 2d square torus jordan michael culp abstract. we consider a model for an n × n lattice network of weakly coupled neural oscillators with periodic boundary conditions (2d square torus), where the coupling between neurons is assumed to be within a von neumann neighborhood of size r, denoted as von neumann r-neighborhood. using the phase model reduction technique, we study the existence of cluster solutions with constant phase differences (ψh, ψv) between adjacent oscillators along the horizontal and vertical directions in our network, where ψh and ψv are not necessarily to be identical. applying the kronecker product representation and the circulant matrix theory, we develop a novel approach to analyze the stability of cluster solutions with constant phase difference (i.e., ψh, ψv are equal). we begin our analysis by deriving the precise conditions for stability of such cluster solutions with von neumann 1-neighborhood and 2 neighborhood couplings, and then we generalize our result to von neumann r-neighborhood coupling for arbitrary neighborhood size r ≥ 1. this developed approach for the stability analysis indeed can be extended to an arbitrary coupling in our network. finally, numerical simulations are used to validate the above analytical results for various values of n and r by considering an inhibitory network of morris-lecar neurons. 1. introduction an average neuron forms and receives one to ten thousand synaptic connections for sending and receiving information. since there are at least 1011 neurons in the human brain, there are thus 1014 ∼ 1015 synaptic connections that are formed in the brain. the summation of this input at the cellular level combine to allow neurons to perform complicated information processing and when considered as a whole brain, or neural region, allow for the complex cognitive task that we use to live our lives to be performed. it is assumed that it is the structure and details inherent in these connections that hold the secret to how these tasks manifest. thus, understanding our brains ability to organize and create coherent patterns out of the collection of electrical activity from billions of coupled individual neurons is of much research interest. the modeling of coupled oscillators have been applied to study a number of biological and physical systems, for example [38, 28, 42, 34, 30, 49, 46, 23, 18]. the synchronization and cluster solutions, as defined below, in networks of large populations of neurons play an important role in various brain functions [43, 9, 41, 22, 16, 6, 27, 48, 14, 47, 44, 40, 45]. the theory of weakly coupled oscillators [13, 25, 31] is a classical tool for using dynamical systems theory to study oscillations in neural networks, where the phase reduction method has been utilized to reduce a model of a weakly coupled neural network to a phase model. this phase model is used to study the existence and stability of cluster solutions in the network of oscillators. these cluster solutions are received by the editors 1 august 2021; accepted 13 september 2021; published online september 19 2021. 2010 mathematics subject classification. 92b20,92b25,34d20. key words and phrases. phase models, coupled oscillators, synchronization, phase-locking, clustering solutions, stability. 194 cluster solutions on a 2d square torus 195 phase locked solutions to the phase model where oscillators separate into subgroups. the oscillators within each subgroup are synchronized with zero phase difference, while those in different subgroups are phase locked. the existence and stability of cluster solutions in networks of weakly coupled oscillators has been studied within the context of various network topologies and coupling schemes, see, e.g., [2, 24, 32, 38, 28] and the references therein. in [13], oscillators are modeled as a nonlinear system with stable limit cycle with bidirectional nearest neighbor coupling, where that the existence and necessary conditions for the stability of phase locking behavior in the network are established. in [26] oscillators are also modeled as a nonlinear system with stable limit cycle with bidirectional nearest neighbor coupling on a ring of oscillators. a hopf bifurcation on the ring network topology is also studied to obtain different types of stable oscillators. a review on the stability of cluster solutions of oscillators can be found in see [2, 5]. general networks of identical oscillators are considered in [15] where network symmetries are examined in order to obtain different types of phase locking behavior. the existence and stability of cluster solutions in linear array (chains) and ring networks with unior bi-directional coupling have been shown in [29, 2, 5, 11, 13]. examples of analysis of phase models with time delayed coupling can be found in [7, 8]. moreover, the study of cluster solutions in models with all to all coupling can be found in [2, 32, 37, 17, 21]. the known results for the analytical study of cluster solutions on 1d network topologies are more extensive than the results for the 2d network topologies. the study of two dimensional lattices has been studied in the case of finite square lattice [35, 33]. in [33], the existence and stability of rotating wave solutions exist on finite square lattices with “no flux” boundary conditions and nearest neighbor coupling is studied. in [35], phaselocked behavior is studied, as the two-dimensional (and higher dimensional) arrays, with nearest neighbor coupling, can be decomposed into two one-dimensional problems, if some conditions on the intrinsic frequencies are met. in [3], rotating wave solutions on infinite lattices are considered. in [4], sufficient conditions for local asymptotic stability of phase-locked solutions in coupled phase models on infinite lattices are considered. in this work, we aim to extend the known analytical results concerning the existence and stability of cluster solutions in a two dimensional network topologies to include two dimensional finite lattices with periodic boundary conditions. our work differs mainly from the previously mentioned results in that we derive general conditions for the existence and stability of cluster solutions on our network topology without specificity to a particular type of pattern such as stable rotating solutions (spiral waves), and for larger (greater than first) nearest neighbor coupling. furthermore, we are able to find a convenient representation of our jacobian matrix by utilizing results from circulant matrix theory and the kronecker product that allows for a novel approach in the analysis of the stability of the cluster solutions on a 2d network topology. the results in this work are shown for a general n×n lattice of weakly coupled oscillators with periodic boundary conditions, with a von neumann 1 and 2 neighborhood coupling size. we allow for further generalization of this model by considering situations where the horizontal and vertical phase differences are not equal, and where the neighborhood size can be extended to a general size r. 2. notations in this section we provide some preliminary mathematical theory and notation to be used in this work. 196 j. culp 2.1. matrices. define a circulant matrix generated by constants c1,c2,c3, · · · ,cn−1,cn as circ(c1,c2,c3, · · · ,cn−1,cn ) =   c1 c2 c3 · · · cn−1 cn cn c1 c2 · · · cn−2 cn−1 cn−1 cn c1 . . . cn−3 cn−2 ... ... . . . . . . . . . ... c3 c4 · · · cn c1 c2 c2 c3 · · · cn−1 cn c1   likewise we will define an m×m block circulant matrix generated by the n1 ×n2 matrices, m1,m2, m3, · · · ,mm−1,mm as bcirc(m1,m2, · · · ,mm−1,mm) =   m1 m2 m3 · · · mm−1 mm mm m1 m2 · · · mm−2 mm−1 mm−1 mm m1 . . . mm−3 mm−2 ... ... . . . . . . . . . ... m3 m4 · · · mm m1 m2 m2 m3 · · · mm−1 mm m1   the kronecker product of n1 ×n2 matrix m1 and m1 ×m2 matrix m2 is denoted as m1 ⊗m2 =   m11m2 m12m2 · · · m1,n2m2 ... ... ... mn1,1m2 mn1,2m2 · · · mn1,n2m2   for more information on known results and properties of circulant matrices and the kronecker product used thorough out this work, please refer to [10]. now let us introduce the notation for some special n ×n matrices used throughout the paper. let a1 = circ(0, 1, 0, · · · , 0), a−1 = circ(0, 0, 0, · · · , 1). it is easy to see that a−1 = a n−1 1 = (a1) −1. moreover, as a1 is a circulant matrix, its spectrum is known explicitly. specifically, the eigenvalues of a1 are {λ0,λ1, · · · ,λn−1} and the corresponding eigenvectors are {u0,u1, · · · ,un−1}, where λj = ω j, uj = 1 √ n (1,ωj,ω2j, · · · ,ω(n−1)j)t , 0 ≤ j ≤ n − 1 with ω = exp(2π √ −1/n) is the primitive nth root of the unity. since a−1 = a n−1 1 , a−1uj = a n−1 1 uj = λ n−1 j uj. note that λj = ω j and ωn = 1. we have λn−1j = ω −j = λ−1j and hence a−1uj = λ −1 j uj. it is straightforward to see that a −k 1 uj = λ −k j uj for k ∈ n. lemma 2.1. let p,q ∈{−(n − 1), · · · ,−2,−1, 0, 1, 2, · · · ,n − 1} be given. then (a p 1 ⊗a q 1)(ui ⊗uj) = ω pi+qjui ⊗uj, 0 ≤ i,j ≤ n − 1. proof. by the commutative property of the kronecker product, (a p 1 ⊗a q 1)(ui ⊗uj) = (a p 1ui) ⊗ (a q 1uj) = (λ p i ui) ⊗ (λ q juj) = (λ p i λ q j)ui ⊗uj. for all 0 ≤ i,j ≤ n − 1. since λi = ωi for all 0 ≤ i ≤ n − 1, λ p i λ q j = ω pi+qj. � cluster solutions on a 2d square torus 197 this result shows that, for any p,q ∈{−(n−1), · · · ,−2,−1, 0, 1, 2, · · · ,n−1}, ap1⊗a q 1 has the same eigenvectors {ui⊗uj : 0 ≤ i,j ≤ n−1} and the corresponding eigenvalues are {ωpi+qj : 0 ≤ i,j ≤ n−1}. in particular, (in ⊗a1)(ui ⊗uj) = ωjui ⊗uj, (in ⊗a−1)(ui ⊗uj) = ω−jui ⊗uj, (a1 ⊗ in )(ui ⊗uj) = ωiui ⊗uj, (a−1 ⊗ in )(ui ⊗uj) = ω−iui ⊗uj. (2.1) 2.2. mapping the 2d lattice into a 1d array. figure 1 shows how we map the indices of a two dimensional n ×n lattice into an one dimensional array from 1, . . . ,n2. here, for 0 ≤ i,j ≤ n − 1, we use the mapping f : z+ × z+ → n defined by f(i,j) = jn + (i + 1). clearly f is a one-to-one and onto mapping from {0, 1, 2, · · · ,n − 1}×{0, 1, 2, · · · ,n − 1} to {1, 2, . . . ,n2}. as we will see later, this mapping is used in the construction of our circulant coupling matrix, and allows us to conveniently represent a two dimensional network structure in one dimension. (0, 0) (n − 1, 0) (n − 1, 1)(0, 1) (0, 2) (0,n − 1) (n − 1,n − 1) f−→ 1 n 2nn + 1 2n + 1 (n − 1)n + 1 n2 figure 1. mapping the 2d lattice into a 1d array 2.3. von neumann neighborhoods. in this work, the von neumann r-neighborhood on a 2d square lattice is defined as the set of rth adjacent nodes of a central node. as compared to the classical definition, the center is the common node is removed for convenience. note that a von neumann 1neighborhood is all the nodes at a manhattan distance of 1, and that an extension consists of taking the set of points at a manhattan distance of r > 1. in figure 2, an example of von neumann 2-neighborhood around node p is plotted. the number of nodes in a von neumann r-neighborhood is r2 + (r + 1)2 −1. in our example, r = 2 and we have 22 + (2 + 1)2 − 1 = 13 − 1 = 12, with 4 cells at distance 1 and 8 cells at distance 2. 2.4. phase reduction method. in this subsection, we review the phase reduction method by which a general network model of identical weakly coupled oscillators can be reduced to a phase model. consider a two dimensional network model of identical, weakly coupled oscillators on an n × n lattice with periodic boundary conditions. dyij dt = f(yij) + � n−1∑ ĩ,j̃=0 c(i,j),(̃i,j̃)g(yij,yĩj̃), 0 ≤ i,j ≤ n − 1. (2.2) here yij ∈ rm, 0 < � � 1 is the coupling strength, f is the vector field of the isolated oscillator, g is a function describing the coupling between oscillators and (c(i,j),(̃i,j̃)) is the coupling matrix, where as c(i,j),(̃i,j̃) ≥ 0 denotes the coupling between the oscillators yij and yĩj̃. furthermore, we assume that for an isolated oscillator, the solution to dyij dt = f(yij) exhibits an exponentially asymptotically t−periodic 198 j. culp 1 2 2 1 2 2 1 2 2 1 2 2 p figure 2. the von neumann 2-neighborhood of node p stable solution, denoted by x̂ij(t) on the domain 0 ≤ t ≤ t. here, the frequency of the ijth isolated oscillator with period t is given by ω = 2π t . under the assumption of sufficiently weak coupling between oscillators, the theory of weakly coupled oscillators allows us to reduce the number of equations representing the dynamics of each oscillator to a single differential equation representing the change of the phase of each oscillator along the corresponding t-periodic limit cycle. this phase model will take the form, dθij dt = ω + � n−1∑ ĩ,j̃=0 c(i,j),(̃i,j̃)h(θĩj̃ −θij), 0 ≤ i,j ≤ n − 1. where h is the interaction function with period 2π that satisfies h(θ) = 1 t ∫ t 0 z(t) ·g ( x̂(t),x̂(t + θ/ω) ) dt. here z is known as the phase response curve of the isolated oscillator, which is the unique periodic solution of the linearized adjoint system: dz dt = −{df(x̂)}tz with the normalization 1 t ∫ t 0 z(t) ·f(x̂)dt = 1 where df(x̂) is the jacobian of f with respect to x evaluated at x = x̂. the phase solution of the above phase model is θij(t) = ωt + φij(t). (2.3) here φij(t) represents the relative (initial) phase of the ij th oscillator. under the assumption of weak coupling, the dynamics of the relative phase of each oscillator satisfies the differential equation (to the first order of �): dφij dt = � n−1∑ ĩ,j̃=0 c(i,j),(̃i,j̃)h(θĩj̃ −θij), (2.4) note now that for two identical oscillators, θĩj̃(t),θij(t), by using the definition in (2.3) we would have that θĩj̃(t) −θij(t) = φĩj̃(t) −φij(t). hence, in our work we will frequently refer to the phase model in cluster solutions on a 2d square torus 199 terms of dφij dt . a more rigorous mathematical discussion on the theory of weakly coupled oscillators can be found at [1, 19, 20, 39]. 3. phase model analysis in this section, we study clustering dynamics of a neural network model on a square torus with von neumann r-neighborhood coupling. particularly, we analyze the existence of cluster solutions with constant phase difference and derive the precise condition for the stability of these solutions. note that cr := {(k,l) ∈ z+ × z+ : 0 < |k| + |l| ≤ r} is the set of indices in the von neumann r-neighborhood on a square lattice, for r ∈ n. throughout the paper, all the 2d indices are defined under modulo n unless otherwise stated. consider a network model consisting of n × n identical weakly coupled neural oscillators on a torus with von neumann r-neighborhood coupling: dθi,j dt = ω + � ∑ 0<|k|+|l|≤r k,l∈z wk,lh(θi+k,j+l −θi,j), 0 ≤ i,j ≤ n − 1 (3.1) where the coupling is assumed to be relative-location dependent, and wk,l = c(i+k,j+l),(i,j) is the coupling between the ijth oscillator and (i + k,j + l)th oscillator for all (i,j). 3.1. existence of cluster solutions. assume that the phase difference only depends on the relative location; that is, the phase difference between (i + k,j + l)th oscillator and the ijth oscillator is ψk,l = θi+k,j+l −θi,j, ∀ 0 ≤ i,j ≤ n − 1. (3.2) by this assumption, there are two types of phase differences ψ1,0 and ψ0,1, where the first one is in the horizontal direction and the second one is in the vertical direction. for simplicity, we define ψh = ψ1,0 and ψv = ψ0,1. so in general ψk,l = kψh + lψv. in view of (3.1) and (3.2), dψk,l/dt = 0 for all (k,l). on the other hand, it follows from the periodic boundary condition that the horizontal and vertical phase differences should satisfy nψh = nψv = 0 mod 2π. then nψh = 2khπ, nψv = 2kvπ for some kh,kv ∈ {0, 1, 2, · · · ,n − 1}. if ψh = ψv = 0 (i.e., kh = kv = 0), it gives a synchronization or one-cluster solution. if ψh = 0, ψv > 0 (i.e., kh = 0, kv ∈ n), it gives a solution of horizontal stripes. if ψh > 0, ψv = 0 (i.e., kh ∈ n, kv = 0) , it gives a solution of vertical stripes. otherwise, suppose that ψh > 0, ψv > 0 (i.e., kh, kv ∈ n). define ph = gcd(n,kh) and pv = gcd(n,kv). let nd = n/pd and md = kd/pd with d = h,v. then the solution with phase difference (ψh, ψv) gives us a cluster solution that is of period nh and nv along the horizontal and vertical directions, respectively, where ψh = 2mhπ/nh and ψv = 2mvπ/nv. it is clear that the period of this cluster solution is n := lcm(nh,nv). this solution defines an n-cluster solution with phase difference (ψh, ψv). let nn = {2, 3, · · · ,n}. it leads to the following existence result. theorem 3.1. the system (3.1) admits four types of cluster solutions with constant horizontal and vertical phase differences between adjacent oscillators. (i) there exists a synchronization solution with phase difference (ψh, ψv) = (0, 0). (ii) assume that nv is a factor of n with nv ∈ nn . there exists an nv-cluster solution of horizontal stripes with phase difference (ψh, ψv) = (0, ψv), where ψv = 2mvπ/nv for some mv ∈ n satisfying mv < nv and gcd(mv,nv) = 1. (iii) assume that nh is a factor with n with nh ∈ nn . there exists an nh-cluster solution of vertical stripes with phase difference (ψh, ψv) = (ψh, 0), where ψh = 2mhπ/nh for some mh ∈ n satisfying mh < nh and gcd(mh,nh) = 1. 200 j. culp (iv) assume that both nv and nh are factors of n with nh,nv ∈ nn . define n = lcm(nh,nv). there exists an n-cluster solution with phase difference (ψh, ψv), where ψd = 2mdπ/nd for some md ∈ n satisfying md < nd and gcd(md,nd) = 1 for d = h,v. . furthermore, if ψh = ψv = ψ, cluster solutions will have constant phase difference ψ between adjacent oscillators in the horizontal and vertical directions. we denote such a solution as a cluster solution with constant phase difference ψ. the following result. corollary 3.2. the system (3.1) admits two types of cluster solutions with constant phase difference ψ. (i) there exists a synchronization solution with ψ = 0. (ii) assume that n is a factor of n with n ∈ nn . there exists an n-cluster solution with constant phase difference ψ, where ψ = 2mπ/n for some m ∈ n satisfying m < n and gcd(m,n) = 1. 3.2. stability of cluster solutions with constant phase difference. in this subsection, we study the stability of cluster solutions and our investigation will be focused on cluster solutions with constant phase difference ψ (between adjacent cells in the horizontal and vertical directions). 3.3. von neumann 1-neighborhood coupling: r = 1. first, let’s consider the case where the coupling is in the von neumann 1-neighborhood. this is the case of the nearest neighbor coupling. accordingly, model (3.1) becomes dθij dt = ω + � ∑ |k|+|l|=1 k,l∈z wk,lh(θi+k,j+l −θij) = ω + � [ h−1h(θi−1,j −θij) + h1h(θi+1,j −θij) + v1h(θi,j+1 −θij) + v−1h(θi,j−1 −θij) ] , (3.3) where h±1 := w±1,0 and v±1 := w0,±1 for 0 ≤ i,j ≤ n − 1. suppose that n ≥ 3 and n divides n. consider an n-cluster solution with constant phase difference ψ θ̄ (1) ij (t) = (ω + �ω (1) c )t + (i + j)ψ, 0 ≤ i,j,≤ n − 1 (3.4) for some ωc > 0. substituting (3.4) into (3.3), we find that ω(1)c = (h−1 + v−1)h(−ψ) + (h1 + v1)h(ψ). let θij(t) = θ̄ (1) ij (t) + yij(t). linearizing (3.3) at θ̄ (1) ij (t), we have dyij dt = � { h−1 [ h′(θi−1,j −θij)yi−1,j −h′(θi−1,j −θij)yij ] + h1 [ h′(θi+1,j −θij)yi+1,j −h′(θi+1,j −θij)yij ] + v1 [ h′(θi,j+1 −θij)yi,j+1 −h′(θi,j+1 −θij)yi,j ] + v−1 [ h′(θi,j−1 −θij)yi,j−1 −h′(θi,j−1 −θij)yi,j ]} . (3.5) in what follows and throughout the rest of the paper, we assume that indices k,l ∈ z and we will omit this assumption over all the related summations. since θi+k,j+l −θij = (k + l)ψ, we can rewrite (3.5) cluster solutions on a 2d square torus 201 by using the notation wk,l for coupling strength as follows: dyij dt = � ∑ |k|+|l|=1 wk,l [ h′((k + l)ψ)yi+k,j+l −h′((k + l)ψ)yij ] = �   ∑ |k|+|l|=1 wk,lh ′((k + l)ψ)yi+k,j+l − ∑ |k|+|l|=1 wk,lh ′((k + l)ψ)yij   . (3.6) in view of the map f defined in section 2.2, for any p ∈ {1, 2, · · · ,n2}, there exists a unique (i,j) ∈ {0, 1, 2, · · · ,n−1}×{0, 1, 2, · · · ,n−1} such that p = f(i,j). hence we define xp = yij with p = f(i,j). let x = (x1,x2, · · · ,xn2 )t and a(1) = ∑ |k|+|l|=1 wk,lh ′((k + l)ψ) = (h−1 + v−1)h ′(−ψ) + (h1 + v1)h′(ψ). so we can express (3.6) in terms of 1d indices, which is given by dx dt = �(m(1) −a(1)in2 )x. (3.7) here m(1) is a block circulant matrix and m(1) = bcirc ( h1h ′(ψ)a1 + h−1h ′(−ψ)a−1,v1h′(ψ)in, 0n, · · · , 0n,v−1h′(−ψ)in ) , where a1 and a−1 are defined in section 2.1 and ip is the p×p identity matrix for p ∈ n. using kronecker products, one can easily see that m(1) =in ⊗ ( h1h ′(ψ)a1 + h1h ′(−ψ)a−1 ) + ( v1h ′(ψ)a1 + v−1h ′(−ψ)a−1 ) ⊗ in. then m(1) =h1h ′(ψ)in ⊗a1 + h−1h′(−ψ)in ⊗a−1 + v1h′(ψ)a1 ⊗ in + v−1h′(−ψ)a−1 ⊗ in = ∑ |k|+|l|=1 wk,lh ′((k + l)ψ)(a1) l ⊗ (a1)k where the last equality is obtained by using wk,l notations for the coupling strengths, and (a1) −1 = a−1. it follows from lemma 2.1, particularly, equations in (2.1), that m(1)ui ⊗uj = ∑ |k|+|l|=1 wk,lh ′((k + l)ψ)ωli+kjui ⊗uj. this shows that the eigenvalues of m are {∑ |k|+|l|=1 wk,lh ′((k + l)ψ)ωli+kj : 0 ≤ i,j ≤ n − 1 } . so the eigenvalues of m −a1in2 are λ (1) ij = ∑ |k|+|l|=1 wk,lh ′((k + l)ψ)ωli+kj −a(1) = ∑ |k|+|l|=1 wk,lh ′((k + l)ψ)(ωli+kj − 1), 0 ≤ i,j ≤ n − 1. then <(λ(1)ij ) = ∑ |k|+|l|=1 wk,lh ′((k + l)ψ)[cos(2(li + kj)π/n) − 1] = −2 ∑ |k|+|l|=1 wk,lh ′((k + l)ψ) sin2((li + kj)π/n). 202 j. culp note that sin2((li + kj)π/n) > 0 for all (k,l) ∈c1. so the cluster solution is stable if [w1,0h ′(ψ) + w−1,0h ′(−ψ)] sin2(jπ/n) + [w0,1h′(ψ) + w0,−1h′(−ψ)] sin2(iπ/n) > 0 for (i,j) 6= (0, 0). this leads to the following result. theorem 3.3. let n ≥ 3 and n be a factor of n with n ∈ nn . for our n ×n torus network model with von neumann 1-neighborhood coupling, the n-cluster solution with constant phase difference ψ is stable if [w1,0h ′(ψ) + w−1,0h ′(−ψ)] sin2(jπ/n) + [w0,1h′(ψ) + w0,−1h′(−ψ)] sin2(iπ/n) > 0 (3.8) for (i,j) 6= (0, 0), 0 ≤ i,j ≤ n − 1. let hodd(θ) = h(θ)−h(−θ) 2 , which is the odd part of the interconnection function h. hence with the nearest neighbor coupling, we have the following result: remark 3.1. (i) if ψ = 0, it is the case of 1-cluster solution and the syncrhonization solution stable if h′odd(0) > 0, as h′odd(0) = h ′(0). (ii) if ψ = π, it leads to an antiphase solution (2-cluster solution). for all the possible (k,l) in the nearest neighbor we have either h′(π) or h′(−π). by h′(π) = h′(−π) and h′odd(π) = h ′(π), the antiphase solution is stable is h′odd(π) > 0. (iii) if wkl ≡ w0 is constant for all k,l, the coupling is homogeneous and (3.8) becomes 2w0h ′ odd(ψ)[sin 2(jπ/n) + sin2(iπ/n)] > 0 for 0 ≤ i,j ≤ n−1 with (i,j) 6= (0, 0). in this case, this cluster solution is stable if h′odd(ψ) > 0. this precise stability condition on our 2d torus is in consistent with the result on the 1d ring [29]. 3.4. von neumann 2-neighborhood coupling: r = 2. next, we consider the coupling is in the von neumann 2-neighborhood, and we analyze the stability of the cluster solution with constant phase difference ψ. dθij dt = ω + � ∑ 0<|k|+|l|≤2 k,l∈z wk,lh(θi+k,j+l −θij), 0 ≤ i,j ≤ n − 1. (3.9) suppose that n ≥ 5 and n divides n. consider an n-cluster solution with constant phase difference ψ θ̄ (2) ij (t) = (ω + �ω (2) c )t + (i + j)ψ, 0 ≤ i,j,≤ n − 1. (3.10) here ω(2)c = ∑ 0<|k|+|l|≤2 wk,lh((k + l)ψ) which can be obtained by plugging (3.10) into (3.9). let θij(t) = θ̄ (2) ij (t) + yij(t). linearizing (3.9) at θ̄ (2) ij (t) to the first order gives dyij dt = � ∑ 0<|k|+|l|≤2 wk,l [ h′((k + l)ψ)yi+k,j+l −h′((k + l)ψ)yij ] = �   ∑ 0<|k|+|l|≤2 wk,lh ′((k + l)ψ)yi+k,j+l − ∑ 0<|k|+|l|≤2 wk,lh ′((k + l)ψ)yij   . cluster solutions on a 2d square torus 203 let x = (x1,x2, · · · ,xn2 )t . convert the indices from 2d to 1d. the above system can be written as dx dt = �(m(2) −a(2)in2 )x, where m(2) = bcirc(m0,m1,m2, 0n, · · · , 0n,mn−2,mn−1) and a(2) = ∑ 0<|k|+|l|≤2 wk,lh ′((k + l)ψ) with m0 = w1,0h ′(ψ)a1 + w2,0h ′(2ψ)a21 + w−2,0h ′(−2ψ)a2−1 + w−1,0h ′(−ψ)a−1 m1 = w0,1h ′(ψ)in + w1,1h ′(2ψ)a1 + w−1,1h ′(0)a−1 m2 = w0,2h ′(2ψ)in m3, . . . ,mn−3 = 0n mn−2 = w0,−2h ′(−2ψ)in mn−1 = w0,−1h ′(−ψ)in + w1,−1h′(0)a1 + w−1,−1h′(−2ψ)a−1. here 0n is the zero matrix of size n. since any block circulant matrix can be expressed in terms of kronecker products, m(2) = in ⊗m0 + a1 ⊗m1 + a21 ⊗m2 + a n−2 1 ⊗mn−2 + a n−1 1 ⊗mn−1 = in ⊗m0 + a1 ⊗m1 + a21 ⊗m2 + (a−1) 2 ⊗mn−2 + a−1 ⊗mn−1 = ∑ 0<|k|+|l|≤2 wk,lh ′((k + l)ψ)(al1 ⊗a k 1). by lemma 2.1, (al1 ⊗ak1)(ui ⊗uj) = ωli+kjui ⊗uj and hence m(2)ui ⊗uj = ∑ 0<|k|+|l|≤2 wk,lh ′((k + l)ψ)ωli+kjui ⊗uj, 0 ≤ i,j ≤ n − 1. it shows that the spectrum of m(2) is given by  ∑ 0<|k|+|l|≤2 wk,lh ′((k + l)ψ)ωli+kj : 0 ≤ i,j ≤ n − 1   . so, the eigenvalues of m −a1in2 are λ (2) ij = ∑ 0<|k|+|l|≤2 wk,lh ′((k + l)ψ)ωli+kj −a(2) = ∑ 0<|k|+|l|≤2 wk,lh ′((k + l)ψ)(ωli+kj − 1), 0 ≤ i,j ≤ n − 1. hence, <(λ(2)ij ) = ∑ 0<|k|+|l|≤2 wk,lh ′((k + l)ψ)[cos(2(li + kj)π/n) − 1] = −2 ∑ 0<|k|+|l|≤2 wk,lh ′((k + l)ψ) sin2((li + kj)π/n). 204 j. culp note that sin2((li + kj)π/n) > 0 for all (k,l) ∈ c2, when (i,j) 6= (0, 0) . thus, the cluster solution is stable if [w2,0h ′(2ψ) + w−2,0h ′(−2ψ)] sin2(2jπ/n) + [w0,2h′(2ψ) + w0,−2h′(−2ψ)] sin2(2iπ/n) +[w1,1h ′(2ψ) + w−1,−1h ′(−2ψ)] sin2((i + j)π/n) + [w1,−1 + w−1,1]h′(0) sin2((i− j)π/n) +[w1,0h ′(ψ) + w−1,0h ′(−ψ)] sin2(jπ/n) + [w0,1h′(ψ) + w0,−1h′(−ψ)] sin2(iπ/n) > 0 for (i,j) 6= (0, 0) and 0 ≤ i,j ≤ n − 1. this leads to the following result. theorem 3.4. let n ≥ 5 and n be a factor of n with n ∈ nn . for our n ×n torus network model with von neumann 2-neighborhood coupling, the n-cluster solution with constant phase difference ψ is stable if [w2,0h ′(2ψ) + w−2,0h ′(−2ψ)] sin2(2jπ/n) + [w0,2h′(2ψ) + w0,−2h′(−2ψ)] sin2(2iπ/n) +[w1,1h ′(2ψ) + w−1,−1h ′(−2ψ)] sin2((i + j)π/n) + [w1,−1 + w−1,1]h′(0) sin2((i− j)π/n) +[w1,0h ′(ψ) + w−1,0h ′(−ψ)] sin2(jπ/n) + [w0,1h′(ψ) + w0,−1h′(−ψ)] sin2(iπ/n) > 0 (3.11) for (i,j) 6= (0, 0), 0 ≤ i,j ≤ n − 1. thus, for the von neumann 2-neighborhood coupling in our network model, we have the following result. again h′odd(0) = h ′(0) and h′odd(π) = h ′(π). remark 3.2. (i) if ψ = 0, it is the case of 1-cluster solution and the synchronization solution stable if h′odd(0) > 0. (ii) if ψ = π, it leads to an antiphase solution (2-cluster solution). for all the possible (k,l) in the nearest neighbor we have h′(π), h′(−π), h′(2π) and h′(0). so a stronger sufficient condition to guarantee that the antiphase solution is stable is h′odd(π) > 0 and h ′ odd(0) > 0. (iii) if wkl ≡ w0 is constant for all k,l, the coupling is homogeneous and (3.11) is equivalent to h′odd(ψ)[sin 2(iπ/n) + sin2(jπ/n)] +h′odd(2ψ)[sin 2(2iπ/n) + sin2(2jπ/n) + sin2((i+j)π/n)] + h′odd(0) sin 2((i − j)π/n) > 0 for 0 ≤ i,j ≤ n − 1 with (i,j) 6= (0, 0). hence, in this case, a stronger sufficient condition to guarantee that this cluster solution is stable is that h′odd(0) > 0, h′odd(ψ) > 0 and h ′ odd(2ψ) > 0. 3.5. von neumann r-neighborhood coupling: r ∈ n. in this subsection, we extend the stability result to the general von neumann r-neighborhood coupling for any given r ∈ n. by the similar method as that used in section 3.5, it is straightforward to establish the result as follows. theorem 3.5. let n ≥ 2r + 1 and n be a factor of n with n ∈ nn . for our n ×n torus network model with von neumann r-neighborhood coupling, the n-cluster solution with constant phase difference ψ is stable if∑ 0<|k|+|l|≤r k,l∈z wk,lh ′((k + l)ψ) sin2((li + kj)π/n) > 0, (i,j) 6= (0, 0), 0 ≤ i,j ≤ n − 1. (3.12) remark 3.3. (i) if ψ = 0, it is the case of 1-cluster solution and the synchronization solution stable if h′odd(0) > 0. (ii) a stronger sufficient condition to guarantee that this cluster solution is stable is h′((k+l)ψ) > 0 for (k,l) ∈cr. cluster solutions on a 2d square torus 205 (iii) if wkl ≡ w0 is constant for all k,l, the coupling is homogeneous and (3.12) is equivalent to∑ 0<|k|+|l|≤r k,l∈z h′odd((|k + l|)ψ) sin 2((li + kj)π/n) > 0 for 0 ≤ i,j ≤ n − 1 with (i,j) 6= (0, 0). hence, in this case, a stronger sufficient condition to guarantee that this cluster solution is stable is that h′odd((|k + l|)ψ) > 0 for (k,l) ∈cr. (iv) the stability result can be extended to an arbiturary coupling, where the set of coupling is denoted as c. suppose that n is sufficiently large such that c ⊂ {0, 1, 2, · · · ,n − 1} × {0, 1, 2, · · · ,n − 1}. theorem 3.6. let n be a factor of n with n ∈ nn . for our n ×n torus network model with coupling c, the n-cluster solution with constant phase difference ψ is stable if∑ (k,l)∈c wk,lh ′((k + l)ψ) sin2((li + kj)π/n) > 0, (i,j) 6= (0, 0), 0 ≤ i,j ≤ n − 1. (3.13) 4. numerical results in this section we apply our analytical results to a network of n2 identical morris-lecar oscillators. we will run simulations for various values of n and various values of our coupling radius r to validate our analytical results. 4.1. model and parameter analysis. we are using the dimensionless formulation of the morrislecar model, [36, 7, 8], as given by the following system, where i ∈ [1, . . . ,n2],  v′i = iapp −gcam∞(vi)(vi −vca) −gkwi(vi −vk) −gl(vi −vl) −gsyn� n2∑ j=1 wijsj(vi −vsyn) w′i = φλ(vi)(w∞(vi) −wi) s′i = αh(vi)(1 −si) − si τs m∞(vi) = 1 2 ( 1 + tanh ( vi −v1 v2 )) w∞(vi) = 1 2 ( 1 + tanh ( vi −v3 v4 )) λ(vi) = cosh ( vi −v3 2v4 ) h(vi) = 5 exp(−(vi −vpre)/.1) here we use the parameter values from [8], with the exception of iapp,�,gsyn, all of which are given in table 1. in the absence of coupling, each oscillator in the network has an exponentially asymptotically stable limit cycle with period ≈ 11.93 and frequency ≈ 0.53. 4.2. calculating h and hodd. the interaction function h can be calculated from the uncoupled single cell version of our morris-lecar model. the calculation of the interaction function h and it’s corresponding odd part, hodd, were done in xppaut with the parameters given in table 1. data was exported from xppaut and imported to python. see the text [12] for more information and tutorials on using xppaut. in python, an univariate spline was then calculated for the function h and hodd from the given data. from the univariate spline we can approximate the derivative of h and hodd over one period. 206 j. culp parameter name value vca calcium equilibrium potential 1 vk potassium equilibrium potential -0.7 vl leak equilibrium potential -0.5 vsyn synaptic reversal potential -0.625 vpre pre-synaptic membrane potential -0.1 gk potassium ionic conductance 2 gl leak ionic conductance 0.5 gca calcium potential conductance 1 φ potassium rate constant 1/3 v1 calcium activation potential -0.01 v2 calcium reciprocal slope 0.15 v3 potassium activation potential 0.1 v4 potassium reciprocal slope 0.145 iapp applied current 0.123 gsyn synaptic conductance 0.025 � coupling strength 1 2r(r+1) α synaptic activation constant 1.0 τs synaptic decay constant 1.0 table 1. parameter values 4.3. numerical simulations. here we begin by considering the case when n = 5 with nearest and second nearest neighbor coupling. in each case, we would expect the existence of a synchronous solution, ψ = 0, along with the 5-clusters ψ = 2πk 5 , k = 1, 2, 3, 4. the values of h′(ψ),h′(−ψ) and h′odd(ψ) for ψ = 2πk 5 , k = 0, 1, 2, 3, 4 are given in table 2. stability analysis using some these values for the nearest and second nearest neighbor case will be provided below. k ψ h′(ψ) h′(−ψ) h′odd(ψ) 0 0 -0.5494149289091872 -0.5494149289091872 -0.6448920358548498 1 2π 5 -0.29701606374660317 0.18635559734182014 -0.05424507664335855 2 4π 5 0.12050167876589174 0.3801261880992415 0.24992749301022554 3 6π 5 0.3801261880992415 0.12050167876589174 0.24993264907886742 4 8π 5 0.18635559734182014 -0.2970160637466032 -0.05424962016118515 table 2. h′(ψ),h′(−ψ),h′odd for ψ = 0, 2π 5 , 4π 5 , 6π 5 , 8π 5 4.3.1. von neumann 1-neighborhood coupling. in table 3, we see that in the case of homogeneous coupling our model predicts unstable and stable 5-clusters for various ψ with n = 5. the firing groups consist of the following clusters of cells c1 = {1, 10, 14, 18, 22},c2 = {2, 6, 15, 19, 23},c3 = {3, 7, 11, 20, 24}, c4 = {4, 8, 12, 16, 25},c5 = {5, 9, 13, 17, 21} cluster solutions on a 2d square torus 207 ψ predicted stability verified numerically 0 unstable yes 2π 5 unstable yes 4π 5 stable yes 6π 5 stable yes 8π 5 unstable yes table 3. prediction under homogeneous coupling, n = 5,r = 1,w = 1 in the case of ψ = 4π 5 , stability was verified for the firing orders (c1,c4,c2,c5,c3), (c2,c5,c3,c1,c4), (c3,c1,c4,c2,c5), (c4,c2,c5,c3,c1), (c5,c3,c1,c4,c2) recall, the stability condition for the nearest neighbor case with n = 5 is given by( w1,0h ′(ψ) + w−1,0h ′(−ψ) ) sin2 ( πj 5 ) + ( w0,1h ′(ψ)v1 + w0,−1h ′(−ψ) ) sin2 ( πi 5 ) > 0 (4.1) for 0 ≤ i,j ≤ 4, where i,j are both not 0. for i,j = 1, 2, 3, 4, we have that sin2(π 5 ) = sin2( 4π 5 ) and sin2( 2π 5 ) = sin2( 3π 5 ), so that a sufficient condition for stability is that( w1,0h ′(ψ) + w−1,0h ′(−ψ) ) > 0 and ( w0,1h ′(ψ)v1 + w0,−1h ′(−ψ) ) > 0 (4.2) from table 2, we can see that for k = 2, 3 both h′(ψ),h′(−ψ) > 0. therefore our simplified stability conditions will be positive for any choice of w±1,0,w0,±1. on the other hand, for k = 1, 4, h ′(ψ) and h′(−ψ) have alternate signs. in the case of homogeneous coupling the conditionality of the inequalities in (4.2) are completely dependent on the value of h′odd(ψ) in table 2, which is negative in either case so that we would expect an unstable solution. we are interested in cases where stability of our cluster solution could change given a change in the coupling strength, particularly from unstable to stable, so we will focus on cases of heterogenous coupling with k = 1, 4. let us define the function f(w1,w−1) = w1h ′(ψ) +w−1h ′(−ψ) where w1 takes the place of w1,0,w0,1 and w−1 takes the place of w−1,0,w0,−1. the plot of f(w1,w−1) yields planar graphs which easily shows that there are many choices of w1,w−1 that would satisfy our inequalities in either case. we will focus on the case when k = 4, ψ = 8π 5 . here, we consider the cases for when w1,0 = w0,1 = 1, w−1,0 = w0,−1 = α, and vary α. a summary of values used in simulations are provided in table 4, along with the smallest and largest value of the stability condition in (4.1). in each case in table 4, a stable 5-cluster could not be verified. case smallest value largest value α 1 ≈ 0.0131199 ≈ 0.068697 1 2 2 ≈ 0.038730 ≈ 0.202793 1 4 3 ≈ 0.057937 ≈ 0.3033663 1 16 4 ≈ 0.063940 ≈ 0.3347953 1 256 table 4. smallest and largest value of the stability condition, (4.1), for the four cases with nearest neighbor coupling, r = 1,n = 5, ψ = 8π 5 208 j. culp 4.3.2. von neumann 2-neighborhood coupling. a similar analysis as done in the nearest neighbor coupling case will also be applied in our second nearest neighbor case by using our stability condition in (3.11), for n = 5, with 0 ≤ i,j ≤ 4, both not zero. in table 5, we have that in the case of homogeneous coupling our model predicts unstable 5-clusters for all possible ψ. we will consider then the case of heterogeneous coupling in which we show a change in stability of our cluster solutions. ψ predicted stability verified numerically 0 unstable yes 2π 5 unstable yes 4π 5 unstable yes 6π 5 unstable yes 8π 5 unstable yes table 5. prediction under homogeneous coupling, n = 5,r = 2,w = 1 we will consider the case when ψ = 4π 5 , so that h′(0) ≈−0.5491, h′(ψ) ≈ 0.1204, h′(−ψ) ≈ 0.3798, h′(2ψ) ≈ 0.1862, h′(−2ψ) ≈−0.2965 (4.3) from (3.11) and (4.3), we can see which coupling weights are associated with negative and positive values of h′. we can thus choose the corresponding coupling weights so that our stability condition, (3.11), is satisfied for all i,j and our cluster solution is predicted to be stable. one such configuration is to assign the coupling weights associated with second nearest neighbor coupling (wk,l, |k| + |l| = 2) the same value, α1 and the coupling weights associated with nearest neighbor coupling, (wk,l, |k|+ |l| = 1), the same value, α2. one such assignment, α1 = 1 16 ,α2 = 1, will allow us to satisfy the conditions in (3.11). in figure ??, we have diagrams representing the change in sign of the real part of the eigenvalues with our assigned values versus the homogeneous coupling case. the corresponding raster plots for the beginning and end of these simulations are given in figure 7, where darker colors indicate larger values. in case 1, we can see that with strictly negative real parts of our eigenvalues where (3.11) is satisfied, we have a stable 5-cluster, likewise, in case 2, where the conditions of (3.11) are not satisfied, we have an unstable cluster solution, as predicted. 4.3.3. third nearest neighbor coupling. here will consider an 8 × 8 lattice of oscillators with third (r = 3) nearest neighbor coupling. in table 6, we have a summary of predicted cluster solutions. notice that in the case of ψ = π 2 ,π, 3π 2 we were unable to verify the predicted stability. in figure 4, we have a diagram giving the real part of the eigenvalues and the beginning and end of a simulation in the case with ψ = 3π 2 , the case for ψ = π 2 is similar. here we see that the simulation appears to produce a stable cluster as a result, in contrast to the unstable prediction. we also note that almost all the real parts of the eigenvalues are negative and much larger in magnitude than the two eigenvalues with positive real part, ≈ 0.1322. in figure 5, we see the real part of the eigenvalues and the beginning and end of the simulation in the unverified unstable case with homogeneous coupling and ψ = π. here, the simulation appears to be stable although there is a mixture of positive and negative real parts of the eigenvalues, with the some large negative eigenvalues. in figure 6, we consider the case with heterogenous coupling with values wk,l = 1/4 when |k| + |l| = 3; wk,l = 1 when |k| + |l| = 2; wk,l = 1/4 when |k| + |l| = 1. in this case, where most eigenvalues are positive and large in magnitude, we have an unstable solution, as would have been predicted. cluster solutions on a 2d square torus 209 (a) fitness function (b) anti-predation response level α figure 3. the fitness function f1(α,v1) and f2(α,v2) and convergent dynamics of anti-predation response level α(t) on the two patches which are not connected by dispersals. ψ predicted stability verified numerically 0 unstable yes π 4 unstable yes π 2 unstable no 3π 4 unstable yes π unstable no 5π 4 unstable yes 3π 2 unstable no 7π 4 unstable yes table 6. prediction under homogeneous coupling, n = 8,r = 3,w = 1 in figure 8, we show 5 firing diagrams across one period of our simulation with the coupling values given in wk,l = 1/16 when |k| + |l| = 3; wk,l = 1/8 when |k| + |l| = 2; wk,l = 1 when |k| + |l| = 1. the cell numberings are given within each cell in figure 8. cells that fire together share the same color in each diagram, and darker colors imply larger values. we expect 4 subgroups of 16 oscillators each, were the predicted grouping are given in (4.4). from figure 8, there appears to be a traveling wave firing pattern which agrees with the firing order given in (4.4). c1 ={1, 5, 12, 16, 19, 23, 26, 30, 33, 37, 44, 48, 51, 55, 58, 62} c4 ={4, 8, 11, 15, 18, 22, 25, 29, 36, 40, 43, 47, 50, 54, 57, 61} c3 ={3, 7, 10, 14, 17, 21, 28, 32, 35, 39, 42, 46, 49, 53, 60, 64} c2 ={2, 6, 9, 13, 20, 24, 27, 31, 34, 38, 41, 45, 52, 56, 59, 63} (4.4) 4.3.4. cluster solutions in a larger network. here we consider a 18 × 18 lattice of 324 oscillators. under the assumption of homogeneous coupling weights, we will begin with the case of nearest neighbor 210 j. culp (a) real part of the eigenvalues (b) beginning and end of simulation figure 4. unverified third nearest neighbor homogenous coupling case with n = 8, ψ = 3π 2 (a) real part of the eigenvalues (b) beginning and end of simulation figure 5. unverified third nearest neighbor homogenous coupling case with n = 8, ψ = π coupling and increase our neighborhood radius, r, and study the change in stability. here we find that for the case of nearest neighbor coupling the only predicted stable cluster solutions are for ψ = 5π 9 , 2π 3 , 7π 9 , 8π 9 ,π, 10π 9 , 11π 9 , 4π 3 , 13π 9 (4.5) when we move to the case with second nearest neighbor homogeneous coupling, we have that there are no predicted stable cluster solutions, but when we move to the third nearest neighbor case we have that ψ = 4π 9 , 14π 9 are the only stable cluster solutions. therefore, for homogeneous coupling weights, there are no cluster solutions that are predicted to stable for the nearest neighbor and remain stable as our neighborhood radius increases. cluster solutions on a 2d square torus 211 (a) real part of the eigenvalues (b) beginning and end of simulation figure 6. unverified third nearest neighbor heterogeneous coupling case with n = 8, ψ = π under the values of ψ in (4.5), with nearest neighbor coupling, we find that for second nearest neighbor coupling with the coupling weights; wk,l = 1/4 when |k| + |l| = 2; wk,l = 1 when |k| + |l| = 1, that only ψ = 5π 9 , 2π 3 , 4π 3 , 13π 9 (4.6) are predicted to be stable. when we furthermore extend our analysis to third nearest neighbor coupling with the coupling weights; wk,l = 1/8 when |k| + |l| = 3; wk,l = 1/4 when |k| + |l| = 2; wk,l = 1 when |k| + |l| = 1 for (4.6), we find that stability is only predicted in the case for ψ = 5π 9 , 13π 9 . if we change our coupling weights to be 1, 1 2 for 1, 1 4 in the case of second nearest neighbor coupling, then there are no predicted stable cluster solutions. likewise, if we change the coupling weights in the third nearest neighbor case to 1, 1 2 , 1 4 for 1, 1 4 , 1 8 , then there are no predicted stable clusters. given in figure 9 are the plots of the stable cluster solutions, with ψ = 5π 9 , for the third nearest neighbor heterogeneous coupling case. the plots for the first and second nearest neighbor coupling cases are similar and will be omitted. 4.3.5. numerical results for the case with ψh 6= ψv. here we examine the firing patterns for a 16×16 lattice of oscillators with various values of ψh 6= ψv and third nearest neighbor coupling. in figure 10, we use the coupling weights; wk,l = 1/8 when |k|+|l| = 3; wk,l = 1/4 when |k|+|l| = 2; wk,l = 1 when |k| + |l| = 1, and kh = 4,kv = 12. using these coupling weights and choice of kh,kv, we predict 4 stable clusters of 64 oscillators each, where that ψh = π 2 , ψv = 3π 2 . in figure 11, we plot our stable cluster solutions over one period (17 frames, where frame 0 and 16 are equivalent) with the same coupling weights as above and kh = 9,kv = 8. here we predict 16 stable clusters of 16 oscillators each, where that ψh = 9π 8 , ψv = π. in figure 10, we have a traveling wave pattern that appears to move through from the upper left to bottom right of the plot. this pattern is opposed to the traveling wave pattern in figure 8, in which the wave appears to travel from the upper right to bottom left of the plot. in figure 11, there appears to be a traveling vertical checkerboard pattern. 212 j. culp (a) stable solution (b) unstable solution figure 7. effect of varying coupling values on stability of cluster solutions in second nearest neighbor coupling case 5. discussion in this work we considered a 2d lattice of n ×n weakly coupled oscillators with period boundary conditions and von neumann neighborhood r coupling. for our analytical results we derived conditions cluster solutions on a 2d square torus 213 figure 8. firing sequence over one period, t for the existence and stability of cluster solutions in our network under various coupling situations, while taking advantage of known results, involving circulant matrices and the kronecker product, so that our stability analysis can be easily performed. we would like to extend our analytical results to include existence and stability analysis for an arbitrary n1 × n2,n1 6= n2 lattice of oscillators with periodic boundary conditions. this extension would dramatically increase the generality of the model, although it is not clear to what extent our representation techniques could be applied. additionally, in the case for non-square lattices, it seems likely from our work with ψh 6= ψv, that the existence of any cluster solutions would require a condition of the kind in which n1,n2 are multiples of each other. expanding our 2d network topology results to 3d, where a von neumann neighborhood is also defined, is also possible. there are additional extensions to our model that could also be investigated. it will be interesting to consider a modification of our model to include an explicit time delay. this modification has biology significance and been studied in [7] for a ring topology. there are also certainly a large diversity of interesting coupling structures possible in larger networks, such as a randomly connected network, that we could incorporate into our network topology. similarly as above, it would be uncertain as to which our convenient representation technique would be limited under these coupling assumptions. in table 6, we were not able to numerically verify the case with ψ = π 2 ,π, 3π 2 . recalling, figure 4 in our numerical section, in the case of ψ = π 2 , 3π 2 , we have a majority negative real part of the eigenvalues, which are large in magnitude compared to the two small positive eigenvalues. it is possible 214 j. culp figure 9. 18 × 18 lattice of coupled oscillators with third nearest neighbor coupling figure 10. firing sequence over one period, 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[49] h.g. winful and s.s wang, stability of phase locking in coupled semiconductor laser arrays, applied physics letters 53 (1988), no. 20, 1894–1896. 218 j. culp department of cell biology & anatomy, university of calgary, calgary, alberta, canada current address: same e-mail address: jordan.culp@ucalgary.ca mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 2, number 2, june 2021, pp.103-122 https://doi.org/10.5206/mase/13569 seasonal dynamics of a generalist and a specialist predator on a single prey noah bolohan, victor leblanc, and frithjof lutscher abstract. in ecological communities, the behaviour of individuals and the interaction between species may change between seasons, yet this seasonal variation is often not represented explicitly in mathematical models. as global change is predicted to alter season length and other climatic aspects, such seasonal variation needs to be included in models in order to make reasonable predictions for community dynamics. the resulting mathematical descriptions are nonautonomous models with a large number of parameters, and are therefore challenging to analyze. we present a model for two predators and one prey, whereby one predator switches hunting behaviour to seasonally include alternative prey when available. we use a combination of temporal averaging and invasion analysis to derive simplified models and determine the behaviour of the system, in particular to gain insight into conditions under which the two predators can coexist in a changing climate. we compare our results with numerical simulations of the temporally varying model. 1. introduction in predator-prey communities, population dynamics strongly depend on the way in which a prey responds to a predator, i.e., how many prey are killed and how frequently [9, 23]. it is therefore important to understand the behaviour of the predator in a system in order to understand the dynamics of the system. species behaviour often varies in response to seasonal phenomena [3, 21]. for example, resource availability may alter how a predator hunts for food. in this paper, we consider two distinct behaviours of predators. when a predator specializes in hunting a single species and other food sources are negligible, we classify them as specialists. there are also instances where a predator may prefer a certain species if available, but as this species becomes hard to find, the predator will switch to alternative food sources. we classify these predators as generalists. changes in season length due to climate change and global warming is expected to have significant impact on prey availability, and consequently on the ecological dynamics of predator-prey systems, especially in the presence of generalist predators. a change in predation behaviour of the great horned owl (bulbo virginialis ) in the boreal forest, an area which is susceptible to climate change [1], is documented empirically [17]. the authors observe that gut content of the great horned owl indicates a change from a specialist in the winter to a generalist in the summer. in the winter, the snowshoe hare (lepus americanus ) represents a high percentage of the diet, and therefore the owl is a specialist predator of the hare in this season. in the summer, the received by the editors 16 january 2021; accepted 14 april 2021; published online 16 april 2021. 2000 mathematics subject classification. primary 92d40, 34c23; secondary 34a36. key words and phrases. predator-prey model, seasonal variation, temporal averaging, qualitative analysis, bifurcation, invasion analysis. v. leblanc was supported in part by a discovery grant from the natural resources and engineering council of canada (rgpin-2016-04318). f. lutscher was supported in part by a discovery grant from the natural resources and engineering council of canada (rgpin-2016-04795). 103 104 n. bolohan, v. leblanc, and f. lutscher hare is still prominent in the diet of the owl, but there is a significant amount of other species present. this indicates that the behaviour of the owl has switched to that of a generalist predator. this scenario of a behaviour switch has previously been modelled by [29]. the authors divided the year into two seasons and modelled the owl as a generalist in the summer and a specialist in the winter. they found that relatively small changes in summer season length can have a profound impact on the system. in particular, the predator can drive prey to extinction, there can be coexisting stable steady states, and there can be large-amplitude limit cycles coexisting with a stable steady state. in this paper, we extend the two-species model of [29] to include the most important predator of the hare, the canadian lynx (lynx canadensis ). the lynx is a specialist predator of the hare whose behaviour does not vary throughout the year [17]. therefore, in the seasonal model the lynx will behave identically in both seasons. we model the three-species system with a set of ordinary differential equations (odes), where we express the rate of change in a species density as a function of the density itself. to accurately model the seasonal dependence of predation, we create one set of odes for the summer season and a separate set for the winter season. the differences between these sets of equations are mainly due to the predators and their behaviour: the owl is a generalist in the summer and a specialist in the winter. we model the different behaviours by using two different functional responses [12]. the functional response represents the rate of decrease of the prey population due to predation. the commonly used holling type ii (for specialist predators) and iii (for generalist predators) were derived mathematically and compared to empirical data ([12, 13] and [22], respectively). the seasonally dependent model we obtain is difficult to study analytically. the model is essentially non-autonomous, periodically and discontinuously forced. this renders any type of analytic study extremely challenging. to compensate, we take the annual average of the seasonal equations and study the resulting autonomous model. in particular, we are interested in stable coexistence of the species. in the framework of odes, this corresponds to a stable equilibrium or limit cycle of the system in which all components are strictly positive. although it is possible to derive formulas to determine regions in parameter space in which we have stable coexistence, these are too complicated to understand the biological implications. therefore, we use a different technique that can predict when there is stable coexistence, but does so in a way that sheds light on the biological mechanisms at play, namely invasion analysis. rather than studying coexistence in the three-species model directly, we ask whether one of the predators can invade (i.e., increase when rare) if the other two species coexist stably, for example at a steady state or a periodic orbit. with linear stability analysis of odes, we find invasion conditions, i.e., conditions under which each predator can simultaneously invade the other two species. this scenario is called mutual invasion. it is a longstanding tenet in population biology that “mutual invasion implies coexistence” [27], which has found numerous applications; see, e.g., [5, 24, 28]. however, explicit scenarios have been found in which mutual invasion does not imply coexistence [4, 26]. we will test this hypothesis in our model. we note that the species can coexist stably at a steady state or at an oscillatory state, and the invasion conditions may not specify which coexistence we obtain. in most of our analysis, we highlight and distinguish the parameter t , which represents in our model the proportion of the year corresponding to summer, and we discuss the effects of this parameter on such ecologically relevant issues as species coexistence and extinction. in this way, it is possible to make predictions on the potential effects of global warming and climate change on these issues. most climate change scenarios predict that the length of the summer season will increase, in particular in northern climates where our study system is located. the paper is organized as follows. in section 2, we derive our model and discuss the significance of the parameters, as well as the well-posedness of the model. in section 3, we use the tools and techniques of seasonal dynamics 105 the qualitative theory of odes, including invasion analysis, to derive information regarding equilibria, periodic orbits, stability and bifurcations in our model. in section 4 we complement our theoretical analysis from section 3 with numerical analysis of the model. in particular we will present results on the positivity of the coexistence steady state, its stability, and relate these issues to predictions made from our invasion analysis. we will include a numerical comparison between the averaged and the seasonal model dynamics in our discussion in section 5. we model the population dynamics of the snowshoe hare (lepus americanus ) and two of its predators, the canadian lynx (lynx canadensis ) and the great horned owl (bubo virginialis ). we extend the model in [29] that did not include the lynx, which is the most important predator of the showshoe hare. because the hunting behaviour of the owl changes seasonally, we divide the year into two distinct seasons (summer and winter) and we define appropriate systems of differential equations for each season. we denote time by τ and the population density of the hare by n(·), that of the lynx by p(·) and that of the owl by q(·). the hare is assumed to grow logistically throughout the summer, whereas it does not reproduce in the winter. the lynx is a specialist predator year round. we use the standard holling type ii functional response [12, 13] to model its impact on the prey and its resulting growth rate. in the absence of the hare, the lynx declines exponentially. the owl is a specialist predator in the winter and a generalist predator in the summer. we model its impact and growth in the winter also by a holling type ii functional response, but in the summer, we use a type iii functional response, which better captures the behaviour of generalist predators [22, 29]. during the winter, the owl declines exponentially in the absence of the hare. during the summer, the owl has an additional growth term that reflects the presence of alternative prey. hare mortality in the winter occurs exclusively through predation. we denote subsequent years by n,n + 1, . . . and the fraction of the year that is summer by t. then τ ∈ [n,n + t) corresponds to the summer in year n while τ ∈ [n + t,n + 1) corresponds to winter. our seasonal model in dimensional form is given by summer : τ ∈ [n,n + t), n ∈ z, dn dτ = rn ( 1 − n k ) − cnp b + n − an2q b2 + n2 , dp dτ = f cnp b + n −mp, dq dτ = h an2q b2 + n2 + sq 1 + vq −uq, winter : τ ∈ [n + t,n + 1), n ∈ z, dn dτ = − cnp b + n − αnq β + n , dp dτ = f cnp b + n −mp, dq dτ = h αnq β + n −uq. (1.1) population densities are continuous from the end of one season to the beginning of the next. parameter r is the intrinsic hare growth rate and k the carrying capacity. parameters c and α are the kill rates in the type ii functional responses; b and β are the corresponding half-saturation constants. the kill rate of the type iii functional response is a; the corresponding half-saturation constant is b2. the conversion efficiency of prey into predator biomass is f for the lynx and h for the owl. parameter s is the maximal growth rate of the owl on their alternative prey, and v is the corresponding density-dependence. finally, m and u are per capita death rates for the lynx and owl, respectively. we summarize these parameters, their interpretations and their units in table 1. it is clear that this model, while still only a crude approximation to reality, is too complex for a complete qualitative analysis. the model is temporally periodic, it contains a large number of parameters (namely 15), and the vector field is not continuous at the switches of the seasons. our first step 106 n. bolohan, v. leblanc, and f. lutscher parameter description units t proportion of the year that is summer dimensionless r hare growth rate 1/year k hare carrying capacity hare c lynx maximum kill rate hare/lynx/year b lynx half-saturation constant hare f lynx conversion efficiency lynx/hare m lynx death rate 1/year a owl maximum kill rate (summer) hare/owl/year b owl half-saturation constant (summer) hare α owl maximum kill rate (winter) hare/owl/year β owl half-saturation constant (winter) hare h owl conversion efficiency owl/hare u owl death rate 1/year s owl growth rate on alternative prey 1/year v owl self-interference on alternative prey 1/owl table 1. parameters in the seasonal model (1.1). at reducing model complexity is a standard nondimensionalization of (1.1) by means of the following change of variables: x = n k , y = cp rk , z = aq rk , t = rτ, b = b k , b = b2 k2 , a = α a , d = β k , f = fc rk , m = m r , h = ha r , g = hα r , s = sa r2k , v = a vkr , u = u r . we drop the overbars to simplify notation and write the rescaled system for hare (x), lynx (y) and owl (z) with the remaining 12 parameters as summer t ∈ [rn,r(n + t)), n ∈ z, ẋ = x(1 −x) − xy b + x − x2z b + x2 , ẏ = fxy b + x −my, ż = hx2z b + x2 + sz v + z −uz, winter t ∈ [r(n + t),r(n + 1)), n ∈ z, ẋ = xy b + x − axz d + x , ẏ = fxy b + x −my, ż = gxz d + x −uz. (1.2) while the reduction of parameters helps, the model is clearly still too complex for a complete qualitative analysis. the summer model alone (i.e., setting t = 1) contains a two-dimensional invariant subsystem of hare and owl dynamics (i.e., y = 0) that was independently analyzed in [8]; see [19] for a related model. it exhibits periodic orbits, bistability, and nonlocal bifurcations. seasonal dynamics 107 nonetheless, our goal is to gain insights into the dynamics of the three populations in a seasonally varying environment. we continue to simplify model (1.2) by taking a weighted average of summer and winter equations, with respective weights rt (length of summer) and r(1 −t) (length of winter). averaging theory is, of course, well developed and applies to periodic vector fields with a small parameter [10]. formally, we do not identify a small parameter in our model. however, previous studies have shown that this seasonal averaging can often predict non-averaged dynamics quite well [15, 29], although these results certainly depend on chosen parameter values. tyson and lutscher found that for biologically reasonable parameter values in the hare–owl model (with the lynx absent), their averaged model either slightly over-predicted averaged densities, or it was virtually indistinguishable from averaged densities [29]. we speculate that one reason for the unreasonable effectiveness of temporal averaging here is that the birth-death dynamics of the two predators are typically somewhat slower than the yearly time scale, although not necessarily much smaller. after averaging, we obtain the equations ẋ = rt ( x(1 −x) − xy b + x − x2z b + x2 ) + r(1 −t) ( − xy b + x − axz d + x ) , ẏ = r ( fxy b + x −my ) , ż = rt ( hx2z b + x2 + sz v + z −uz ) + r(1 −t) ( gxz d + x −uz ) . (1.3) this averaged model is the main focus of our analysis here. we will use several strategies to determine as much as possible of the qualitative behaviour of this model. we then numerically compare how well the averaged model predicts the dynamics of the seasonal model (1.2). before we start the qualitative analysis of the averaged model, we establish some basic properties of the averaged system. lemma 1.1. system (1.3) is well-posed and preserves positivity. furthermore, if u > max{ h b+1 , g d+1 } then there is a compact, forward invariant, attractive set in the positive orthant. the x-axis, as well as the planes {y = 0} and {z = 0} are invariant for the dynamics. proof. the vector field is smooth in the positive orthant, hence, solutions exist at least locally and are unique. when x = 0 then ẋ = 0, and the same is true for the other two variables. hence the nonnegative orthant is invariant. since ẏ = 0 whenever y = 0 and ż = 0 whenever z = 0, each of the coordinate planes, where one of the state variables is zero, is invariant. when x > 1, then ẋ < 0. hence, for any δ > 0 and any nonnegative initial condition, the x-coordinate of the solution satisfies limt→∞x(t) ≤ 1 + δ. in the invariant plane {z = 0}, the system is the well-known rosenzweig– macarthur model. hence, there is some positive constant y (depending on parameters but not on initial conditions), so that each nonnegative solution in this plane satisfies limt→∞y(t) ≤ y . since the first equation in (1.3) is decreasing in z and the second equation in (1.3) is increasing in x, the first two components of the solution of (1.3) with any nonnegative initial condition (x0,y0,z0) are bounded above by the solution with initial condition (x0,y0, 0). hence, limt→∞y(t) ≤ y holds not only in the {z = 0} plane but also in the orthant. finally, we turn to the equation for z. since x is bounded by unity and the middle term in the first parentesis is bounded independently of z, the conditions in the lemma are sufficient for there to exist some z > 0 such that for z > z, we have ż ≤ 0. hence, the region where 0 ≤ x ≤ 1, 0 ≤ y ≤ y and 0 ≤ z ≤ z is compact and forward invariant. � 2. qualitative analysis of the averaged model at first glance, the qualitative analysis of system (1.3) should simply follow established techniques. in fact, many ecological models exist for three-species dynamics; see, e.g., [11, 14]. one difficulty arises 108 n. bolohan, v. leblanc, and f. lutscher from the still large number of parameters that reflects having two seasons. another difficulty arises from the dynamics in one of the invariant planes alone being quite complex [8]. we begin our analysis with some relatively simple observations and then use reduction to the invariant planes combined with invasion analysis (see introduction) to obtain insights into the qualitative behaviour of the model. 2.1. the coexistence state and its stability. clearly, the trivial state (x∗,y∗,z∗) = (0, 0, 0) is a steady state of system (1.3). linearization at this state yields a diagonal jacobi matrix with eigenvalues λ1 = rt > 0, λ2 = −rm < 0, and λ3 = r ( ts v −u ) . (2.1) hence, the zero state is unstable. more specifically, the first eigenvalue is related to the x coordinate. since it is positive, the hare can grow when all species are at low densities. the second eigenvalue is related to the y coordinate. since it is negative, the lynx cannot grow when all species are at low densities. from the third eigenvalue, which corresponds to the z coordinate, we see that the owl can grow when all species are at low densities, provided that t > t∗ = uv s . (2.2) in terms of invasion analysis, we say that the hare will successfully invade when the two other species are absent, the lynx will never successfully invade, and the owl can invade provided (2.2) holds. a second, relatively simple steady state is the hare-only state (x∗,y∗,z∗) = (1, 0, 0). the associated jacobi matrix is upper triangular with eigenvalues λ1 = −rt, λ2 = r ( f b + 1 −m ) and λ3 = rt ( h b + 1 + s v ) + r(1 −t)g d + 1 −ru. hence, this state is locally asymptotically stable if and only if f < m(b + 1) (2.3) and t < t∗∗ = u− g d+1 h b+1 + s v − g d+1 . (2.4) these two conditions can again be interpreted by invasion analysis. if (2.3) is reversed, then the lynx can invade the hare-only state, and if (2.4) is reversed, then the owl can invade that state. we will come back to these conditions in sections 2.2 and 2.3, respectively, where we will study steady states where two of the three species are present. here, we continue with the case that all three species have nonzero densities. lemma 2.1. system (1.3) has a unique steady state where all densities are nonzero. it is given by x∗ = mb f −m , y∗ = (b + x∗) ( t ( 1 −x∗ − x∗z∗ b + (x∗)2 ) − (1 −t)az∗ d + x∗ ) , z∗ = −ts t h(x∗)2 b+(x∗)2 + (1−t )gx∗ d+x∗ −u −v. (2.5) seasonal dynamics 109 the computations that lead to this result are fairly straight forward. equally straightforward is the condition that x∗ > 0 if and only if f > m. formal conditions for positivity of the other two components can be written down but are much more cumbersome and do not lead new insights. later, we will evaluate the conditions numerically. we call this state the coexistence state. the linearization of (1.3) at the coexistence state is given by the jacobi matrix  j11 − rx ∗ b+x∗ − rt (x ∗)2 b+(x∗)2 − r(1−t )ax ∗ d+x∗ rfby∗ (b+x∗)2 r ( fx∗ b+x∗ −m ) 0 2rt bhx∗z∗ (b+(x∗)2)2 + r(1−t )gdz∗ (d+x∗)2 0 j33   , (2.6) where j11 = rt ( 1 − 2x∗ − 2bx∗z∗ (b + (x∗)2)2 ) − r(1 −t)adz∗ (d + x∗)2 − rby∗ (b + x∗)2 , j33 = rt ( h(x∗)2 b + (x∗)2 + sv (v + z∗)2 ) + r(1 −t)gx∗ d + x∗ −ru. if parameter values were known, we could evaluate whether the coexistence state has all positive components and whether it is stable. unfortunately, there are too many parameters and for some of those, reasonable ranges are too large or unknown. we also know that the invariant two-dimensional subsystem of hare and owl only can have complicated dynamics, including bistability [8, 29], so that linear stability analysis will not give us the complete picture. for those reasons, we will combine invasion analysis with the study of the two invariant two-species submodels to obtain insights about the full three-species model. more precisely, we will consider each of the two-species subsystems at its attractor and use invasion analysis to find conditions under which the missing species can successfully invade, i.e., grow when rare. if invasion conditions for all species are satisfied simultaneously, we say the system exhibits mutual invasion. it is an old tenet of theoretical ecology that mutual invasion implies coexistence [27], but it is known that this approach may fail [2]. we will see whether and to what extent this principle holds in our system. moreover, we investigate how the dynamics of the two-species subsystems influence the qualitative behaviour of the full system. since the hare-lynx system is well understood, we begin by analyzing the owl-invasion conditions in that system. 2.2. owl invasion. in the absence of the owl, system (1.3) becomes the classical rosenzweig–macarthur model [23] ẋ = rtx(1 −x) −r xy b + x , ẏ = r ( fxy b + x −my ) , z = 0, (2.7) whose dynamics are well understood. if 0 < mb f −m < 1, (2.8) there exists a unique positive steady state for system (2.7), given by (x∗,y∗, 0) = ( mb f −m ,t(1 −x∗)(b + x∗), 0 ) . (2.9) we note that the y-component is positive precisely when condition (2.3) is satisfied. therefore, as the hare-only state loses stability via a transcritical bifurcation, the steady state (2.9) enters the positive quadrant of the (x,y)–plane. 110 n. bolohan, v. leblanc, and f. lutscher the steady state (2.9) is stable when f(1 − b) −m(1 + b) < 0. (2.10) when the inequality is reversed, the positive steady state is unstable and there exists a unique positive and globally stable periodic orbit [16] that arises through a supercritical hopf bifurcation where the inequality turns into an equality. hence, there are only two scenarios at which we have to perform our invasion analysis for the owl: at the state in (2.9) and at a limit cycle. case 1: owl invasion at steady state. when condition (2.10) is satisfied, the steady state (2.9) is stable in the hare-lynx plane of (2.7). after we linearize, the equation for the owl decouples from the other two. it is given by ż = [ rt ( h(x∗)2 b + (x∗)2 + s v ) + r(1 −t)gx∗ d + x∗ −ru ] z. (2.11) for successful owl invasion, we require that z = 0 be unstable in (2.11). hence, the owl invasion condition is t ( h(x∗)2 b + (x∗)2 + s v ) + (1 −t)gx∗ d + x∗ −u > 0. (2.12) case 2: owl invasion at a periodic orbit. when (2.10) is reversed, there is a globally stable periodic orbit of the form ( x∗(t),y∗(t), 0 ) with some period t . invasion analysis at a periodic orbit asks the exact same question as invasion analysis at a steady state, namely, whether a species (in this case the owl) can grow when at low density in the existing community (in this case hare and lynx). since the existing community is oscillatory, the linearized equation for the species at low density is a periodically forced equation. whether the invading species can grow depends on the conditions that it encounters during one period. if the invading species can grow when rare, one expects a periodic orbit with all species present to emerge. when we linearize our model along the hare-lynx orbit, the owl equation again decouples. it is given by ż = [ rt ( h(x∗(t))2 b + (x∗(t))2 + s v ) + r(1 −t)gx∗(t) d + x∗(t) −ru ] z ≡ a(t)z. (2.13) we evaluate the stability of z = 0 in (2.13) by means of the floquet multiplier [6] µ = 1 t ∫ t 0 a(t)dt. (2.14) the owl can invade the periodic orbit of the hare-lynx system if µ = 1 t ∫ t 0 [ rt ( h(x∗(t))2 b + (x∗(t))2 + s v ) + r(1 −t)gx∗(t) d + x∗(t) −ru ] dt > 0. (2.15) 2.3. lynx invasion. in the absence of the lynx, system (1.3) becomes ẋ = rt ( x(1 −x) − x2z b + x2 ) + r(1 −t) ( − axz d + x ) , y = 0, ż = rt ( hx2z b + x2 + sz v + z −uz ) + r(1 −t) ( gxz d + x −uz ) , (2.16) seasonal dynamics 111 which, up to a scaling of parameters, is the hare-owl model of tyson and lutscher [29]. the nonzero steady states of this equation are given by the roots of a complicated cubic polynomial, so that explicit calculations are tedious and uninformative [29]. instead, we use invasion analysis on this twodimensional submodel to find conditions for the existence of steady states. the trivial state is unstable as in the three-dimensional model. the hare-only state is stable if t < t∗∗; see (2.4), but unstable to owl invasion if t > t∗∗. the owl-only state (0,z∗(t)) = ( 0, s u (t −t∗) ) , t∗ = uv s (2.17) is stable if t − a d z∗(t)(1 −t) < 0 (2.18) and unstable to hare invasion if the inequality is reversed. this invasion condition is a quadratic equation in t , which can be solved explicitly to find the range of values of t , where the hare can invade the owl-only state. the result of invasion analysis is summarized by the following theorem. theorem 2.2. define t∗ as in (2.2) and t∗∗ as in (2.4). then t∗∗ < t∗. the roots of (2.18) are given by t± = du 2as  −(1 − a(s + uv) du ) ± √( 1 − a(s + uv) du )2 − 4a2sv d2u   . (2.19) if t± are real, we have t ∗∗ < t∗ < t− < t+. if 0 < t < t ∗∗, the hare-only state is globally asymptotically stable in the hare-owl system (2.16). if t > t∗∗, the hare-only state is unstable and the owl can invade it. if t > t∗, both species can invade the trivial state independently. for t∗∗ < t < t−, each species can invade the other at the single-species steady state. for t− < t < t+, the hare cannot invade the owl-only state. for t+ < t < 1, we have again mutual invasion. this theorem is a significant extension of the results in tyson and lutscher [29]. we illustrate its statements in figure 1, where we plot the numerically calculated steady-state values for hare and owl. there are two cases. in the right plot, there is a stable coexistence state for all values t > t∗∗; in the left plot, the hare goes extinct for some t inside the interval (t−,t+). tyson and lutscher [29] did not find the scenario in the right plot, nor did they realize that there is a second branch of coexistence for large enough t in the left plot. the plots show regions of bistability: the owl-only state is stable for t− < t < t+, yet there exists a positive stable coexistence state at least for some values of t in that interval. in the right plot, stable coexistence is possible for all t > t∗∗. in the left plot, there is a pair of saddle-node bifurcations where the coexistence state disappears and the reappears as t increases. the critical value between these two scenarios cannot be calculated explicitly, but a combination of algebraic geometry and bifurcation analysis can be used to obtain implicit equations that determine the critical case where the two saddle-node bifurcations touch. we can, however, give one simple condition to ensure that there is no bistability possible in the hare-owl system, namely when the solutions (2.18) are not real. this is the case if the discriminant in (2.19) is negative, i.e., if( 1 − a(s + uv) du )2 < 4a2sv d2u . (2.20) we now come to the question of lynx invasion into the hare-owl system. in addition to steady states and bistability, tyson and lutscher found limit cycles in the hare-owl model, and even bistability between a limit cycle and a steady state [29]. therefore, we consider two cases of invasion conditions of the lynx in the hare-owl model. 112 n. bolohan, v. leblanc, and f. lutscher figure 1. steady states of model (2.16) as a function of t. blue and green solid curves represent the hare and owl densities at the coexistence state(s), respectively. the green dashed line represents the owl-only steady state. for t < t∗∗, the only nontrivial steady state is the hare-only state (1, 0). fixed parameters are b = 0.0625, a = 0.2, d = 0.08, h = 0.07, u = 0.5, g = 0.07, v = 0.3333. we set s = 0.75 in the left plot and s = 0.74 in the right. case 1: lynx invasion at a steady state. when we linearize the system at the steady state (x∗, 0,z∗), the linearized lynx equation decouples and reads ẏ = ( fx∗ b + x∗ −m ) y. (2.21) hence, the lynx invasion condition at a steady state is fx∗ b + x∗ −m > 0, (2.22) which is the same as (2.3), where x∗ = 1. case 2: lynx invasion at a periodic orbit. when we linearize the system at a periodic orbit (x∗(t),0,z ∗(t)) with period t , the lynx equation decouples and reads ẏ = ( fx∗(t) b + x∗(t) −m ) y. (2.23) we use floquet theory to obtain the lynx invasion conditions at a periodic orbit as 1 t ∫ t 0 [ fx∗(t) b + x∗(t) −m ] dt > 0. (2.24) in the next section, we use these results and insights about the two-species subsystems to study the three-species model in the positive orthant. we choose the two parameters t and s for illustration purposes. as mentioned previously, the impact of t is of great interest when interpreting the results in terms of global change since it represents the fraction of the year that is summer. we chose s as our second parameter for a number of reasons. the hare-lynx system is independent of t and s, so that the dynamics in that invariant subspace do not change throughout 2-parameter plane. also, s is the parameter that we have the least information on, while for many other parameter values, estimated ranges can be found in the literature [29]. it also turns out that we can express the curves of t∗, t∗∗, seasonal dynamics 113 figure 2. regions in the t-s parameter plane according to threshold values (2.25) and (2.26). the blue dashed curve represents t∗∗. below it, the owl cannot invade the system under any circumstances. even if introduced at high density, it will asymptotically approach extinction. above the blue dashed curve, the owl can invade the hare-only state, but not the hare-lynx state nor the trivial state. above the black curve, which represents s̄, the owl can invade the hare-lynx state, but not trivial state. above the blue solid curve, which represents t∗, the owl can invade all possible steady states in the hare-lynx plane. above the red curve, which represents t±, the hare cannot invade the owl-only state, so that there is bistability in the hare-owl plane. condition (2.20) corresponds to a horizontal line that touches the minimum of the red curve. and t± explicitly for s = s(t), which makes plotting these quantities easy. the explicit expressions are s∗ = uv t , s∗∗ = v t ( u− g(1 −t) d + 1 − th b + 1 ) , s± = uv t + du a(1 −t) , (2.25) respectively. furthermore, with the hare density at the hare-lynx steady state (2.9) denoted as x∗ = mb f−m < 1, the owl invasion condition (2.12) at the hare-lynx state for s as a function of t is s̄ = v t ( u− gx∗(1 −t) d + x∗ − th(x∗)2 b + (x∗)2 ) . (2.26) from the expressions, it is obvious that s∗∗ < s̄ < s∗ < s±. we illustrate these quantities in figure 2. while the absolute locations of these curves change with parameter values, their relative locations do not. in the next section, we take this figure as the blueprint and we superimpose the region of three-species coexistence and its stability properties for difference scenarios. 3. numerical results and comparison with the seasonal model in this section, we evaluate the invasion conditions, the steady-state expression and its stability conditions numerically and display them in plots similar to that in figure 2. we are particularly interested in whether mutual invasion implies coexistence and how the behaviour in the two-species subsystems in the invariant planes affects the behaviour of the three-species system in the positive orthant. we also compare the behaviour of the averaged model with simulations of the seasonal model. we consider different scenarios, depending on the dynamics of the two-species subsystems. we continue to use t and s as our two bifurcation parameters. 114 n. bolohan, v. leblanc, and f. lutscher figure 3. invasion conditions (left), coexistence region (middle) and stability of the coexistence state (right) in the case that the hare-lynx submodel has a globally stable coexistence state. left: the lynx invades the hare-owl state (where it exists) in regions a and c. the owl invades the hare-lynx state in regions a and b. middle: the threespecies coexistence state has all positive components in the black region. only the owl term is non-positive in the magenta region, and only the lynx term is non-positive in the cyan region. right: all three eigenvalues have negative real part in the black region, while only two have negative real part in the blue region. fixed parameters are r = 1, b = 0.0625, a = 0.2, d = 0.08, h = 0.07, u = 0.5, g = 0.07, v = 1 3 , b = 0.3, f = 3.2 and m = 2. 3.1. globally stable coexistence steady state in the hare-lynx system. we begin with the simplest case where (2.8) and (2.10) are satisfied so that the hare-lynx subsystem has a globally stable coexistence state as in (2.9). the qualitative dynamics of the hare-lynx subsystem are independent of parameters t and s. we choose s small enough so that the hare-owl subsystem does not show bistability according to theorem 2.2. in addition to the owl invasion condition at the hare-lynx state (2.25) we calculate the the lynx invasion condition at the hare-owl state; see figure 3. for the chosen parameters, the owl cannot invade when t and s are low (region c), and the lynx cannot invade when t and s are high (region b), but there is mutual invasion in some intermediate range (region a). the green curve in this plot is exactly the same as the black curve in figure 2. the coexistence state (2.5) has all positive components in the black region in the middle panel of figure 3. this region exactly equals region a in the left plot. finally, the eigenvalues at the coexistence state have all negative real parts in the black region in the right panel of figure 3. again, this region exactly equals region a in the left plot. (we observed that the regions are identical by superimposing the plots.) hence, we see that mutual invasion implies (stable) coexistence here. we conjecture that the coexistence state is globally stable if it is positive, but we cannot prove this. 3.2. stable periodic orbit in the hare-lynx system. in the next simplest case, (2.8) is satisfied but (2.10) is not so that the hare-lynx subsystem has a globally stable limit cycle, independent of t and s. as above, we choose s small enough so that the hare-owl subsystem does not show bistability according to theorem 2.2. we calculate the respective invasion conditions, evaluate whether the threespecies coexistence state has all positive components, and calculate the eigenvalues at that state; see figure 4. we do not display the invasion conditions separately since they look qualitatively exactly like in figure 3, left plot. however, we plot the two curves into the steady state (left) and stability (right) plots here. the region of mutual invasibility (i.e., the lynx invades the hare-owl steady state and the owl invades the hare-lynx periodic orbit) corresponds exactly to the region where a positive coexistence state exists (black region in the left plot). the positive coexistence state is unstable and seasonal dynamics 115 figure 4. in the left plot, mutual invasion occurs between the red and grey curves, and the steady state is positive in the black region. in the right plot, mutual invasion occurs between the cyan and magenta curves, and the steady state is stable in the black region. fixed parameters are r = 1, b = 0.0625, a = 0.2, d = 0.08, h = 0.07, u = 0.5, g = 0.07, v = 1 3 , b = 0.3, f = 3.2 and m = 1.6. has two complex conjugate eigenvalues with positive real part in the green region in the right plot. we expect a positive periodic orbit in this region. this orbit disappears in a hopf bifurcation at the boundary between the green and black regions. the positive coexistence state is stable in the black region and disappears in a transcritical bifurcation at the boundary with the blue region where one positive real eigenvalue emerges. 3.3. more complex dynamics in the hare-owl plane. all the cases considered thus far had very simple steady-state dynamics in the hare-owl plane, yet, it is known that more complex scenarios are possible. tyson and lutscher found that the hare-owl system alone can have multistability (three positive steady states of which two are locally stable) or a homoclinic loop bifurcation to a large limit cycle; see figure 5b in [29]. since that system is two dimensional, chaotic behaviour cannot occur. steady states and periodic orbits in the hare-owl plane can both undergo transcritical bifurcations if the lynx invasion conditions (2.22) or (2.24) are satisfied. these conditions can be evaluated numerically. rather than attempting a complete analysis of all potential scenarios (which is beyond the scope of this work because of the sheer number of parameters), we consider several interesting cases. we begin with the case where the hare-owl subsystem shows multiple steady states (figure 5, left plot). the hare nullcline (blue) and the owl nullcline (red) intersect three times in the positive quadrant. the black curves show solutions that approach the left and the right steady state and reveal the saddle structure of the middle state. the lynx can invade the largest hare-owl state if its death rate is small enough, here m < 0.27. it can invade the largest two if m < 0.093 and all three if m < 0.011. however, we know that there can be at most one coexistence state with all three coordinates positive. if the lynx can invade all three states, what happens to the other two? the lynx death rate does not affect the hare-owl subsystem. it turns out that the lynx component of the coexistence state changes sign in this range of m (figure 5, right plot). we illustrate what happens to lynx invasion along the upper positive branch of the coexistence state in figure 5, right plot. this corresponds to the case where 0.093 < m < 0.27, i.e., the lynx can invade 116 n. bolohan, v. leblanc, and f. lutscher 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 hare o w l −0.5 0 0.5 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 lynx o w l figure 5. left plot: multistability in the hare-owl phase plane. the hare nullcline (blue) intersects the owl nullcline (red) three times in the positive quadrant. the smaller and the larger intersection are locally stable, the middle intersection is a saddle. right plot: the owl and lynx components of the coexistence state as a function of lynx death rate m. the intersections with the line where the lynx component is zero correspond to the three positive steady states of the hare-owl subsystem. the lynx component is zero when the lynx invasion condition is an equality, i.e., for m ≈ 0.27 (top), m ≈ 0.093 (middle) and m ≈ 0.011 (bottom). parameters are b = 0.0025, a = 0.2, d = 0.08, h = 0.07, u = 0.5, g = 0.07, v = 1/3, b = 2, f = 1, s = 1/3, t = 0.65. all three positive hare-owl states. for m = 0.25, the lynx invades from low density and the system settles at the stable coexistence point (figure 6, top left). for m = 0.18, the system reaches the stable coexistence point by decaying oscillations. for m = 0.173, the lynx invades from low density, decreases the hare density so low that the lynx itself cannot survive. the hare-owl system settles at its lower stable state (bottom left). however, for the same value of m, all three species can persist in the system if initial conditions are chosen close to the coexistence state. the coexistence state can be stable or unstable, in which case a limit cycle exists (bottom right). decreasing m further increases the amplitude of the limit cycle, so that eventually the hare and lynx drop below the threshold where the lynx can persist, and the hare-owl system settles at its lower stable state as in the bottom left plot. hence, the region of multistability in the hare-owl subsystem creates a region of multistability in the three-species system. there are two cases: two locally stable states may coexist (one with and one without the lynx), or one locally stable state (without the lynx) may coexist with a locally stable limit cycle (of all three species). the second case that we consider also includes multistability in the hare-owl subsystem, but this time between a periodic orbit and a steady state (figure 7, left plot). the nullclines intersect only once in the positive quadrant, and the steady state is locally stable. in addition, there is a locally stable limit cycle that arose through a homoclinic loop bifurcation; for details, see [8, 29]. the hare density at the steady state is approximately 0.0417, so that the invasion condition for the lynx at this state requires m < 0.02. the lynx can invade the limit cycle according to condition (2.24). numerical evaluation gives a floquet multiplier of almost 6. hence, we observe a limit cycle of coexistence of all three species in the absence of a coexistence steady state with positive densities for all species. in addition, we have multistability since the semitrivial state where the lynx is absent and the hare and owl are at their steady state, is also locally stable. as in the previous case, initial conditions can lead to the system moving from one basin of attraction to another and thereby lead to a surprising outcome. if the lynx is introduced in low density near the hare-owl steady state, it will decline to extinction because its death rate (m = 0.4) is much higher than the threshold for invasion (m = 0.02). however, if the lynx is seasonal dynamics 117 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 time d e n s it ie s 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 time d e n s it ie s 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 time d e n s it ie s 0 100 200 300 400 500 0.2 0.25 0.3 0.35 0.4 0.45 0.5 time d e n s it ie s figure 6. time course of the system as the lynx invades a hare-owl state for m = 0.25 (top left), m = 0.18 (top right), m = 0.173 (bottom). the lynx initial conditions for all plots except bottom right are 0.02. at the bottom right, initial conditions are near the coexistence state. parameters are as in figure 5 introduced at a very high density, it will decimate the hare initially and thereby push the entire system into the basin of attraction of the large limit cycle. at that limit cycle, the lynx can invade, and it will therefore stay in the system. for the given parameter values, the hare-lynx system (in the absence of the owl) will approach a globally stable positive steady state. the owl can invade this steady state from low density. therefore, the owl will never be excluded from the system in this scenario. this latter example shows that three species may coexist indefinitely even in the absence of a positive steady state. in other words, we would not have found this coexistence periodic orbit from a local stability analysis of the (explicitly available) steady-state expression. this finding illustrates the power of invasion analysis. we note that periodic coexistence of a two-predator–one-prey system in the absence of a coexistence steady state has been proven before in a different system [25]. 4. discussion mathematical models for population dynamics in a seasonally varying environment are becoming more prominent and more important for studying the effects of a changing climate on biological communities. a particularly interesting aspect arises when the interaction between species changes with seasons. our study is inspired by the snowshoe hare, canadian lynx and great horned owl system in western canada, where the owl acts as a generalist predator in the summer and as a specialist in the winter, while the lynx is a specialist predator year round [17]. our model is an extension of the two-species model in [29], where the lynx was not considered, even though it is the dominant predator of the showshoe hare. 118 n. bolohan, v. leblanc, and f. lutscher 0 0.2 0.4 0.6 0.8 0 0.02 0.04 0.06 0.08 0.1 hare o w l 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 time d e n s it ie s figure 7. left plot: multistability in the hare-owl plane. right plot: successful lynx invasion leads to limit cycle coexistence of the three species. parameters are b = 0.0025, a = 0.2, d = 0.1, h = 0.05, u = 0.4, g = 0.1, v = 1/3, b = 2, f = 1, m = 0.05, s = 1/3 and t = 0.414. figure 8. comparing solutions of the averaged model (1.3) to those of the seasonal model (1.2). averaged solutions are plotted in black, and seasonal dynamics are plotted in blue, red and green, representing hare, lynx and owl, respectively. condition (2.10) is satisfied in the first row, yielding a steady state in the averaged model. the condition is reversed in the second row, yielding a periodic solution. fixed parameters are r = 1, b = 0.0625, a = 0.2, d = 0.08, h = 0.07, u = 0.5, g = 0.07, v = 1 3 , b = 0.3, f = 3.2, s = 0.4 and t = 0.5. we set m = 2 in the first row and m = 1.7 in the second. even though we represented a continuously varying environment by simply distinguishing two seasons, our model is still a periodically forced system of equations. we use temporal averaging to derive an autonomous model that still is complicated since the model has in some sense “more” parameters than “normal” models due to the fact that two seasons are represented. we could show that our model has seasonal dynamics 119 figure 9. comparing solutions of the averaged model (1.3) to those of the seasonal model (1.2). averaged solutions are plotted in black, and seasonal dynamics are plotted in blue, red and green, representing hare, lynx and owl, respectively. b = 0.0025, a = 0.2, d = 0.08, h = 0.07, u = 0.5, g = 0.07, v = 1/3, b = 2, f = 1, s = 1/3, t = 0.65. m = 0.1753 in the first row, m = 0.17513 in the second and m = 0.1749 in the third. initial conditions for all rows are (x0,y0,z0) = (0.75, 0.01, 0.168). a unique coexistence steady state (with all three components nonzero), but the precise conditions for positivity and stability require numerical evaluation. instead, we used the structure of the system with two invariant planes (the hare-owl and the hare-lynx subsystem) in combination with invasion analysis to obtain a number of more general qualitative results on our system. in particular, we considered the principle that “mutual invasion implies coexistence” [27] to find conditions of coexistence of the two predators. we found that the length of the summer season (t) can have a profound effect on system dynamics with various bifurcations occurring as t changes. general conclusions, however, are difficult to draw since the dynamics overall can be hugely varying. a longer summer season tends to benefit the generalist predator because it has additional food sources then. in particular, there is a threshold value of t, above which the generalist does not require the focal prey for persistence. it is unclear how likely this 120 n. bolohan, v. leblanc, and f. lutscher scenario is in reality, both in terms of t ever getting that large and winter mortality being modeled by a linear process. on the other hand, for the parameter values chosen, the specialist predator has an advantage at short season lengths. intermediate values of t seem to promote coexistence between the two species. it was not our goal to analyze system behavior in all of parameter space since this would be impossible. instead, the techniques that we presented translate to other seasonal population models. if and when estimates of parameter are available, our method allows a relatively inexpensive analysis of the averaged model, which can then be followed up with simulations of the seasonal model in parameter ranges that the averaged model identified as interesting. it is an old tenet in population biology that two predator species cannot persist at a stable steady state on a single limiting resource [18], but they can coexist in a stable limit cycle [2]. since one of our predators has an alternate food source during the summer, this tenet does not apply to our model, and we found indeed that coexistence of the two species is possible at a stable steady state as well as a stable limit cycle. we also found that mutual invasion implied coexistence when the dynamics in each of the two invariant subsystems were simple enough. when the dynamics in the hare-owl system showed bistability, some ecologically surprising outcomes were possible. in particular, an invasion attempt by the lynx could be successful in the short run but cause the system to move to the basin of attraction of another state, at which the lynx could not invade, so that it would eventually go extinct. vice versa, an invasion attempt with a sufficient number of lynx could push the system from a non-invasible state to the basin of attraction of an invasible state and the lynx could persist in the long run. such an initial invasion at large numbers may not be expected to occur naturally but could be used a a management tool (e.g., through a captive breeding and release program) to resettle a species that has become locally extirpated. while our model included some variability between seasons, it did not include all that is possible. for example, it is conceivable that the specialist predation success varies with season when snow cover gives refuges to the snowshoe hare. we also did not include any winter mortality of the prey. such extensions can easily be included into the model, and the process of averaging can still be applied. the analysis of the resulting model is complicated not only by the increased number of parameters but also by the fact that the dynamics on the invariant hare-lynx plane then depend on season length t as well. the most obvious question that remains from our analysis is to what extent the averaged model captures and represents the dynamics of the seasonal model. from an abstract point of view, if there is a small parameter that represents the separation of time scales, averaging theory can be applied. if no obvious small parameter exists, there are no general theorems about the applicability of any kind of averaging. however, it is a typical observation in averaging theory that the averaged model provides a good approximation even beyond the case where it can be mathematically proven to be valid. while we cannot analytically identify a small parameter in our model, biological considerations indicate different time scales. while the hare population dynamics are relatively fast with several reproductive events within a single summer, lynx and owls reproduce only once a year so that their dynamics could be considered slow compared to seasonal changes. when we compared numerical simulations of the averaged model with the seasonal model, we found a good agreement for the simpler dynamics from sections 3.1 and 3.2; see figure 8. the solution of the averaged model (in black) is very close to the solution of the seasonal model (in colour). the corresponding analysis of the averaged model is in figures 3 and 4. with some more complex dynamical behaviour, the correspondence between the averaged and the seasonal model is not as perfect; see figure 9 and corresponding figure 5. as lynx mortality m decreases, the system passes through a hopf bifurcation, but as limit cycles grow in amplitude, solutions get trapped into the semitrivial state where the lynx is extinct. here, the solution to the averaged model seasonal dynamics 121 captures the solution to the seasonal model at least for some transient period but tends to collapse to the semitrivial state earlier. the discrepancy might not be satisfying for mathematicians, yet, decent short-term predictions and transients are still highly valuable in applications in ecosystem management. these findings are in line with previous results [15, 29]. it is clear, however, that this method is limited and cannot work always. for example, it is known that 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[29] r. tyson and f. lutscher, seasonally varying predation behaviour and climate shifts are predicted to affect predator– prey cycles, am nat 188 (2016), 539–553. department of mathematics and statistics, university of ottawa, ottawa, on, k1n 6n5, canada e-mail address: nbolo094@uottawa.ca department of mathematics and statistics, university of ottawa, ottawa, on, k1n 6n5, canada e-mail address: vleblanc@uottawa.ca corresponding author, department of mathematics and statistics, and department of biology, university of ottawa, ottawa, on, k1n 6n5, canada e-mail address: flutsche@uottawa.ca mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 2, june 2020, pp.181-206 https://doi.org/10.5206/mase/10739 on mechanisms of trophic cascade caused by anti-predation response in food chain systems yang wang and xingfu zou abstract. motivated by a recent field study [nat. commun. 7(2016), 10698] on the impact of fear of large carnivores on the populations in a cascading ecosystem of food chain type with the large carnivores as the top predator, in this paper we propose two model systems in the form of ordinary differential equations to mechanistically explore the cascade of such a fear effect. the models are of the lotka-volterra type, one is three dimensional and the other four dimensional. the 3-d model only considers the cost of the anti-predation response reflected in the decrease of the production, while the 4-d model considers also the benefit of the response in reducing the predation rate, in addition to the cost by reducing the production. we perform a thorough analysis on the dynamics of the two models. the results reveal that the 3-d model and 4-model demonstrate opposite patterns for trophic cascade in terms of the dependence of population sizes for each species at the co-existence equilibrium on the anti-predation response level parameter, and such a difference is attributed to whether or not there is a benefit for the anti-predation response by the meso-carnivore species. 1. introduction predator-prey interactions have attracted the great attention of both ecologists and mathematical biologists, not only because of their vast existence in nature but also because of their diversified forms and rich consequences in the real world. mathematically, if only considering direct interaction through predation, a classic predator-prey model can be generally described by a system of ordinary differential equations of the form:   du dt = f1(u(t)) −p(u(t),v(t))v(t), dv dt = f2(v(t)) + cp(u(t),v(t))v(t). (1.1) see, e.g., [16, 17, 28, 29]. here u(t) and v(t) are the populations of the prey and predator respectively, f1(u) and f2(v) denote growth functions of the prey and the predator respectively, p(u,v) is the functional response which accounts for the predation rate and biomass transfer from the prey to the predator after predation, and the constant c explains the efficiency in biomass transfer. predator-prey ode models of the above form only consider interactions of two species with direct effect reflected by the predation term. however, since 1990s, more and more ecologists have realized the existence of indirect effects (e.g., fear effect), and observed impact of such effects; see, e.g. [1, 14, 21, 26]. recent field experiments have found the presence of predator itself, can have significant influence on prey’s population through changes in reproduction [15, 35], habitat selection [4, 27] and physiology [3, 5, 34]. in contrast, as far as mathematical modeling is concerned, indirect effects have been largely (if received by the editors 25 may 2020; accepted 16 june 2020; online first 18 june 2020. 2000 mathematics subject classification. 34k20, 92b05. key words and phrases. predator-prey, food chain, anti-predation strategy, stability. supported by nserc of canada (rgpin-2016-04665). 181 182 y. wang and x. zou not all) ignored in those existing models describing predator-prey interactions and those on conservation and management of the ecosystem. motivated by the field study in [35] which observed an as high as 40% decrease in prey’s reproduction rate when the prey perceived a risk of predator coming from the playback of the predator’s voice, [30] formulated a mathematical model in the form of the following ordinary differential equations   du dt = f(k,v(t)) r0 u(t) −du(t) −au2(t) −p(u(t)) v(t), dv dt = cp(u(t)) v(t) −mv(t). (1.2) here u is the population of the prey species and v is the population of predator species, and the prey’s growth follows a logistic growth with the intrinsic growth rate being split into the net growth rate r0 and natural death rate d, given by r0 −d. to mimic the scenario of the field experimental study in [35] in which predation actually did not occur due to the use of electronic fence, in (1.2) the fear effect is only incorporated into the production term by the function f(k,v(t)) accounting for a cost. the term au2 = (au)u reflects the self-limiting mechanism of u (due to intra-species competition) and c is the biomass transform efficiency constant. the function p(u) is the functional response which is assumed to depend on the prey population only. analysis of (1.2), both analytical and numerical, have revealed some interesting dynamics that would have not occurred without considering the fear effect. since [30], there have been some follow-up modelling works that extend the model (1.2) to accommodate various aspects of fear effect. for example, [31] considered age structure and discussed different impacts of fear effect on different age stages; [32] explored the fear effect reflected thought the dispersals and its impact on the pattern formation. [8] incorporated an extra food source for the predator in (1.2) and also added a white noise to the death rates of the prey and predator, and analyzed the resulting stochastic model. [18] further considered a digestion delay in addition to the cost of fear, white noise in the death rates, and extra food for the predator but ignored the benefit of the anti-predation response. more recently, in addition to a digestion delay and a cost in prey’s production due to the anti-predation response, [33] further incorporated a benefit term from the anti-predation response, and explored the joint impact of the fear effect and the digestion delay on the population dynamics for both predator and prey. in the real world, it is quite common that a species can predate on one species and in the mean time, it can be a prey of other species. this leads to occurrence of food chains between multiple species, consisting of cascading predator-prey interactions. one may naturally ask how a fear effect arising from one or more species in the chain will affect the dynamics of the whole cascaded populations? in a more recent field experimental study [25], the authors tested a meso-predator cascade by manipulating the large carnivores playback, which resulted in a decrease in the population of meso-carnivore and increase in the population of its prey. that is, the fear effect on the top species in that chain of three species actually affects every species in the ecosystem explicitly or implicitly. to better understand the above mentioned propagation of the fear effect from top layer to the bottom layer in that food chain ecosystem reported in [25], we incorporate a fear effect on the top species into a lotka-volterra type food chain model, formulated by the following system of ordinary differential equations anti-predation response in food chain systems 183   dn1 dτ = n1 (r1 −a11n1 −a12n2) , dn2 dτ = n2 (r2 −a22n2 −a23n3 + a21n1) , dn3 dτ = n3 (b(α) −d −a33n3 + a32n2) , n1(0) ≥ 0 , n2(0) ≥ 0 , n3(0) ≥ 0. (1.3) here, n3 is the population of the meso-carnivore (e.g., racoon) which is affected by the large carnivore’s (e.g., wolf, bear) playback; n2 is the prey (e.g., crab) of the meso-carnivore n3, and n1 is the prey of n2. each species is assumed to follow a logistic growth with growth rates r1, r2 and b(α)−d respectively. the population of the large carnivores does not appear in the system because in the field study [25], only their voices are played, and hence they only have fear (indirect) effect on the meso-carnivores represented by b(α), where the net birth (production) rate b(α) depends on a parameter α standing for the meso-carnivore’s anti-predation response level. by its biological meaning, b(α) is assumed to satisfy b′(α) < 0, b(0) = b3 > 0 and lim α→∞ b(α) = 0. (1.4) the constant d is the natural death rate of n3, aii (i = 1, 2, 3) are the intra-species competition coefficients. a12 and a23 are the predation rates, while a21 and a32 are the conversion rate of the biomass from n1 to n2 and from n2 to n3 respectively; thus, a12/a21 and a23/a32 actually account for the efficiency of biomass transfer from the predations. we point out that this type of three dimensional food chain models have been intensive and extensively studied by some researchers, see, e.g., [9, 11, 12, 13] and the references therein. as far as fear effect in food chains is concerned, two recent papers [19, 20] have also followed line of [30] to consider fear effect in food chain of three species; their scenario is different from ours: they considered other types of functional responses, they incorporated fear effects in the bottom and middle species, and they assumed that the top and middle species are specialist predators. the first goal of this paper is to explore the dynamics of (1.3), particularly, the impact of the meso-carnivore’s anti-predation response level α on the dynamics. in (1.3), only the cost for the meso-carnivore’s anti-predation response is considered. for such a response, in addition to cost, there should also be a benefit (see, e.g., [6, 26]), typically reflected by the decrease in the chance of being predated. a response strategy is expected to seek balance between cost and benefit to achieve certain optimality. in order to also add the benefit into the interplay in the above model system (1.3), we need the term of predation on the meso-carnivore by a large carnivore, which inevitably requires us to add the the population of that large carnivore into the system. this leads to the following four dimensional food chain model  dn1 dτ = n1 (r1 −a11n1 −a12n2) , dn2 dτ = n2 (r2 −a22n2 −a23n3 + a21n1) , dn3 dτ = n3 (b(α,n4) −d3 −a33n3 + a32n2 −a34(α)n4) , dn4 dτ = n4 (−d4 + c̄a34(α)n3) , n1(0) ≥ 0 , n2(0) ≥ 0 , n3(0) ≥ 0 , n4(0) ≥ 0, (1.5) where n4 is the population of the restored top predator (large carnivore) which is assumed to be a specialist predator with the mortality rate d4. now the net growth function of n3 depends not only 184 y. wang and x. zou on the anti-predation response level α but also on the population of its predator n4. the function a34(α) denotes the encounter rate between n3 and n4 which is affected by the protective behaviours of n3 species characterized by its dependence on the anti-predation response level α. by their biological meanings of b(α,n4) and a34(α), they are assumed to satisfy the following conditions: { b(α,n4) is decreasing in α and n4, b(0,n4) = b(α, 0) = b3 > 0, lim α→∞ b(α,n4) = lim n4→∞ b(α,n4) = 0, (1.6) and a34(α) is decreasing, a34(0) = a0 > 0, lim α→∞ a34(α) = 0. (1.7) finally, c̄ is the efficiency of biomass transform. so in this model, we consider both the complex multi-trophic predator-prey structure and the trade-off from anti-predation response. the remainder of this paper is organized as follows. in section 2, we analyze the model system (1.3). we establish the well-posedness of system (1.3) and find the condition for existence and stability for all its equilibrium solutions. we also discuss the relationship between the anti-predation level and the final population size. in the end, some numerical examples, together with some discussions, are given to demonstrate our results. in section 3, we investigate the dynamics of the four dimension model (1.5), including the existence and stability of equilibria as well as the continuously dependence between the final population size with respect to the anti-predation level. we also discuss the difference of the results from those for (1.3) in section 2. in addition, we also present some numerical examples to illustrate that different functional response functions may lead to slightly different dynamical behaviour of the solution. in section 4, we summarize our main results and discuss their biological implications. we also discuss some possible future projects along this direction of anti-predation response in predator-prey interactions. 2. analysis of the model without large carnivores in this section, we analyze the three-species model (1.3). we first show the well-posedness of the system (1.3). then we find all equilibrium solutions and discuss their stability in terms of the parameter values and anti-predation strategy level. as we mentioned in last section, this kind of food chain models have been studied in literatures, and thus, some technical results can be found in existing researches, e.g., [9, 11, 12, 13] and their references. but we need to associate the results to the new parameter α, the anti-predation response level of the species n3 to shy light on influence of the fear effect for this model. 2.1. preliminaries. for mathematical simplification, we first non-dimensionalize the model (1.3). let t = r1τ, x = a11n1 r1 , y = a12n2 r1 , z = a23n3 r1 , then model (1.3) becomes   dx dt = x(1 −x−y), dy dt = y(k −d1y −z + β1x), dz dt = z(f(α) −d2z + β2y), (2.1) anti-predation response in food chain systems 185 where k = r2 r1 , d1 = a22 a12 , d2 = a33 a23 , β1 = a21 a11 , β2 = a32 a12 , f(α) = b(α) −d r1 , f(0) = b3 −d r1 , lim α→∞ f(α) = − d r1 . by the basic theory of ode systems, we can easily show that the initial value problem for (2.1) has a unique solution; moreover, the solution is nonnegative (positive) with nonnegative (positive) initial conditions because each equation in (2.1) is of gaussian type. now we show that the solution to system (2.1) is bounded. from the first equation in system (2.1) and by the non-negativity of y(t), we have dx dt = x(1 −x−y) ≤ x(1 −x). by the comparison theorem [24], we can obtain limt→∞ sup x(t) ≤ 1. therefore, for any �1 > 0, there holds x(t) ≤ 1 + �1 for large t. incorporating this estimate for large t into the second equation in (2.1) results in dy dt = y(k −d1y −z + β1x) ≤ y(k + β1(1 + �1) −d1y), for large t. applying the comparison theorem again, we then obtain lim t→∞ sup y(t) ≤ k + β1(1 + �1) d1 . since �1 > 0 is arbitrary small, the above inequality actually implies lim t→∞ sup y(t) ≤ k + β1 d1 . incorporating the above inequality into the third equation in (2.1) and by the same argument, we can obtain lim t→∞ sup z(t) ≤ max ( f(α)d1 + β2(k + β1) d1d2 , 0 ) . combining the above, we have proved that the solution (x(t),y(t),z(t)) to (2.1) is bounded. 2.2. existence and stability of the boundary equailibria. in this section, we find all boundary equilibrium solutions and give the condition for their existence and stability. for the result to be biologically meaningful, we are only interested in equilibria with nonnegative components. an equilibrium of (2.1) solves the following system  x(1 −x−y) = 0, y(k −d1y −z + β1x) = 0, z(f(α) −d2z + β2y) = 0. there are seven boundary equilibrium solutions. e0 = (0, 0, 0) is the trivial equilibrium solution which always exists. e1 = (1, 0, 0), e2 = (0,k/d1, 0) and e3 = (0, 0,f(α)/d2) are the equilibria representing the scenario that only one species survives; e1 and e2 always exist, while e3 exists only when f(α) > 0. 186 y. wang and x. zou there are also other three possible equilibria corresponding to the scenario of two species coexisting, and they are given by e12 = ( 1 − k + β1 d1 + β1 , k + β1 d1 + β1 , 0 ) , e13 = ( 1, 0, f(α) d2 ) , e23 = ( 0, d2k −f(α) d1d2 + β2 ,k − d1d2k −d1f(α) d1d2 + β2 ) . by the nonnegative requirement, e12 exists when k < d1, e13 exists when f(α) > 0 and e23 exists when d2k > f(α) > −β2k/d1. the local stability of an equilibrium is obtained by linearization at the equilibrium. the jacobian matrix at equilibrium (x∗,y∗,z∗) is given by j (e = (x∗,y∗,z∗)) =   1 − 2x∗ −y∗ −x∗ 0 β1y ∗ k − 2d1y∗ + β1x∗ −z∗ −y∗ 0 β2z ∗ f(α) − 2d2z∗ + β2y∗   . (2.2) at e0 = (0, 0, 0), the jacobian is given by j(e0) =   1 0 0 0 k 0 0 0 f(α)   , therefore e0 is unstable, as there are positive eigenvalues λ1 = 1 and λ2 = k. similarly, at e1 = (1, 0, 0), the jacobian is given by j(e1) =   −1 −1 0 0 k + β1 0 0 0 f(α)   , therefore e1 is unstable, as there is a positive eigenvalue λ = k + β1. at e2 = (0,k/d1, 0), the jacobian is given by j(e2) =   1 − k d1 0 0 β1k d1 −k − k d1 0 0 f(α) + β2k d1   , therefore e2 is asymptotically stable if and only if 1 −k/d1 < 0 and f(α) < −β2k/d1. anti-predation response in food chain systems 187 at e3 = (0, 0,f(α)/d2), the jacobian is given by j(e3) =   1 0 0 0 k − f(α) d2 0 0 β2f(α) α2 −f(α)   , hence, e3 is unstable since there is a positive eigenvalue λ = 1. at e12, the jacobian becomes j(e12) =   k + β1 d1 + β1 − 1 k + β1 d1 + β1 − 1 0 β1k + β 2 1 d1 + β1 − d1k + d1β1 d1 + β1 − k + β1 d1 + β1 0 0 f(α) + β2k + β2β1 d1 + β1   , thus, e12 is asymptotically stable if and only if f(α) < − β2k + β2β1 d1 + β1 . at e13 = (1, 0,f(α)/d2), the jacobian reduces to j(e13) =   −1 −1 0 0 k + β1 − f(α) d2 0 0 β2f(α) α2 −f(α)   , therefore e13 is asymptotically stable if and only if f(α) > d2(k + β1). at e23 = ( 0, d2k −f(α) d1d2 + β2 ,k − d1d2k −d1f(α) d1d2 + β2 ) , the jacobian is given by j(e23) =   1 − d2k −f(α) d1d2 + β2 0 0 β1d2k −β1f(α) d1d2 + β2 − d1d2k −d1f(α) d1d2 + β2 − d2k −f(α) d1d2 + β2 0 β2k − β2d1d2k −β2d1f(α) d1d2 + β2 −d2k + d1d 2 2k −d1d2f(α) d1d2 + β2   , therefore e23 is asymptotically stable if and only if f(α) > d2k −d1d2 −β2. we summarize the above analysis in the following proposition. proposition 2.1. for system (2.1), the following statement hold. (i) equilibrium e0 and e1 always exist and are unstable. (ii) equilibrium e2 always exists; it is asymptotically stable if and only if k > d1 and f(α) < −β2k/d1. 188 y. wang and x. zou (iii) when f(α) > 0, e3 exists; it is unstable. e13 exists and is asymptotically stable if and only if f(α) > d2(k + β1). (iv) when k < d1, e12 exists; it is asymptotically stable if and only if f(α) < −β2(k +β1)/(d1 +β1). (v) when d2k > f(α) > −β2k/d1, e23 exists; it is asymptotically stable if and only if f(α) < d2k −d1d2 −β2. 2.3. existence and stability of a positive equilibrium solution. there is a unique positive equilibrium solution e∗ = (x∗,y∗,z∗) if d2(k + β1) > f(α) > max ( d2k −d1d2 −β2, −β2(k + β1) d1 + β1 ) . (2.3) indeed, e∗ = (x∗,y∗,z∗) is given by   x∗ = 1 −y∗, y∗ = d2(k + β1) −f(α) d2(d1 + β1) + β2 , z∗ = k + β1 − (d1 + β1)y∗. (2.4) the conditions in (2.3) directly comes from the formulas in (2.4). the jacobian matrix at the positive equilibrium can be simplified as j(e∗) =   −x∗ −x∗ 0 β1y ∗ −d1y∗ −y∗ 0 β2z ∗ −d2z∗   , thus, the corresponding characteristic equation is λ3 + (d1y ∗ + d2z ∗ + x∗)λ2 + (d1d2y ∗z∗ + β1x ∗y∗ + β2y ∗z∗ + d1x ∗y∗ + d2x ∗z∗)λ + (β1d2 + d1d2 + β2)x ∗y∗z∗ = 0. where   a1 = d1y ∗ + d2z ∗ + x∗ > 0, a2 = d1d2y ∗z∗ + β1x ∗y∗ + β2y ∗z∗ + d1x ∗y∗ + d2x ∗z∗ > 0, a3 = (β1d2 + d1d2 + β2)x ∗y∗z∗ > 0, and a1a2 −a3 = d21d2(y ∗)2z∗ + d1d 2 2y ∗(z∗)2 + β1d1x ∗(y∗)2 + β2d1(y ∗)2z∗ + β2d2y ∗(z∗)2 + d21x ∗(y∗)2 + 2d1d2x ∗y∗z∗ + d22x ∗(z∗)2 + β1(x ∗)2y∗ + d1(x ∗)2y∗ + d2(x ∗)2z∗ > 0. by routh-hurwitz criterion, the positive equilibrium is locally asymptotically stable. moreover, we can prove that it is actually globally asymptotically stable as long as it exists (i.e., (2.3) holds). theorem 2.1. if (2.3) holds, then the positive equilibrium e∗ is globally asymptotically stable. proof. consider the lyapunov function v (x,y,z) = β1β2(x−x∗ −x∗ ln x x∗ ) + β2(y −y∗ −y∗ ln y y∗ ) + (z −z∗ −z∗ ln z z∗ ), then dv dt = −β1β2(x−x∗)2 −β2(y −y∗)2 − (z −z∗)2 ≤ 0 anti-predation response in food chain systems 189 and dv dt = 0 if and only if (x,y,z) = (x∗,y∗,z∗). by lasalle’s invariant principle, we conclude e∗ is globally asymptotically stable. � for readers’ convenience, we summarize the analytical results on the dynamics of the model (2.1) obtained above in the following table 1. table 1. condition of existence and stability of the equilibria in model (2.1) equilibrium solution existence stability e0 always unstable e1 always unstable e2 always d1 < k and f(α) < −β2k/d1 e3 f(α) > 0 unstable e12 d1 > k f(α) < −β2(k + β1)/(d1 + β1) e13 f(α) > 0 f(α) > d2(k + β1) e23 d2k > f(α) > −β2k/d1 f(α) < d2k −d1d2 −β2 e∗ d2(k + β1) > f(α) > max ( d2k −d1d2 −β2,− β2(k + β1) d1 + β1 ) gas from theorem 2.1 , we know that the positive equilibrium is always globally asymptotically stable as long as it exists (i.e., (2.3) holds), implying that the populations of all three species will converge to co-existence state at the respect levels x∗, y∗ and z∗. thus, it is worthwhile to investigate how the response strength α will affect these levels. indeed, direct calculations give  dy∗ dα = −f′(α) d2(d1 + β1) + β2 > 0, dx∗ dα = − dy∗ dα < 0, dz∗ dα = −(d1 + β1) dy∗ dα < 0. that is, within the range of α that guarantees (2.3), with the increase of the anti-predation response strength α, the final population sizes of the meso-carnivore, its prey and the prey’s prey will decrease, increase and decrease respectively, demonstrating an alternative pattern for the fear effect in the cascade, which was observed in the field study [25]. therefore, the model (1.3) does provide a mechanism that can explain the phenomenon of trophic cascade caused by a fear of large carnivores reported in [25]. 2.4. numerical simulations. in this subsection, we present some numerical simulations to illustrate the analytical results obtained above. for this purpose, we choose a particular form for the function b(α) given by b(α) = r3 1 + cα , 190 y. wang and x. zou and this sends (1.3) to the following system:  dn1 dτ = n1 (r1 −a11n1 −a12n2) , dn2 dτ = n2 (r2 −a22n2 −a23n3 + a21n1) , dn3 dτ = n3 ( r3 1 + cα −d −a33n3 + a32n2 ) , n1(0) ≥ 0 , n2(0) ≥ 0 , n3(0) ≥ 0, (2.5) we fix the parameters r1 = 1, a11 = 1, a12 = 0.4, r2 = 1, a22 = 0.2, a23 = 0.5, a21 = 0.5, r3 = 3, d = 1, a33 = 0.5, a32 = 0.05, c = 0.4, (2.6) and demonstrate how α impacts the population dynamics. in this case, k = r2/r1 = 1 and d1 = a22/a12 = 1/2 so k > d1. according to proposition 2.1, there are three threshold values for α, denoted by α∗1, α ∗ 2 and α ∗ 3, which are given by  f (α∗1) = d2(k + β1), f (α∗2) = d2k −d1d2 −β2, f (α∗3) = − β2k d1 . using the parameter values in (2.6), we obtain α∗1 = 0.5, α ∗ 2 ≈ 2.954545455 and α∗3 = 7.5. by proposition 2.1, when 0 < α < 0.5, the equilibrium e13 is stable (as demonstrated in figure 1-(a) for α = 0.4); when 0.5 < α < 2.954545455, the coexistence equilibrium e∗ is stable (as demonstrated in figure 1-(b) for α = 2); when 2.954545455 < α < 7.5 (destroying (2.3), hence e∗ no longer exists), the equilibrium e23 is stable (as demonstrated in figure 1-(c) for α = 5); when α > 7.5, the equilibrium e2 is stable (as demonstrated in figure 1-(d) for α = 10). the bifurcation diagram with respect to α is given in figure 2. now, we change r1 to r1 = 3 and a33 to a33 = 0.1 and keep other parameters the same as in (2.6). then k = r2/r1 = 1/3 and d1 = a22/a12 = 1/2 leading to the scenario of k < d1. according to proposition 2.1, there are two threshold values for α, denoted by α1 and α2, which are given by  f (α1) = d2(k + β1), f (α2) = − β2(k + β1) d1 + β1 . using the parameter values in (2.6), we obtain α1 = 2.5 and α2 ≈ 8.409090911. by proposition 2.1, when 0 < α < 2.5, the equilibrium e13 is stable (as demonstrated in figure 3-(a) for α = 2); when 2.5 < α < 8.409090911, the coexistence equilibrium e∗ is stable (as demonstrated in figure 3-(b) for α = 6); when α > 8.409090911, the equilibrium e12 is stable (as demonstrated in figure 3-(c) for α = 10). the bifurcation diagram with respect to α is given in figure 4. we find that depending on the difference between k and d1, we have two kinds of bifurcation. in both cases, when the anti-predation response α passes a threshold, it leads to a transcritical bifurcation and we can always observe a meso-predator cascade inside the coexistence region when increasing α: the population of meso-predator is decreasing, the population of its prey is increasing and the population of the prey’s prey (bottom prey) is decreasing. however, this pattern will be dramatically changed when we restore large carnivores instead of only manipulating their playback to induce fear. in the next section, we will model the case when we also introduce the large carnivores back into the food chain, leading to a 4-d model. anti-predation response in food chain systems 191 0 20 40 60 80 100 time 0 1 2 3 4 p o p lu a ti o n solution of the three dimension system = 0.4 level-1 level-2 level-3 (a) 0 20 40 60 80 100 time 0 0.5 1 1.5 2 2.5 p o p lu a ti o n solution of the three dimension system = 2 level-1 level-2 level-3 (b) 0 20 40 60 80 100 time -1 0 1 2 3 4 5 p o p lu a ti o n solution of the three dimension system = 5 level-1 level-2 level-3 (c) 0 20 40 60 80 100 time 0 1 2 3 4 5 6 p o p lu a ti o n solution of the three dimension system = 10 level-1 level-2 level-3 (d) figure 1. population dynamics of (2.5) when d1 < k. (a) when 0 < α = 0.4 < 0.5, the equilibrium e13 is stable, (b) 0.5 < α = 2 < 2.954545455, the coexistence equilibrium e∗ is stable, (c) when 2.954545455 < α = 5 < 7.5, the equilibrium e23 is stable, (d) when α = 10 > 7.5, the equilibrium e2 is stable. 3. model with restoring large carnivores in this section, we analyze (1.5) which has the population of large carnivores incorporated together with a benefit in preventing predation of the meso-carnivore by the large carnivores, in addition to the cost in the meso-carnivore’s production. parallel to section 2, we first establish the well-posedness of the 4-d model (1.5), discuss all possible equilibrium solutions and find the condition for their existence and stability in terms of the parameter values and anti-predation strategy level. 3.1. well-posedness. for mathematical simplification, we still apply non-dimensionalization for our model (1.5) which is a natural expansion of the non-dimensionalization for (1.3) in section 2, given by t = r1τ, x = a11n1 r1 , y = a12n2 r1 , z = a23n3 r1 , w = a0n4 r1 , 192 y. wang and x. zou 0 2 4 6 8 10 0 1 2 3 4 5 p o p lu a ti o n bifurcation diagram when k>d 1 level-1 level-2 level-3 (a) 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 p o p lu a ti o n meso-predator cascade in the coexistence region level-1 level-2 level-3 (b) figure 2. (a) bifurcation diagram of (2.5) when k > d1, (b) in the region when the coexistence equilibrium e∗ is stable, we can observe a meso-predator cascade. with a0 = a34(0). then model (1.5) becomes  dx dt = x(1 −x−y), dy dt = y(k −d1y −z + β1x), dz dt = z (f(α,w) −d2z + β2y −p(α)w) , dw dt = w(−m + cp(α)z) (3.1) where k = r2 r1 , d1 = a22 a12 , d2 = a33 a23 , β1 = a21 a11 , β2 = a32 a12 , m = d4 r1 , c = c̄a0 a23 . for the transformed and rescaled functions f(α,w) and p(α), the conditions (1.6) and (1.7) are transformed to   f(α,w) = b(α,n4) −d3 r1 , f(α, 0) = f(0,w) = f0 = b3 −d3 r1 , ∂f ∂α < 0, ∂f ∂w < 0, lim α→∞ f(α,w) = lim w→∞ f(α,w) = −d3 r1 , p(α) = a34(α) a0 , dp dα < 0, p(0) = 1 and lim α→∞ p(α) = 0. (3.2) anti-predation response in food chain systems 193 0 20 40 60 80 100 time 0 2 4 6 8 p o p lu a ti o n solution of the three dimension system = 2 level-1 level-2 level-3 (a) 0 20 40 60 80 100 time 1 2 3 4 5 6 p o p lu a ti o n solution of the three dimension system = 6 level-1 level-2 level-3 (b) 0 20 40 60 80 100 time 0 1 2 3 4 5 6 p o p lu a ti o n solution of the three dimension system = 10 level-1 level-2 level-3 (c) figure 3. population dynamics of (2.5) when d1 > k. (a) when 0 < α = 2 < 2.5, the equilibrium e13 is stable, (b) 2.5 < α = 6 < 8.409090911, the coexistence equilibrium e∗ is stable, (c) when 8.409090911 < α = 10, the equilibrium e12 is stable. by the fundamental theory of odes, we easily see that the initial value problem associated with (3.1) has a unique solution; moreover, the solution is nonnegative (positive) with nonnegative (positive) initial conditions. next we show that the solution to system (3.1) is bounded. firstly, by the same argument as in subsection 2.2, we can obtain the same estimates for x(t) and y(t): lim t→∞ sup x(t) ≤ 1 and lim t→∞ sup y(t) ≤ k + β1 d1 . for z and w, we consider p = cz + w. then we have dp dt = cz (f(α,w) −d2z + β2y) −mw. 194 y. wang and x. zou 0 2 4 6 8 10 0 5 10 15 20 p o p lu a ti o n bifurcation diagram when k<d 1 level-1 level-2 level-3 (a) 2 3 4 5 6 7 8 0 1 2 3 4 5 6 p o p lu a ti o n meso-predator cascade in the coexistence region level-1 level-2 level-3 (b) figure 4. (a) bifurcation diagram of (2.5) when d1 > k, (b) in the region when the coexistence equilibirum e∗ is stable, we can observe a meso-predator cascade. now for any given � > 0, there holds y(t) ≤ (k + β1)(1 + �)/d1 for large t. then for some µ ∈ (0,m) and large t, we have dp dt + µp = cz (µ + f(α,w) −d2z + β2y) − (m−µ)w, ≤ cz ( µ + f0 + β2(k + β1)(1 + �) d1 −d2z ) =: cz(k −d2z) (3.3) where k = µ + f0 + β2(k + β1)(1 + �) d1 . if k ≤ 0, since z(t) ≥ 0, we can conclude that dp dt + µp ≤ 0, leading to p ≤ p0e−µt, where p0 = cz0 + w0 comes from the initial condition. thus, lim t→∞ sup p(t) ≤ 0. if k > 0, then dp dt + µp ≤ ck 2 4d2 . by the comparison theorem [24], we can obtain p ≤ p0e−µt + ( 1 −e−µt ) ck2 4µd2 . which implies lim t→∞ sup p(t) ≤ ck 2 4µd2 . thus, in both cases, p is bounded, implying that z and w are both bounded. therefore, all four components of the solution are bounded. anti-predation response in food chain systems 195 3.2. existence and stability of the boundary equailibrium solutions. as in section 2, we first analyze the boundary equilibria of the model (3.1). again, we are only interested in the equilibria with nonnegative components. by solving the system   x(1 −x−y) = 0, y(k −d1y −z + β1x) = 0, z (f(α,w) −d2z + β2y −p(α)w) = 0, w(−m + cp(α)z) = 0, we find that there are eleven possible boundary equilibria and they are described below. e0 = (0, 0, 0, 0) is the trivial equilibrium solution which always exists, e1 = (1, 0, 0, 0), e2 = (0,k/d1, 0, 0) and e3 = (0, 0,f0/d2, 0) are the equilibria that correspond to the case of only one species surviving: e1 and e2 are always exist while e3 exists only when f0 > 0. there are also four possible equilibria accounting for the scenario of two species coexisting and they are e12 = ( d1 −k d1 + β1 , k + β1 d1 + β1 , 0, 0 ) , e13 = ( 1, 0, f0 d2 , 0 ) , e23 = ( 0, d2k −f0 d1d2 + β2 , kβ2 + d1f0 d1d2 + β2 , 0 ) , e34 = ( 0, 0, m cp(α) ,w ) where w satisfies the equation f(α,w) − d2m cp(α) −p(α)w = 0. (3.4) due to the requirement of non-negativity, e12 exists when k < d1, e13 exists when f0 > 0, and e23 exists when d2k > f0 > −β2k/d1. for the existence and uniqueness of e34, we denote f1(w) = f(α,w) − d2m cp(α) −p(α)w. then it is obvious that f1(w) is a decreasing function with respect to w and limw→∞f1(w) = −∞. thus, the sufficient and necessary condition for e34 to exist is f1(0) > 0, that is f0 > d2m/cp(α). there are three possible equilibria for the case of three species coexisting, given by e123 = ( d1d2 −d2k + β2 + f0 β1d2 + d1d2 + β2 , β1d2 + d2k −f0 β1d2 + d1d2 + β2 , β1β2 + β1f0 + β2k + d1f0 β1d2 + d1d2 + β2 , 0 ) , e134 = ( 1, 0, m cp(α) ,w ) , e234 = ( 0, cp(α)k −m d1cp(α) , m cp(α) , ŵ ) , where w̄ is as in (3.4) and ŵ satisfies the equation f(α,ŵ) − d2m cp(α) + β2cp(α)k −β2m d1cp(α) −p(α)ŵ = 0. (3.5) thus, e123 exists when d2(k + β1) > f0 > max ( d2k −d1d2 −β2,− β2(k + β1) d1 + β1 ) , 196 y. wang and x. zou the condition for e134 to exist is the same as for the existence of e34, that is, when f0 > d2m/cp(α). condition for e234 to exist is cp(α)k −m > 0 and f0 > d2m cp(α) − β2cp(α)k −β2m d1cp(α) . in order to discuss the local stability, we calculate the jacobian matrix at equilibrium e = (x∗,y∗,z∗,w∗) as j(e) =   1 − 2x∗ −y∗ −x∗ 0 0 β1y ∗ k − 2d1y∗ + β1x∗ −z∗ −y∗ 0 0 β2z ∗ j33 j34 0 0 cp(α)w∗ −m + cp(α)z∗   , (3.6) where j33 = f (α,w ∗) − 2d2z∗ + β2y∗ −p(α)w∗, and j34 = z ∗ ∂f(α,w) ∂w ∣∣∣∣ w=w∗ −p(α)z∗ < 0 for z∗ > 0, w∗ > 0. at e0 = (0, 0, 0, 0), the jacobian becomes j(e0) =   1 0 0 0 0 k 0 0 0 0 f0 0 0 0 0 −m   , thus, e0 is unstable. at e1 = (1, 0, 0, 0), the jacobian reduces to j(e1) =   −1 −1 0 0 0 k + β1 0 0 0 0 f0 0 0 0 0 −m   , hence e1 is unstable. anti-predation response in food chain systems 197 at e2 = (0,k/d1, 0, 0), the jacobian is given by j(e2) =   1 − k d1 0 0 0 β1k d1 −k − k d1 0 0 0 f0 + β2k d1 0 0 0 0 −m   , so e2 is asymptotically stable if and only if 1 −k/d1 < 0 and f0 < −β2k/d1. at e3 = (0, 0,f0/d2, 0), the jacobian is now j(e3) =   1 0 0 0 0 k − f0 d2 0 0 0 β2f0 α2 −f0 j34 0 0 0 −m + cp(α)f0 d2   , therefore e3 is unstable. at e12, the jacobian is reduced to j(e12) =   k −d1 d1 + β1 k −d1 d1 + β1 0 0 β1k + β 2 1 d1 + β1 − d1k + d1β1 d1 + β1 − k + β1 d1 + β1 0 0 0 f0 + β2k + β2β1 d1 + β1 0 0 0 0 −m   , therefore e12 is asymptotically stable if and only if f0 < −(β2k + β2β1) d1 + β1 . 198 y. wang and x. zou at e13, the jacobian is j(e13) =   −1 −1 0 0 0 k + β1 − f0 d2 0 0 0 β2f0 α2 −f0 j34 0 0 0 −m + cp(α)f0 d2   , consequently, e13 is asymptotically stable if and only if md2 cp(α) > f0 > d2(k + β1). at e23, the jacobian is given by j(e23) =   1 − d2k −f0 d1d2 + β2 0 0 0 β1d2k −β1f0 d1d2 + β2 − d1d2k −d1f0 d1d2 + β2 − d2k −f0 d1d2 + β2 0 0 kβ22 + d1f0β2 d1d2 + β2 − kβ2d2 + d1f0d2 d1d2 + β2 j34 0 0 0 −m + cp(α) (kβ2 + d1f0) d1d2 + β2   , therefore e23 is asymptotically stable if and only if f0 > d2k −d1d2 −β2 and p(α) < m (d1d2 + β2) c (kβ2 + d1f0) . at e34 = ( 0, 0, m cp(α) ,w ) , the jacobian becomes j(e34) =   1 0 0 0 0 k − m cp(α) 0 0 0 β2m cp(α) −d2m cp(α) j34 0 0 cp(α)w 0   , therefore e34 is unstable. anti-predation response in food chain systems 199 at e123 = (x1,y1,z1, 0), the jacobian is given by j(e123) =   −x1 −x1 0 0 β1y1 −d1y1 −y1 0 0 β2z1 −d2z1 0 0 0 0 −m + cp(α)z1   , where x1, y1 and z1 denote the three positive components in e123. in section 2, for the three dimensional model (2.1), we have proved the principle 3 × 3 sub-matrix of j(e123) only has negative eigenvalues. therefore e123 is asymptotically stable if and only if −m + cp(α)z1 < 0, that is p(α) < m (β1d2 + d1d2 + β2) c (β1β2 + β1f0 + β2k + d1f0) . at e134 = ( 1, 0, m cp(α) ,w ) , the jacobian is given by j(e134) =   −1 −1 0 0 0 k + β1 − m cp(α) 0 0 0 β2m cp(α) − d2m cp(α) j34 0 0 cp(α)w 0   , since j34 < 0, e123 is asymptotically stable if and only if k + β1 − m/cp(α) < 0, that is p(α) < m/c (k + β1). at e234, the jacobian becomes j(e234) =   1 − cp(α)k −m d1cp(α) 0 0 0 β1 (cp(α)k −m) d1cp(α) − d1 (cp(α)k −m) d1cp(α) − cp(α)k −m d1cp(α) 0 0 β2m cp(α) − d2m cp(α) j34 0 0 cp(α)ŵ 0   , the lower 3 × 3 principle sub-matrix can be written as a =   −d1y2 y2 0 β2z2 −d2z2 j34 0 cp(α)ŵ 0   , 200 y. wang and x. zou where y2 = cp(α)k −m d1cp(α) and z2 = m cp(α) . then the characteristic polynomial of matrix a is λ3 + (d1y2 + d2z2)λ 2 + (−cp(α)ŵj34 + d2z2d1y2 + β2z2y2) λ− cp(α)j34d1ŵy2 = 0, where   a1 = d1y2 + d2z2 > 0, a2 = −cp(α)ŵj34 + d2z2d1y2 + β2z2y2 > 0, a3 = cp(α) −j34d1ŵy2 > 0, and a1a2 −a3 = −cp(α)j34d2ŵz2 + d21d2y 2 2 z2 + d1d 2 2y2z 2 2 + β2d1y 2 2 z2 + β2d2y2z 2 2 > 0. therefore, by the routh-hurwitz criterion, the sub-matrix a only has negative eigenvalues. thus, e234 is asymptotically stable if and only if 1 − cp(α)k −m d1cp(α) < 0. 3.3. existence and stability of the positive equilibrium. by solving the system  1 −x−y = 0, k −d1y −z + β1x = 0, f(α,w) −d2z + β2y −p(α)w = 0, −m + cp(α)z = 0, (3.7) we can find the expression of the possible positive equilibrium solution e∗ = (x∗,y∗,z∗,w∗) where  x∗ = cp(α) (d1 −k) + m cp(α) (β1 + d1) , y∗ = cp(α) (β1 + k) −m cp(α) (β1 + d1) , z∗ = m cp(α) , and w∗ is given by the equation f(α,w∗) −d2z∗ + β2y∗ −p(α)w∗ = 0. therefore, there exists a unique positive equilibrium if and only if the following inequalities hold  cp(α) (d1 −k) + m > 0 cp(α) (β1 + k) −m > 0, f0 − d2m cp(α) + cp(α)β2 (β1 + k) −β2m cp(α) (β1 + d1) > 0, (3.8) anti-predation response in food chain systems 201 the jacobian of the positive equilibrium is given by j(e∗) =   −x∗ −x∗ 0 0 β1y ∗ −d1y∗ −y∗ 0 0 β2z ∗ −d2z∗ j34 0 0 cp(α)w∗ 0   , the corresponding characteristic equation is λ4 + a1λ 3 + a2λ 2 + a3λ + a4 = 0, where   a1 = d1y ∗ + d2z ∗ + x∗ > 0, a2 = −j34cp(α)w∗ + d1d2y∗z∗ + β1x∗y∗ + β2y∗z∗ + d1x∗y∗ + d2x∗z∗ > 0, a3 = −j34cp(α)d1w∗y∗ −j34cp(α)w∗x∗ + β1d2x∗y∗z∗ + d1d2x∗y∗z∗ + β2x∗y∗z∗ > 0, a4 = −j34β1cp(α)w∗x∗y∗ −j34cp(α)d1w∗x∗y∗ > 0. calculation gives a1a2 −a3 = −j34cp(α)d2w∗z∗ + d21d2(y ∗)2z∗ + d1d 2 2y ∗(z∗)2 + β1d1x ∗(y∗)2 + β2d1(y ∗)2z∗ + β2d2y ∗(z∗)2 + d21x ∗(y∗)2 + 2d1d2x ∗y∗z∗ + d22x ∗(z∗)2 + β1(x ∗)2y∗ + d1(x ∗)2y∗ + d2(x ∗)2z∗ > 0 and a1a2a3 −a21a4 −a 2 3 = d2x ∗z∗ (cp(α)w∗j34 + β1x ∗y∗) 2 + d1d2y ∗z∗ (β1x ∗y∗ + cp(α)w∗j34) 2 + positive terms > 0. therefore, by the routh-hurwitz criterion, the positive equilibrium is locally asymptotically stable. for readers’ convenience, we summarize the results obtained above about the existence and stability of all the equilibria in table 2. as was done to rescaled model (1.5) in section 2, we can also examine the relationship how the population size for each species at the stable positive equilibrium depends on the anti-predation level α. indeed, we can calculate to obtain dz∗ dα = −m cp2(α) dp dα > 0. by using implicit differentiation on the system (3.7), we can also determine dx∗ dα > 0 and dy∗ dα < 0. however, we are not able to determine the sign of dw∗ dα . from the above discussion, we see that after incorporating the benefit obtained by the meso-predator’s anti-predation response in reducing the predation by the large carnivores, the final population sizes of the meso-predator, its prey and its prey’s prey are increasing, decreasing, and increasing respectively with respect to the response strength α. this alternating pattern is totally opposite to the one obtained in section 2 on this context. 202 y. wang and x. zou table 2. condition of existence and stability of the equilibria in model (3.1) equilibrium solution existence stability e0 always unstable e1 always unstable e2 always 1 −k/d1 < 0 and f0 < −β2k/d1 e3 f0 > 0 unstable e12 k < d1 f0 < − β2k + β2β1 d1 + β1 e13 f0 > 0 md2 cp(α) > f0 > d2(k + β1) e23 d2k > f0 > −β2k/d1 f0 > d2k −d1d2 −β2 and p(α) < m (d1d2 + β2) c (kβ2 + d1f0) e34 f0 > d2m/cp(α) unstable e123 d2(k + β1) > f0 > max ( d2k −d1d2 −β2,− β2(k + β1) d1 + β1 ) p(α) < m (β1d2 + d1d2 + β2) c (β1β2 + β1f0 + β2k + d1f0) e134 f0 > d2m/cp(α) p(α) < m c (k + β1) e234 cp(α)k −m > 0 and f0 > d2m cp(α) − β2cp(α)k −β2m d1cp(α) . 1 − cp(α)k −m d1cp(α) < 0 e∗ condition (3.8) stable 3.4. numerical simulations. in this part, we present some numerical simulations to illustrate the analytical results obtained above. to this end, we choose b(α,n4) = r3 1 + c1αn4 and a34(α) = 1 1 + c2α in (1.5), leading to the following system   dn1 dτ = n1 (r1 −a11n1 −a12n2) , dn2 dτ = n2 (r2 −a22n2 −a23n3 + a21n1) , dn3 dτ = n3 ( r3 1 + c1αn4 −d3 −a33n3 + a32n2 − n4 1 + c2α ) , dn4 dτ = n4 ( −d4 + cn3 1 + c2α ) , n1(0) ≥ 0 , n2(0) ≥ 0 , n3(0) ≥ 0 , n4(0) ≥ 0, (3.9) we fix the parameters r1 = 3, a11 = 1, a12 = 0.4, r2 = 1, a22 = 0.2, a23 = 0.5, a21 = 0.5, r3 = 3, d3 = 1, a33 = 0.5, a32 = 0.05, c1 = 0.4,d4 = 0.1,c = 0.5,c2 = 0.2, (3.10) and illustrate how α impacts the population dynamics. for the above set of parameter values, k = 1/3 < d1 = 1/2, the bifurcation diagram with respect to α is given in figure 5. there is a transcritical bifurcation between e∗ and e123 where the critical value α∗ is given by f0 − d2m cp(α∗) + cp(α∗)β2 (β1 + k) −β2m cp(α∗) (β1 + d1) = 0. using the parameter values in (3.10), we can solve this equation to obtain α∗ ≈ 97.77777780. we can observe a trophic cascade in figure 5 inside the coexistence region with respect to increment of anti-predation response in food chain systems 203 0 50 100 150 0 1 2 3 4 5 6 p o p lu a ti o n bifurcation diagram p( )=1/(1+0.2 ) level-1 level-2 level-3 level-4 figure 5. bifurcation diagram of (3.9) when k < d1. 0 50 100 150 0 1 2 3 4 5 p o p lu a ti o n bifurcation diagram p( )=1/(1+0.2 ) level-1 level-2 level-3 level-4 figure 6. bifurcation diagram of (3.9) when k > d1. α. contrary to the previous section, this cascade shows an increasing population in odd level and decreasing population in even level. next, we change r1 from r1 = 3 to r1 = 1 in (3.10) and keep the same values for other parameters. we then have k = r2/r1 = 1 and d1 = a22/a12 = 1/2 so that k > d1. for this case, we observe more complicated dynamical behaviours: there are three critical values for α, denoted by α1, α2 and α3, which are given by   cp(α1) (d1 −k) + m = 0 cp(α2) (β1 + k) −m = 0, md2 cp(α3) = f0. using the parameter values in (3.10), we obtain α1 = 20, α2 = 70 and α3 = 95. in figure 6, when 0 < α < α1, e234 is stable; when α1 < α < α2, e ∗ is stable; when α2 < α < α3, e134 is stable; when α3 < α, e13 is stable. we can also observe trophic cascade in this case as is shown in figure 6. 204 y. wang and x. zou 0 5 10 15 0 1 2 3 4 5 6 p o p lu a ti o n bifurcation diagram p( )=1/(1+2 ) level-1 level-2 level-3 level-4 (a) 0 1 2 3 4 5 0 1 2 3 4 5 6 p o p lu a ti o n meso-predator cascade in the coexistence region level-1 level-2 level-3 level-4 (b) figure 7. (a) bifurcation diagram of (3.9) when c2 = 2, (b) in the region when the coexistence equilibrium e∗ is stable, we can observe a meso-predator cascade with non-monotonically change on top predator. in the last two cases, we can see that population size of large carnivores at the positive equilibrium is monotonically decreasing. we point out that this is dependent on the choice of the benefit reflecting term p(α) = a34(α)/a0. to see this, we change c2 from 0.2 to 2 (corresponding to a more significant benefit to n3 and disadvantage to n4), then, as is shown in figure 7-(a), we can see a transcritical bifurcation from e∗ to e123 at threshold value α̃ = 10.43750000, and before this value, w ∗(α) is not monotone: it increases first and then decreases. figure 7-(b) is an enlargement of (a) in which, one can more clearly see that while the lower trophic still follow the cascade, the top predator (w∗) increases first and then decreases with respect to the increment of α. thus, the response function p(α) has an impact of the trophic cascade. 4. conclusion and discussions a recent experimental study [25] in fields observed a phenomenon of trophic cascade in a food chain population system consisting of three species, i.e., meso-carnivore on top, its prey in the middle and the prey’s prey in the bottom, caused by the fear of virtual large carnivores which is implemented by playback of the large carnivores. this phenomenon, together with some recent works on fear effect in two species predator-prey models, has motivated us to theoretically explore the mechanisms for such trophic cascade in this paper. to this end, we have proposed two models, (1.3) and (1.5), with (1.3) directly corresponding to the scenario of field study in [25], and (1.5) being an extension to include a benefit in the meso-carnivore from the anti-predation response, in addition to a cost, as in (1.3). in order to incorporate the benefit term into the model, we have to add the population of the large carnivores into the interplay, making (1.5) a 4-d system. we have thoroughly analyzed the two models, using the approach of dynamical systems. for each of the two models, we have obtained complete structure of the equilibria, and established their stability/instability in terms of the model parameters, in the form of thresholds for certain parameters. for model (1.3) our results show that an anti-predation response at lower level is beneficial to the top and anti-predation response in food chain systems 205 bottom species (n3 and n1); while at higher level, it is beneficial to the middle species (n2) but disadvantageous to n3 and n1, confirming the phenomenon of trophic cascade reported in that experimental study. for model (1.5), our results show that there are now three threshold values for the response level α, distinguishing ranges for α accounting for various combinations of co-existence among the four species. particularly, within certain range of parameters, the model also demonstrates the phenomenon of trophic cascade but with an opposite alternating pattern for the three species (the meso-carnivore, its prey and its prey’s prey): increasing the response level α is beneficial to n3 and n1 but disadvantageous to n2. this change is attributed to the effect of the benefit of the anti-predation response in reducing the predation and its balance with cost of such a response. our results can have ecological implications as they may suggest practical strategies of management/control for maintaining biodiversity. for example, in some ecosystems, populations of some meso-predators have been observed to increase significantly due to the loss/extinction of larger carnivores, and this has in turn put some pressure on the meso-predators’ preys for their survivals. our results on model (1.3) suggest that by creating certain virtual situations (e.g, vocal) mimicking the large carnivore predators, one may expect to reduce the populations of the meso-predators, and consequently relax the pressure on the meso-predators’ preys. on the other hand, if the large (top) predators of the meso-predators are present, their predation on the meso-predators poses a major threat to the meso-predators. in such a case, by out results on model (1.5), creating the aforementioned virtual situations may stimulate the meso-predators to increase their anti-predation response level, which will then reduce the predation risk by the top predators. this way, the benefit of anti-predation response of the meso-predators in reducing the predation risk may outplay the cost in production, and thus, enhance the survival probability of the meso-predators. such a net benefit in the meso-predators can then be passed on to the lower level species in an alternating fashion. therefore, the risk events such as fear effect in some species in an ecosystem may actually offer a management tool in shifting the structure of ecosystem and help conserve the biodiversity. note that in our model, we have used the mass action or holling type i functional responses as the predation mechanism. for some species between which the predation involves foraging, this mechanism is not suitable and other types of functional responses should be adopted. it would be interesting and worthwhile to investigate the population dynamics in models like (1.3) and (2.1) with such replaced functional responses. we also point out that in our models, we have only considered fear effect of meso-carnivore species n3 against the large carnivores n4. such fear effect may also exist in n2 against n3 and in n1 against n2. modeling fear effect in those or in all levels would also be interesting but would be very challenging mathematically. we remark that for predator-prey interactions between two species only, the recent works mentioned in the introduction may also suggest some possible extensions and expansions of the two models in this paper. for example, one may also incorporate age structure, spatial structure, digestion delay, extra food, stochastic noise, as was done in [8, 18, 31, 32, 33]. efforts on all these lines will greatly enhance our understanding of predator-prey interactions, and enrich the theory in this area. references [1] r. boonstra, d. hik, g.r. singleton and a. tinnikov, the impact of predator-induced stress on the snowshoe hare cycle, ecol. monogr. 68(1998) , 371-394. 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[35] l. y. zanette, a. f. white, m. c. allen and m. clinchy, perceived predation risk reduces the number of offspring songbirds produce per year, science 334 (6061)(2011), 1398-1401. corresponding author. department of applied mathematical sciences, university of western ontario london, on, canada n6a 5b7 e-mail address: ywan342@uwo.ca department of applied mathematical sciences, university of western ontario london, on, canada n6a 5b7 e-mail address: xzou@uwo.ca mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 4, december 2020, pp.383-402 https://doi.org/10.5206/mase/10855 on the focusing generalized hartree equation anudeep kumar arora, svetlana roudenko, and kai yang abstract. in this paper we give a review of the recent progress on the focusing generalized hartree equation, which is a nonlinear schrödinger-type equation with the nonlocal nonlinearity, expressed as a convolution with the riesz potential. we describe the local well-posedness in h1 and ḣs settings, discuss the extension to the global existence and scattering, or finite time blow-up. we point out different techniques used to obtain the above results, and then show the numerical investigations of the stable blow-up in the l2-critical setting. we finish by showing known analytical results about the stable blow-up dynamics in the l2-critical setting. 1. introduction in this paper we give a review of recent progress on a schrödinger-type equation with nonlocal potential, the focusing generalized hartree (ghartree) equation, iut + ∆u + ( 1 |x|n−γ ∗ |u|p ) |u|p−2u = 0, (x,t) ∈ rn ×r. (1.1) here, u(x,t) is a complex-valued function, ∗ denotes the convolution operator in rn , and the convolution with 1 |x|n−γ is associated to the riesz potential iγ of order γ given by iγ(x) = c(n,γ) 1 |x|n−γ , 0 < γ < n, where c(n,γ) = γ(n−γ 2 ) γ(γ 2 ) 2γ πn/2 . typically, p ≥ 2, however, it is also possible to consider powers 1 < p < 2. the equation (1.1) is a generalization of the standard hartree equation with p = 2, i.e., iut + ∆u + ( 1 |x|n−γ ∗ |u|2 ) u = 0, (x,t) ∈ rn ×r, (1.2) which, for example, can be considered as a model for many-body quantum systems in non-relativistic setting; it also arises in the study of long range interactions between the molecules. the work on the mean-field limit of many-body quantum systems, where the number of bosons is very large, but the interactions between them are weak, goes back to hepp [30], also see [58], [9], [8], [18]. lieb and yau [42] mentioned it in the context of chandrasekhar theory of stellar collapse, which says that after the death of a star, depending on its mass, the stellar remnants can take one of the three forms: neutron stars, white dwarfs and black hole. lieb and thirring [41] conjectured that the collapse for boson stars can be predicted by a hartree-type equation. a special case of the riesz potential with γ = 2 in r3 is received by the editors 1 july 2020; accepted 1 december 2020; published online 16 december 2020. 2010 mathematics subject classification. primary 35q55, 35q40; secondary 37k05. key words and phrases. hartree equation, choquard-pekar equation, convolution nonlinearity, global well-posedness, blow-up, dynamic rescaling. 383 384 anudeep k. arora, svetlana roudenko, and kai yang known as the coulomb potential, which goes back to work of lieb [39] and has been intensively studied, see reviews [22], [21]. it also appears as a model of a boson star in the pseudo-relativistic setting (for example, see [19], [20]), given by iut − √ −∆ + m2 u + ( 1 |x| ∗ |u|2 ) u = 0, (x,t) ∈ r3 ×r. (1.3) as mentioned before, the distinct feature of the hartree equation (1.2) is that it models systems with long-range interactions. possible experimental realizations of such (repulsive) interactions, where the power in the convolution changes, include the interaction of ultracold rydberg atoms that have large principal quantum numbers [44]. these interactions between atoms in highly excited rydberg levels are long range and dominated by dipole-dipole-type forces (the strength of the interaction between rb atoms is about 1012 times stronger than that between rb atoms in the ground state [57]). the spatial dependence of interactions may be 1/|x|3 for small |x| and 1/|x|6 for larger |x|. other powers such as 1/|x|2 are also possible, see [53]. the equation (1.1) can be written as an electrostatic version of the maxwell-schrödinger system, describing the interaction between the electromagnetic field and the wave function related to a quantum non-relativistic charged particle (see, for instance, [11] and [40]){ iut + ∆u + v |u|p−2u = 0 −∆v = (n − 2)|sn−1| |u|p, (1.4) which can be viewed as the schrödinger poisson system for the wave function u and the potential v ; here, sn−1 is the sphere in rn , and |sn−1| stands for its volume. the aim of this paper is to survey the main results from a unified point of view of the generalized hartree equation and show the current developments in the global existence and finite time blow-up in the ghartree equation. the paper is organized as follows. in section 2, we review the necessary background such as invariances of the equation and conserved quantities, then some useful tools such as strichartz estimates. in section 3, we state the known results on the local well-posedness in h1 (available for 0 < s < 1) and at the critical regularity ḣs (available for s > 0, with certain restrictions in some cases). the local wellposedness is then extended to either global existence or finite-time blow-up in section 4, depending on initial conditions. in the same section we review results about ground states in the ghartree equation. in section 5, we discuss initial conditions that predict blow-up in finite time, and can be used in various cases (including energy-supercritical case, s > 1). in section 6 we show the numerical results on the blow-up in the l2-critical case, followed by the discussion of spectral property in section 7.1. finally, in section 7 we explain what is known about the stable blow-up dynamics in ghartree equation and open questions. acknowledgments. all three authors on this project were partially supported by the nsf grant dms-1815873/1927258 (pi: roudenko). 2. preliminaries we start with the duhamel formulation (for example, see [62]), where the solution u : i ×rn → c to the equation (1.1) is written in the integral form u(t) = eit∆u0 + i ∫ t 0 ei(t−t ′)∆ ( 1 |x|n−γ ∗ |u|p ) |u|p−2u(t′) dt′ (2.1) for all t ∈ i ⊂ r. the interval i is known as the lifespan of u. if i = r, the solution u is said to be global. the first question to understand is whether the equation (1.1), or equivalently, (2.1), can survey of ghartree equation 385 have local solutions. before stating the results about the local well-posedness, i.e., existence of a unique local-in time solution satisfying (2.1) that lies in some sobolev space and continuous dependence on the initial data, we review conserved quantities and other useful properties. during their lifespans, solutions to (1.1) conserve the mass, energy (hamiltonian) and momentum, namely, for any t ∈ r m[u(t)] def = ∫ rn |u(x,t)|2 dx = m[u0], e[u(t)] def = 1 2 ∫ rn |∇u(x,t)|2 dx− 1 2p ∫ rn ( 1 |x|n−γ ∗ |u( · , t)|p ) |u(x,t)|p dx = e[u0], p [u(t)] def = im ∫ rn ū(x,t)∇u(x,t) dx = p[u0]. the equation (1.1) enjoys several invariances, among them is the scaling invariance: if u(x,t) solves (1.1), then so does uλ(x,t) = λ γ+2 2(p−1) u(λx,λ2t). (2.2) this implies that ḣsc norm is invariant under the above scaling provided the critical scaling index sc is sc = n 2 − γ + 2 2(p− 1) . (2.3) the equation (1.1) is referred to as the ḣs-critical if for given n,γ,p in (1.1) the ḣs norm is invariant under the scaling (2.2) with s = sc, defined by (2.3). in particular, • if sc = 0, or p = 1 + γ+2n , the equation (1.1) is referred to as the mass-critical (or l 2-critical). • if sc = 1, or p = 1 + γ+2n−2 , the equation is called the energy-critical (or ḣ 1 critical). • if 0 < sc < 1, the equation is intercritical. • if sc > 1, the equation is said to be energy-supercritical. we define the linear schrödinger evolution from initial data u0 as follows u(x,t) = eit∆u0(x) = 1 (4πit)n/2 ∫ rn ei |x−y|2 4t u0(y) dy. then by the l2-isometry and l∞−l1 estimate, one can obtain the time decay estimate for 2 ≤ r ≤∞, 1 r + 1 r′ = 1, ‖eit∆f0(x)‖lrx(rn ) . |t| −n 2 ‖f0‖lr′x (rn ) (2.4) for all t 6= 0. for the local well-posedness we need estimates in both, space and time. this space-time integrability is demonstrated by strichartz estimates. in what follows, we will always consider the case 0 ≤ s < n 2 . 2.1. strichartz estimates. definition 2.1. the pair (q,r) is called l2-admissible pair if n ≥ 1 and 2 q + n r = n 2 , 2 ≤ q,r ≤∞ provided (q,r,n) 6= (2,∞, 2). remark 2.1. one can also define the ḣs-admissibility for n ≥ 1 and s ≥−1 by 2 q + n r = n 2 −s. (2.5) definition 2.2 (see [17]). the pair (q,r) is said to be acceptable if n ≥ 1 and 1 ≤ q,r ≤∞ and 1 q < n ( 1 2 − 1 r ) , or (q,r) = (∞, 2). 386 anudeep k. arora, svetlana roudenko, and kai yang remark 2.2. for s ≥ 0, every ḣs-admissible pair is acceptable. we now recall the well-known strichartz estimates (see [59], [35], [10]). lemma 2.1. if (q,r) is an ḣs-admissible pair for s ≥ 0 then the following linear estimate holds ‖eit∆f‖lqtlrx(r×rn ) . ‖f‖ḣsx(rn ). (2.6) we next consider the inhomogeneous estimate (see [17]). lemma 2.2. let 1 ≤ q, q̃,r, r̃ ≤∞. if the pairs (q,r) and (q̃, r̃) are acceptable, satisfy the condition 1 q + 1 q̃ = n 2 ( 1 − 1 r − 1 r̃ ) and verify the following conditions: • n = 2, we require that r, r̃ < ∞, • n > 2, we classify two cases; – non sharp case: 1 q + 1 q̃ < 1, (2.7) n − 2 n ≤ r r̃ ≤ n n − 2 ; (2.8) – sharp case: 1 q + 1 q̃ = 1, (2.9) n − 2 n < r r̃ < n n − 2 , (2.10) 1 r ≤ 1 q , 1 r̃ ≤ 1 q̃ . (2.11) then the following estimate holds∥∥∥∥ ∫ t 0 ei(t−t ′)∆f(t′) dt′ ∥∥∥∥ l q tl r x + ∥∥∥∥ ∫ ∞ t ei(t−t ′)∆f(t′) dt′ ∥∥∥∥ l q tl r x . ‖f‖ l q̃′ t l r̃′ x . (2.12) we are now ready to review the results on the local well-posedness of the ghartree equation. 3. local well-posedness we discuss the local well-posedness results in two settings, one with the finite energy and finite mass initial data, thus, considering the h1 space (the equation (1.1) would be energy-subcritical or critical at most). the second variant of the local well-posedness is established at the critical regularity ḣs, which allows to have local-wellposendess in the energy-supercritical cases. both of these cases consider p ≥ 2. recently, some results on wellposedness for p < 2 were obtained in [3]. survey of ghartree equation 387 3.1. local well-posednes in h1. we start with considering the initial data in h1 space, u0 ∈ h1(rn ), so that we have a finite hamiltonian (or finite energy) solutions. we consider the integral equation (2.1) with the power p as follows{ 2 ≤ p ≤ 1 + γ+2 n−2, if n ≥ 3 2 ≤ p < ∞, if n = 1, 2. (3.1) we mention that the local existence of h1 solutions in the standard hartree equation (1.2) (p = 2) is available from the work of ginibre and velo [24], see also cazenave [10]. in the general setting (p ≥ 2) the cauchy problem for the equation (1.1) with the initial data u0 ∈ h1 was investigated by the first and second authors in [6], showing the local well-posedness in h1, provided sc < 1. this is guaranteed by (3.1) (note that the nonlinearity in this case is always h1-subcritical). proposition 3.1. for p as in (3.1) and u0 ∈ h1(rn ), there exists t > 0 and a unique solution u(x,t) of the integral equation (2.1) on the time interval [0,t] with u ∈ c([0,t]; h1(rn )) ∩lq([0,t]; w 1,r(rn )), (3.2) where (q,r) is an l2-admissible pair given by (q,r) = ( 2p 1 + sc(p− 1) , 2np n + γ ) . in the energy-subcritical case p < 1 + γ+2 n−2 , the time t = t(‖u0‖h1,n,p,γ) > 0. in the energy-critical case p = 1 + γ+2 n−2 (or sc = 1) an additional assumption of smallness of ‖u0‖h1x is required. the proof relies on a fixed point argument, which can be achieved by showing that the operator φu0 (u) = e it∆u0 + i ∫ t 0 ei(t−t ′)∆ ( 1 |x|n−γ ∗ |u|p ) |u|p−2u(t′) dt′ defines a contraction on x = { u ∈ l∞t ([0,t]; h 1 x(r n )) ∩lqt ([0,t]; w 1,r(rn )) : ν(u) ≤ m } , for some m > 0, where ν(u) = max { sup t∈[0,t] ‖u‖h1x, ‖u‖lq1t w1,r1x } . 3.2. local well-posedness in ḣsc. one can also ask for the local well-posedness at the critical regularity ḣsc for sc ≥ 0, which we state below. the proof can be found in [5]. proposition 3.2. let 0 < γ < n and p ≥ 2 so that sc ≥ 0. assume in addition that if p is not an even integer, then sc < p − 1. let u0 ∈ ḣsc(rn ). then there exists a unique solution u(x,t) of the equation (1.1) with data u0 defined on [0,t] for some t > 0, and such that (1) for sc = 0 and n ≥ 1, u ∈ c([0,t]; l2x) ∩ lq([0,t]; lrx), where (q,r) = ( 2p, 2np n+γ ) is the l2-admissible pair and x ∈ rn , (2) for 0 < sc < 1 and n ≥ 1, u ∈ c([0,t]; ḣscx ) ∩l q1 ([0,t]; ẇsc,r1x ) ∩l q2 ([0,t]; ẇsc,r2x ), where (q1,r1) = ( 2p 1 + sc(p− 1) , 2np n + γ ) , (q2,r2) = ( 2p 1 −sc , 2np n + γ + 2scp ) are the l2-admissible pairs and x ∈ rn , (3) for sc = 1 and n ≥ 3, u ∈ c([0,t]; ḣ1x) ∩ lq([0,t]; ẇ 1,rx ), where (q,r) = ( 2, 2n n−2 ) is the l2-admissible and x ∈ rn , 388 anudeep k. arora, svetlana roudenko, and kai yang (4) for sc > 1, u ∈ c([0,t]; ḣscx ) ∩lq([0,t]; ẇsc,rx ), where x ∈ rn and (a) for p = 2 (thus, n ≥ 5), (q,r) = ( 3, 6n 3n−4 ) is the l2-admissible pair, (b) for p > 2 (thus, n ≥ 3) and 0 < γ < min ( n, 2p p−2 ) , the l2-admissible pair is (q,r) =( 2, 2n n−2 ) . moreover, for all 0 < t̃ < t , the continuous dependence upon the initial data holds. the proof of this proposition is also done via the fixed point argument in the spaces given in each of the cases (1)-(4) above via the corresponding strichartz estimates. 4. global existence and scattering after establishing the local well-posedness either in h1 and ḣsc, a natural question to ask is whether it is possible to extend local in-time existence to larger time intervals. it turns out that the local existence can be extended to obtain global solutions for small data, which is the next statement. its proof is in [6]. proposition 4.1 (small data theory in h1). let p ≥ 2 satisfy (3.1) with 0 < γ < n and u0 ∈ h1(rn ). suppose ‖u0‖h1 ≤ a. there exists δ = δ(a) > 0 such that if ‖eit∆u0‖s(ḣsc) ≤ δ, then there exists a unique global solution u of (1.1) in h1(rn ) such that ‖u‖ l 2p 1−sc t l 2np n+γ x ≤ 2‖eit∆u0‖ l 2p 1−sc t l 2np n+γ x and ‖|∇|scu‖ l 2p 1+sc(p−1) t l 2np n+γ x ≤ 2 c‖u0‖h1, where c depends on constants from the gagliardo-nirenberg interpolation estimate and the strichartz inequality. a similar result is available in ḣsc, which makes it possible to extend the local existence to the larger time intervals. this is proved in [5]. proposition 4.2 (small data theory in ḣsc). let γ,n,p be as in proposition 3.2 so that sc ≥ 0. assume in addition that if p is not an even integer, then sc < p − 1. let u0 ∈ ḣsc(rn ) with ‖u0‖ḣsc ≤ a. there exists δ = δ(a) > 0 such that if ‖eit∆u0‖wsc ≤ δ, then one can find a unique global solution u of (1.1) in ḣsc(rn ) such that ‖u‖wsc ≤ 2‖eit∆u0‖wsc , and ‖|∇|scu‖s0 ≤ 2 c1 ‖u0‖ḣsc . here, ‖u‖wsc =   ‖u‖ l 2p 1−sc t l 2np n+γ x , for 0 < sc < 1, ‖u‖ l∞t l 2n n−2 x , for sc = 1, max ( ‖u‖ l3tl 6n 3γ+2 x ,‖u‖ l∞t l 2n γ+2 x ) , for sc > 1 and p = 2, ‖u‖ l∞t l 2n(p−1) γ+2 x , for sc > 1 and p > 2, survey of ghartree equation 389 and ‖u‖s0 =   ‖u‖ l 2p 1+sc(p−1) t l 2np n+γ x , for 0 < sc < 1, ‖u‖ l2tl 2n n−2 x , for sc = 1, max ( ‖u‖ l3tl 6n 3n−4 x ,‖u‖l∞t l2x ) , for sc > 1 and p = 2, ‖u‖ l2tl 2n n−2 x , for sc > 1 and p > 2 now that we have some global solutions, one can ask about their asymptotic behavior as t → ±∞. specifically, if solutions eventually behave as a linear evolution (or approach a linear evolution), which is called scattering, or exhibit a nonlinear behavior. a global solution u(t) to (1.1) is said to scatter in hs(rn ) as t → +∞, if there exists u+ ∈ hs(rn ) such that lim t→+∞ ‖u(t) −eit∆u+‖hs(rn ) = 0. global existence, asymptotic behavior of solutions and scattering theory for the standard hartree equation (1.2) goes back to work of ginibre and velo [24], where the local wellposedness is established and the authors also prove asymptotic completeness for a repulsive potential (that is, the sign in front of the convolution term in (1.2) is negative, or often called the defocusing case). hayashi and tsutsumi [29] obtained the asymptotic completeness of wave operators in hm ∩lp(|x|βdx). related results were established in various settings, for example, see ginibre and ozawa [23], ginibre and velo [25]-[26], and hayashi, naumkin and ozawa [28]. the following scattering result in h1(rn ) is proved in [6]. proposition 4.3 (h1 scattering). let u(t) be a global solution to (1.1) with initial data u0 ∈ h1(rn ). if ‖u‖ l 2p 1−sc t l 2np n+γ x < +∞ (globally finite ḣsc strichartz norm) and supt∈r+ ‖u(t)‖h1 ≤ b (uniformly bounded h1(rn ) norm). then u(t) scatters in h1(rn ) as t → +∞, i.e., there exists u+ ∈ h1(rn ) such that lim t→+∞ ‖u(t) −eit∆u+‖h1 = 0. the next question is if the small data global existence can be extended to the global existence for large solutions, or if there is a threshold for global existence. in [6] we showed a dichotomy for global existence and scattering vs. finite time blow-up solutions, provided the initial data is in h1. the threshold was given by a combination of the mass-energy and the gradient comparison to that of the ground state. for the ḣs data, it is a more difficult question as the conserved quantities at the ḣs level are not available (unless s = 0 or s = 1). in order to characterize the sharp threshold for the dichotomy, one needs a notion of a ground state, which we review next. 4.1. ground state solutions. the equation (1.1) in the case when sc < 1 admits standing wave solutions of the form u(x,t) = eitq(x), where q the nonlinear nonlocal elliptic equation −q + ∆q + ( 1 |x|n−γ ∗ |q|p ) |q|p−2q = 0. (4.1) 390 anudeep k. arora, svetlana roudenko, and kai yang (in the energy-critical case the above equation reduces to the one without the linear term.) the equation (4.1) is known as the nonlinear choquard or choquard-pekar equation. a special case of (4.1) when n = 3, p = 2, and γ = 2, ∆q−q + ( 1 |x| ∗ |q|2 ) q = 0,x ∈ rn, (4.2) appeared back in 1954 in the work of s. i. pekar [54] describing the quantum mechanics of a polaron at rest. lieb in [39] mentions it in the context of the hartree-fock theory of plasma, pointing out that p. choquard proposed investigating minimization of the corresponding functional in 1976. in 1996 r. penrose proposed equation (4.2) as a model of self-gravitating matter, in which quantum state reduction is understood as a gravitational phenomenon, see [50]. the existence and uniqueness of the positive solutions to (4.2) was first proved by lieb [39]. the general existence result of positive solutions along with the regularity and radial symmetry of solutions to (4.1) was shown by moroz and van schaftingen [51] (see also a review by moroz and van schaftingen [52] and references therein). the uniqueness proof of lieb in r3 for p = 2 with γ = 2 was extended to the dimension n = 4 by krieger, lenzmann and raphaël in [37]; the uniqueness in the pseudo-relativistic 3d setting (1.3) was established by lenzmann [38]. in [6, appendix] the proof of uniqueness is written for 2 < n < 6 (and p = 2,γ = 2). (the uniqueness and nondegeneracy of the ground state for γ = 2 and p = 2 + �, i.e., when p is sufficiently close to 2 in r3 was shown in [63].) in general, the uniqueness is still open. in the case of ghartree equation when the uniqueness is known, we denote this unique positive solution, or the ground state, by q. when it is not available, it is sufficient to use the minimizer of the gagliardo-nirenberg inequality of convolution type∫ rn ( 1 |x|n−γ ∗ |u|p ) |u|p dx ≤ cgn‖∇u‖ np−(n+γ) l2 ‖u‖n+γ−(n−2)p l2 . (4.3) and the unique value of the sharp constant, expressed via ‖q‖l2 , see [6]. we next show how large the initial data can be taken to continue enjoying the property of global existence and scattering. 4.2. dichotomy: global vs blow-up solutions. we state a dichotomy result for global vs. finite time solutions under the so-called mass-energy threshold, which also shows the h1 scattering for the global solutions. this result was proved in [6], following the concentration-compactness and rigidity road map of kenig and merle [36]. this is in the spirit of [33], [14], [27], [34] for the focusing nls equation, given as iut + ∆u + |u|p−1u = 0, (x,t) ∈ rn ×r. (4.4) as in [32] and [33] for the nls equation, we observe that the quantities ‖u0‖1−scl2(rn ) ‖∇u0‖ sc l2(rn ) and m[u0] 1−sc e[u0] sc are scale-invariant in the ghartree equation (1.1), and for sc > 0 with θ = 1−sc sc we define • renormalized mass-energy: me[u] = m[u]θe[u] m[q]θe[q] , • renormalized gradient (which depends on t): g[u(t)] = ‖u‖θ l2(rn )‖∇u(t)‖l2(rn ) ‖q‖θ l2(rn )‖∇q‖l2(rn ) . for simplicity, we state the version with the zero momentum; the full version can be found in [6]. survey of ghartree equation 391 theorem 4.4. let u0 ∈ h1(rn ) with p [u0] = 0 and let u(t) be the corresponding solution to (1.1) with the maximal time interval of existence (t∗,t ∗). suppose that me[u0] < 1. (1) if g[u0] < 1, then (a) the solution exists globally in time with g[u(t)] < 1 for all t ∈ r, and (b) u(t) scatters in h1, in other words, there exists u± ∈ h1 such that lim t→±∞ ‖u(t) −eit∆u±‖h1(rn ) = 0. (2) if g[u0] > 1, then g[u(t)] > 1 for all t ∈ (t∗,t∗). moreover, if (a) |x|u0 ∈ l2(rn ) (finite variance) or u0 is radial, then the solution blows up in finite time, (b) u0 is of infinite variance and nonradial, then either the solution blows up in finite time or there exits a sequence of times tn → +∞ (or tn →−∞) such that ‖∇u(tn)‖l2(rn ) →∞. in a recent work [12], dodson and murphy presented a simplified proof of theorem 4.4 part (1) for (4.4) with p = 3 in r3 that avoids concentration-compactness route. they used a scattering criterion introduced by tao in [61], which together with the radial sobolev embedding and virial/morawetz estimate was sufficient to prove scattering (in the radial setting). in [13], they extended the above approach to the non-radial case, avoiding the concentration-compactness. the first author of this paper generalized the method of dodson and murphy in the radial case to the inter-critical range of the nonlinear schrödinger equation (4.4) and also showed that it can be applied in the case of the nonlocal potential such as the ghartree equation (1.1), see [1]. 5. blow-up criterion a similar question about the global existence for large data at the critical regularity ḣsc can be considered, or if there is a threshold for global existence. for the ḣsc data, this is a more difficult question to answer, since the conserved quantities at the ḣsc level are not available (unless sc = 0 or sc = 1). what is possible to answer is to show that large data may blow-up in finite time. we give a sufficient condition, blow-up criterion, for the blow-up in finite-time in the generalized hartree equation (1.1), which follows the ideas in [31, 15, 43, 44] except that now we have to find a bound for the convolution term. to state the result we define the variance, v (t) def = ‖xu(t)‖2 l2(rn ), and note that similar to the nls case, finite variance solutions with negative energy blow up in finite time in ghartree equation by a similar virial (or convexity-type) argument modified to the ghartree case. theorem 5.1. let u0 ∈ h1 if sc ≤ 1 and u0 ∈ hsc if sc > 1. assume also v (0) < ∞ and e[u] > 0. the following is a sufficient condition for the blow-up in finite time for the solutions to the ghartree equation (1.1) with initial data u0 in the mass-supercritical case (sc > 0): ∂t v (0) ωm[u0] < 4 √ 2 f ( e[u0]v (0) (ωm[u0])2 ) , (5.1) where ω2 = n2(n(p− 2) + b− 2) 8(n(p− 2) + b) and the function f is defined as (here, k = sc(p− 1)) f(x) =   √ 1 kxk + x− 1+k k if 0 < x < 1 − √ 1 kxk + x− 1+k k if x ≥ 1. (5.2) 392 anudeep k. arora, svetlana roudenko, and kai yang the proof of theorem 5.1 can be found in [5], where the authors show examples of gaussian data with thresholds in various cases (such as the energy-subcritical, critical and supercritical cases). those examples play an essential role in studying the actual dynamics of stable finite time blow-up. in [66] the dynamics of stable blow-up is investigated (including rates and profiles) for the ghartree equation in the l2-critical and supercritical cases, and it was compared with the known stable blow-up dynamics of the (local) nonlinear schrödinger equation. we discuss that next. 6. numerical investigation we have shown that there are solutions which blow-up in finite time in the case s ≥ 0. the next question is to understand the dynamics of such solutions, in particular, how the blow-up happens. for that we need to separate the l2-critical and supercritical case (similar to the nls equation). in this section we show the numerical investigations of the stable l2-critical blow-up and that it happens in the self-similar regime. we point out that the minimal mass blow-up in the l2-critical setting occurs at the threshold m[u0] = m[q] and is similar to the nls (see, for example, [37]), however, the minimal mass blow-up is not stable and is not possible to observe numerically. in what follows we consider the initial mass larger than m[q] (again, note that this quantity is uniquely defined), afterwards we discuss the analytical methods available and challenges in them for blow-up studies. 6.1. direct numerical simulation for blow-up solutions. to investigate the blow-up behavior numerically, we note that a blow-up solution can behave as a “delta” function, and hence, standard numerical methods cannot be applied. thanks to the scaling invariance (2.2), one can apply the dynamic rescaling method to investigate the blow-up dynamics. we refer to [60] as well as [65], [66] for details on this method. recalling the scaling (2.2), we write u(x,t) = 1 l2/(p−1) v(ξ,τ) with ξ = x l , τ = ∫ t 0 1 l2(s) ds. substituting the above into the ghartree equation (1.1) yields ivτ + ia(τ)( 2 p−1v + ξvξ) + ∆v + (iγ ∗ |v| p) |v|p−2v = 0, (6.1) where a(τ) = −llt = − d(ln l) dτ and we choose l(t) = ( 1 ‖u(·, t)‖∞ )(p−1)/2 (for the choice on l(t) see [65]). understanding the behavior of the parameter l(t) as t approaches the blow-up time t , or equivalently, l(τ) as τ →∞, will reveal the rates of the blow-up as well as the convergence to a blow-up profile. to study the self-similar profile in the blow-up, we separate variables v(ξ,τ) = eiτq(ξ) and obtain ∆ξq−q + ia(τ) ( 2 p− 1 q + ξqξ ) + (iγ ∗ |q|p)|q|p−2q = 0, (6.2) here, ∆ξ := ∂ξξ + d−1 ξ ∂ξ denotes the laplacian with radial symmetry. it was shown that a(τ) converges to a constant a (and in the l2-critical case a = 0, however, the convergence is very slow), thus, instead of (6.2) we study  ∆ξq−q + ia ( 2 p− 1 q + ξqξ ) + (iγ ∗ |q|p)|q|p−2q = 0, qξ(0) = 0, q(0) ∈ r, q(∞) = 0. (6.3) survey of ghartree equation 393 the first condition for q indicates that the local maximum is at zero. the second condition on q shows that we fix the phase of the solutions, since the equation is phase invariant; the last condition means that q(ξ) → 0 as ξ →∞. actually, in the l2-critical case one advantage is that the profile solution will be a ground state solution from (4.1), since a = 0. thus, (6.3) is simply reduced to (4.1). (the parameter a is non-zero in the l2-supercritical case, see [66] or [65], and we need to study the non-zero a case in the l2-critical case, since that allows us to track the blow-up rate with the logarithmic corrections.) we mention that numerically we study only γ = 2 case (the convolution is then inverse laplancian up to a dimensional constant, or in other words, a fundamental solution of the poisson equation), and solving (6.3) with a = 0 numerically produces a unique ground state solution q to (4.2) (iterations always converge to the same q, see remark 6.1 in [66]). we return to the equation (6.1) and note that it is well-defined for τ > 0, and thus, can be solved with a standard numerical method with respect to ξ and τ, for details refer to [66]. we investigate the blow-up dynamics in the l2-critical ghartree equation (with γ = 2) in dimensions 3 ≤ n ≤ 7), the snapshots while tracking the blow-up solution in the 4d case (n = 4, p = 2) is shown in figure 1. in other dimensions, the snapshots look similar. one can note that the solution converges to the rescaled ground state q slowly (recall that in 4d the ground state q from (4.2) is proved to be unique). figure 1. the 4d hartree (p = 2,γ = 2): snapshots of the blow-up dynamics, converging to the ground state q at different time t. the snapshots are given in pairs: the left figure is a rescaled solution v from (6.1) and the right is the actual solution compared to the rescaled q, note the height on the vertical axis and concentration on the horizontal axis. next, we study the blow-up rate. for that we track the quantities l(t) and a(τ) in figure 2. the left subplot in figure 2 shows that ln l(t) depends on ln(t − t) linearly with the slope 0.50, which means 394 anudeep k. arora, svetlana roudenko, and kai yang l(t) ∼ √ t − t. we check the convergence of the parameter a(τ) to see if there are any corrections to the rate. the middle subplot in figure 2 shows that a(τ) decays (very) slowly to zero. this affects the rate of the blow-up, or convergence to the blow-up profile q, hence, we investigate further the dependence of a(τ) on possible logarithmic corrections. the right subplot in figure 2 shows a(τ) decays at least at a rate of 1/ ln(τ), possibly with further corrections. it is quite challenging to track an extra logarithmic correction, however, we do functional fitting (see [66]) as well as the asymptotic analysis. this leads to the conclusion that the square root rate has a “log-log” correction term, similar to the nls equation, and in the l2 cases that we have tracked, the dynamics of blow-up is very similar to the nls. we refer the interested reader to [66] for further details on asymptotic analysis. figure 2. the 4d hartree (p = 2,γ = 2). left: the slope of l(t) vs. t − t on a log scale. middle: the behavior of a(τ), indicating a very slow decay to zero. right: the fitting a(τ) vs. 1/ ln(τ) a(τ) decays as 1/ ln(τ). 7. stable blow-up dynamics numerical simulations and asymptotic analysis in section 6 show that stable blow-up dynamics in the l2-critical ghartree equation (in the considered cases of γ = 2 and dimensions 3 ≤ n ≤ 7) follows the log-log regime, similar to the known results in the l2-critical nls equation, which had an interesting history. we mention some of it. in the l2-critical nls, the numerical and heurestical investigations of stable blow-up solutions go back to 1970’s and attracted an enormous amount of attention (see [64] for a review). the search of the correct blow-up rate was especially involved, as it has a correction and it is a challenging task to understand what the correction should be (numerically it is not possible to track double logarithm correction). the first rigorous analytical proof of the stable log-log blow-up regime was done at the turn of this century by galina perelman [55] for the 1d quintic nls equation, which was followed by a systematic study in a series of papers by merle and raphaël [45, 46, 47, 48, 49], obtaining a detailed description of the stable blow-up dynamics for solutions with mass slightly higher than the mass of the ground state solution. the proof requires certain coercivity properties on some bilinear forms, often referred to as the spectral property (see section 7.1, also [47, section 4.4(d)] or [64]). in the 1d case, the spectral property is proved analytically, since the ground state in the nls equation is explcit (a rescaled version of sech1/2 x), for example see [47, appendix a]. in higher dimensions the available proofs are numerically-assisted due to the fact that q is not explicit as well as certain signs of the inner products are also computed numerically (and since the signs are robust to perturbations, it is suffient for the validity of the spectral property). for dimensions 2 ≤ n ≤ 5, see [16]; for dimensions 2 ≤ n ≤ 12, see [64]. there is very little known about the blow-up dynamics (how it happens, what rates, profiles and other characterizations) for the other forms of nonlinearities, in particular, nonlocal, convolution-type survey of ghartree equation 395 nonlinearity. we mention that understanding the blow-up dynamics for the convolution nonlinearity, as it is in the hartree, or ghartree equation, is relevant for the development of theories for a gravitational collapse of, for example, boson stars (as mentioned in the introduction) modeled by the equation (1.3). fröhlich and lenzmann [19] proved the existence of finite time blow-up solutions in the pseudorelativistic hartree equation (1.3) in regards to the theory of gravitational collapse. in [4] there is a first attempt to analytical study of the stable blow-up dynamics for the l2-critical ghartree equation. most of the results in that paper hold for the general l2-critical ghartree equation (with γ = 2), however, the spectral property we were able to verify only in r3, see subsection 7.1 and [7]. it is an open question to prove analytically the log-log blow-up dynamics in other dimensions in the l2-critical setting of the ghartree equation, for example, as shown in figures 1 and 2 (or for other examples, see [66]). as mentioned in section 6 we take γ = 2 (allowing us to write the convolution as the inverse laplacian). then the l2-critical exponent for (1.1) is p = 1 + 4 n , and the equation (1.1) becomes iut + ∆u + ( 1 |x|n−2 ∗ |u|1+ 4 n ) |u| 4 n −1u = 0. (7.1) the corresponding ground state equation is −q + ∆q + ( 1 |x|n−2 ∗q1+ 4 n ) q 4 n = 0. (7.2) we gave numerical confirmation in section 6 to the following conjecture (originally stated in [66]) and in the rest of this survey we will give the sketch of the proof of this conjecture in the 3d case with the one log correction rate. conjecture 7.1. a stable blow-up solution to the l2-critical ghartree equation has a self-similar structure and comes with the rate lim t→t ‖∇u(·, t)‖l2x = ( ln | ln(t − t)| 2π(t − t) )1 2 as t → t, known as the log-log rate. the solution blows up in a self-similar regime with profile converging to a rescaled profile q, which is a ground state solution of (7.2), namely, u(x,t) ∼ 1 l(t) d 2 q ( x−x(t) l(t) ) eiγ(t) with time depending parameters l(t), x(t) and γ(t), converging when t → t as follows: x(t) → xc (the blow-up center), γ(t) → γ0 (for some γ0 ∈ r) and l(t) ∼ ( 2π(t − t) ln | ln(t − t)| )1 2 . thus, the stable blow-up dynamics in the l2-critical ghartree equation is similar to the stable blow-up dynamics in the l2-critical nls equation. in [4] the following blow-up result is proved (see also [2]). theorem 7.2. let n = 3 and consider the l2-critical ghartree equation (7.1) with p = 7 3 iut + ∆u + ( 1 |x| ∗ |u| 7 3 ) |u| 1 3 u = 0. (7.3) consider u0 ∈ h1(r3) such that m[q] < m[u0] < m[q] + α, for some α > 0, (7.4) 396 anudeep k. arora, svetlana roudenko, and kai yang and w [u0] < 0, im (∫ r3 ū0∇u0 dx ) = 0. let u(t) be the corresponding solution to (7.3). then there exist α0 > 0 such that for all α < α0 (1) there exist time depending parameters (λ(t),x(t),γ(t)) ∈ r×r3 ×r such that u(t) = 1 λ(t)3/2 ( q + ε )(x−x(t) λ(t) ) eiγ(t), and ‖ε(t)‖h1(r3) . √ α0, (2) u(t) blows up in finite time, i.e., there exists 0 < t < +∞ such that lim t→t ‖∇u(t)‖l2(r3) = +∞; and (3) for t close to the blow-up time t , we have ‖∇u(t)‖l2(r3) ≤ c ( | ln(t − t)| t − t )1 2 , (7.5) for some universal constant c > 0. we outline the strategy of the proof below and refer readers to [4] for a detailed analysis. note that we start with the general setting (in any dimension), and we point out where the proof is only possible to carry in 3d. the variational characterization of the ground state q along with (7.4) and conservation laws allows us to decompose the solution u(x,t) to (7.1) around q ε(y,t) = eiγ(t)λ(t)n/2u(λ(t)y + x(t), t) −q(y), (7.6) where ‖ε‖h1(rn ) ≤ δ(α0), λ(t) > 0, x(t) ∈ rn and γ(t) ∈ r are c1 functions of time. we rescale the time variable by ds dt = 1 λ(t)2 , and write ε = ε(y,s), observe that in this time rescaling we have s ∈ [0,∞). this decomposition allows us to transform the analysis to ε = ε1 + iε2. before we write the equations for each component ε1 and ε2, we define the scaling generator λf = n 2 f + x ·∇f. (7.7) we obtain the following equations (ε1)s −l−ε2 = λs λ λq + xs λ ·∇q + λs λ (λε1) + xs λ ·∇ε1 + γ̃sε2 −r2(ε), (ε2)s + l+ε1 = −γ̃sq + λs λ (λε2) + xs λ ·∇ε2 − γ̃sε1 + r1(ε), where γ̃s = −s−γs, the operators l± are defined by l+ε1 := −∆ε1 + ε1 − 4 n ( |y|−(n−2) ∗q1+ 4 n ) q 4 n −1ε1 − ( 1 + 4 n )( |y|−(n−2) ∗ ( q 4 n ε1 )) q 4 n , l−ε2 := −∆ε2 + ε2 − ( |y|−(n−2) ∗q1+ 4 n ) q 4 n −1ε2, and the remainders r1,r2 are quadratic in ε. we choose the modulation parameters λ(s), x(s), γ(s) such that ε satisfies the following orthogonality conditions ε1 ⊥ yjq, ε1 ⊥ λq + λ2q and ε2 ⊥ λ2q. (7.8) survey of ghartree equation 397 now to perform the blow-up analysis one needs to have the virial identity which is given by d dt ∫ |x|2|u(x,t)|2dx = 4 im (∫ ūx ·∇u ) = −16 ∣∣e[u0]∣∣t + c. however, we rewrite the above virial identity for the ε, which will be given by calculating the time derivative (in s) of the quantity ψ(ε(s)) = im (∫ ε̄x ·∇ε ) (s). thus, evaluating ψ(u(t)) via the decomposition (7.6), we get ψ(u(t)) = im (∫ (ε + q) y ·∇(ε + q) ) = −4 ∣∣e[u0]∣∣t + c 4 , which is equivalent to ψ(ε(s)) − 2 ∫ ε2 λq = −4 ∣∣e[u0]∣∣t + c 4 . taking the derivative of above expression with respect to s and using dt ds = λ2(s), we obtain (ψ(ε))s(s) = 2(ε2, λq)s(s) − 4λs(s) ∣∣e[u0]∣∣. thus, for the virial identity in ε we compute (ε2, λq)s and obtain the following (ε2, λq)s = h(ε,ε) + 2λ 2|e0|− γ̃s(ε1, λq) − λs λ (ε2, λ 2q) − xs λ (ε2,∇(λq)) + g(ε), where g(ε) is cubic in ε and h(ε,ε) is given by (7.16) (or (7.17)). the next step would be to show coercivity of the bilinear form h and then proceed with the bounds on (ε, λq)s, which will allow us to obtain the blow-up rate with the log correction. this is a point, where the spectral property is needed to proceed. we pause the proof here, and discuss the spectral property in a separate subsection. before we give the sketch of the proof, we discuss the spectral property needed to prove theorem 7.2, which will indicate why we only consider the 3d case. 7.1. spectral property. we recall the scaling generator λf = n 2 f + x · ∇f. we define the two operators l1 and l2 as l1f = 1 2 [l+(λf) − λ(l+f)] ; (7.9) l2f = 1 2 [l−(λf) − λ(l−f)] . (7.10) definition 7.1. let n > 2. given l1,2 and a skew-adjoint operator λ, consider the two real schrödinger operators l1,2 = −∆ + v1,2, defined by the commutator relations l1,2f = 1 2 [l1,2(λf) − λ(l1,2f)] . let the real quadratic form for z = (u,v)t ∈ h1r ×h1r with radial symmetry be b(z, z) = b1(u,u) + b2(v,v). the system is said to satisfy a spectral property with radial symmetric assumption on the subspace u ∈ h1r ×h1r if there exists a universal constant δ0 > 0 such that ∀z ∈u, b(z, z) ≥ δ0 ∫ ( |∇z|2 + e−|y||z|2 ) dy. we establish the following results (for the proofs with numerical assistence see [7]). 398 anudeep k. arora, svetlana roudenko, and kai yang theorem 7.3. the spectral property holds for the 3d generalized hartree equation (1.1) for (f,g)t ∈ u ⊂ l2 ×l2 specified by the following orthogonality conditions 〈f,q〉 = 0, 〈f, λ2q〉 = 0; (7.11) 〈λq,g〉 = 0, 〈λ2q,g〉 = 0, (7.12) where λq = n 2 q + x ·∇q as in (7.7), and λ2q = λ(λq). theorem 7.4. if we treat the dimension n as a parameter, since we are under the radial symmetric assumption, we have the following results: 1. let the dimensions α1 ≤ n ≤ α2 and assume the subspace u ⊂ l2r × l2r with the orthogonal conditions 〈f,q〉 = 0, 〈f, λq〉 = 0; 〈λq,g〉 = 0, 〈λ2q,g〉 = 0. (7.13) then, the spectral property holds for (f,g)t ∈u with α1 ≈ 2.02 and α2 ≈ 2.6. 2. let the dimensions α3 ≤ n ≤ α4 and assume the subspace u ⊂ l2r × l2r with the orthogonal conditions 〈f,q〉 = 0, 〈f,λ2q〉 = 0; 〈λq,g〉 = 0, 〈λ2q,g〉 = 0. (7.14) then, the spectral property holds for (f,g)t ∈u with α3 ≈ 2.7 and α4 ≈ 3.1. note that in the above theorem the only acceptable integer is n = 3 (between α3 and α4). for the purpose of analytical proof later, we need a modified version of the above spectral property to incorporate the span of λq, which we state next. theorem 7.5. the spectral property holds for the 3d generalized hartree equation for (f,g)t ∈ u ⊂ l2 ×l2 in the space orthogonal to the spans 〈f,q〉 = 0, 〈f, λq + αλ2q〉 = 0; 〈λq,g〉 = 0, 〈λq,g〉 = 0, (7.15) with α in the range α < α∗1 or α > α ∗ 2, where α ∗ 1 ≈−0.44601 and α∗2 ≈ 0.69022. remark 7.1. theorem 7.5 actually holds for 2.8 ≤ n ≤ 3.1 with slightly different values of α∗1 and α∗2 depending on the value of n. we point that the 3d case is of the most interest (as this is the only integer dimension that fits the above spectral property. the numerically-assisted proof of the above theorems consists of the following steps: 1. identify the number of negative eigenvalues (indices) of l1 and l2. 2. show that the indices of l1 and l2 are stable under perturbations. 3. justify that the chosen orthogonal conditions produce the negative spans. we refer the readers to [7] for further details. note that the reason that we cannot consider the case n = 4 is due to the fact that in 4d the potentials in definition 7.1 decay as c|x|2 with a large constant c, which leads to infinitely many negative eigenvalues, and thus, in the step (1.) we would get infinitely many directions (or orthogonal conditions) to deal with, see [56]. thus, reformulating the above spectral property in terms of the bilinear form h, we have two realvalued operators l1 and l2, defined in (7.9) and (7.10), and the associated real-valued quadratic form h(ε,ε) for ε = ε1 + iε2 ∈ h1(r3) defined as h(ε,ε) = h1(ε1,ε1) + h2(ε2,ε2) (7.16) = (l1ε1,ε1) + (l2ε2,ε2). (7.17) survey of ghartree equation 399 then there exists a universal constant δ0 > 0 such that for any ε ∈ h1(r3), the quadratic form h is positive, or more precisely, h(ε,ε) ≥ δ0 ∫ ( |∇ε|2 + |ε|2e−2 −|y| ) dy, provided (ε1,q) = (ε1, λq + λ 2q) = 0 and (ε2, λq) = (ε2, λ 2q) = 0. 7.2. completing the 3d blow-up rate proof. we are now ready to finish the sketch of the proof of the stable log-log blow-up in the 3d case. we show why the choice of the orthogonality condition λq + λ2q comes into play. we fix the dimension n = 3 and using the spectral property discussed above we proceed as follows. let ε ∈ h1(r3) with (ε1,yiq) = (ε1, λq + λ 2q) = (ε2, λ 2q) = 0 (i.e., ε satisfies (7.8), which implies that it verifies the modulation theory). we set ε̃ = ε−aq− bλq− icλq. observe that (ε̃1,yiq) = (ε̃2, λ 2q) = 0. also, (ε̃1,q) = 0 and (ε̃1, λq + λ 2q) = 0 with a = (ε1,q) ‖q‖2 l2(r3) = b. similarly, (ε̃2, λq) = 0 with c = (ε2, λq) ‖λq‖2 l2(r3) . hence, ε̃ now satisfies both the spectral property and the modulation theory. we evaluate h(ε,ε) =h(ε̃, ε̃) + 2a(ε̃1,l1q) + 2b(ε̃1,l1(λq)) + a 2h1(q,q) + b 2h1(λq, λq) + 2ab(l1q, λq) + 2c(ε̃2,l2(λq)) + c 2h2(λq, λq) ≥ δ̃0 ∫ |∇ε̃|2 − c (a2 + c2) ≥ δ0 ∫ |∇ε|2 − 1 δ0 ( (ε1,q) 2 + (ε2, λq) 2 ) (7.18) for some fixed universal constant δ1 > 0 small enough. here, we have used the fact that h1(q,q) < 0, h1(λq, λq) = 0, h2(λq, λq) < 0 and (l1q, λq) < 0. we now give a maximum principle type property, which gives the sign structure of the quantity (ε2, λq), which says that there exists a unique s0 ∈ r such that for all s < s0, (ε2, λq)(s) < 0, for all s > s0, (ε2, λq)(s) > 0 and (ε2, λq)(s0) = 0.. this together with the relation involving scaling parameter and the quantity (ε2, λq) of the form λs λ ∼−(ε2, λq) yields the monotonicity of the scaling parameter λ(t), i.e., for all s2 ≥ s1 ≥ s0, λ(s2) < 2λ(s1). using the monotonicity property of scaling parameter, we establish the preliminary weaker upper bound on the blow-up rate, given by ‖∇u(t)‖l2(r3) ≤ c√ |e[u(0)]|(t − t) . we then use the fact that the quadratic form h(ε,ε) has a non-trivial kernel to establish a refined version of the the localized virial relation, which then allows us to prove a superior control on the scaling parameter, namely, λ2(s) ≤ exp ( − c (ε2, λq)(s) ) , or equivalently, (ε2, λq)(s) ≥ c∣∣ ln(λ(s))∣∣. 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[66] k. yang, s. roudenko and y. zhao, stable blow-up dynamics in the l2-critical and l2-supercritical generalized hartree equation, stud. appl. math., 145 (2020), 647-695. corresponding author, department of mathematics, statistics & computer science, university of illinois at chicago, chicago, il, usa e-mail address: anudeep@uic.edu department of mathematics & statistics, florida international university, miami, fl, usa e-mail address: sroudenko@fiu.edu department of mathematics & statistics, florida international university, miami, fl, usa e-mail address: yangk@fiu.edu mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 4, december 2020, pp.333-354 https://doi.org/10.5206/mase/10842 a predator-prey model in the chemostat with holling type ii response function tedra bolger, brydon eastman, madeleine hill, and gail s. k. wolkowicz abstract. a model of predator-prey interaction in a chemostat with holling type ii predator functional response of the monod or michaelis-menten form, is considered. it is proved that local asymptotic stability of the coexistence equilibrium implies that it is globally asymptotically stable. it is also shown that when the coexistence equilibrium exists, but is unstable, solutions converge to a unique, orbitally asymptotically stable periodic orbit. thus the range of the dynamics of the chemostat predator-prey model is the same as for the analogous classical rosenzweig-macarthur predator-prey model with holling type ii functional response. an extension that applies to other functional responses is also given. 1. introduction in this paper, we analyze a predator-prey model in the chemostat with holling type ii predator response function of monod form and prey response function of mass action form. the chemostat is a widely-used apparatus used in the study of microbial biology. it is helpful for the study of microbial growth and interactions under nutrient limitation in a controlled environment. chemostats can be used as a guide for identifying the dynamical nature of population interactions that may be present in a more complex system such as a lake. there are many articles related to the study of the chemostat, both from the experimental and the modelling point of views (see for example [6], [15], [29] and [37]). here, we look at a model of a basic chemostat setup in which a single, essential, non-reproducing nutrient is supplied to a growth chamber from a nutrient reservoir at a constant rate. a population of microorganisms, designated as the prey, lives in the growth chamber and it feeds on the nutrient and a predator population predates on the prey population. the growth chamber is assumed to be well-stirred. it is assumed that the inflow rate of nutrient is the same as the outflow rate from the growth chamber to the waste reservoir so that the volume of the growth chamber remains constant and all of the contents of the growth chamber are removed in proportion to their amount in the growth chamber. how the amounts of the nutrient, prey population, and predator populations change as time changes are all modelled. the more familiar predator-prey models, introduced in rosenzweig and macarthur [34], involve models of predator-prey interactions in which the prey population reproduces and involves only two equations, one for the prey and one for the predator. in [34], models for which the prey nullcline can have at most a single local extremum, a local maximum, were considered. such a model is often referred to as the rosenzweig-macarthur model, and it has been well studied (see, for example, [26] and [33]). the case where the prey nullcline has both a local maximum and a local minimum is received by the editors 30 june 2020; accepted 1 december 2020; published online 22 december 2020. 2010 mathematics subject classification. primary 34c25, 34c60, 34d05, 34d23 secondary 92b05, 34c23, 92d40. key words and phrases. hopf bifurcation, global dynamics, predator-prey system, chemostat, monod form. g.s.k.w was supported in part by a natural science and engineering council of canada (nserc) discovery grant with accelerator supplement. 333 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/10842 334 t. bolger, b. eastman, m. hill, and g.s.k. wolkowicz possible and has also been considered as a generalized version of the rosenzweig-macarthur model (see for example, [1], [13], [14], [36] and [38]). collectively, the rosenzweig-macarthur model and its generalizations are known as the classical predator-prey models. the mathematical expressions needed for the response functions in these models are not usually known in practice. biologists typically collect sets of data that can be fitted by many functions. following the work of holling [20], modellers have used three main forms to describe response functions in predator-prey models. holling type i refers to a mass-action response function, that is linearly increasing. holling type ii responses are increasing and concave down. holling type iii are sigmoidal. the dynamics of predator-prey models are directly influenced by the types of response functions chosen. recently, in fussmann and blasius [14], it was shown that the range of possible dynamics can be different for classical predator-prey models modelled by different forms of holling type ii response functions. here, we determine the global dynamics of a predator-prey model in the chemostat with predator response function of holling type ii (monod form) and prey response function of mass action form. we then compare the dynamics of this predator-prey model in the chemostat with that of the analogous classical predator-prey model. harrison [16] considered a wide class of classical predator-prey models and obtains sufficient (but not necessary) conditions for the global stability of the coexistence equilibrium. he also proved that when the coexistence equilibrium is unstable, at least one periodic orbit exists. in the special case that the prey grow logistically in the absence of predators and the predator response function is of holling type i form, hsu [21] proved that the coexistence equilibrium is globally asymptotically stable whenever it exists. he also proves that when the predator response function is of monod form, then the coexistence equilibrium is globally asymptotically stable whenever it is locally asymptotically stable, and liou and cheng [28] and kuang and freedman [27] proved that when the coexistence equilibrium exists and is unstable, it is surrounded by a unique periodic orbit. in this paper, we focus on the analogous model of predator-prey interaction in a chemostat. in wolkowicz [39], a food web in a chemostat was considered. in the special case that the model studied in [39] is a food chain that includes one resource, and only a single prey, and a single predator population, the model is of the basic form studied here. a lyapunov function was used to prove that the coexistence equilibrium is globally asymptotically stable whenever it exists, under the assumption that the predator response function is holling type i and the prey response function is either holling type i or holling type ii (of monod form). in the model considered in this paper, we assume instead that the prey response function is holling type i and the predator response function is holling type ii (of monod form). in this case, we prove that the dynamics are more complicated. in particular, we prove that whenever the coexistence equilibrium is locally asymptotically stable, it is globally asymptotically stable, and that whenever the coexistence equilibrium is unstable, there is a unique orbitally asymptotically stable periodic orbit. we also prove that the change in stability occurs by means of a hopf bifurcation that is always supercritical. we thus show the similarity between the dynamics of the classical predator-prey model in which the prey is assumed to grow logistically in the absence of the predator population and its analogous chemostat predator-prey counterpart model. the chemostat model is of one dimension higher, since the resource that the prey population consumes in order to grow is also modelled and the growth of the prey in the absence of the predator population depends instead on the abundance of the resource. this paper is organized in the following manner. the predator-prey model in a chemostat is described in section 2, where three equivalent lower dimensional limiting systems are also derived. the limiting system that most resembles the classical predator-prey model is then chosen to be the focal system. properties of the prey nullcline of this system are derived in section 3. preliminary analytic predator-prey chemostat model: holling type ii response 335 results appear in section 4, where it is also determined that the system undergoes a supercritical hopf bifurcation. in section 5, we prove that the coexistence equilibrium is globally asymptotically stable whenever it is locally asymptotically stable, and also that when the coexistence equilibrium is unstable, there is a unique, orbitally asymptotically stable periodic orbit. finally, a discussion of the similarities between chemostat models and their analogous, classical predator-prey models is given in section 7. appendices provide a modification of a theorem due to huang [22], an extension of a theorem due to hsu [21], and the hopf bifurcation analysis. the bifurcation diagrams were done using xppaut [12] and the simulations were done using matlab [32]. 2. the model let s(t) denote the concentration of the nutrient in the growth chamber at time t, and let x(t) and y(t) denote the density of prey (that feeds off this nutrient) and predator populations, respectively. we consider the following system of autonomous ordinary differential equations as a model of predator-prey interaction in a well-stirred chemostat: s′(t) = (s0 −s)d −xp(s) x′(t) = x(−d + cp(s)) −yq(x) y′(t) = y ( −d + γq(x) ) s(0) ≥ 0, x(0) ≥ 0, y(0) ≥ 0. (2.1) where s0 denotes the concentration of the nutrient in the nutrient reservoir, and c and γ are yield constants (respectively, to the prey’s consumption of the nutrient and the predator’s consumption of prey). to ensure that the volume of this vessel remains constant, d denotes both the rate of inflow from the nutrient reservoir to the growth chamber, as well as the rate of outflow from the growth chamber. we assume the species specific death rates are insignificant with respect to the flow rates and they are ignored. it is also assumed that the functions p and q are continuously differentiable (additional smoothness assumptions are given below) and that s0,d,c, and γ are all positive constants. the rate of conversion of nutrient to biomass is given by the function p(s), and is assumed to satisfy p(0) = 0, p(s) > 0 for s > 0 and p′(s) > 0 for s ≥ 0. (2.2) a function of the form p(s) = ms where m is a positive constant, satisfies all of the above properties. for the remaining of this paper, we consider p to be of this form. the function q(x) denotes the predator response function. it is assumed that q(x) has properties similar to p(s). in particular, q(0) = 0, q′(x) > 0 for x ≥ 0 and q′′(x) < 0 for x ≥ 0. (2.3) the monod functional response q(x) = ax b+x satisfies these properties and will be the focus in the remainder of this paper. note that the following inequality holds for that form: q(x) −xq′(x) > 0 for all x > 0. (2.4) to simplify (2.1), the yield constants c and γ can be scaled out by performing a change of variables. let ŝ = σs, x̂ = ξx, ŷ = ηy and t̂ = τt. then, taking c = σ ξ , γ = ξ η , ŝ0 = σs, d̂ = d τ , m̂ = m ξτ , â = aξ τη , and b̂ = ξb, and removing hats to simplify notation, reduces system (2.1) to: 336 t. bolger, b. eastman, m. hill, and g.s.k. wolkowicz s′(t) = (s0 −s)d −msx x′(t) = x(−d + ms) −y ax b + x y′(t) = y ( −d + ax b + x ) s(0) ≥ 0, x(0) ≥ 0, y(0) ≥ 0. (2.5) remark 2.1. since the system has four variables (s, x, y and t), we are able to scale out two more parameters for a total of four. however, this is not needed to complete our analysis. lemma 2.1. the solutions of s(t), x(t) and y(t) of (2.5) are non-negative and bounded. proof. it is important to first note that since the vector field is c1, existence and uniqueness of solutions hold. s(t) is positive for all t > 0 since s(τ) = 0 for any τ ≥ 0 implies that s′(τ) > 0. furthermore, x(t) > 0 and y(t) > 0 for all t > 0 since the (s, 0,y)-plane and (s,x, 0)-plane are invariant with respect to solutions of (2.5). hence, by the uniqueness of solutions, they cannot be reached in finite time by trajectories for which x(0) > 0 or y(0) > 0, respectively. next consider the sum of our differential equations: s′ + x′ + y′ = (s + x + y)′ = d ( s0 − (s + x + y) ) (2.6) is a first order ode that has the solution s(t) + x(t) + y(t) = e−dt (( s(0) + x(0) + y(0) ) −s0 ) + s0. (2.7) thus s(t) + x(t) + y(t) ≤ max{s(0) + x(0) + y(0),s0}. since all three components are non-negative, this implies each component is bounded above. � nonetheless, (2.5) is still a 3d system of equations, and so is harder to analyze than the classical predator-prey model. from the above proof, we can see that the sum of solutions s(t) + x(t) + y(t) converges to s0 exponentially as t tends to infinity. this tells us that for any point (s,x,y) in the omega limit set, ω = { (s,x,y) ∈ r3+ : ∃{tn} ∞ n=1 , tn →∞ as n →∞, ( s(tn),x(tn),y(tn) ) → (s,x,y) as n →∞ } , it follows that s(tn) + x(tn) + y(tn) → s + x + y = s0. hence, ω is restricted to the simplex{ (s,x,y) : s + x + y = s0 } , a two dimensional set. we can then obtain three equivalent limiting systems by eliminating one variable in our system using the fact that s(t) + x(t) + y(t) = s0. in this case, it is most useful to eliminate s, as this yields a 2d predator-prey system with nullclines that resemble those of the classical predator-prey model. in doing so, if we define f(x) = −dx + mx(s0 −x) mx + q(x) , we obtain the limiting system, x′(t) = x ( −d + m(s0 −x−y) ) −y ax b + x , ( mx + ax b + x )( f(x) −y ) y′(t) = y ( −d + ax b + x ) x(0) ≥ 0, y(0) ≥ 0. (2.8) predator-prey chemostat model: holling type ii response 337 since (2.8) closely resembles the classical predator-prey model, it will be the main system we analyze throughout this paper. that it has the same dynamics as the 3d systems (2.5) and therefore (2.1), is justified in section 6. the predator nullcline is the vertical line x = q−1(d) = bd a−d and is in the first quadrant if and only if a > d. we will assume, unless otherwise stated, that a > d so that the coexistence equilibrium point exists in the interior of the first quadrant. the prey nullcline, f(x), is studied in detail in the next section. 3. the prey nullcline the prey nullcline is given by the continuously differentiable function f(x) = −dx+mx(s0−x) mx+q(x) . the properties of this function play a key role in our analysis. in this section, we determine the properties of f(x) when the predator response function is of monod form, q(x) = ax b+x . define f(0) = lim x→0+ f(x) = (ms0 −d)b a + mb . (3.1) we will assume that ms0 > d, so that f(0) > 0. the slope of the prey nullcline is given by f ′(x) = −(b + x)2 m2 −a (b + 2 x− s0 ) m−ad (m (b + x) + a) 2 . (3.2) define k such that f(k) = 0, namely k = ms 0−d m > 0. then, f ′(k) = − s0m + bm−d s0m + bm + a−d < 0 (3.3) and f ′(0) = −b2m2 + s0am−abm−ad (bm + a)2 , (3.4) with the sign determined by the sign of the numerator. for x ≥ 0, the second derivative of the prey nullcline is: f ′′(x) = − 2ma(s0m + bm + a−d) (bm + mx + a)3 < 0, (3.5) and the third derivative is: f ′′′(x) = 6m2a(s0m + bm + a−d) (bm + mx + a)4 > 0. (3.6) though not discussed any further, it is clear that the higher order derivatives will alternate in sign. since f is concave down for x ≥ 0, it has at most one hump on [0,k]. define m such that f(m) = max x∈[0,k] f(x). remark 3.1. it follows immediately that (a) if f ′(0) ≤ 0, then m = 0, and (b) if f ′(0) > 0, then 0 < m < k and f ′(m) = 0. it is useful to know where m lies in this case. lemma 3.1. assume ms0 > d. then, m < ms 0−d 2m . proof. if we assume that m = 0, the result follows, since then, ms 0−d 2m > 0 under our assumptions. next assume that m > 0. then, f ′(m) = 0. by (3.2), the derivative of f at ms 0−d 2m is f ′ (ms0 −d 2m ) = − ( (ms0−d)2 4m )( m + q′ ( ms0−d 2m )) ( ms0−d 2 + q ( ms0−d 2m ))2 < 0 338 t. bolger, b. eastman, m. hill, and g.s.k. wolkowicz (a) f ′(0) < 0 (b) f ′(0) = 0 (c) f ′(0) > 0 fig. 1. permissible configurations of the prey nullcline. in plots (a) and (b) f(x) is strictly decreasing for all x ∈ [0,k], whereas in plot (c) f(x) has a single local extremum, a local maximum, at m ∈ (0,k). since q′(x) > 0 for x > 0. thus, since f is concave down on [0,k] and f ′(m) = 0, then m < ms0−d 2m . � lemma 3.1 tells us that m is always less than half the distance to k. furthermore, with a monod response function, one can actually find an explicit expression for m, namely: m = −mb−a + √ s0am + abm + a2 −ad m . (3.7) fig 1 illustrates how the shape of the graph of f depends on the sign of f ′(0). in section 4, we select s0 as the bifurcation parameter and investigate how changes in s0 affect the shape of the graph of f . since, ∂ ∂s0 f(x) = m(b + x) bm + mx + a > 0 for x > 0, (3.8) f is an increasing function of s0 for each fixed x. f ′ is also an increasing function of s0 for each fixed x > 0 since ∂ ∂s0 f ′(x) = ma (mb + mx + a)2 > 0. (3.9) remark 3.2. by equations (3.8) and (3.9), as s0 increases, the local maximum of f moves up and to the right. it also is important to note that since f ′(x) increases with s0, one can find a fixed s0 value so that f ′(x) = 0, namely: s0 , s0(x) = (b + x) 2 m2 + a (b + 2 x) m + ad am , (3.10) for any fixed x > 0. if we take x = x∗ in s0(x), we get: s0crit , s 0(x∗) = (abm + ad −d2)(bm + a−d) m(a−d)2 . (3.11) then, taking s0 = s0crit forces f ′(x∗) = 0. taking s0 = s0crit will prove especially useful in our analysis. predator-prey chemostat model: holling type ii response 339 4. local analysis in this section, we consider some of the properties of the different invariant sets associated with (2.8). it is first important to note that non-negativity and boundedness of solutions for system (2.8) follows immediately from the non-negativity and boundedness of solutions of the 3d system (2.5). nonnegativity and boundedness of solutions is a prerequisite of any reasonable model of the chemostat. there are three possible critical points of (2.8). the first two, (0, 0) and (k, 0), always exist. define x∗ = q−1(d) = bd a−d > 0 since we are assuming that a > d, and let y ∗ = f(x∗) > 0. then the interior equilibrium, (x∗,y∗), exists only if x∗ < k. if x∗ = k, then (x∗,y∗) coalesces with (k, 0). the following is the stability analysis for these critical points. the jacobian matrix of (2.8) is j(x,y) = ( f ′(x) ( q(x) + mx ) + ( q′(x) + m )( f(x) −y ) − ( mx + q(x) ) yq′(x) −d + q(x) ) . (4.1) evaluating it at the zero equilibrium, we obtain: j(0, 0) = (( q′(0) + m ) f(0) 0 0 −d ) . (4.2) since q(x) is an increasing function and f(0) > 0, the diagonal entries have opposite signs. hence, (0, 0) is always a saddle. similarly for (k, 0), j(k, 0) = ( f ′(k) ( q(k) + mk ) − ( mk + q(k) ) 0 −d + q(k) ) . (4.3) one can see that if x∗ > k, then q(k) −d = q(k) − q(x∗) < 0 and hence, (k, 0) would be a global attractor since f ′(k) < 0 as well. however, there is no coexistence equilibrium in this case. therefore we assume that x∗ < k, which means the diagonal entries of (4.3) have opposite signs, and hence, (k, 0) is also a saddle. finally, we consider the coexistence equilibrium point. note once again, the coexistence equilibrium exists if and only if x∗ < k. in this case the jacobian is given by a , j(x∗,y∗) = ( f ′(x∗)(d + mx∗) −(mx∗ + d) y∗q′(x∗) 0 ) , (4.4) which has characteristic equation: λ2 −λf ′(x∗)(d + mx∗) + y∗q′(x∗)(d + mx∗) = 0. (4.5) since d + mx∗ > 0, the constant term is positive and the roots of (4.5) have negative real part if and only if f ′(x∗) < 0. thus, (x∗,y∗) is locally asymptotically stable when f ′(x∗) < 0 and unstable when f ′(x∗) > 0. note that unless x∗ = k or f ′(x∗) = 0 all critical points are hyperbolic. when x∗ = k, we determine that the equilibrium point is asymptotically stable using standard phase plane analysis. proposition 4.1. system (2.8) has a supercritical hopf bifurcation occurring when f ′(x∗) = 0. proof. since system (2.8) is planar, any periodic orbit must surround an equilibrium point by the poincaré-bendixson theorem [3]. by the non-negativity of this system, the only equilibrium a periodic orbit can surround is (x∗,y∗). from phase plane analysis, any periodic orbit would lie in the set {(x,y) : 0 < x < k and y > 0}. the eigenvalues of the variational matrix about (x∗,y∗) are λ(s0) = f ′(x∗)(d + mx∗) ± √( f ′(x∗)(d + mx∗) )2 − 4y∗q′(x∗)(d + mx∗) 2 (4.6) when the coexistence equilibrium exists, these eigenvalues are ply imaginary if and only if f ′(x∗) = 0. the condition f ′(x∗) = 0 can be achieved by fixing s0 = s0crit from (3.11). also, the imaginary part of 340 t. bolger, b. eastman, m. hill, and g.s.k. wolkowicz fig. 2. bifurcation diagram of system (2.8) with parameters a = 2, b = 0.5, m = 2, and d = 1.3788. solid lines correspond to stable equilibria, dashed lines correspond to unstable equilibria, filled circles correspond to stable periodic orbits and empty circles correspond to unstable periodic orbits. as s0 increases there is a transfer of stability from (0,0) to (k,0) to (x∗,y∗) by means of transcritical bifurcations, and finally a transfer of stability from (x∗,y∗) to a stable periodic orbit by means of a hopf bifurcation. the eigenvalues is non-zero. furthermore, the transversality condition holds, since the derivative with respect to s0 of the real part of the eigenvalue at the hopf bifurcation is positive by (3.9). thus, the eigenvalues are complex in a neighbourhood of s0crit and cross the imaginary axis at s 0 = s0crit, implying that a hopf bifurcation occurs there. the direction and stability of the bifurcating periodic orbit is determined by the sign of the following quantity, called the vague attractor condition: w = − 4am ( (b + x∗)2(b + 1 2 x∗)m2 + 2(a− 1 4 d)(b + x∗)2m + a ( (a + d)b + dx∗ ))( (b + s0)m + a−d ) ( (b + x∗)m + a )4 (b + x∗)2 , (4.7) which was determined using the algorithm in marsden and mccracken [31] as outlined in appendix c. under our assumptions on the parameters, w < 0, hence, the hopf bifurcation is always supercritical. � the bifurcation diagram in fig 2 summarizes each of these local stability results. 5. global analysis to establish the global dynamics of system (2.8), first examine periodic orbits. the following lemma together with the poincaré criterion will be used to show that when periodic orbits exist, they must surround the local maximum, ( m,f(m) ) . lemma 5.1. let γ be any periodic orbit of (2.8). then c , ∮ γ div(x′,y′) dt = ∮ γ ( mx + q(x) ) f ′(x) dt. predator-prey chemostat model: holling type ii response 341 proof. let g(x) = −d+m(s0−x). then f(x) = xg(x) mx+q(x) , and thus, f ′(x) = g(x)+xg′(x) mx+q(x) − xg(x) ( m+q′(x) )( mx+q(x) )2 . this means that c = ∮ γ div(x′,y′) dt = ∮ γ [( xg′(x) + g(x) −y ( m + q′(x) )) + ( −d + q(x) )] dt = ∮ γ [( xg′(x) + g(x) + (x′ −xg(x) mx + q(x) )( m + q′(x) )) + y′ y ] dt = ∮ γ [( mx + q(x) ) f ′(x) + x′ ( m + q′(x) mx + q(x) ) + y′ y ] dt = ∮ γ [( mx + q(x) ) f ′(x) + d dt ln ( mx + q(x) ) + d dt ln(y) ] dt = ∮ γ ( mx + q(x) ) f ′(x) dt. � proposition 5.2. any periodic orbit of (2.8) must surround the local maximum of f(x). proof. assume f ′(x) > 0 for the entire portion of f inside of γ. then c > 0 by lemma 5.1, implying that γ is an unstable periodic orbit by the poincaré criterion [9]. since any periodic orbit must surround (x∗,y∗), it follows that f ′(x∗) > 0 and so (x∗,y∗) would also be unstable, which is impossible. using a similar argument, it also follows that f ′(x) < 0 for the entire portion of f inside of γ is impossible. thus, the slope of the portion of the prey nullcline inside any periodic orbit cannot be entirely of the same sign, i.e., it must change sign, and therefore any periodic orbit must surround the local maximum of f(x), ( m,f(m) ) . � global stability of the coexistence equilibrium point for the classical predator-prey model was studied by harrison [16]. if one denotes the coexistence equilibrium point as (x∗,y∗), he proved that this equilibrium is globally asymptotically stable, if it is locally asymptotically stable, i.e., in phase space it lies on prey nullcline, f(x), where its slope is negative (f ′(x∗) < 0), and as well, it lies below the vertical line y = f(0), i.e., y∗ < f(0). hence, he only obtained a sufficient condition for the global stability. hsu [21] theorized more generally that if the prey nullcline is concave down, then wherever the interior equilibrium is locally stable, it is also globally stable. although this is not true for all systems (some counterexamples were found in [19]), we prove for (2.8), that local asymptotic stability implies global stability by using the dulac criterion and the poincaré-bendixson theorem, that the equilibrium destabilizes via a supercritical hopf bifurcation, and that if a periodic orbit exists it is unique. let (x∗,y∗) be locally asymptotically stable, i.e. x∗ ∈ [m,k]. since the hopf bifurcation at x∗ = m is supercritical, (x∗,y∗) is also locally asymptotically stable there. we start using a similar argument to the proof given for theorem 3.3 in hsu [21]. however, it was pointed out in [8], that the proof in hsu [21] is “not rigorously correct” and so we make modifications. choose h(x,y) = ( mx+q(x) )−1 yβ−1 as the auxiliary function to use with the dulac criterion, where the value β > 0 will be determined. here, the function h(x,y) is defined in the interior of the first quadrant. let f = ( x′ y′ ) . then, ∆ ,∇· (hf) = yβ−1h(x) mx + q(x) (5.1) 342 t. bolger, b. eastman, m. hill, and g.s.k. wolkowicz where h(x) = ( mx + q(x) ) f ′(x) + β ( −d + q(x) ) . (5.2) in the interior of the first quadrant, yβ−1 > 0 and mx + q(x) > 0, hence, ∆ changes sign if and only if h(x) changes sign. as opposed to the argument in [21], it is important to note that f ′(x) ≥ 0 and −d + q(x) < 0 for x ∈ [0,m], while f ′(x) < 0 and −d + q(x) ≥ 0 for x ∈ [x∗,k]. therefore, the choice of β is critical in order to ensure that h(x) does not change sign. as a consequence, we have the following proposition. proposition 5.3. let β(x) = − ( mx+q(x) ) f′(x) −d+q(x) . if there exits β > 0 such that max x∈[0,m] β(x) ≤ β ≤ min x∈[x∗,k] β(x), then h(x) ≤ 0, for all x ∈ [0,k]. remark 5.1. any positive β would work for x ∈ [m,x∗], since both f ′(x) ≤ 0 and −d + q(x) ≤ 0. first consider when x∗ ∈ (m,k]. for system (2.8) with a monod functional response, the function β(x) is given by: β(x; s0) = ( (b + x)2m2 + a(b + 2x−s0)m + ad ) x ((a−d)x− bd)(m(b + x) + a) . (5.3) note that β(x) is parameterized by s0. we next look to see how changes in s0 affect β(x). since, ∂ ∂s0 β(x; s0) = − amx (a−d)(x−x∗)(bm + mx + a) , (5.4) β(x) is increasing with respect to s0 if x < x∗ (and hence, if x < m) and decreasing with respect to s0, if x > x∗. using (3.10), we can find the value of s0 that forces x∗ = m, namely s0crit from (3.11). since x∗ > m with the original s0, s0crit > s 0 by remark 3.2. thus, β(x; s0crit) > β(x; s 0) when x < m and β(x; s0crit) < β(x; s 0) when x > x∗. the function β(x; s0crit) is given by: β(x; s0crit) = mx(2abm + amx−dbm−dmx + 2a2 − 2ad) (bm + mx + a)(a−d)2 . (5.5) note that the function β(x; s0crit) in (5.5) is a continuously differentiable function. let βcrit be the value of β(x) with s0 = s0crit so that x = x ∗ = m, namely: βcrit = β(x ∗; s0crit) = 2dbm (a−d)2 > 0. (5.6) lemma 5.4. if x∗ > m, then max x∈[0,m] β(x; s0) < βcrit < min x∈[x∗,k] β(x; s0). proof. consider the derivative with respect to x of β(x; s0crit): β′(x; s0crit) = (2a−d)b2m3 + 2(a−d)bm3x + (a−d)m3x2 (bm + a)(a−d)2 + abm2(4a− 3d) + 2am2x(a−d) + 2a2m(a−d) (bm + a)(a−d)2 (5.7) > 0 for x ≥ 0, hence, β(x; s0crit) is an increasing function on [0,k]. this, together with (5.4) and remark 3.2, tells us that βcrit > β(x; s 0 crit) > β(x; s 0) for x < m, and βcrit < β(x; s 0 crit) < β(x; s 0) for x > x∗. finally at the boundaries, β(m; s0) = 0 < βcrit, and lim x→x∗+ β(x; s0) = ∞ > βcrit. predator-prey chemostat model: holling type ii response 343 (a) s0 < s0crit (b) s 0 > s0crit fig. 3. graphs of β(x;s0) (solid), β(x;s0crit) (dashed), βcrit (dash-dotted) and the vertical asymptote x = x∗ (dotted) with parameters a = 2, b = 0.5, m = 2, and d = 1.3788. plot (a) illustrates when s0 < s0crit, i.e., x ∗ > m, and hence, the coexistence equilibrium is locally asymptotically stable, the function β(x;s0) can be used to show that if β = βcrit, that the function h(x) defined in (5.2) is of one sign (see proposition (5.3), remark 5.1, and lemma 5.4)), and hence, can be used to prove global stability of the coexistence equilibrium using the dulac criterion. plot (b) illustrates that when s0 > s0crit, and hence, the coexistence equilibrium is unstable, as expected, the function h(x) changes sign for any choice of β. thus, altogether, max x∈[0,m] β(x; s0) < βcrit < min x∈[x∗,k] β(x; s0). � finally, we consider the case when x∗ = m. then β(x; s0) is precisely β(x; s0crit), an increasing function with respect to x, and thus, max x∈[0,m] β(x; s0) = βcrit = min x∈[x∗,k] β(x; s0). fig 3 demonstrates that the function β(x; s0) can be used to determine when the dulac criterion can be used to obtain global stability of the coexistence equilibrium using the function h(x) defined in (5.2). in particular, a value βcrit can be chosen so that the dulac criterion can be used to prove global stability of the coexistence equilibrium if s0 < s0crit, i.e., x ∗ < m, which is precisely when the coexistence equilibrium is locally asymptotically stable, but when s0 > s0crit, i.e., x∗ < m, and the coexistence equilibrium is unstable, no value βcrit can be chosen. theorem 5.5. consider system (2.8). if (x∗,y∗) is locally asymptotically stable, then it is globally asymptotically stable. if (x∗,y∗) is unstable, then there exists a unique, stable periodic orbit that surrounds the point (m,f(m)). proof. since βcrit satisfies proposition 5.3, h(x) ≤ 0 and consequently ∆ ≤ 0 for 0 ≤ x ≤ k. therefore, since ∆ does not change sign, by the dulac criterion it follows that system (2.8) has no nontrivial closed orbits lying entirely in the first quadrant. thus, since we have proved that all orbits are bounded (see lemma 2.1), by the poincaré-bendixson theorem, (x∗,y∗) is globally asymptotically stable. to determine uniqueness of the periodic orbit when x∗ < m, apply a modified version of huang’s theorem from [22], stated as theorem a.1 in appendix a by taking φ(x) = mx+q(x), ψ(x) = q(x)−d, π(y) = y and ρ(y) = y. then conditions (i)-(iii) of theorem a.1 are satisfied. for condition (iv) of theorem a.1, notice that the function h(x) is identical to β(x) from (5.3). we had β′(x) > 0 from (5.7), (5.4), and remark 3.2, so we also have h′(x) > 0. thus, condition (iv) of theorem a.1 is satisfied, and so when x∗ < m, there is a unique periodic orbit and it is orbitally asymptotically stable. � 344 t. bolger, b. eastman, m. hill, and g.s.k. wolkowicz (a) s0 < x∗ + d m (b) x∗ + d m < s0 < s0crit (c) s0 > s0crit fig. 4. sample trajectories of (2.8) with parameters a = 2, b = 0.5, m = 2, and d = 1.3788. plot (a) depicts globally asymptotically stable convergence to (k,0), when x∗ > k. plot (b) depicts convergence to the globally asymptotically stable interior equilibrium, when it is to the right of the local maximum. plot (c) depicts an orbitally asymptotically stable periodic orbit surrounding the unstable interior equilibrium, when it is to the left of the local maximum. fig 4 illustrates convergence to the globally asymptotically stable equilibrium (k, 0) when the coexistence equilibrium does not exist and the existence of the two types of dynamics that are possible when a coexistence equilibrium point does exist: convergence to a globally asymptotically stable coexistence equilibrium point or convergence to an orbitally asymptotically stable periodic orbit that attracts all solutions with positive initial conditions except the unstable coexistence equilibrium point. in appendix b, we extend theorem 5.5 to a more general class of predator-prey models, and give examples of systems that satisfy the hypotheses so that a β can be found in order to prove global stability of the coexistence equilibrium. 6. dynamics of the 3d system the dynamics of the 2d system (2.8) have been analyzed in great detail. we next justify why studying this system is equivalent to studying the 3d system (2.5), and hence, the original system (2.1). any point (x,y) in the 2d system (2.8) corresponds to a point (s0 −x−y,x,y) in the 3d system (2.5), and so solutions of the 2d system correspond to solutions of the 3d system that lie on the 2d simplex s = {(s,x,y) ∈ r3 : s,x,y > 0,s + x + y = s0}. thus, the two systems share some of the same properties. the equilibrium points (0, 0), (k, 0) and (x∗,y∗) of the 2d system correspond to (s0, 0, 0), (s0 −k,k, 0) and (s0 −x∗ −y∗,x∗,y∗), respectively, for the 3d system. in terms of local predator-prey chemostat model: holling type ii response 345 stability, the additional eigenvalue of the jacobian matrix is negative, since solutions of the 3d system converge exponentially to s. thus, the local stability of the 3d equilibrium points is the same as for the corresponding 2d equilibrium point. since all periodic orbits of the 3d system must lie on s, the existence and number of periodic orbits of the two systems is the same. from theorem 5.5 and standard methods for asymptotically autonomous systems (see smith and waltman [37], or using the butler-mcgehee lemma [5] directly), we obtain the following theorem. theorem 6.1. consider system (2.5). if (x∗,y∗) is locally stable for the 2d system (2.8), then it is globally stable, and (s0 − x∗ − y∗,x∗,y∗) is globally asymptotically stable for the 3d system (2.5). if (x∗,y∗) is unstable in 2d system (2.8), then there is a unique periodic orbit that lies on the s+x+y = s0 simplex and it is orbitally asymptotically stable. corollary 6.2. consider the original system (2.1). if the coexistence equilibrium exists and is stable, then it is globally asymptotically stable, and if it is unstable, then there is a unique periodic orbit that lies on the {(s,x,y) ∈ r3 : σs,ξx,ηy > 0,σs + ξx + ηy = σs0} simplex, and it is orbitally asymptotically stable. 7. discussion a system of odes modeling predator-prey interactions in a chemostat was analyzed assuming a predator response function of monod form. it was shown that whenever the coexistence equilibrium is locally asymptotically stable, it is also globally asymptotically stable, and whenever the coexistence equilibrium is unstable, there is a unique, orbitally asymptotically stable periodic orbit. these results are consistent with the dynamics of the analogous classical predator-prey model with monod predator response function, in whichthe resource and how the prey grows based on the amount of resource available is not modelled, but instead the prey is assumed to grow logistically in the absence of the predator population. this classical model has been studied extensively. see for example, [8,27,28,36] the resource is not modelled, but instead the prey is assumed to grow logistically in the absence of the predator. these authors found, just as for the predator-prey model in the chemostat studied in this paper, that whenever the coexistence equilibrium is locally asymptotically stable, it is globally asymptotically stable. it was also shown in cheng [7] that when a periodic orbit exists around an unstable coexistence equilibrium in the classical predator-prey model with a monod functional response, it is unique, and hence any nontrivial periodic orbit is asymptotically stable. however, there was a gap in the proof, that was later corrected by liou and cheng [28]. another proof of the uniqueness of the limit cycle for the classical model in the case of the monod functional response was also given in kuang and freedman [27]. that the range of dynamics of the model studied here, and of the analogous classical predator-prey model is basically the same, is not entirely surprising. for example, after hastings and powell [17] showed that a three-species food chain with monod response functions in which the population at the lowest tropic level grows logistically in the absence of a predator population could have chaotic dynamics, daoussis [10] showed this was also the case for the analogous chemostat food-chain model. classical predator-prey models have been shown to be sensitive to the mathematical form used to model the predator response function, even when the forms have the same qualitative shape by fussmann and blasius [14]. they considered three mathematical forms: the monod form, the ivlev form [23], and the hyperbolic tangent form [24], all of which are monotone increasing and concave down and are nearly indistinguishable when appropriate parameters are chosen (see fig 5). fussmann and blasius provided a similar figure and demonstrated that the qualitative and quantitative dynamics predicted by models with these response functions can be quite different. the model with the three 346 t. bolger, b. eastman, m. hill, and g.s.k. wolkowicz fig. 5. graphs of the three mathematical forms considered in [14]: the monod form fm (x) = am bm +x (dashed), the ivlev form fi(x) = ai(1 − e−bi x) (solid) and hyperbolic tangent form ft (x) = at tanh(bt x) (dotted). parameters were chosen by nonlinear least-squares fits to ivlev’s response functions with ai = 1 and bi = 2 (i.e. am = 1.14, bm = 0.37, at = 0.99 and bt = 1.48). functions with such a shape are said to be of holling type ii form [20]. different response function forms was studied in more detail using a bifurcation theory approach in seo and wolkowicz [36]. in the case of the classical model with the hyperbolic tangent response function, seo and wolkowicz [36] proved that the hopf bifurcation is always supercritical when it occurs at the local maximum of the prey nullcline, but can be either super or subcritical when it occurs at the local minimum. they also studied the dynamics in more detail for all three forms of the response functions using a one and two parameter bifurcation approach, and found that in the hyperbolic tangent case, two limit cycles surrounding a stable coexistence equilibrium can arise through a saddle-node bifurcation of limit cycles when the hopf bifurcation at the local minimum is subcritical. seo and wolkowicz [35] also considered the classical predatory-prey model, with a functional response of arctan form and proved that when the coexistence equilibrium is locally asymptotically stable, more than one limit cycle is possible, providing a counterexample to a result in attili and mallak [2]. the classical predator-prey model with ivlev response functions was analyzed by kooij and zegeling [25]. they proved that model has a similar range of dynamics as the model with monod response function. the analogous chemostat models in the cases of the hyperbolic tangent and the arctan as the response function was studied in eastman [11] and the analogous model with the ivlev response function was studied in bolger [4]. it was also shown that in these cases the analogous chemostat models have a similar range of dynamics when compared with their analogous classical predator-prey models. acknowledgement: this research is based on collaboration with t.b, m.h., and b.e. during their graduate studies at mcmaster university under the supervision of g.s.k.w. appendix appendix a modified huang’s theorem we use a slightly adapted version of huang’s theorem that better applies to system (2.8) by accounting for invariance of the first quadrant. theorem a.1 (huang, 1988). consider a system of the form: dx dt = φ(x) (f(x) −π(y)) , (a.1) dy dt = ρ(y)ψ(x). predator-prey chemostat model: holling type ii response 347 if (i) ∃ k > x∗ such that f(k) = 0 and (x − k)f(x) < 0 for x 6= k, ∃ 0 < x∗ < k such that ψ(x∗) = 0, i.e. there is a positive equilibrium point (x∗,y∗), (ii) all functions in (a.1) are c1 in the interior of r3+, and f ′(x) is continuous in the interior of r3+, (iii) φ(0) = π(0) = ρ(0) = 0, φ′(x) > 0 and ψ′(x) > 0 for x > 0, ρ′(y) > 0 and π′(y) > 0 for y > 0, and (iv) h(x) = −f ′(x)φ(x)/ψ(x) is non-decreasing for 0 < x < x∗ and x∗ < x < k. then, system (a.1) has at most one limit cycle in the first quadrant, and, if it exists it is stable. appendix appendix b extension of hsu’s theorem consider the following predator-prey system: x′(t) = ξ(x) ( f(x) −γ(y) ) y′(t) = η(y) ( −d + q(x) ) x ≥ 0, y ≥ 0. (b.1) (b.1) is a generalized version of the system that hsu studied in [21]. here, the prey nullcline is given by the function γ−1 ( f(x) ) , and the interior equilibrium is the unique point (x∗,y∗) satisfying q(x∗) = d and γ(y∗) = f(x∗). hsu [21, theorem 3.3] conjectured that if (x∗,y∗) is stable and the prey nullcline is concave down, then (x∗,y∗) is globally stable. since this conjecture is not true, as demonstrated by the counter example given by hofbauer and so [19], we state the following theorem that was shown to be satisfied in seo and wolkowicz [36] for the classical predator-prey model with a hyperbolic tangent response function. the following theorem can also be shown to be satisfied if, for example, ξ(x) = q(x), γ(y) = η(y) = y, f(x) = x(1− x k ) q(x) , and q(x) = a tanh(bx). theorem b.1. assume there exists an equilibrium (x∗,y∗) with positive components satisfying q(x∗) = d and η(y∗) = f(x∗), and that the following assumptions hold: (i) ∃k > 0 such that f(k) = 0, f(x) > 0 for 0 ≤ x ≤ k, f(x) < 0 for x > k, and f(x) ∈ c1 ( [0,k] ) . (ii) f has at most one local extremum, a local maximum, for x ∈ [0,k] assumed to occur at x = m. (iii) η(y) ∈ c1(r+), η(0) = 0 and η′(y) > 0 for y > 0. (iv) γ(y) ∈ c1(r+), γ(0) = 0 and γ′(y) > 0 for y > 0. (v) ξ(x) ∈ c1(r+), ξ(0) = 0 and ξ′(x) > 0 for x > 0. (vi) q(x) ∈ c1(r+), q(0) = 0, and q′(x) > 0 for x > 0. (vii) f ′(x∗) ≤ 0. then (x∗,y∗) is globally stable provided there exists a β > 0 such that: max x∈[0,m] − ξ(x)f ′(x)( −d + q(x) ) inf y≥0 η′(y) ≤ β ≤ min x∈[x∗,k] − ξ(x)f ′(x)( −d + q(x) ) sup y≥0 η′(y) . (b.2) remark b.1. if m = 0 then the lyapunov function in harrison [16] can be used to show global stability. proof. to show that (x∗,y∗) is globally stable, we will use a similar argument as hsu. that is, we will show that there are no closed orbits in the first quadrant using the dulac criterion. define the auxiliary 348 t. bolger, b. eastman, m. hill, and g.s.k. wolkowicz function to be h(x,y) = ξ(x)−1η(y)β−1, where β > 0. the resulting divergence is then ∆ ,∇· (hf) = η(y)β−1h(x) ξ(x) (b.3) where f = ( x′ y′ ) and h(x) is given by: h(x) = ξ(x)f ′(x) + β ( −d + q(x) ) η′(y). (b.4) in the first quadrant, η(y)β−1 > 0 and ξ(x) > 0 by assumptions (iii) and (v), hence, ∆ will only change sign if h(x) changes sign. in hsu’s proof, he tried to show that h(x) ≤ 0 by choosing an appropriate β > 0. observe that d dx γ−1 ( f(x) ) = f ′(x) γ′ ( γ−1 ( f(x) )). (b.5) since the denominator is positive by assumption (iv), f ′(x) has the same sign as d dx γ−1 ( f(x) ) . the problem with the β hsu chose is that it neglects the fact that f ′(x) ≥ 0 for x ∈ [0,m] by assumption (vii). by assumptions (ii) and (vii), we can conclude that x∗ ≥ m. despite that −d + q(x) ≤ 0 in [0,m] by assumption (vi), if β is too small (as the one he chose), h(x) > 0. to guarantee that h(x) ≤ 0 ∀x ∈ [0,m], it is then required that β ≥ max x∈[0,m] − ξ(x)f ′(x)( −d + q(x) ) inf y≥0 η′(y) . (b.6) however, β can not be too large either. since x∗ ≥ m, −d + q(x) ≥ 0 and f ′(x) ≤ 0 for x ∈ [x∗,k], then β ≤ min x∈[x∗,k] − ξ(x)f ′(x)( −d + q(x) ) sup y≥0 η′(y) (b.7) must also hold to guarantee that h(x) ≤ 0 ∀x ∈ [x∗,k]. we do not need to worry about the region [m,x∗] since both f ′(x) ≤ 0 and −d + q(x) ≤ 0 there, so any positive β would work. thus, if there exists β > 0 that satisfies both (b.6) and (b.7), then h(x) ≤ 0. since h does not change sign, ∆ does not change sign. hence, by the dulac criterion, it follows that system (b.1) has no nontrivial orbits lying entirely in the first quadrant. thus by the poincaré-bendixson theorem, (x∗,y∗) is globally stable. � we next include another example of a system where such a β is obtainable under the above assumptions. consider the classical predator-prey model with a monod response function. in this notation, this would be equivalent to letting ξ(x) = q(x), γ(y) = η(y) = y, f(x) = x(1− x k ) q(x) , and q(x) to be of monod form. note that since γ(y) = y, γ−1 ( f(x) ) = f(x) and hence, that the interior equilibrium is given by( x∗,f(x∗) ) . we can then equivalently transform the parameters of the monod response function by a 7→ dm̄ and b 7→ x∗(m̄−1), where m̄ > 1. then q(x) = dm̄x x∗(m̄−1)+x. we first show that these functions satisfy the assumptions of theorem b.1. the prey nullcline is given by the function: f(x) = (1 − x k ) ( x∗(m̄− 1) + x ) dm̄ (b.8) which has the second derivative f ′′(x) = − 2 dm̄k < 0. (b.9) predator-prey chemostat model: holling type ii response 349 since f(x) is concave down for all x ∈ [0,k], it has at most one local extremum, a local maximum, in this region and thus, satisfies assumption (ii). we find the maximum to occur at: m = 1 2 ( k −x∗(m̄− 1) ) . (b.10) it is also clear that f(x) > 0 for all x ∈ [0,k], f(k) = 0, and f(x) < 0 x > k, satisfying assumption (i). furthermore, ξ′(x) = q′(x) = dm̄x∗(m̄−1)( x∗(m̄−1)+x )2 > 0 for x > 0, and ξ(0) = q(0) = 0. thus both (v) and (vi) are also satisfied. finally, γ(0) = η(0) = 0, and γ′(y) = η′(y) = 1 > 0 for y > 0, satisfying assumptions (iii) and (iv). it is important to note that under assumption (i), the interior equilibrium (x∗,y∗) lies in the positive quadrant. moreover since η(0) = 0 and ξ(0) = 0, the point (0, 0) is also an equilibrium point. thus, the x and y axes are both nullclines. to satisfy assumption (vii), we require x∗ ∈ [m,k]. first assume x∗ ∈ (m,k]. since all hypotheses of theorem b.1 are met, we proceed to the find the β as described by (b.2). let β(x) = − ξ(x)f ′(x)( −d + q(x) ) η′(y) = − x ( k − 2x−x∗(m̄− 1) ) dk(m̄− 1)(x−x∗) = − 2x(m −x) dk(m̄− 1)(x−x∗) . (b.11) we can then determine where the maximum and minimum of β(x) occur, by finding its critical values. taking the first derivative, we get: β′(x) = 2(x2 − 2xx∗ + mx∗) dk(m̄− 1)(x−x∗)2 . (b.12) note that the sign of β′(x) depends on the numerator. the roots of x2 − 2xx∗ + mx∗ = 0 are x+,− = x ∗ ± √ x∗(x∗ −m), (b.13) and are both positive. the following two lemmas will be useful in determining which of these values is the local maximum of β (the other being the local minimum), and whether these values lie in our desired regions [0,m] and [x∗,k]. lemma b.2. let β− = β(x−) > 0. then max x∈[0,m] β(x) = β−. furthermore, β ′(x) < 0 for x∗ < x < x+. proof. first consider x−. we proceed using proof by contradiction to show that x− ∈ [0,m]. suppose that x− > m. then x∗ − √ x∗(x∗ −m) > m − √ x∗(x∗ −m) > m −x∗ > 0, which is and x− ∈ [0,m], we can conclude that β− is a local maximum, and hence, that max x∈[0,m] β(x) = β−. (b.14) now consider x+. clearly x+ > x ∗. moreover, β′(x) has a removable singularity at x = x∗ and since sign ( lim x→x∗ β′(x∗) ) = x∗(m −x∗) < 0, (b.15) β(x) is decreasing between x∗ and x+. � 350 t. bolger, b. eastman, m. hill, and g.s.k. wolkowicz lemma b.3. let β+ = β(x+) > 0 and βk = β(k) > 0. if x ∗ ≤ k 2 2k−m , then min x∈[x∗,k] β(x) = β+. otherwise, if x∗ > k 2 2k−m , then min x∈[x∗,k] β(x) = βk. proof. first assume that x∗ ≤ k 2 2k−m . since β(x) is decreasing between x ∗ and x+, if we can show that β(k) ≥ 0, then a local minimum occurs in [x∗,k]. furthermore, since x− is not in this region, it would mean that this local minimum must be β(x+), and hence, x+ ≤ k. indeed, β(k) = 2k2 − 2kx∗ + mx∗ = 2k2 + (x∗)2 − m̄(x∗)2 − 3kx∗ = 2k(k −x∗) + ( −kx∗ + (x∗)2(1 − m̄) ) = 2k(k −x∗) + x∗ ( (1 − m̄)x∗ −k ) = 2k(k −x∗)2 + x∗ ( − (k − 2m) −k ) = 2k(k −x∗) + 2x∗(m −k) = 2[k(k −x∗) + x∗(m −x∗ + x∗ −k)] = 2[(k −x∗)2 + x∗(m −x∗)] ≥ 0 this means that in this case, min x∈[x∗,k] β(x) = β+. (b.16) now assume that x∗ > k 2 2k−m . then, β(k) < 0, and hence, x+ > k. furthermore, since β(x) is decreasing between x∗ and x+, it sly is decreasing on [x ∗,k]. thus the minimum value it attains on that region is at k, namely min x∈[x∗,k] β(x) = βk. (b.17) � lemma b.4. for our positive quantities β−, β+ and βk, (a) β+ > β− when m ≤ x+ ≤ k (b) βk > β− when x+ > k. proof. (a) β+ −β− = 8x∗(x∗ −m) dk(m̄− 1) √ x∗(x∗ −m) > 0 (b.18) (b) βk −β− = 2 ( (x∗ −k)2 + x∗(x∗ −m) )√ x∗(x∗ −m) + 4x∗(k −x∗)(x∗ −m) dk(m̄− 1)(k −x∗) √ x∗(x∗ −m) > 0 (b.19) in any case, there exists a positive β between the two quantities when x∗ ∈ (m,k]. � finally consider when x∗ = m. then β′(x) = 2(x−x∗)2 dk(m̄−1)(x−x∗)2 > 0 for x 6= x ∗ and furthermore lim x→x∗ β′(x) = 2 dk(m̄−1) > 0, hence, β(x) is increasing on [0,k]. define β(x∗) = lim x→x∗ β(x) = lim x→x∗ 2x(x−x∗) dk(m̄− 1)(x−x∗) = 2x∗ dk(m̄− 1) . (b.20) thus, max x∈[0,m] β(x) = β(x∗) = min x∈[x∗,k] β(x). predator-prey chemostat model: holling type ii response 351 remark b.2. our above examples were both the case that η(y) = y. it is conceivable that other increasing functions will work, for instance η(y) = δy + αy γ+y , where α, δ and γ are all positive. then sup y≥0 η′(y) = δ + α γ and inf y≥0 η′(y) = δ, both positive, finite values that would not blow up (b.2). appendix appendix c analysis of the hopf bifurcation computation of (4.7) was done using the computer algebra system maple [30], as provided in the supplementary material [18]. the following is a summary of the algorithm used, highlighting the main results. the formula in marsden and mccracken [31] is localized to where the hopf bifurcation occurs, and thus, we assume that we are near the critical value of our bifurcating parameter, s0crit. to use this formula, we first need matrix (4.4) in real jordan canonical form. that is, we need to find an invertible matrix p so that j = p−1ap = ( α β −β α ) where α± iβ are the eigenvalues of a. lemma c.1. let the eigenvector for α + iβ be pre + ipim. then p = ( pre pim ) . proof. we have: p−1p = p−1 ( pre pim ) = ( p−1pre p −1pim ) . by the identity p−1p = i, this means that p−1pre = ( 1 0 ) and p−1pim = ( 0 1 ) . since pre + ipim is the eigenvector for α+iβ, we know a(pre +ipim) = (α+iβ)(pre +ipim), hence, apre = αpre−βpim and apim = βpre + αpim. thus, j = p−1ap = ( p−1apre p −1apim ) = ( p−1(αpre −βpim) p−1(βpre + αpim) ) = ( αp−1pre −βp−1pim βp−1pre + αp−1pim ) = ( α ( 1 0 ) −β ( 0 1 ) β ( 1 0 ) + α ( 0 1 )) = ( α β −β α ) . � near s0crit, the eigenvalues of (4.4) are: l+,− = (mx∗ + d)f ′(x∗) ± i √ − ( (mx∗ + d)f ′(x∗) )2 + 4q′(x∗)y∗(mx∗ + d) 2 . (c.1) thus by lemma c.1, we can construct the transformation matrix p by taking the real and imaginary parts of the eigenvector corresponding to l+, namely p =  f′(x∗)(mx∗+d) 2y∗q′(x∗) √ (mx∗+d) ( −f′(x∗)2(mx∗+d)+4y∗q′(x∗) ) 2y∗q′(x∗) 1 0   . (c.2) 352 t. bolger, b. eastman, m. hill, and g.s.k. wolkowicz with this p , we can rewrite system (2.8) so that the linear part is in real jordan canonical form as follows: ( x y )′ = a ( x y ) + h.o.t.’s p−1 ( x y )′ = p−1a ( x y ) + p−1 h.o.t.’s p−1 ( x y )′ = p−1app−1 ( x y ) + p−1 h.o.t.’s p−1 ( x y )′ = jp−1 ( x y ) + p−1 h.o.t.’s and thus, by letting ( u v ) = p−1 ( x y ) , we transform system (2.8) to: ( u v )′ = j ( u v ) + h.o.t.’s , ( f(u,v) g(u,v) ) . (c.3) in our case, p−1 ( x′ y′ ) =   y ( −d + q(x) ) 2y∗q′(x∗) ( q(x)+mx )( f(x)−y ) −f′(x∗)(mx∗+d)y ( −d+q(x) ) √ (mx∗+d) ( −f′(x∗)2mx∗−f′(x∗)2d+4y∗q′(x∗) )   = (f(x,y) g(x,y) ) , (c.4) where x(u,v), and y(u,v). one can find x and y in terms of u and v by inverting the change of variables, ( x y ) = p ( u v ) =  f′(x∗)(mx∗+d)u+ √ (mx∗+d) ( −f′(x∗)2mx∗−f′(x∗)2d+4y∗q′(x∗) ) v 2y∗q′(x∗) u   . (c.5) thus, we finally compute the vague attractor condition (4.7) using the chain rule: w ′′′(x∗,y∗,s0crit) = 3π 4|β| ( fuuu + fuvv + guuv + gvvv ) + 3π 4|β|2 ( −fuv(fuu + fvv) + guv(guu + gvv) + fuuguu −fvvgvv ) = (mx∗ + d) ( 2f ′′(x∗)q′(x∗)2 + 2f ′′(x∗)q′(x∗)m + f ′′′(x∗)q′(x∗)d ) y∗q′(x∗)2 − (mx∗ + d) ( q′′(x∗)f ′′(x∗)mx∗ −f ′′′(x∗)q′(x∗)mx∗ + q′′(x∗)f ′′(x∗)d ) y∗q′(x∗)2 . (c.6) since only the sign of w ′′′ matters, one can factor the positive constant mx ∗+d y∗q′(x∗) , obtaining w = (mx∗ + d)f ′′′(x∗) − (mx∗ + d)f ′′(x∗)q′′(x∗) q′(x∗) + (2q′(x∗) + 2m)f ′′(x∗), (c.7) which is consistent with the expression found in wolkowicz [38] for, in their notation, p(x) = mx+q(x). at last, substituting the monod response function yields (4.7). predator-prey chemostat model: holling type ii response 353 references [1] c. aldebert and d. b. stouffer, community dynamics and sensitivity to model structure: towards a probabilistic view of process-based model predictions, j. r. soc, interface 15 (2018), 20180741. [2] b. s. attili and s. f. mallak, existence of limit cycles in a predator-prey system with a functional response of the form arctan(ax), commun. math. anal. 1 (2006), no. 1, 33–40. [3] i. bendixson, sur les courbes définies par des équations différentielles, acta math. 24 (1901), no. 1, 1–88. 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[39] , successful invasion of a food web in a chemostat, math. biosci. 93 (1989), 249–268. department of mathematics and statistics, mcmaster university, on, canada l8s 4k1 current address: 425 parkside dr, waterdown on, canada l8b0y6 e-mail address: tedra.bolger@gmail.com department of mathematics and statistics, mcmaster university, on, canada l8s 4k1 current address: department of applied mathematics, university of waterloo, on, canada n2l 3g1 e-mail address: b2eastma@uwaterloo.ca department of mathematics and statistics, mcmaster university, on, canada l8s 4k1 current address: 404 king st. w apt. 420 kitchener, on, canada n2g 4z9 e-mail address: madeleine 0715@hotmail.com corresponding author, department of mathematics and statistics, mcmaster university, on, canada l8s 4k1 e-mail address: wolkowic@mcmaster.ca 1. introduction 2. the model 3. the prey nullcline 4. local analysis 5. global analysis 6. dynamics of the 3d system 7. discussion appendix a. modified huang's theorem appendix b. extension of hsu's theorem appendix c. analysis of the hopf bifurcation references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 2, number 1, march 2021, pp.32-54 https://doi.org/10.5206/mase/11087 a level set approach for a multi-scale cancer invasion model thomas carraro, sven e. wetterauer, ana victoria ponce bobadilla, and dumitru trucu abstract. the quest for a deeper understanding of the cancer growth and spread process focuses on the naturally multiscale nature of cancer invasion, which requires an appropriate multiscale modeling and analysis approach. the cross-talk between the dynamics of the cancer cell population on the tissue scale (macroscale) and the proteolytic molecular processes along the tumor border on the cell scale (microscale) plays a particularly important role within the invasion processes, leading to dramatic changes in tumor morphology and influencing the overall pattern of cancer spread. building on the multiscale moving boundary framework proposed in trucu et al. (multiscale model. simul. 11(2013), 309-335), in this work we propose a new formulation of this process involving a novel derivation of the macro scale boundary movement law based on micro-dynamics, involving a transport equation combined with the level set method. this is explored numerically in a novel finite element macro-micro framework based on cut-cells. 1. introduction involving a wide range of cross-related processes occurring on several spatio-temporal scales, cancer cell invasion in human tissue is one of the hallmarks of cancer [36], playing a crucial role in the overall development of a growing malignant tumour. taking advantage of the heterotypic character of the tumour microenvironment (which includes immuno-inflamatory cells, stromal cell, fibroblasts, endothelial cells, macrophages), complex molecular processes facilitate intense interactions between the cancer cell population and the extracellular matrix (ecm)[29, 36, 39, 40, 58]. these interactions lead to a cascade of specific developmental patterns and behaviours of the growing tumours, most notable stages including the degradation of the ecm, the local progression of the tumour, followed by the tumour angiogenesis process and the subsequent metastatic spread of the cancer cells in the human body. the alteration and remodelling of the ecm by the matrix degraded enzymes (mdes) such as matrix metalloproteinases mmps or the urokinase plasminogen activator (upa) play a key role in local tumour progression. alongside cell-adhesion and multiple taxis processes (including haptotaxis and chemotaxis), the matrix degrading enzymes processes degrade various components of the surrounding ecm that leads to further tumour progression. however, as the full mechanisms involved in these complex processes is yet to be deciphered biologically, over the past two decades or so cancer invasion received extensive mathematical modelling attention, in which systems of reaction-diffusion-taxis partial differential equations [3, 7, 8, 9, 13, 15, 19, 20, 25, 33, 35, 49, 50, 54, 55, 74] as well as nonlocal integro-differential systems [10, 18, 26, 34] were derived and proposed to deepen the understanding, validate and create new experimental hypothesis. furthermore, to capture various heterotypic aspects received by the editors 15 october 2020; accepted 11 march 2021; published online 15 march 2021. 2000 mathematics subject classification. 65m60, 35k57, 35q92. key words and phrases. cancer invasion; multiscale modelling; level set; fem; cut-cells. avpb was funded by the heidelberg graduate school of mathematical and computational methods for the sciences (hgs mathcomp), founded by dfg grant gsc 220 in the german universities excellence initiative. 32 a level set approach for a multi-scale cancer invasion model 33 and related processes within tumour invasion, several multiphase models based on the theory of mixtures [14, 17, 21, 31, 56, 61, 69, 77, 78] were derived (by exploring the mass and momentum balances as well as the inner multiphase constitutive laws). a particularly important role in cancer invasion is played by the mdes (such as the mmps) that are secreted from the outer proliferating rim and released within the tumour peritumoural microenvironment. this gives the cancer invasion a moving boundary character, and to that end several level set approaches were recently proposed to study the tumour progression both in homogeneous environments [32, 43, 44, 45, 80] and in complex heterogeneous tissues [46]. despite recent advances, the multiscale modelling of the processes involved in cancer invasion remains an open problem. although this is a truly multiscale process, most mathematical models were offering a one-scale perspective, whether that is from a purely macro scale (tissue scale) or an exclusively micro scale (cell scale) stand point. however, recently a novel 2d multiscale moving boundary modelling platform for cancer invasion was proposed in [73]. this explores in an integrated manner the tissuescale cell population dynamics and relevant cell scale molecular mechanics together with the permanent link between these two biological scales. this addresses directly the dynamics of the mdes proteolytic processes occurring at the tumour boundary (i.e., at the invasive edge of the tumour) that are sourced from within outer proliferating rim of the tumour and facilitate the complex molecular transport and ecm degradation within the peritumoural region. the tissue-scale progression of tumour morphology is captured here in a multiscale moving boundary approach where the contribution arriving from the cell scale activity to the cancer invasion pattern is realised by the micro scale mdes dynamics (occurring along the tumour invasive edge), which, for its part, is induced by the cancer macro-dynamics. this was recently applied to the extended context in which, rather than the mmps dynamics, the upa is considered as the proteolytic system, and has led to biologically relevant results [53]. in this work we present a new formulation of the link that connects the two scales of the tumour dynamics that was considered in the initial multiscale moving boundary modelling approach presented in [73] as well as in the multiscale cancer invasion modelling developments that followed [4, 5, 6, 71, 52, 62, 63, 64, 65, 67, 72]. in particular, the movement of the tumor boundary is here defined by a velocity field instead of a displacement of the interface. therefore, the moving interface is defined and implemented in a different way. furtheremore, the discretization is based on finite elements instead of finite differences as in the previous work. the new model is based on a level set approach in which the moving domain is defined as the zero level of a level set function. the reason for this choice of the problem setting is twofold: on one hand, all components of the model can be described by partial differential equations at the continuum level allowing the complete separation between modelling and discretization; on the other hand, it is better suited for an extension to the three-dimensional case since the formulation of all components of the model is dimension independent and the use of a dimension independent implementation of the discretization, like in our case using the finite element method (fem) package deal.ii [11], facilitates the realization of the code. the level set method was first introduced in [51] for tracking moving interface with complex deformations. this method was developed starting from the notion of weak solutions for evolving interfaces. the main aspect of this method is that an interface or a domain boundary is defined through the embedding of the interface as the zero level set of a higher dimensional function. furthermore, the velocity of the interface is also embedded to the higher dimensional function. we avoid handling a sharp interface, i.e. a lower dimensional manifold in the computational domain, but the velocity needs to be extended from the interface to the rest of the domain. while the original setting, with the sharp interface, poses several numerical difficulties due to its lagrangian approach, the later setting, using an eulerian approach, can exploit techniques developed for hyperbolic problems. 34 t. carraro, s.e. wetterauer, a.v. ponce bobadilla, and d. trucu other works have presented a level set approach for moving the tumour interface. in [43] the authors use a level set function to define the boundary of a tumour mass and extend the velocity orthogonally to the interface using a filter technique to damp numerical noise coming from the extension procedure. the velocity at the interface is defined as a function of the gradient of a computed quantity (the pressure). this work nevertheless does not link different model scales. in [80] an adaptive finite element combined with a level set approach is used to solve a model that considers tumour necrosis, neo-vascularization and tissue invasion. the model is composed of a continuum part and a hybrid continuum-discrete part. the velocity of the interface is the cell velocity. therefore, the velocity does not need to be extended into the neighbourhood of the interface. in [47] a level set approach with a ghost-cell method is applied to tumour growth of glioblastioma. the velocity of the interface depends on solutions of linear and nonlinear equations with curvature-dependent boundary conditions. since the velocity is only defined at the interface the authors extend it beyond the interface and use a narrow band/local level technique to update the interface velocity and level set function only in the vicinity of the interface. our approach uses a level set method with an extension of the velocity. while the coupling between the macroscopic and microscopic scales was originally introduced in [73] considering a lagrangian approach to move the nodes of the discrete approximation of the interface, we introduce here a continuous link between the two scales defined by the velocity at the interface at the continuum level. this changes the formulation of the multiscale coupling that goes with the definition of the velocity starting from heuristic arguments. the scope of this work is to present as a whole the new formulation of the tumour invasion model explaining the possible advantages that this approach can have for future developments and the numerical aspects that need further attention and further development. the paper is organized as follows. in section 1 we state the problem setting and describe the different components of the model: the macroscopic and microscopic components and the description of the moving boundary. in section 3.2 we introduce the weak formulation of the model which is needed for the approximation of the continuum problem with a finite element method. we introduce the discretization of the problem using cut-cells for the approximation of the cancer region domain. in section 4 we present some numerical results showing the interplay of the different parts of the multiscale model. finally we present an outlook and some concluding remarks in section 5. 2. the two-scale tumour dynamics we present a two-scale model for cancer invasion that involves a double feedback loop to link the dynamics occurring at two different spatial scales explored by the following two modelling components: a macroscopic component describing the population of cancer cells and extracellular matrix at tissue-scale and a microscopic component describing the dynamics of a generic matrix-degrading enzyme molecular population whose cell-scale takes place at the leading edge of the tumour. both scales are considered at the continuum level, and we assume that possible stochastic effects (in regions where the continuum assumption is not valid) can be neglected. nevertheless, besides the establishment of a new approach to linking the scales, our aim includes also the derivation of a flexible numerical framework that would allow to extend the model with a stochastic part (e.g. at the interface of the domain) that would require to a hybrid formulation. the cancer cells population mixed with the extracellular matrix density exercises its macro-scale interacting dynamics within a tumour invading domain ω(t) that changes its size and morphology in time during the invasion process within a reference tissue domain y . its boundary ∂ω(t) will also be referred to as interface because this represents the tumour interface separating the healthy region with zero cancer cells from the region with a distribution of cancer cells. a level set approach for a multi-scale cancer invasion model 35 assumption 2.1 (scale separation). we assume here scale separation in space and time between the tissue-scale tumour macro-dynamics (involving cancer cells population mixed with ecm) and the cell-scale molecular boundary micro-dynamics (involving matrix degrading enzymes). the model is of two-scale nature, since, due to the different physical dimensions of tissue and tumour constituents that are involved in the process (such as of cells, ecm, and matrix degrading enzymes), cancer invasion is genuinely a multiscale biological process [75], which in its most basic setting involves a cancer cell population mixed with ecm density (for the macroscopic part) and a population of matrix degrading enzymes (for the microscopic part) that have their dynamics occurring at separated spatial scales (namely, at macroand microscale, respectively). hence, the natural assumption of scale separation enables us to derive the multiscale model and to address the dynamics at each of the two scales on correspondingly different scale domains. finally, the characteristic length l for the macroscopic part of the model relates to the diameter of the cancer region and is considered here as in [8, 34], ranging between 0.1cm and 1.0cm. furthermore, the characteristic length ` of the microscopic part related to the region where the matrix degrading enzymes are spatially transported is considered to be of the order of 10−3cm, [48]. the ratio between the scales is denoted ε = `/l, and so in the non-dimensional setting of the model we propose here ε will represent the cell-scale size (i.e., the micro-scale size). during the macro-dynamics, the cancer cels from the outer proliferating rim secrete matrix degrading enzymes (such as mmps), providing this way a source of mdes along the leading edge of the tumour [36, 75], i.e., in the cell-scale proximity of the tumour boundary. once secreted, these mdes cell-scale exercise a diffusive transport across the tumour interface in the peritumoural region, causing degradation of the ecm that they meet in the immediate proximity of the tumour and this way enabling further tumour progression [36, 75]. proceeding in a similar manner as in [73], the boundary micro-dynamics of the matrix degrading enzymes population is explored here on a bundle of micro-domains εy , of cell-scale size � > 0, whose union provides a cell-scale neighbourhood for the interfacial points in ∂ω(t). using the scale separation, we explore the micro-dynamics within a cell-scale neighbourhood of each of the boundary points x ∈ ∂ω(t) (i.e., at each point of the macroscopic interface) that is enabled by acorresponding εy micro-domain centred at x. in brief, adopting here a multiscale modelling perspective similar to the one proposed in [4, 5, 6, 71, 52, 62, 63, 64, 65, 67, 72, 73], the coupling of two-scale dynamics of cancer invasion is captured as follows: • the “top-down” macroscopic-to-microscopic coupling is realised via the source of the matrixdegrading enzymes, which is induced by the tumour macro-dynamics and is formed as a collective contribution of the cancer cells that arrive during the macro-dynamics within an appropriate distance from any boundary point x ∈ ∂ω(t), see equation (2.8); • the “bottom-up” microscopic-to-macroscopic coupling determines the movement of the macroscopic tumour interface that is induced by the dynamics of the micro-spatial distribution of mdes within the cell-scale neighbourhood of the tumour interface (enabled by an appropriate union of boundary micro-domains, see figure 1) in the form of a velocity field (see equation (2.10)) that drives a transport process for boundary relocation at macro-scale (see equation (2.6)) . the rate of cancer cells invasion into the surrounding tissues is therefore driven by the velocity of the interface that depends on the boundary mde micro-dynamics microscopic enzyme dynamics. furthermore, the tumour interface is described here by the zero-level of a level set function and its spatial dynamics is governed by a transport equation. to account for all interactions between the different parts of the problem, the tissue macro-domain y is assumed to be sufficiently large such that the complete dynamics happen inside it. 36 t. carraro, s.e. wetterauer, a.v. ponce bobadilla, and d. trucu in the following we proceed with the multiscale model description in three parts: macroscopic, microscopic and a transport process for the tumour boundary which is induced by the micro-scale. focussing first on the macro-scale, at this level the model considers cancer cells and extracellular matrix (ecm) dynamic interaction as well as a micro-scale-induced transport process that will ultimately describe the movement of the macro-scale tumour boundary. 2.1. tumour macro-dynamics. let c(x,t) and v(x,t) denote the cancer and the extracellular matrix distributions at (x,t) ∈ ω(t) × (0,t), respectively. proceeding as in [73], the dynamics at macroscopic scale is given by the following pde system: ∂c ∂t = random motility︷ ︸︸ ︷ d1∆c − haptotaxis︷ ︸︸ ︷ η∇· (c∇v) + proliferation︷ ︸︸ ︷ µ1 c (1 − c−v), (2.1) ∂v ∂t = − αcv︸︷︷︸ degradation + µ2(1 − c−v)︸ ︷︷ ︸ ecm remodelling , (2.2) with boundary condition (d1∇c−ηc∇v) ·n = 0 on ∂ω × (0,t). (2.3) and initial conditions c(x, 0) = c0(x) on ω(0) (2.4) v(x, 0) = c0(x) on ω(0). (2.5) where d1 is the diffusion coefficient for the cancer cells, η is the advection coefficient, µ1 is the proliferation coefficient, α a degradation coefficient and µ2 a coefficient for the remodelling of ecm. all these coefficients are considered constant and their typical values are included in table 2. it is assumed that the cancer cells are zero outside ω(t) and that there is no transport of cells through the boundary ∂ω(t), see boundary condition (2.3). furthermore, under the presence of these boundary and initial conditions, for the case of constant proliferation rate µ1, the results in [68, 76, 37] explore the local and global existence of system (2.1)-(2.2). the initial distribution of cancer cells c0(x) and extracellular matrix v0(x) are given in the larger domain y . in the next section we introduce the part of the model used to update the tumour domain in time. 2.2. two-scale tumour boundary movement: the macro-scale transport process induced by the mdes boundary micro-dynamics. as mentioned above, the movement of the tumour interface is directly governed by the matrix degrading enzymes (mde) dynamics occurring in a cell scale neighbourhood of the tumour interface ∂ω(t). the pattern of degradation of the peritumoural ecm by the advancing front of mdes drives the invasion of the tumour cells in the surrounding tissues and determines the movement of the tumour boundary ∂ω(t). therefore, the movement of the timedependent macro domain ω(t) is enabled by a velocity field defined on the points of the interface x ∈ ∂ω(t), which is determined by the micro-dynamics occurring on a small micro-domain εy centred at x. hence, the velocity field generated in this way is induced directly by the micro-dynamics of the mde molecular distribution m(y,τ) over a suitable micro-spatio-temporal domain εy ×(0, ∆t) (which will be detailed in section 2.3). we denote this velocity field by v (m). since v (m) is defined only on points at the interface, we consider an extension of the velocity to the whole domain y . this allows us to describe the cancer region boundary by a level set approach. the interface is defined as the zero-level of the level set function φ which satisfies the following transport equation: ∂φ ∂t + v (m) ·∇φ = 0, in y × (0,t). (2.6) a level set approach for a multi-scale cancer invasion model 37 for later purposes, we introduce the notation l0(t) = {x ∈ y : φ(x,t) = 0} (2.7) for the zero level of the level set function that defines the interface ∂ω(t). a natural extension of the velocity is the constant continuation of the velocity at the boundary in normal direction [23]. in section 3.6 more details about this point are given. 2.3. mdes boundary micro-dynamics. due to the scale separation assumption 2.1 we can describe the micro-dynamics of the mdes within a cell-scale neighbourhood of the tumour interface by exploring this on a bundle of micro-domains εy defined and centred at each macroscopic interface point x ∈ ∂ω(t). assuming for convenience that the maximal domain y is centred at origin of the space, the micro scale coordinates y of the micro-scale problem on a εy centred at x0 are obtained by an appropriate scaling and translation of y (given by the transformation y = x0 + ε(x−x0)), see figure 1 for an illustration of these micro-domains on the ∂ω(t). as argued in [73], collectively, the cancer cells that get to be point on the macroscopic tumour boundary microscopic domains �y ∂ω(t) �y �y figure 1. sketch of micro-dynamics sampling at the macroscopic tumour boundary ∂ω(t) distributed during their dynamics within a certain radius rm > 0 from the interface x ∈ ∂ω(t) secrete matrix degrading enzymes, giving rise this way to a source of mdes within each boundary microdomain εy . this micro-scale source mdes (which is induced by the macro-dynamics) can therefore be formalised mathematically as fx,t(c)(y) :=   1 |b| ∫ b c(ξ,t) dξ y ∈ εy ∩ ω(t), 0 otherwise. (2.8) where b := {ξ ∈ y : ‖ξ−x‖≤ rm}. therefore, in the presence of source (2.8), the cross-interface microdynamics of mde molecular distribution m(y,τ), which takes place on the boundary micro-domain εy over a time interval (0, ∆t), is governed by the following reaction diffusion equation ∂m ∂t (y,τ) = d2∆m(y,τ) + fx,t(c) in εy × (0, ∆t), m(y, 0) = 0 in εy, ∂m ∂n (y,τ) = 0 in ∂εy × (0, ∆t), (2.9) with ∆t> 0 representing here the micro scale time perspective and serving also later on as natural time interval for the coupling between the microscopic and macroscopic stages of the multiscale model. 38 t. carraro, s.e. wetterauer, a.v. ponce bobadilla, and d. trucu therefore, the pattern of peritumoural ecm degradation caused by the advancing fronts of mde molecules (which are transported across the tumour interface in the immediate proximity within the appropriate microscale region) gives rise to a boundary velocity that can be described by v (m) = cvel ∆t |εy | ∫ ∆t 0 ∫ εy m∇m dydτ, (2.10) where |εy | = ∫ εy 1 dt and cvel is a tuning scaling factor, see table 2. specifically, this form of v is based on the following main considerations: • the term ∇m takes into consideration the assumption that the cancer boundary moves following the gradient with respect to the mde; • further, by multiplying it by m(y,τ), we are taking into account the influence of the amount of enzymes over their given gradient direction at each spatio-temporal micro-node (y,τ), enabling an appropriate weighting of its “strength” (magnitude); • finally, by considering the average contribution of mde microdynamics over εy × [0, ∆t ] by simply accounting upon the mean-value in time of the revolving weighted mde gradient spatial direction [0, ∆t] 3 τ 7→ 1 |εy | ∫ εy m∇m dy, we ultimately obtain the definition of the velocity given in (2.10), where v (m) is taken as being proportional to this spatio-temporal mean value, with proportion constant cvel > 0. 2.4. schematic summary of our multiscale moving boundary modelling. the new model that we introduced here falls in the class of heterogeneous multiscale models that were developed over the past two decades not only for multiscale biological processes but also for other multiscale processes arising in material science or fluid-structure interactions [1, 4, 5, 6, 22, 71, 27, 28, 30, 52, 62, 63, 64, 65, 67, 72, 73]. schematically, the two-scale dynamics of our cancer invasion model is coupled across the scales as depicted in figure 2, and its progression can be summarised in the following three steps: (1) at a time t∗> 0, the macro-scale cancer distribution (i.e., the solution of the macro-dynamics) on the domain ω(t∗) induces in a non-local manner the enzymatic source for the mdes boundary micro-dynamics. (2) the boundary micro-dynamics is explored on each boundary micro-domain �y . the time-space average of the microscale mdes spatio-temporal distribution over the micro-domain �y and a fixed but arbitrarily small time range of size ∆t> 0 is used to determine pointwise the tumour interface velocity, which will ultimately result in describing the direction and displacement magnitude of the macroscopic tumour boundary relocation. (3) the interfacial velocity obtained from the boundary mdes micro-dynamics is then set into a transport equation that finally determines the the position of the tumour boundary at time t∗ + ∆t . therefore, a new tumour macro-domain ω(t∗ + ∆t) is defined, and this becomes the new playground for the cancer macro-dynamics, which continues now its evolution on this newly expanded domain, progressing again through the stages described in (1)-(3). 3. multiscale computational approach 3.1. definition of the computational microscopic problem. we consider the microscopic problem in a bundle of boundary microdomains εy , with y := (y1,y2) being the standard local microscale reference system within a given εy and τ denoting the time at micro scale. as will be explained below in section 3.5, in our finite element approach the macroscopic dynamics will be considered on an appropriately defined macroscopic domain ωh(t) with a linearized boundary a level set approach for a multi-scale cancer invasion model 39 boundary dynamics b o tt o m − u p to p − d o w n transport process for tumor macro−dynamics macro−dynamics mde micro−dynamics φ(t∗ + ∆t)φ(t∗) ω(t∗) ω(t∗ + ∆t) figure 2. sketch of the coupling (including the top-down and bottom-up links) between the macroand microdynamics of our two-scale moving boundary model for tumour invasion. ∂ωh(t). the microscopic dynamics is then explored within a microdomain εy centred at a macro scale boundary point x ∈ ∂ωh(t) and eventually appropriately rotated so that this is positioned with two edges parallel to the linearized boundary (in direction y1) and two edges orthogonal to it (in direction y2) as shown in figure 1. this simplifies the setting of the microscopic problem. in fact, since we consider the linearized boundary ∂ωh(t), the right hand side fx,t in equation (2.9) does not depend on y1. furthermore, we would like to note that, due to scale separation, the quantity m does not depend on the macroscopic variable x. nevertheless, since each microscopic domain is centred at a different point x on the boundary, a potentially different mdes micro-source is induced by the macro-scale for each micro-dynamics on each microscopic domain �y . the value of cancer cells concentration c in one single microscopic domain is constant because (due to scale separation) no oscillations in y are considered for c. in addition, since on the boundaries of the quadrilateral domain no flux conditions are prescribed, it follows that the solution is constant in y1 direction. in fact, it is straightforward to show that the 1d solution is also solution of the 2d problem. due to uniqueness of both problems this constant property is given. therefore, we can consider the following simplified one-dimensional microscopic problem for the quantity m, which is the integral of m along y1 (giving the amount of enzyme molecules per unit of length) ∂m ∂τ (y2,τ) = d2∆m(y2,τ) + fx,t(y2) in (0,εl) × (0, ∆t), m(y2, 0) = 0 in (0,εl), ∂m ∂n (y2,τ) = 0 in ∂(0,εl) × (0, ∆t), (3.1) where fx,t is fx,t integrated over y1. since fx,t and m do not depend on y1, the solution m of (2.9) is the constant extension of the solution m of (3.1) in y1 direction. we introduce now a scaling of the domain to the interval (0, 1) through the following tranformation y2 = εlz with z ∈ (0, 1), 40 t. carraro, s.e. wetterauer, a.v. ponce bobadilla, and d. trucu then we get after the rescaling the transformed system ∂m̂ ∂t (z,τ) = d2ε −2l−2∆m̂(z,τ) + f̂x,t(z) in (0, 1) × (0, ∆t), m̂(z, 0) = 0 in (0, 1), ∂m̂ ∂n (z,τ) = 0 in ∂(0, 1) × (0, ∆t), (3.2) with f̂x,t(z) :=   1 |b| ∫ b c(ξ,t) dξ z ∈ [0, 1/2], 0 otherwise, (3.3) note that the coordinate ξ is a macroscopic quantity. notice furthermore that a solution of (3.1) is a solution of (3.2) by m̂(z,τ) = m(εz,τ). remark 3.1 (limit ε → 0). in case of ε → 0 we have in (3.2) a large diffusion coefficient, therefore a fast redistribution process of the solution occurs, leading to negligible spatial variations of the solution. the only relevant parameter of the problem at the limit becomes the time. the limit problem becomes an ordinary differential equation (ode). even if we consider scale separation in this model, we do not consider the limiting case ε → 0. in that case the velocity has to be defined in a different way since the term ∇m becomes the zero vector. the parameter ε in our model has always a finite value bounded below ε ≥ ε cell > 0, where ε cell is assumed here to be a minimal microscale size of the order of a cell length. thus, using (3.2), we obtain that the velocity can be further expressed as v (m) = cvel ∆tε2 ∫ [0,ε]2 ∫ ∆t 0 m∇ym dτ dy = cvel ∆tε2 ε ∫ ε 0 ∫ ∆t 0 m∇ym dy dτ, = cvel ∆t ε ∫ 1 0 ∫ ∆t 0 m̂∇zm̂ dz dτ, (3.4) where m̂ indicates the transformed function on the unit domain. 3.2. weak formulation of the two-scale tumour invasion model. to describe the model in the setting needed for the fem we introduce the following weak formulation. we use the notation (·, ·) to define the usual l2 scalar product of lebesgue square integrable functions. the space h1 is the hilbert space of square integrable functions with square integrable (weak) first derivative and h∗ is its dual space, i.e. the space of bounded linear functional on h1. furthermore, we use bochner spaces of the form u = {u ∈ l2(0,t; h1) : ∂tu ∈ l2(0,t; h∗)} to introduce the weak formulation for the dynamics at each of the two scales. specifically, we have the following: • at macro-scale: – for the weak formulation of the dynamics of the cancer cells population (2.1) as well as for the dynamics of the ecm (2.2), we consider the space um = {u ∈ l2(0,t; h1(ω(t))) : ∂tu ∈ l2(0,t; h∗(ω(t)))} – for the boundary transport process (2.6), we consider the space ut = {φ ∈ l2(0,t; h1(y )) : ∂tφ ∈ l2(0,t; h∗(y ))}; • at micro-scale, for the mdes micro-dynamics (3.2), we consider: um = {m ∈ l2(0, ∆t ; h1((0, 1))) : ∂τm ∈ l2(0, ∆t; h∗((0, 1)))} a level set approach for a multi-scale cancer invasion model 41 thus, at macro-scale the weak formulation for the tumour macro-dynamics (2.1)-(2.2) is as follows: find (c,v) ∈ um ×um such that, for almost all t ∈ (0,t), it satisfies(∂c ∂t ,ϕ ) + ( d1∇c,∇ϕ ) − ( ηc∇v,∇ϕ ) − ( µ1(v) c(1 − c−v),ϕ ) = 0 ∀ϕ ∈ h1(ω(t)), (3.5a)(∂v ∂t ,ϕ ) + ( αcv,ϕ ) − ( µ2(1 − c−v),ϕ ) = 0 ∀ϕ ∈ h1(ω(t)), (3.5b) c(x, 0) = c0 in ω(0), (3.5c) v(x, 0) = v0 in ω(0). (3.5d) note that the initial conditions for cancer cells, c0, and ecm distributions, v0, are defined on the larger maximal domain y and that this formulation implies the natural zero flux condition for the cancer cells and ecm, i.e. ∂nc = ∂nv = 0 on ∂ω(t). similarly, the weak formulation of the macro-scale transport equation for interface dynamics (2.6) is: find φ ∈ ut such that, for almost all t ∈ (0,t) it satisfies(∂φ ∂t ,ϕ ) + ( v (m) ·∇φ,ϕ ) = 0 ∀ϕ ∈ h1(y ), (3.6a) φ(x, 0) = φ0 in y, (3.6b) with φ0 being the level set function at the initial time. using the notation ϕ+ := max{ϕ, 0} to define the positive part of a function ϕ, we have that supp(φ+0 ) describes the initial support region of the cancer cells. finally, at micro-scale, the weak formulation of the mde micro-dynamics associated with each of the boundary micro-domains �y is: find m̂ ∈ um such that for almost all τ ∈ (0, ∆t) it satisfies(∂m̂ ∂τ ,ϕ ) + ( d2ε −2l−2∇m̂,∇ϕ ) = ( f̂x,t,ϕ ) ∀ϕ ∈ h1((0, 1)), (3.7a) m̂(z, 0) = 0 in (0, 1), (3.7b) where f̂x,t is defined as in (3.3) and the natural condition ∂nm̂ = 0 is implicitly defined. 3.3. discretization. the model is first discretized in time by the implicit euler method and then discretized in space by the fem. the discretized system is defined on a regular mesh mh composed of quadrilateral cells k ∈ mh of the same dimension. this mesh has the advantage that it can be generated starting from an initial square that is successively refined to achieve a given cell diameter. in this work we use global refinement to generate the mesh. since the macroscopic domain ω(t) is time-dependent, the discrete space domain would need to be remeshed at every time step, if a fitted fem formulation is used. in case of large deformations, the procedure of remeshing has to deal with the possible loss of shape regularity of the mesh. to avoid these complications related to remeshing, we use an unfitted approach by using so called cut-cells. these are a special realization of the fem as described below. in particular, these are finite elements with shape functions with a support on a subdomain of the cells that is defined by the intersection of the interface with the cells. let us consider the space of bi-linear polynomials q1 defined on a unit cell k̂ = [0, 1] 2, i.e. q1 = span(1,x,y,xy) and the space of linear functions p1 = span(1,x) 42 t. carraro, s.e. wetterauer, a.v. ponce bobadilla, and d. trucu defined on the one-dimensional unit cell k̂ = [0, 1]. the finite element space is defined as uhm (t) = {u ∈ c(ω(t)) :u|k ◦tk ∈ q1 if k ∩∂ω(t) = ∅; u|k∩ω(t) = ψ|ω(t), ψ ◦tk ∈ q1 if k ∩∂ω(t) 6= ∅}, where tk is a bijective transformation from the unit cell k̂ to the physical cell k. since the mesh is non-fitted, the shape functions u (in case of cut-cells) are defined only on the portion of cell that is intersected by ω(t), while outside of ω(t) they need not to be defined. in figure 3 the restriction of two shape functions ϕ1 and ϕ2 on the cut-cell is shown. for the transport and microscopic problems 0 1cut−cell ϕ1 ϕ2 figure 3. shape functions on a unit cut-cell we use the following spaces uht = {u ∈ c(y ) : u|k ◦tk ∈ q1} and uhm = {u ∈ c((0, 1)) : u|k ◦tk ∈ p1}. 3.4. approximation of the solution on the moving domain. in every time step of the time discretization scheme the domains at tn and tn+1, ω(tn) and ω(tn+1), are defined by the level set function at the two times tn and tn+1. at the n th time step the solution is known only in ω(tn). to determine the solution at tn+1 the variation of the domain in time should be taken into account in the formulation of the problem. one possible formulation of the problem would be to consider a reference domain ω(t0) for a given t0 in each time step and to use a (time dependent) mapping to transform the solution from the domain ω(tn) to the reference domain ω(t0) and then to the domain ω(tn+1). however, if the time step is small enough the combination of these two transformations can be approximated with the identity and, instead of implementing a complicated formulation, an extension of the solution uh(tn) from the old domain ω(tn) to the new domain ω(tn+1) could be used. this procedure introduces an approximation error that for small enough time steps can be neglected in comparison to other sources of error (such as space-time discretization of the solution, or splitting error etc.). in the following, we describe the extension used in this work and advice that this should be defined properly depending on the finite element ansatz used. a level set approach for a multi-scale cancer invasion model 43 0 1st cut 2nd cut extension 1 tn tn+1 figure 4. one dimensional sketch of the extension of the macroscopic solution in case of a cell cut twice 0 01st cut 1 1 extension 2nd cut tn+1tn figure 5. one dimensional sketch of the extension of the macroscopic solution in case the interface cuts two neighbour cells at tn and tn+1. following the above construction, the fully discrete formulation of the macroscopic problem becomes( cn+1h ,ϕ ) ω(tn+1) + k [( d1∇cn+1h ,∇ϕ ) ω(tn+1) − ( η cn+1h ∇vh,∇ϕ ) ω(tn+1) − ( µ1(v n+1 h ) c n+1 h (1 − c n+1 h −v n+1 h ),ϕ ) ω(tn+1) ] = ( c̃nh,ϕ ) ω(tn+1) , ∀ϕ ∈ uhm (t n+1), (3.8a) ( vn+1h ,ϕ ) ω(tn+1) + k [( αcn+1h v n+1 h ,ϕ ) ω(tn+1) − ( µ2 (1 − cn+1h −v n+1 h ),ϕ ) ω(tn+1) ] = ( ṽnh,ϕ ) ω(tn+1) , ∀ϕ ∈ uhm (t n+1), (3.8b) where k = tn+1 −tn is the macroscopic time step, c̃nh and ṽ n h are the extensions from ω(tn) to ω(tn+1). in fact, the two components ch(tn) and vh(tn) are defined only in ω(tn). therefore an extension in the region ω(tn+1)\ω(tn) needs to be defined. we have chosen a continuous extension using the prescribed values c0(x) and v0(x) for x ∈ y . in particular, we have considered two cases: case (i) the cell where we need to define the extension is cut at time tn and at time tn+1, see figure 4 and case (ii) the cell is cut at time tn and uncut at time tn+1, see figure 5. in case (ii) the cut goes to the neighbour cell at time tn+1. in case (i) both components are extended up to the new cut using the values of all degrees of freedom of the considered cell (also those lying outside the domain ω(t)) with a bilinear nodal interpolation. note that the bilinear nodal interpolation is justified only within the domain ω(t) in the cut-cell formulation of the problem, because the integrals in the weak formulation are computed only in the inner part of the cells. in this sense, we are “extrapolating” the values ch and vh outside the region of validity of the finite element interpolation. it is known that the problem formulation using cut-cells can lead to instabilities due to the lost of coercivity outside of the physical domain ω(tn). to overcome to this 44 t. carraro, s.e. wetterauer, a.v. ponce bobadilla, and d. trucu problem special stabilization techniques can be used, e.g. the ghost penalty method [12]. to implement stabilization techniques extra boundary integrals have to be implemented increasing the complexity of the code. we have therefore decided to use a heuristic solution. we have set a threshold of 1% on the volume to be considered for extrapolation. cells, whose volume is cut by 99%, are eliminated from the mesh. the value of 1% has been chosen after testing different values and we have seen that also with larger values we got the same boundary, meaning that this threshold is accurate enough. in fact, neglecting the contribution of small cells can be interpreted as a quadrature error. by keeping small the threshold for the suppression, we keep small this quadrature error. stabilization of the haptotaxis term. due to the large difference between the diffusion coefficient d1 and the haptotactic coefficient η, see table 2, the macroscopic problem is transport dominant and therefore must be numerically stabilized. typically, upwind techniques are used in the finite element framework to stabilize the calculations [41]; see, for example, [66] for an application of these techniques to a chemotaxis problem. in this work, we have chosen a streamline diffusion stabilization [81] that adds an artificial diffusion term only in the direction of the ecm distribution gradient. hence, the weak formulation (3.5a) of the macroscopic problem becomes(∂c ∂t ,ϕ ) + ( d1∇c,∇ϕ ) − ( ηc∇v,∇ϕ ) + δc ( η∇v ·∇c,∇v ·∇ϕ ) − ( µ1 c(1 − c−v),ϕ ) = 0, (3.9) with a stabilization parameter δc that has been heuristically choosen testing a range of values until the spurious oscillations has been reduced on the choosen refinenemnt level of the computational meshes. the transport equation is defined on y . since this is a hyperbolic equation, a suitable discretization is needed. here we have chosen the streamline diffusion approach for its easy implementation and good performance. we have used an artificial diffusion in the streamline direction scaled with a parameter δ > 0, whose value can be found in table 2. therefore, the discretisation of the weak formulation (3.6) for the transport equation is given by( φn+1h ,ϕ ) y + k ( v nh ·∇φ n+1 h ,ϕ + δ (v n h ·∇ϕ) ) y = ( φnh,ϕ ) y ∀ϕ ∈ uht , φ0h = φ0 ∀x ∈ y, (3.10) where φ0 is the initial level set function, k the time step and v n h is the discrete velocity defined as v nh := cvel ∆t ε ix ( iτ ( m̂h,n∇m̂h,n )) , (3.11) where ix and iτ are two quadrature formulas for the approximation of the integral in space and time (see expression (3.4)) and m̂h,n is the discrete solution of the microscopic problem as defined below. since we assume scale separation, in each point of the macroscopic boundary ∂ω(t) we need to solve a microscopic problem that defines the local velocity. in the discrete version, we define the microscopic problem in a finite number of points at the interface and we discuss later the issue of how to use these pointwise defined velocities to solve the transport problem. then the discretisation of weak formulation of the microscopic problem (3.7) reads:( m̂l+1h,n,ϕ ) + kτ ( d2ε −2l−2∇m̂l+1h,n,∇ϕ ) = ( m̂lh,n,ϕ ) + ( f̂x,n,h,ϕ ) ∀ϕ ∈ uhm, m̂0h,n = 0 in (0, 1), (3.12) where kτ = τn+1 − τn is the time step and m̂lh,n indicates the discrete microscopic solution for the macroscopic step n (note that the right hand side depends on the macroscopic solution at time tn), with l being the time step of the time variable τ, i.e. m̂lh,n = m̂h,n(τl). the term f̂x,n,h is an approximation of f̂x,tn in which c is substituted by its discrete counterpart and the integral over b is a level set approach for a multi-scale cancer invasion model 45 approximated by a quadrature rule f̂x,n,h(z) :=   ib(c n h) ib(1) z ∈ [0, 1/2], 0 otherwise, (3.13) where ib(·) is a quadrature rule that approximates the integral of the argument over b ib(f) ≈ ∫ b f(ξ) dξ. 3.5. cut-cells finite element approach in approximating the macroscopic tumour interface. as introduced previously, we discretize in time the tumour macro-dynamics and the transport equation for the tumour boundary relocation. therefore, in the semidiscrete formulation we have terms that are defined at time t = tn+1 and terms defined at time t = tn. since the domain is time dependent, the integrals of these terms are defined on different domains. therefore, in each time step we need to consider two configurations defined by the position of the boundary ∂ω(t) in two subsequent time steps. in particular, we have to consider the case in which the boundary cuts the same finite element cell in both time steps, see in figure 6a the cell at the bottom left, and the case in which the boundary cuts one cell at time tn and it goes over to the neighbour cells at time tn+1 leaving the previous cell uncut at time tn+1, see in figure 6a the cell at bottom right. for the discretized version of the system of equations, we consider the linearized domain ωh(t), which is defined by the piece-wise linear boundary ∂ωh(t) := l0,h(t), (3.14) where the linearized zero level l0,h(t) is defined by the polygonal line that connects all intersections of the zero level l0(t) with the mesh cells boundaries shown in figure 6b. cell cut twice ωh(tn+1) ωh(tn) (a) ∂ωh(t) ∂ωh(t) ∂ω(t) (b) figure 6. left: sketch of the linearized zero level l0,h. right: cell cut twice by the interface at two subsequent time steps. using the linearized boundary ∂ωh(t) we can apply the quadrature rule described in [16] to integrate the terms of the model on cut-cells with a single cut. furthermore, if a cell is cut twice, i.e. at time tn and at time tn+1, we apply the previous quadrature rule recursively. 46 t. carraro, s.e. wetterauer, a.v. ponce bobadilla, and d. trucu approximation of the nonlocal term. the nonlocal term (3.13) is approximated by a quadrature rule. we use a further level set function to define the distance from the macroscopic point x, which determines the domain of integration b. also in this case we introduce a piece-wise linear approximation of this level set function and integrate the cells using the quadrature rule only for the cell portion contained in b. 3.6. extension of the micro-scale induced velocity. the velocity induced by the boundary mde micro-dynamics is determined using formula (3.4). it is computed at the macroscopic point x on the linearized boundary ∂ωh(t) and then extended to the rest of the domain ωh(t). in this work, we consider only one point x per finite element cell. this is taken at the midpoint of the segment of the interface l0,h that intersects the cell, as shown in figure 1. we set the velocity computed at this point x to all cells which center lies at the closest distance from x. therefore, the velocity is approximated as a piecewise constant function. for finite element cells that lie in the cancer region, i.e. k ∩ ωh(t) 6= 0, at a distance larger than a prescribed radius of influence ρ> 0, we set the velocity to zero. this enables us to avoid the transport of numerical pollution from the center of the domain due to the singularity of the level set in the point that we take as reference to compute the distance function. this definition of the velocity extension can lead to regularity problems in the transport of the interface if two parts of the boundary approach each other. this happens because the velocity of cells that lie at the same distance from the two approaching boundary parts is not well defined. this is a typical problem in level set approaches that can be overcome using a fast marching method [2]. 3.7. overall numerical solution process. we sketch the overall solution process underlying the coupling between the different parts of the model. algorithm 1 overall solution process 1: set n = 0 and choose the splitting time step ∆t 2: set φ0(x), c0(x) and v0(x) in y 3: define l0(t n) as in (2.7) and linearize it to get ωh(t n) 4: solve macroscopic part (3.8) for (x,t) ∈ ωh(tn) × (tn, tn + ∆t) 5: compute f̂x,n,h(z), see (3.13) 6: solve microscopic part (3.12) for (x,τ) ∈ εy × (0, ∆t) 7: compute velocity v nh , see (3.11) 8: extend velocity on all y 9: solve transport problem (2.6) for (x,t) ∈ y × (tn, tn + ∆t) 10: if tn + ∆t = t then 11: stop 12: else 13: set tn = tn + ∆t 14: goto 3 the macroscopic system is solved with an implicit euler scheme. at each time step a nonlinear system of the type (3.8a-3.8b) has to be solved. we use an exact jacobian and no damping for the newton method, which converges generally in 2 steps to an accuracy lower than 10−6. the linear system arising in each newton step is solved by the direct solver umfpack [24]. the system (3.10) is a linear system and is computationally much cheaper than the macroscopic problem. the direct solver is used in every time step. finally, the microscopic problem (3.12) is a linear one dimensional parabolic problem solved with an implicit euler method and the direct solver in each time step. a level set approach for a multi-scale cancer invasion model 47 4. numerical results in this section we show some numerical results obtained with the numerical method explained above. we have used the following initial condition for cancer cells c0(x) =   r−‖(x1,x2) − (2, 2)‖2 10 r if ‖(x1,x2) − (2, 2)‖2 < r, 0 otherwise, (4.1) (4.2) where r is the initial radius of the cancer region and the point (2, 2) is the center of the computational domain, see table 2 for the numerical parameters. for the extracellular matrix distribution we consider two cases: • case 1: v0(x) = 0.3 sin (3π‖(x1,x2) − (0, 0)‖3) + 0.5. • case 2: v0(x) = 0.15 (sin (2π‖(x1,x2) − (0, 0)‖3) + sin (2π‖(x1,x2) − (4, 0)‖3)) + 0.75. in figure 7 are shown the results for case 1. in figures 7a, 7c and 7e is depicted the distribution of cancer cells at time t = 0, t = 5 and t = 10 respectively, while in figures 7b, 7d and 7f is depicted the ecm distribution at the corresponding time points. the initial tumour mass is located at a valley of the ecm distribution. the white circle in figure 7b shows the initial cancer cells mass position with respect to the ecm distribution. the coordinates of the center of the initial distribution are in table 1. we observe that the cancer cells population haptotaxis biases the macro-dynamics of cancer cells towards regions with higher ecm levels, which in turn leads to the formation of stronger mdes sources for the micro-process taking place on the boundary micro-domains εy situated on the part of the tumour interface facing those elevated ecm regions, leading to a stronger mdes boundary micro-dynamics that ultimately translates into a larger magnitude velocity that is induced from the micro-scale and is fed into the transport equation which governs the tumour boundary movement, that results into progression of the tumour in those regions. this is a truly multi-scale characteristic of the actual cancer invasion process that our model is able to capture, resulting in this pronounced lobular progression of the tumour. it can be observed at time t = 5 that the cancer cells distribution moves towards the maximum of ecm, and then it follows the path along the two ecm ridges besides the starting position, which confirms in silico the well-known process of durotaxis observed experimentally [38, 42, 57, 59, 60, 70, 79]. in figure 8 are shown the results of case 2. in the figures 8a, 8c and 8e is depicted the distribution of cancer cells at time t = 0, t = 5 and t = 10 respectively, while in figures 8b, 8d and 8f is depicted the ecm distribution at the corresponding time points. also here the initial cacer cells distribution is set at the valley of the ecm distribution, see table 1. in this case, in the vicinity of the starting position there are four peaks of ecm distribution and the haptotactic term induces a transport of cancer cells distribution towards all four peaks in a nonsymmetric manner, because the initial position is not equidistant to the peaks. therefore, the results in this figure with changed surrounding tissue structure (induced by the different ecm initial pattern) confirm the same multiscale character of the tumour progression that our multiscale moving boundary model is able to capture. again here, we observe this pronounced lobular progression underscoring again the durotaxis behaviour observed experimentally for the cancer cell migration. the behavior of the model is consistent with the expected behavior, in which an interplay of the microscopic and macroscopic processes determine the transport of the cancer cell distribution in a nonuniform environment with a nonconstant distribution of the ecm. specifically, it can be observed that the haptotactic movement of the cancer cell distribution at macro-scale leads higher levels of sources of mdes that the macro-dynamics induces at micro-scale for the boundary proteolytic processes, which 48 t. carraro, s.e. wetterauer, a.v. ponce bobadilla, and d. trucu (a) (b) (c) (d) (e) (f) figure 7. case 1: distribution of c2ncer cells at time t=0 (7a), t=5 (7c) and t=10 (7e). distribution of ecm at time t=0 (7b), t=5 (7d) and t=10 (7f). in turn induces a movement of the macroscopic tumourinterface progression in pronounced lobular manner toward the higher values of the ecm distribution. a level set approach for a multi-scale cancer invasion model 49 (a) (b) (c) (d) (e) (f) figure 8. case 2: distribution of cancer cells at time t=0 (7a), t=5 (7c) and t=10 (7e). distribution of ecm at time t=0 (7b), t=5 (7d) and t=10 (7f). 50 t. carraro, s.e. wetterauer, a.v. ponce bobadilla, and d. trucu final time t 10 microdynamics time range ∆t 0.1 initial radius of cancer distribution r 0.3 scale factor ε 0.01 diffusion cancer cells d1 0.00043 haptotactic coefficient η 0.2 proliferation coefficient µ1 0.25 ecm remodelling µ2 0.15 degradation α 1.5 diffusion mde d2 0.001 center of init. cancer distr. case 1 (2.3,2.2) center of init. cancer distr. case 2 (2.0,1.9) table 1. model parameters in our numerical experiments time step k 0.1 mesh size h 0.0078125 stream-line stabilization transport problem δ 0.5 stream-line stabilization macroscopic problem δc 0.004 total tissue domain y (0, 4) × (0, 4) table 2. details of the numerical setting 5. conclusions we presented a new formulation of a two-scale model for the simulation of cancer invasion. this included a new derivation of the tumor boundary motion law by considering the contributions of mde micro-dynamics within a transport equation (2.6) (via the velocity field v (m) that is induced by the micro-scale mdes processes at tumour interface), the solution of which provides the level set function indicating the cancer boundary progression on the macro scale. for the computational implementation we used an unfitted regular mesh with uniform cell diameters and a cut-cell finite element formulation to avoid the problem of re-meshing in case of large deformations. we have shown that the presented framework is highly flexible to study different aspects of the cancer invasion process. in particular, since it is important to study the interplay between the two scales, the presented 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[81] g. zhou, how accurate is the streamline diffusion finite element method?, mathematics of computation 66 (1997), 31–44. 54 t. carraro, s.e. wetterauer, a.v. ponce bobadilla, and d. trucu corresponding author. faculty of mechanical engineering, applied mathematics, helmut schmidt university / university of the federal armed forces hamburg, holstenhofweg 85, 22043 hamburg, germany. e-mail address: carraro@hsu-hh.de institute for applied mathematics, heidelberg university, im neuenheimerfeld 205, 69120 heidelberg, germany. institute for applied mathematics, heidelberg university, im neuenheimerfeld 205, 69120 heidelberg, germany. division of mathematics, university of dundee, dundee, dd1 4hn, united kingdom. e-mail address: trucu@maths.dundee.ac.uk mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 3, number 2, 2022, pp.106-118 https://doi.org/10.5206/mase/14625 modeling road surface potholes within the macroscopic flow framework gabriel o. fosu, joseph m. opong, bright e. owusu, and samuel m. naandam abstract. the continual wearing of road surfaces results to crack and holes called potholes. these road surface irregularities often elongate travel time. in this paper, a second-order macroscopic traffic model is therefore proposed to account for these road surface irregularities that affect the smooth flow of vehicular traffic. though potholes do vary in shape and size, for simplicity the paper assumes that all potholes have conic resemblances. the impact of different sized potholes on driving is experimented using fundamental diagrams. besides, the width of these holes, driver reaction time amid these irregularities also determine the intensity of the flow rate and vehicular speed. moreover, a local cluster analysis is performed to determine the effect of a small disturbance on flow. the results revealed that the magnitude of amplification on a road surface with larger cracks is not as severe as roads with smaller size holes, except at minimal and jam density where all amplifications quickly fade out. 1. introduction traffic flow models have evolved from simple microscopic to complex macroscopic and mesoscopic models. macroscopic models are either of first-order or second-order. the model by [22, 30] is known to be the inaugural offshoot of the first-order macroscopic branch. the second-order type was introduced by payne[25], and whitham[36], to fill the modeling gap of the first-order equation. the classical payne’s model was not invariant with respect to direction. thus, in the early 2000s some second-order models were developed to address the backward wave traveling phenomena of payne’s model [4, 16, 29, 38]. subsequently, extensive work has been done in relation to second-order macroscopic traffic models [9, 12, 19, 31]. all second-order macroscopic traffic models have followed payne’s conceptualization. an aspect of traffic modeling is the impact of road surface conditions on vehicular flow. some authors have developed a number of mathematical models to explain this connective conjecture [7, 32, 33]. in [7], the authors modeled the effect of best and worst road conditions on flow using explicit equations; one equation to model best road condition and the other to explain the worst road condition. in another paper[33] the authors developed a car-following model with consideration of varying road condition based on empirical data. an advantage of the model was the ability to qualitatively describe the effect of road condition on the micro driving behavior. a similar model was formulated using macroscopic properties [32], while [14] proposed a model to describe the effect of road, institutional and weather conditions on the traffic flow. this model used the data-driven approach. another observable road surface phenomena not yet accounted for within the macroscopic modeling domain is potholes. potholes are significant feature of many parts of the highway network in thirdworld countries. beside cause driving discomfort, its adverse effect is not limited to sudden deflation received by the editors 24 january 2022; accepted 5 may 2022; published online 15 may 2022. 2010 mathematics subject classification. 35q35, 76m20. key words and phrases. higher-order traffic models, upwind finite difference, lwr model, bad roads, cluster effect, speed-density equation. 106 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14625 modeling road surface potholes 107 of pneumatic tires, speed fluctuations cause passenger jerking, and steering misalignment [21]. these potholes can be a nightmare for road-users. they are usually formed as a direct result of snow, ice, and prolonged periods of rain. potholes frequently begin as undetectable microscopic cracks in the road surface. they are created when water penetrates these tiny cracks on the road-cracks that are usually caused by traffic. when this water penetrates, it expands and widens the cracks as traffic thumps down on these spaces. extreme weather conditions, poor drainage, and heavy traffic can all cause that surface to loosen and wear away. moreover, this deterioration causes unnecessary delays in traffic flow, distorts pavement aesthetics, and also causes road traffic accidents [23]. even though potholes are observable road characteristic, the relation between potholes and macroscopic traffic characteristics is still a grey research area. it is imperative to know that literature on potholes is related to its detection [3, 6, 8, 15] but not to its modeling as is the intent of this research. therefore, the paper seeks to derive a new second-order macroscopic model that accounts for these road surface irregularities. the classical second-order macroscopic model is extended to characterize pothole effect. the subsequent sections are organized as follows. a new macroscopic traffic model is derived in section two to consider vehicular road potholes. in sections three, the model is subjected to realistic data values to examine the effect of these holes on the fundamental plot. following-up with section four, we explore how a small disturbance propagates on the road with diverse irregularities. the final section presents a general summary of the entire work. 2. second-order pothole model the two main branches of macroscopic traffic models are first-order and second-order classes of models. second-order models are oftentimes referred to as higher-order models. macroscopic models have a continuous property with respect to space and time as opposed to discrete models. macroscopic models aggregate its quantities and possess continuous feature along both temporal and spatial axes. the three main aggregate variables required for macroscopic analysis are traffic density k(x,t), average speed u(x,t), and traffic flow rate q(x,t). two classical traffic equations relate these variables; the hydrodynamic equation, and the continuity or lwr equation [1, 11]. the hydrodynamic equation is expressed mathematically as: q(x,t) = k(x,t) ·u(x,t) (2.1) the second relation, called the lwr, is also expressed as: kt + qx = 0 → kt + (ku)x = 0 (2.2) in reducing the boredom of continual writing the bracketed spatial and temporal notations (x,t), it is often omitted as in the case of equation (2.2), and will follow suit in all other subsequent formulations. the continuity equation is famous because of its simplicity and realistic efficiency. compared to other macroscopic models, the lwr is known to be simplest based on the parsimony principle. however, the lwr model is handicapped for its inability to explain non-equilibrium traffic. this shortfall leads to the development of higher-order models. higher-order models supplement the continuity equation with a dynamic velocity equation. a classical speed-gradient second-order model is of the form[16]: kt + (ku)x = 0 (2.3) ut + uux = u(ke) −u τ + coux (2.4) u(ke) is the steady state speed, τ is the relaxation time, and co is the propagation speed of a small disturbance. 108 g. o. fosu, j. m. opong, b. e. owusu, and s. m. naandam as emphasized in the introduction, an additional source term is to be added to the dynamic velocity equation to characterize the effect of potholes on vehicular traffic. from [13, 26, 27], a road surface irregularity r(θ) can be generated using equation (2.5). r(θ) =  − α 2 ( 1 − cos 2πθ β ) , 0 ≤ θ ≤ β 0, θ < 0,θ > b (2.5) where α is the depth of the pothole; α < 0 corresponds to bulge road surface. β > 0 is the width of the pothole, θ is any location within the pothole measured starting from the leftmost endpoint. unique depth and width values will result in unique pothole sizes. figures 1 and 2 show different sizes of potholes differing by width and depth. 0 0.5 1 1.5 2 2.5 3 3.5 pothole width -0.15 -0.1 -0.05 0 po th ol e d ep th width=0.50 width=1.00 width=1.50 width=2.00 width=2.50 width=3.00 width=3.50 figure 1. different pothole sizes with variable depth 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 pothole width -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 p ot ho le d ep th depth = 0.00 depth = 0.05 depth=0.10 depth=0.15 depth= 0.20 depth=0.25 depth=0.30 figure 2. different pothole sizes with variable depth figure 1 shows seven different potholes with a width ranging from 0.5 meters to 3.5 meters, having a constant depth of 0.15 meters. potholes with absurd widths (β > 1.5) are characteristic of feeder roads. in figure 2 are potholes having fixed width of two meters and a depth ranging from zero to 0.30 meters. zero depth denotes a normal/smooth road surface devoid of any form of cracks or holes. it could be observed that figures 1 and 2 have close similitude to a hollow cone. this assumption is associated with some margin of error considering each hole as a microscopic entity. nonetheless, the errors may wane or extinct within the macroscopic framework due to variable aggregation. therefore, to quantize these road surface irregularity, all potholes are assumed to be cone-like, though with a shorter vertical height. on this premise, a given pothole size (sp) can be approximated using equation (2.6). sp = − 1 2 πβ √ β2 4 + α2 (2.6) modeling road surface potholes 109 as stated earlier, road surface irregularity is known to have some impact on vehicular flow, but the intensity of this impact depends much on the density of the traffic as well. for highly dense traffic, the effect of potholes is not so substantial compared to fairly moving traffic. the speed changes resulting from some road surface irregularity on a near empty road vast exceed that of dense traffic. this to say that the overall effect of these irregularities on flow during jam traffic may be insignificant. this analogy is incorporated in the derivation, yielding the modified form of (2.6) as: tp =  − 1 2 πβt √( β2 4 + α2 )( 1 − k kcrit ) , k ≤ kcrit 0, otherwise (2.7) kcrit denote the critical density, and t is the driver physiological reaction time. the latter accounts for the varying response time among different type of drivers/vehicles. aggressive drivers have little respect for potholes and will often maintain their speed driving through such irregularities, hence they are associated with shorter reaction time. directly opposite to this group of drivers are the sluggish type who will slow down irrespective of the hole size. moreover, vehicles with weaker shock absorbers often slow down driving through potholes. these classes of drivers/vehicles are associated with a greater t value. equation (2.7) is appended to the dynamic velocity equation to yield the new second-order macroscopic pothole traffic model. the proposed equation is expressed as kt + (ku)x = 0 (2.8) ut + uux = u(ke) −u τ + coux − 1 2 πβt √( β2 4 + α2 )( 1 − k kcrit ) (2.9) equation (2.8) is the usual continuity equation, while (2.9) is the modified dynamic speed equation. in the next section, the second-order pothole model is subjected to assumed realistic data values to examine the effect of different sized potholes on vehicular traffic. 3. fundamental diagrams fundamental plots are graphical representation of the pairwise relationship between speed, density, and flow. some graphical illustrations are presented in this section to exemplify the effect of potholes on these macroscopic variables. for uniform flow, traffic characteristics, specifically density and speed, are invariant with respect to space and time. hence, equation (2.9) reduce to: u(ke) −uu τ − 1 2 πβt √( β2 4 + α2 )( 1 − ku kcrit ) = 0 (3.1) uniform speed is denoted by uu, while ku represents uniform density. this equation (3.1) is again simplified as: uu = u(ke) − 1 2 πβtτ √( β2 4 + α2 )( 1 − ku kcrit ) (3.2) an equation for steady-state speed u(ke) is required for this qualitative analysis. several of these equations have been presented in [10]. an appropriate choice among these speed-density models for this analysis is the exponential kk model (3.3) [17]. u(ke) = umax [ 1/ { 1 + exp ( k kmax − 0.25 0.06 )} − 372 × 10−8 ] (3.3) 110 g. o. fosu, j. m. opong, b. e. owusu, and s. m. naandam where umax denote the maximum attainable speed on a freeway, and kmax is the maximum density at jam traffic. the assumed parameter values presented in table 1 are used for the computations. these are chosen to be coherent with realistic traffic flow [18, 34]. table 1. simulation parameters name notation value maximum speed umax 25m/s relaxation time τ 1s critical density kcrit 0.38veh/m jam density kmax 1.0veh/m reaction time t [0.5 6]s these values are adopted to exemplify the pairwise relationship between speed-density and flowdensity, taken into account different pothole widths and reaction times. in relation to sizes, potholes are categorized under three broad domains; namely small, medium, and large potholes. potholes with a width of less than 0.5 meters is classified as small pothole. the width of large pothole category exceed 1.4 meters. sandwich between the smaller and larger categories are the medium type of potholes. similarly, drivers are also grouped into three based on their reaction time. it is assumed that aggressive drivers have a reaction time of 0.5 seconds, while the reaction time for sluggish drivers is 6 seconds. any time seconds within the neighborhood of the mean times of aggressive and sluggish drivers correspond to normal driving. 0 0.2 0.4 0.6 0.8 1 density 0 5 10 15 20 25 s p e e d width = 0.00 width = 0.30 width = 0.60 width = 0.90 width = 1.20 width = 1.50 0 0.2 0.4 0.6 0.8 1 density 0 0.5 1 1.5 2 2.5 3 3.5 f lo w width = 0.00 width = 0.30 width = 0.60 width = 0.90 width = 1.20 width = 1.50 figure 3. impact of a pothole on speed-density plot (left), and flow-density plot (right) with t = 0.5s figure 3 exemplifies the fundamental plot of an aggressive driver for varying pothole sizes. these potholes have a constant depth of 0.15 meters, with varying width from zero to 1.5 meters. starting the simulation from regular road surface (width=0.00), all subsequent simulations have width increment of 0.3 meters. though not apparent, it can be observed that at lower densities, each simulation plot has its corresponding speed value, and then merges after the critical point. for a near empty highway, aggressive drivers decrease their speed slightly below the maximum when driving through these potholes. the speed drop is totally insignificant for driving through holes with a width of 0.3 meters. we only see aggressive drivers slowing down driving through larger potholes. likewise, from the flow-density plot, the effect of aggressive driving on flow rate is again not so remarkable. flow rate is seen to drop from 3.5veh/s to 3.3999veh/s as the width is widened from zero to 1.5 modeling road surface potholes 111 meters. the flow rate on roads with these simulated surface irregularities is practically the same if all drivers drive aggressively. 0 0.2 0.4 0.6 0.8 1 density 0 5 10 15 20 25 s p e e d width = 0.00 width = 0.30 width = 0.60 width = 0.90 width = 1.20 width = 1.50 0 0.2 0.4 0.6 0.8 1 density 0 0.5 1 1.5 2 2.5 3 3.5 f lo w width = 0.00 width = 0.30 width = 0.60 width = 0.90 width = 1.20 width = 1.50 figure 4. impact of a pothole on speed-density plot (left), and flow-density plot (right) with t = 6s figure 4 is related to the case of sluggish driving. the speed drop is clearly evidential in this case. bigger portholes cause sluggish drivers to reduce their speed to the neighborhood 15m/s, but, maintains a speed limit between [20m/s, 25m/s] driving through small and medium potholes. from an empty road to the critical density, the sizes of these road surface irregularities have a significant effect on vehicular speed within the domain of sluggish driving. however, the speed remains the same during jam traffic. this exhibit a practical exemplification of driving on the road with these irregularities. similarly, the flow rate is also influenced by these irregularities. from figure 4, flow is observed to increase, attain maximum value, and then decline. flow rate is totally dependent on the width of these potholes. a clear case for regular driving is shown in figure 5 with an average reaction time of three seconds. this class of drivers are neither not too fast as aggressive nor too slow as sluggish. except for width equal 1.5 meters, they could still attain a speed value greater than 20m/s driving through the simulated potholes. this has a corresponding effect on the flow-density diagram. 0 0.2 0.4 0.6 0.8 1 density 0 5 10 15 20 25 s p e e d width = 0.00 width = 0.30 width = 0.60 width = 0.90 width = 1.20 width = 1.50 0 0.2 0.4 0.6 0.8 1 density 0 0.5 1 1.5 2 2.5 3 3.5 f lo w width = 0.00 width = 0.30 width = 0.60 width = 0.90 width = 1.20 width = 1.50 figure 5. impact of a pothole on speed-density plot (left), and flow-density plot (right) with t = 3s figure 6 shows the fundamental diagrams for road having potholes depths 0.05 meters (left) and 0.3 meters (right). driver reaction time is assumed to be three seconds. it can be observed that the depth 112 g. o. fosu, j. m. opong, b. e. owusu, and s. m. naandam of potholes on vehicular speed is not as substantial as the width exemplification. from the simulation, it can be actualized that the width of potholes highly induces vehicular speed/flow compared to its depth. 0 0.2 0.4 0.6 0.8 1 density 0 5 10 15 20 25 s p e e d width = 0.00 width = 0.30 width = 0.60 width = 0.90 width = 1.20 width = 1.50 0 0.2 0.4 0.6 0.8 1 density 0 5 10 15 20 25 s p e e d width = 0.00 width = 0.30 width = 0.60 width = 0.90 width = 1.20 width = 1.50 figure 6. impact of a pothole on speed-density plot with depth equal 0.05 (left), and depth equal 0.25 (right) these road surface irregularities are characteristic of many roads in ghana, an hour journey could span a minimum of two hours owing to these road surface irregularities [24, 28, 35], hence the essence of this model. this corroborates that improving road infrastructure will enhance the efficiency of vehicular flow. 4. cluster effect the section investigates how a disturbance evolves on roads with differing surface irregularities. we inquire how a local perturbation propagates through space and time amidst such irregularities. following from [16, 32] the finite difference upwind scheme is used to discretize the continuity equation (2.8) and the modified dynamic velocity equation (2.9). first, the continuity equation is simplified as kt + ukx + kux = 0 to ease the discretization. hence, the discrete pothole model is given as: kn+1m = k n m + ∆t ∆x knm ( unm −u n m+1 ) + ∆t ∆x unm ( knm−1 −k n m ) (4.1) the dynamic speed equation has two discrete cases. if the wave disturbance speed is higher than the vehicular speed, then the model is expressed as un+1m = u n m + ∆t ∆x (co −unm) ( unm+1 −u n m ) + ∆t τ (u(knm) −u n m)− ∆t 2 πβt √( β2 4 + α2 )( 1 − knm kcrit ) (4.2) otherwise un+1m = u n m + ∆t ∆x (co −unm) ( unm −u n m−1 ) + ∆t τ (u(knm) −u n m)− ∆t 2 πβt √( β2 4 + α2 )( 1 − knm kcrit ) (4.3) where u(knm) has the form u(knm) = umax [ 1/ { 1 + exp ( knm kmax − 0.25 0.06 )} − 372 × 10−8 ] (4.4) modeling road surface potholes 113 the simulation is subjected to a periodic boundary condition of the form k(l,t) = k(0, t) and u(l,t) = u(0, t) where l is the total length of roadway. the initial density is given by the wave profile (4.5) [17]. k(x, 0) = ko + ∆ko { cosh−2 [ 160 l ( x− 5l 16 )] − 1 4 cosh−2 [ 40 l ( x− 11l 32 )]} (4.5) with v(x, 0) = f(k(x, 0)). ko is initial constant density, and ∆ko is the amplitude of the displacement. the additional parameter values required for these wave amplification are detailed as follows. ∆ko is chosen as 0.20veh/m, while ko will range between [0.10veh/m, 0.42veh/m]. a road length l = 1km is equally divided into 100 grid points. the simulation time is 10 minutes with the time step of 0.1 seconds. the wave disturbance speed co = 9m/s, the relaxation time τ = 3s, and the reaction time t = 1s. kmax,kcrit, and umax remain the same as in table 1. potholes will have the following specific dimensions as in table 2. table 2. pothole groupings pothole type width depth small 0.4m 0.1m medium 1.4m 0.2m large 2.4m 0.3m the following commentaries are based on the graphical presentations 7-9. the amplifications on the road with smaller potholes is presented with figure 7. these amplifications quickly dies-off when the initial density is within the neighborhood of its endpoints. this is clearly evinced when ko = 0.10 and ko = 0.42. these amplifications increase with an increasing number of vehicles on the stretch and then begin to flatten after reaching a critical density value. there are pieces of evidences of dipole and multiple clusters within 0.24veh/m and 0.28veh/m. these are associated with stop-and-go traffic. a similar effect could be realized considering the medium size and larger size potholes. these illustrations are presented in figures 8 and 9, respectively. again, the disruption dissolve quickly at the two extreme density values; ko = 0.10 and ko = 0.42. this implies that the effect of this disturbance does not propagate through space and time for near-empty traffic or close to jam density traffic. for medium densities, the amplification for large potholed road is not as severe as smaller once. the severity of road surface irregularities may cause drivers to slow down first [2, 5, 20, 37], hence reducing the net effect of any disturbance. comparatively, the amplification for large sized potholed roads is quite stable. in summary, this new second-order macroscopic pothole traffic model is useful for examining the dynamics of vehicular flow on roads with cracks and holes. 5. conclusion this study applies mathematical models to describe road surface potholes within the macroscopic flow framework. potholes are created by the continual wearing of road surfaces resulting in cracks and holes. these irregularities have not been explicitly modeled within the ambit of macroscopic models. therefore, we proposed a second-order macroscopic equation to explore the effect of these irregularities of vehicular traffic. to ease the computations, ro ad surface irregularities were categorized into three broad domains (small, medium, and large), and also assumed to be conical. a qualitative solution to the model was presented to illustrate the effect of these irregularities on the speed-density and flow-density plots. from the fundamental diagrams, speed was observed to reduce drastically, driving through these 114 g. o. fosu, j. m. opong, b. e. owusu, and s. m. naandam figure 7. space-time evolution of density on small size pothole road. the initial density profiles are given as ko = 0.10(top left), ko = 0.18(top right), ko = 0.24(middle left), ko = 0.28(middle right), ko = 0.35(bottom left), ko = 0.42(bottom right) potholes as in the case of sluggish driving. the differences in the flow rate were not evident, as in the case of a platoon of aggressive drivers. however, all vehicles-driver type were observed to possess the same speed and flow characteristics beyond a critical density value. showing a clear conceptualization of realistic flow. a further simulation was carried out to determine the effect of a small perturbation on uniform traffic. what is often referred to as the local cluster effect. the amplifications were observed to be negativity related to pothole sizes but within specific density values. the degree of the disturbance increases as the holes become smaller. however, the disruption quickly dies-off when there are either few vehicles on the stretch or close to jam density traffic. these simulations suggest that the proposed model is potent in exploring diverse traffic phenomena on roads with potholes. the possible extension is to analyze road surface irregularity microscopically. modeling road surface potholes 115 figure 8. space-time evolution of density on medium size pothole road. the initial density profiles are given as ko = 0.10(top 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[38] h. m. zhang, a non-equilibrium traffic model devoid of gas-like behavior, transp. res. b: methodol. 36 (2002), 275–290. corresponding author, department of mathematics, kwame nkrumah university of science and technology, kumasi, ghana email address: gabriel.of@knust.edu.gh department of mathematics, presbyterian university college, abetifi, ghana email address: joeopong@presbyuniversity.edu.gh department of mathematics, kwame nkrumah university of science and technology, kumasi, ghana email address: bright.owusu@knust.edu.gh department of mathematics, university of cape coast, cape coast, ghana email address: snaandam@ucc.edu.gh 1. introduction 2. second-order pothole model 3. fundamental diagrams 4. cluster effect 5. conclusion references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 3, number 3, 2022, pp.190-199 https://doi.org/10.5206/mase/15074 the existence and uniqueness of solutions of a nonlinear toxin-dependent size-structured population model yan li and qihua huang abstract. in this paper, we study a toxin-mediated size-structured population model with nonlinear reproduction, growth, and mortality rates. by using the characteristic method and the contraction mapping argument, we establish the existence-uniqueness of solutions to the model. we also prove the continuous dependence of solutions on initial conditions. 1. introduction how do anthropogenic and natural environmental toxins affect population dynamics and ecological integrity? it is an essential question in environmental toxicology [1, 11]. mathematical models (including individual-based models, matrix population models, ordinary differential equation models, and so on) have been widely applied to address this question [4, 5, 6, 7, 9]. in terms of the fact that in a population, individuals of different sizes may have different sensitivities to toxins, huang and wang [8] developed a size-structured population model for a population living in an aquatic polluted ecosystem, which is given by the following system of nonlinear first-order hyperbolic equations:  ut + (g(x,p(t))u)x + µ(x,p(t),v(x,t))u = 0, x ∈ (xmin,xmax), t > 0, vt + g(x,p(t))vx + σ(x,t)v −a(x,t)e(t) = 0, x ∈ (xmin,xmax), t > 0, g(xmin,p(t))u(xmin, t) = ∫ xmax xmin β(x,p(t),v(x,t))u(x,t)dx, t > 0, v(xmin, t) = 0, t > 0, u(x, 0) = u0(x), x ∈ (xmin,xmax), v(x, 0) = v0(x), x ∈ (xmin,xmax) (1.1) where u(x,t) represents the density of individuals of size x at time t; p(t) = ∫xmax xmin u(x,t)dx is the total population biomass at time t, where xmin and xmax denote the minimize size and the maximum size of the population, respectively; v(x,t) denotes the size-dependent body burden — concentration of toxin per unit population biomass. the function g(x,p(t)) represents the growth rate of an individual of size x which depends on the total population biomass due to competition for resources. the function received by the editors 7 july 2022; accepted 26 september 2022; published online 29 september 2022. 2010 mathematics subject classification. 35l60; 35f15; 92f99. key words and phrases. existence-uniqueness; continuous dependence; size-structured population; characteristic method; contraction mapping theorem. yan li is supported by the natural science foundation of shandong province, china (no.s zr2021ma028, zr2021ma025). qihua huang is supported by the national natural science foundation of china (no. 11871060), the venture and innovation support program for chongqing overseas returnees (no. 7820100158). 190 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15074 the existence and uniqueness of solutions 191 µ(x,p(t),v(x,t)) denotes the mortality rate of an individual of size x which depend on the total population biomass and the body burden. the function β(x,p(t),v(x,t)) is the reproduction rate of an individual of size x. the function σ(x,t) is the toxin elimination rate due to the metabolic process of the population. the function a(x,t) represents the toxin uptake rate by the population from the environment. the function e(t) is the concentration of toxin in the environment at time t. see [8] for detailed model derivation. in [8], an explicit finite difference approximation to partial differential equation problem (1.1) was developed. the existence and uniqueness of the weak solution — a solution in integral form with two test functions — were established and convergence of the finite difference approximation to this unique weak solution was proved. the main purpose of this paper is to prove the existence-uniqueness of solutions of problem (1.1) by using the characteristic method and contraction mapping theorem, and the continuous dependence on initial conditions, which are quite different from the method of numerical approximation used in (1.1). 2. existence and uniqueness results throughout the discussion, we let ω1 = (xmin,xmax)×(0,∞) and ω2 = (xmin,xmax)×(0,∞)×(0,∞). we make the following assumptions on the parameters in problem (1.1): (h1) g(x,p) is a strictly positive lipschitz function with respect to x and p in ω1 with a common lipschitz constant lg. (h2) µ(x,p,v) is a nonnegative lipschitz function with respect to x,p, and v in ω2 with a common lipschitz constant lµ. (h3) β(x,p,v) is a nonnegative lipschitz function with respect to x,p,v in ω2 with a common lipschitz constant lβ. furthermore, β(x,p,v) is uniformly bounded in ω2 with 0 ≤ β ≤ βm . (h4) a(x,t) is a nonnegative lipschitz function with respect to x in ω1 with a lipschitz constant la. furthermore, a(x,t) is uniformly bounded in ω1 with 0 ≤ a ≤ am . (h5) σ(x,t) is a nonnegative lipschitz function with respect to x and t in ω1 with a lipschitz constant lσ. (h6) e(t) is a nonnegative continuous function and bounded for 0 < t < ∞ with 0 ≤ e(t) ≤ em . (h7) u0(x) ∈ l1(xmin,xmax) and u0(x) ≥ 0. (h8) v0(x) is a nonnegative lipschitz function with a lipschitz constant lv and bounded for xmin < x < xmax with 0 ≤ v0(x) ≤ vm . we begin with the definition of the solutions of problem (1.1). definition 2.1. a nonnegative function (u(x,t),v(x,t)) on [xmin,xmax] × [0,t), with u(x,t) and v(x,t) integrable, is a solution of (1.1) if p(t) = ∫xmax xmin u(x,t)dx is a continuous function on [0,t) and (u(x,t),v(x,t)) satisfies (1.1)3,4,5,6 and the equations du(x,t) = −µ̃u(x,t), (2.1) dv(x,t) = −[σ(x,t)v −a(x,t)e(t)] (2.2) with du(x,t) = lim h→0 u(x(t + h; x,t), t + h) −u(x,t) h , dv(x,t) = lim h→0 v(x(t + h; x,t), t + h) −v(x,t) h , 192 yan li and qihua huang where t is a positive constant, µ̃(x,p(t),v(x,t)) = gx(x,p(t)) + µ(x,p(t),v(x,t)) and x(t; x0, t0) is the solution of the equation for the characteristic curves given by  dx dt = g(x,p(t)), x(t0) = x0. (2.3) from (h1), we know that the function x(t; x0, t0) is strictly increasing. thus a unique inverse function τ(x; x0, t0) exists. let z(t) = x(t; xmin, 0) be the characteristic through the point (xmin, 0). in what follows, we reduce problem (1.1) to a system of coupled equations for p(t) and b(t) by using the method of characteristics, where b(t) = ∫ xmax xmin β(x,p(t),v(x,t))u(x,t)dx. integrating (2.1) along the characteristics, we have u(x,t) =   u0(x(0; x,t))e − ∫ t 0 µ̃(x(s;x,t),p(s),v(x(s;x,t),s))ds, x ≥ z(t), b(τ(xmin)) g(xmin; p(τ(xmin))) e − ∫ t τ(xmin) µ̃(x(s;xmin,τ(xmin)),p(s),v(x(s;xmin,τ(xmin)),s))ds, x < z(t), (2.4) where τ(xmin) = τ(xmin; x,t). similarly, we have v(x,t) =   v0(x(0; x,t))e − ∫ t 0 σ(x(s;x,t),s)ds + ∫ t 0 a(x(s; x,t),s)e(s)e− ∫ t s σ(x(τ;x,t),τ)dτ ds, x ≥ z(t), 0, x < z(t). (2.5) then p(t) = ∫ xmax xmin u(x,t)dx = ∫ t 0 b(η)e − ∫ t η µ(x(s,xmin,η),p(s),v(x(s;xmin,η),s))dsdη + ∫ xmax xmin u0(ξ)e − ∫ t 0 µ(x(s;ξ,0),p(s),v(x(s;ξ,0)))dsdξ (2.6) and b(t) = ∫ t 0 β(x(t; xmin,η),p(t),v(x(t; xmin,η), t))b(η)e − ∫ t η µ(x(s,xmin,η),p(s),v(x(s;xmin,η),s))dsdη + ∫ xmax xmin β(x(t; xmin,ξ),p(t),v(x(t; xmin,ξ), t))u0(ξ)e − ∫ t 0 µ(x(s;ξ,0),p(s),v(x(s;ξ,0)))dsdξ. (2.7) if p(t) and b(t) are nonnegative continuous solutions of (2.6) and (2.7), then u(x,t) and v(x,t) defined by (2.4) and (2.5) respectively are the solutions of (1.1). on the other hand, if u(x,t) and v(x,t) are the solutions of (1.1), then p(t) and b(t) are nonnegative continuous solutions of (2.6) and (2.7). therefore, in order to obtain the existence and uniqueness results for problem (1.1), we only need to study the solvability of the system consisting of integral equations (2.6) and (2.7). by using the contraction mapping theorem, we first obtain the local existence and uniqueness results for problem (1.1). to this end, let st;k = {f ∈ c[0,t]|f(0) = ‖u0‖l1, 0 ≤ f(t) ≤ k, where k > ‖u0‖l1}, st;h = {f ∈ c[0,t]|0 ≤ f(t) ≤ h, where h > βm‖u0‖l1}. for each p ∈ st;k, the function x(t; x0, t0) is well-defined by the characteristic curve (2.3). thus, there is a unique function v(x,t) determined by (2.5). the existence and uniqueness of solutions 193 define the operator y : st;k × st;h → c[0,t] × c[0,t] by y (p,b) = (φ(p,b), ψ(p,b)) where φ(p,b) and ψ(p,b) are given by the right-hand sides of (2.6) and (2.7) respectively. then, a fixed point of the operator y corresponds to a solution of (2.6) and (2.7). next lemma establishes the existence and uniqueness of a fixed point of the operator y . lemma 2.1. suppose that hypotheses (h1)-(h8) hold. then there exists a value t > 0 for which y has a unique fixed point in st;k ×st;h ⊂ c[0,t] ×c[0,t]. proof. as mentioned above, we just need to show that y has a unique fixed point in st;k ×st;h. for any p,p̂ ∈ st,k, b,b̂ ∈ st,h, let u,û and v, v̂ be given by (2.4) and (2.5) corresponding to b,b̂ and p,p̂ , respectively. we use the following notations to simplify the expressions: µ(x p̂ (s; xmin,η), p̂(s), v̂(s,t)) = µp̂ , µ(xp(s; xmin,η),p(s),v(s,t)) = µp ; β(x p̂ (s; xmin,η), p̂, v̂) = βp̂ , β(xp(s; xmin,η),p,v) = βp ; µ(x p̂ (s; ξ, 0), p̂, v̂) = µ̄ p̂ , µp(x(s; ξ, 0),p,v) = µ̄p ; β p̂ (x(s; ξ, 0), p̂, v̂) = β̄ p̂ , βp(x(s; ξ, 0),p,v) = β̄p . in terms of (2.7), we can conclude that ψ(p,b)(t) ≤ βm ∫ t 0 b(η)dη + βm‖u0‖l1 ≤ βmht + βm‖u0‖l1 ≤ h. by a series of computations, we have |ψ(p,b)(t) − ψ(p̂,b̂)(t)| ≤ tβm‖b − b̂‖∞ + (βmhlµ + hlβ)t(|xp −xp̂ | + |p − p̂ | + |v − v̂|). since xp (t; xmin, 0) and xp̂ (t; xmin, 0) are the solutions of  dx dt = g(x,p(t)) x(0) = xmin and   dx dt = g(x,p̂(t)) x(0) = xmin respectively, we have that |xp −xp̂ | ≤ lg ∫ t 0 (|xp −xp̂ | + |p − p̂ |)ds. (2.8) gronwall’s inequality tells us that |xp −xp̂ | ≤ lgte lgt‖p − p̂‖∞. (2.9) similarly, we get |βp −βp̂ | ≤ lβ(|xp −xp̂ | + |p − p̂ | + |v − v̂|)ds, (2.10) |µp −µp̂ | ≤ lµ(|xp −xp̂ | + |p − p̂ | + |v − v̂|)ds, (2.11) 194 yan li and qihua huang |v − v̂| ≤ vm ∫ t 0 |σ(xp ) −σ(xp̂ )|ds + em ∫ t 0 a(xp ,s) ∫ t s |σ(xp ) −σ(xp̂ )|dsds + |v0(xp ) −v0(xp̂ )| + em ∫ t 0 |a(xp ,s) −a(xp̂ ,s)|ds ≤ (vmtlσ + lv + emamtlσ + emlat)|xp −xp̂ | ≤ (vmtlσ + lv + emamtlσ + emlat)lgtelgt‖p − p̂‖∞. (2.12) thus, |ψ(p,b)(t) − ψ(p̂,b̂)(t)| ≤ tβm‖b − b̂‖∞ + h1(t)t‖p − p̂‖∞, (2.13) where h1(t) = (βmhlµ + hlβ) t [ lgte lgt + 1 +(vmtlσ + lv + emamtlσ + emlat)lgte lgt ] . for the φ component, note that φ(p,b)(t) − φ(p̂,b̂)(t)) = ∫ t 0 (b(η) − b̂(η))e− ∫ t η µp dsdη + ∫ t 0 b̂(η)(e − ∫ t η µp ds −e− ∫ t η µ p̂ ds )dη + ∫ xmax xmin u0(ξ)(e − ∫ t 0 µ̄p ds −e− ∫ t 0 µ̄ p̂ ds)dξ ≤ ∫ t 0 |b(η) − b̂(η)|dη + ∫ t 0 b̂(η) ∫ t η |µp −µp̂ |dsdη + ∫ xmax xmin u0(ξ) ∫ t 0 |µ̄p − µ̄p̂ |dsdξ. (2.14) let f(η) = b(η) − b̂(η), by (2.7), we get |f(t)| ≤βm ∫ t 0 |f(η)|dη + ∫ t 0 b̂(η)|βp −βp̂ |dη + βm ∫ t 0 b̂(η)|µp −µp̂ |dη + ∫ xmax xmin u0(ξ)|β̄pe− ∫ t 0 µ̄p ds − β̄ p̂ e− ∫ t 0 µ̄ p̂ ds|dξ, (2.15) which leads to |f(t)| ≤ βm ∫ t 0 |f(η)|dη + ψ(t), where ψ(t) = ∫ t 0 b̂(η)|βp −βp̂ |dη + βm ∫ t 0 b̂(η)|µp −µp̂ |dη + ∫ xmax xmin u0(ξ)|β̄pe− ∫ t 0 µ̄p ds − β̄ p̂ e− ∫ t 0 µ̄ p̂ ds|dξ. (2.16) the existence and uniqueness of solutions 195 we also find that∣∣∣β̄pe−∫ t0 µ̄p ds − β̄p̂e−∫ t0 µ̄p̂ ds∣∣∣ =|(β̄p − β̄p̂)e−∫ t0 µ̄p ds + β̄p̂ (e−∫ t0 µ̄p ds −e−∫ t0 µ̄p̂ ds)| ≤ |β̄p − β̄p̂| + βm ∫ t 0 |µ̄p − µ̄p̂ |ds. (2.17) from the above analysis, we can conclude that ψ(t) ≤ [βm‖u0‖l1eβmt (lβ + βmlµ) + ‖u0‖l1 (lβ + βmlµt)]· [lge lgtt + 1 + (vmtlσ + lv + emamtlσ + emlat)e lgttlg]‖p − p̂‖∞ =: j(t)‖p − p̂‖∞. thus, |f(t)| ≤ βm ∫ t 0 |f(η)|dη + ψ(t) ≤ βm ∫ t 0 |f(η)|dη + j(t)‖p − p̂‖∞. by gronwall’s inequality, we have that |f(t)| ≤ j(t)‖p − p̂‖∞e ∫ t 0 βm dτ = j(t)‖p − p̂‖∞eβmt. therefore, |φ(p,b)(t) − φ(p̂,b̂)(t)| ≤ tj(t)‖p − p̂‖∞eβmt + (βm‖u0‖l1eβmtt + ‖u0‖l1 )t 2lµ‖p − p̂‖∞ (lge lgtt + 1 + (vmtlσ + lv + emamtlσ + emlat)tlge lgt ) =: th2(t)‖p − p̂‖∞. (2.18) combining (2.13) and (2.18), we obtain ||y (p,b) −y (p,b)|| = ||ψ(p,b) − ψ(p̂,b̂)|| + ||φ(p,b) − φ(p̂,b̂)|| ≤ (th1(t) + th2(t))||p − p̂ ||∞ + tβm||b − b̂||∞ = r(t)(||p − p̂ ||∞ + ||b − b̂||∞), (2.19) where r(t) = max{(th1(t) + th2(t)),tβm}. note that r(0) = 0. therefore, there exists a sufficiently small constant t > 0 such that r(t) ∈ (0, 1). hence, for such a small t , the mapping y is a contractive mapping. by the contracting mapping theorem, y has a fixed point. the proof is completed. � note that the uniqueness of the solution p(t) and b(t) of system (2.6)-(2.7) implies that the uniqueness of the solution to problem (1.1) because each u(x; t), v(x,t) given by (2.4) and (2.5) is uniquely determined by p(t) and b(t). thus, we have the following result on local existence and uniqueness to (1.1). theorem 2.2. suppose that hypotheses (h1)-(h8) hold. then there exists a value t > 0 such that problem (1.1) has a unique solution up to time t . in order to establish the global existence-uniqueness result for problem (1.1), we conclude the following upper bound on p(t) for t ∈ [0,t]. 196 yan li and qihua huang lemma 2.3. let u(x,t) and v(x,t) be a solution of (1.1) up to time t . then for t ∈ [0,t], p(t) satisfies the following bound p(t) ≤‖u0‖l1eβmt. proof. p(t) is differentiable since p(t) = ∫xmax xmin u(x,t)dx and u(x,t) is differentiable by definition 2.1. differentiating (2.6) with respect to t, we get p ′(t) = ∫ xmax xmin (β(x,p(t), ,v(x,t)) −µ(x,p(t), ,v(x,t)))u(x,t)dx ≤ βmp(t), gronwall’s inequality tells us that p(t) ≤‖u0‖l1eβmt. using similar arguments as in the proof of theorem 3 in [2], we are able to derive the following global existence-uniqueness result. � theorem 2.4. suppose that hypotheses (h1)-(h8) hold. then problem (1.1) has a unique solution for t ∈ [0,∞). 3. continuous dependence on initial conditions the purpose of this section is to establish the continuous dependence of solutions on initial conditions. for this purpose, we first show that the fixed point of the operator φ associated with an initial condition depends continuously on initial conditions. lemma 3.1. let p1(t) and p2(t) be the fixed points of (2.6) associated with initial conditions (u01,v01) and (u02,v02), respectively, then |p1(t) −p2(t)| ≤ eβmt 1 −l ‖u01 −u02‖l1, (3.1) where l is the contraction constant of the operator φ. proof. it is easy to see that |p1(t) −p2(t)| ≤ |p1(t) −p3(t)| + |p3(t) −p2(t)|, (3.2) where p3(t) = ∫ t 0 b3(η)e − ∫ t η µ(x2(s,xmin,η),p2(s),v2(x2(s;xmin,η),s))dsdη + ∫ xmax xmin u01(ξ)e − ∫ t 0 µ(x2(s;ξ,0),p2(s),v2(x2(s;ξ,0),s))dsdξ and b3(t) =∫ t 0 β(x2(t; xmin,η),p2(t),v2(x2(t; xmin,η), t))b3(η)e − ∫ t η µ(x2(s,xmin,η),p2(s),v2(x2(s;xmin,η),s))dsdη + ∫ xmax xmin β(x2(t; xmin,ξ),p2(t),v2(x2(t; xmin,ξ), t))u01(ξ)e − ∫ t 0 µ(x2(s;ξ,0),p2(s),v2(x2(s;ξ,0),s))dsdξ. the existence and uniqueness of solutions 197 direct calculations give |p3(t) −p2(t)| = ∫ t 0 (b3(η) −b2(η))e − ∫ t η µ(x2(s,xmin,η),p2(s),v2(x2(s;xmin,η),s))dsdη + ∫ xmax xmin (u01(ξ) −u02(ξ))e− ∫ t 0 µ(x2(s;ξ,0),p2(s),v2(x2(s;ξ,0)))dsdξ ≤ ∫ t 0 |b3(η) −b2(η)|dη + ‖u01 −u02‖l1 and |b3(t) −b2(t)| ≤ βm ∫ t 0 |b3(η) −b2(η)|dη + βm‖u01 −u02‖l1. so we can conclude that |p3(t) −p2(t)| ≤ eβmt‖u01 −u02‖l1. from (3.2), by the contraction mapping theorem, we have |p1(t) −p2(t)| ≤ |p1(t) −p3(t)| + |p3(t) −p2(t)| ≤ l|p1(t) −p2(t)| + |p3(t) −p2(t)| ≤ l|p1(t) −p2(t)| + eβmt‖u01 −u02‖l1, which implies (3.1). � in the following, in virtue of the above estimates (3.1), we can show the continuous dependence of solutions on initial conditions. theorem 3.2. let (u1,v1) and (u2,v2) be the solutions of (1.1) with initial conditions (u01,v01) and (u02,v02), respectively. then for any ε > 0, there exists δ = δ(ε,t,u0i,v0i) > 0 such that if ‖u01 − u02‖l1 + ‖v01 −v02‖l1 < δ, then ‖u1 −u2‖l1 + ‖v1 −v2‖l1 ≤ ε. proof. firstly we estimate the difference between the two characteristics. by (2.8) and gronwall’s inequality, and combining with (3.1), we find that |xp1 −xp2| ≤ lg ∫ t 0 |p1(σ) −p2(σ)|dσelg(t−s) ≤ lge (lg+βm )(t−s) βm (1 −l) ‖u01 −u02‖l1, 198 yan li and qihua huang which implies that when t ≥ s, |xp1 −xp2|→ 0 as ‖u01 −u02‖l1 → 0. we assume that z1(t) ≤ z2(t). by (2.4), direct calculations show that∫ xmax xmin |u1(x,t) −u2(x,t)|dx ≤ ∫ z1(t) xmin ∣∣∣b(τ1(xmin)) g(xmin,p1) − b(τ2(xmin)) g(xmin,p2) ∣∣∣e−∫ tτ1(xmin) µ̃(x1,p1,v1)dsdx + ∫ z1(t) xmin b2(τ(xmin)) g(xmin,p2) (e − ∫ t τ1(xmin) µ̃(x1,p1,v1)ds −e ∫ t τ2(xmin) µ̃(x2,p2,v2)ds)dx + ∫ z2(t) z1(t) ( u01(x1)e − ∫ t 0 µ̃(x1,p1,v1)ds − b(τ2(xmin)) g(xmin,p2) e ∫ t τ2(xmin) µ̃(x2,p2,v2)ds ) dx + ∫ xmax z2(t) u01(x1)(e − ∫ t 0 µ̃(x1,p1,v1)ds −e ∫ t 0 µ̃(x2,p2,v2)ds)dx + ∫xmax z2(t) |u01(x1) −u01(x2)|e− ∫ t 0 µ̃(x2,p2,v2)dsdx + ∫ xmax z2(t) (u01(x2) −u02(x2))e− ∫ t 0 µ̃(x2,p2,v2)dsdx and∫ xmax xmin |v1(x,t) −v2(x,t)|dx = ∫ z2(t) z1(t) ( v01(x1)e − ∫ t 0 σ(x1,s)ds + ∫ t 0 a(x1,s)e(s)e − ∫ t s σ(x1,τ)dτ ds ) dx + ∫ xmax z2(t) ∫ t 0 [a(x1,s) −a(x2,s)]e(s)e− ∫ t s σ(x1,τ)dτ dsdx + ∫ xmax z2(t) ∫ t 0 a(x2,s)e(s)[e − ∫ t s σ(x1,τ)dτ −e− ∫ t s σ(x2,τ)dτ ]dsdx + ∫ xmax z2(t) v01(x1)[e − ∫ t 0 σ(x1,τ)dτ −e− ∫ t 0 σ(x2,τ)dτ ]dsdx + ∫ xmax z2(t) (v01(x1) −v01(x2))e− ∫ t 0 σ(x2,τ)dτ dx + ∫ xmax z2(t) (v01(x2) −v02(x2))e− ∫ t 0 σ(x2,τ)dτ dx, where xi = xpi, (i = 1, 2), µ̃ is defined in definition 2.1. the following proof can be completed by using similar arguments as in the proof of theorem 2 in [2]. � 4. concluding remarks in this paper, by using the method of characteristic and contracting mapping theorem, we proved the existence-uniqueness of solutions to problem (1.1). we also derive the continuous dependence on initial conditions of the solutions. in the future, we plan to study the asymptotic behavior of the population under the influence of environmental toxins. in addition, problem (1.1) assumes that the population growth rate g = g(x,p(t)). this mortality rate, however, may depend on the body burden v. including the dependence of the growth rate on the body burden (i.e., g = g(x,p(t),v(x,t))) will yield new and challenging problems. the existence and uniqueness of solutions 199 references [1] s. m. bartell, r. a. pastorok, h. r. akcakaya, h. regan, s. ferson and c. mackay, realism and relevance of ecological models used in chemical risk assessment. hum. ecol. risk assess. 9 (2003), 907–938. 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[11] d. m. thomas, t. snell and s. jaffar, a control problem in a polluted environment, math. biosci. 133 (1996), 139-163. college of science, china university of petroleum (east china), qingdao 266580, p. r. china email address: liyan@upc.edu.cn corresponding author, school of mathematics and statistics, southwest university, chongqing 400715, p. r. china email address: qihua@swu.edu.cn 1. introduction 2. existence and uniqueness results 3. continuous dependence on initial conditions 4. concluding remarks references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase online first, pp.1-12 https://doi.org/10.5206/mase/14626 macroscopic analysis of the viscous-diffusive traffic flow model gabriel obed fosu, albert adu-sackey, and joseph ackora-prah abstract. second-order macroscopic traffic models are characterized by a continuity equation and an acceleration equation. convection, anticipation, relaxation, diffusion, and viscosity are the predominant features of the different classes of the acceleration equation. as a unique approach, this paper presents a new macro-model that accounts for all these dynamic speed quantities. this is done to determine the collective role of these traffic quantities in macroscopic modeling. the proposed model is solved numerically to explain some phenomena of a multilane traffic flow. it also includes a linear stability analysis. furthermore, the evolution of speed and density wave profiles are presented under the perturbation of some parameters. 1. introduction the main classifications of macroscopic traffic models are the first-order and second-order models [13, 21, 20, 36]. second-order equations were introduced to overcome the shortcoming of the first-order equation [6, 30]. the first-order category is also known as the continuity or lwr equation. the second-order branch encompasses the first-order equation together with a dynamic velocity equation. oftentimes the dynamic velocity equation is also called the acceleration or momentum equation. payne [31] set forth the precedence with his classical dynamic velocity equation with little revision by whitham [37] some few years afterwards to form the payne-whitham (pw) model. the constitutive terms of the acceleration equation are convection, anticipation, relaxation, diffusion, and viscosity. a review of these dynamic terms as either accounted or unaccounted for within a given model formulation is presented in table 1. from the reviewed models in table 1, it was observed that convection, anticipation, and relaxation are always present, but that of diffusion and viscosity are barely modeled. as a novel formulation, all these terms are brought together to form a new momentum equation. the proposed momentum equation is coupled with the continuity equation to form a system of partial differential equation. note that each of these terms under consideration describes some realistic traffic phenomena, and hence the need to examine their contributive role in vehicular traffic analysis. the definition and mathematical representation of these terms are detailed in the ensuing paragraphs. in a traffic sense, the convection or transport term explains the movement of vehicles along with their density/velocity profiles. (a density profile in the case of the continuity equation, and a velocity profile for that of the acceleration equation). these are respectively defined as q′(x,t) ∂k(x,t) ∂x and v(x,t) ∂v(x,t) ∂x (1.1) received by the editors 24 january 2022; accepted 5 july 2022; published online 9 july 2022. 2010 mathematics subject classification. 35q35, 65m06, 35l40. key words and phrases. second-order model, viscosity, multilane flow, diffusion, speed-density profiles. 1 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14626 2 g. o. fosu, a. adu-sackey, and j. ackora-prah table 1. constituents of the acceleration equation author(s) convection anticipation relaxation diffusion viscosity vvx (·)kx or (·)vx (·)(ve −v) (·)vxx (·)vy [31] x x x × × [23] x x x × × [25] x x x x × [28] x x x × × [16] x x x × x [26] x x x × × [29] x x x x × [34] x x x × × [38] x x x x × [39] x x x × × [40] x x x × × [14] x x x × x k(x,t),v(x,t) and q(x,t) denotes traffic density, speed, and flow rate in that order. the term v(x,t) · ∂v(x,t)/∂x is to ensure that vehicles do not adjust their speeds in a haste manner. rather, they are to gradually change their speed to that of the downstream traffic and describes the coordination between the speeds upstream and downstream. the anticipation term describes the dispersion effect of the speed of heterogeneous traffic. that is, how identical driver-vehicle react to flow conditions in a close neighborhood, more specifically frontal effects. drivers usually slow down when approaching heavy traffic downstream. the term is a diverging point for most navier-stokes-like traffic models. in most payne-like models, anticipation is modeled by 1 2τk dve(k) dk ∂k ∂x or − 1 k dp(k) dk ∂k ∂x (1.2) where p(k) = −ve(k)/2τ is the pressure term which describes how drivers perceive ahead and adjust to the density downstream. the anticipation term is the core component contributing to the backward traveling wave of payne-like models. because of this critique by [9], aw and rascle postulated a new pressure term of the form p(k) = kγ with γ > 0 [3, 32]. however, the ar model was found to be ill-posed for a virtual road [27]. besides, [23] in their bid to correct the deficiency of negative wave speed replaced the density gradient term with a velocity gradient component, hence defined anticipation as c(k) ∂v(x,t) ∂x (1.3) the authors specifically used constant c > 0 in place of the functional c(k) to denotes the propagation speed of some perturbations. another important characterization of traffic is through the relaxation term. this is sometimes called the speed adaptation term. it details how vehicles adapt their speed to the steady-state speed. given the average speed v(x,t) and ve(k) the steady-speed as a function of density, then the relaxation term can be defined as ve(k(x,t)) −v(x,t) τ (1.4) this explains how vehicles at average speed could adjust to the situational density-dependent speed. the quantity τ > 0 is the relaxation time for the entire process of adapting to the localized speed. it viscous-diffusive traffic flow model 3 is a constant that does not depend on either speed or density. under normal condition, it is estimated to be between 20 to 30 seconds for highway traffic and something less for urban flow [36]. on certain occasions, the diffusive term d ∂2v(x,t) ∂x2 (1.5) is introduced as part of the macroscopic equations. it was earlier fused into the acceleration equation [25, 29], and later as part of the lwr model [35]. the same expression (equation 1.5) is used in both the lwr and acceleration equations but may differ in the relevant application. the entire term (1.5) explains how vehicles adjust their velocities to surrounding traffic conditions. it was introduced to smooth shocks and abrupt transitions of traffic between different regimes. that is, smoothing the density profiles in the event of a diffusive first-order model, and smoothing the velocity profile for a diffusive acceleration equation. the diffusive term enhances the numerical properties of the model and also eradicate shockwaves if there is the existence of numerical instabilities. the last term to discuss is the lateral viscosity term. in the immediate recent past, [16] introduced lateral viscosity to explain traffic resistance on a multilane highway. the term was deduced from the no-slip condition of fluids and is given as sd µ2 k(x,t) ∂v(x,t) ∂y (1.6) where µ2 is the lateral viscous rate, sd is to model the sensitivity of safe vertical distance between vehicles moving on neighboring lanes in the same direction. the velocity gradient ∂v(x,t)/∂y account for the speed variations with respect to changes in usage of road lanes. all these terms together with the local derivative ∂v(x,t)/∂t yields a generalized equation of the form ∂v(x,t) ∂t + v(x,t) ∂v(x,t) ∂x = f ( c(k) ∂v(x,t) ∂x ,c2o(k) ∂k(x,t) ∂x , ve(k) −v(x,t) τ ,d ∂2v(x,t) ∂x2 ,φ ∂v(x,t) ∂y ) (1.7) f is an arbitrary function, c2o(k) = − 1 2τk(x,t) dve(k) dk and φ = sd µ2 k(x,t) . equation (1.7) is decoupled as ∂v(x,t) ∂t + v(x,t) ∂v(x,t) ∂x = ve(k) −v(x,t) τ − c2o(k) ∂k(x,t) ∂x + d ∂2v(x,t) ∂x2 −φ ∂v(x,t) ∂y (1.8) and ∂v(x,t) ∂t + v(x,t) ∂v(x,t) ∂x = c ∂v(x,t) ∂x + ve(k) −v(x,t) τ + d ∂2v(x,t) ∂x2 −φ ∂v(x,t) ∂y (1.9) equation (1.8) has a density-gradient anticipation term, while (1.9) has a velocity-gradient anticipation term. equation (1.8) together with the continuity equation is the isotropic viscous-diffusive model. it connotes an extension of all payne-like models. equation (1.9) is the anisotropic correspondent. as stated in table 1, earlier formulations are devoid of either the diffusion or viscous term. as such, most recent model presentations and analyses also fellow suite [2, 4, 7, 17, 27, 41] with few considering the effect of stochasticity [4, 42]. concerning traffic stability, the steady-state condition of a viscous second-order macroscopic traffic flow model was performed by [1]. these authors obtained the equilibrium points, the stability criterion, and the phase plane solution of the two velocity difference model (tvdm) [18], which is considered as a simple extension the classical anisotropic macro model. similarly, [5] conducted a similar analysis to determine the global stability and the bifurcation of a 4 g. o. fosu, a. adu-sackey, and j. ackora-prah second-order continuum model. the authors observed the presence of subcritical hopf bifurcation for their derived macroscopic model. on the other hand, [8] established that the impact of viscosity under a stable traffic condition is approximately zero. these authors used a series of proofs to confirm this near-zero assertion. this paper also presents a graphical simulation with a stability analysis to determine the effect of both viscosity and diffusion for a multilane traffic flow. the next section begins with a derivation of the instability criterion of a proposed anisotropic model. specifically, a linearization analysis to determine either stable or unstable traffic flow is presented. it is followed by a numerical simulation; investigating the potency of the model to reproduce some relevant flow phenomena. these simulations are presented within the domain of a multilane infrastructure. the final part of this paper is reserved as the concluding section. however, the analysis concerning the proposed density-gradient model could be considered as future research work. 2. derivation of instability criterion the stability criterion of the proposed model is determined using the linearization technique. assuming a homogeneous solution k(x,y,t) = ke and v(x,y,t) = ve(k). any deviation from these stationary solutions are given as δk = k(x,y,t) −ke, and δv = v(x,y,t) −ve(k) (2.1) consequently, the viscous-diffusive anisotropic macroscopic traffic flow model is linearized as ∂(δk) ∂t + ve ∂(δk) ∂x + ke ∂(δv) ∂x = 0 ∂(δv) ∂t + ve ∂(δv) ∂x − c ∂(δv) ∂x = 1 τ ( dv dk ·δk − δv ) + d ∂2(δv) ∂x2 −φ ∂(δv) ∂y (2.2) by carefully examining vehicle trajectories, it is realized that traffic behaves wave-like. due to this proposition, the propagation of disturbance of flow can be inferred from the theory of waves. therefore, the underlying simple wave functions (2.3) is adopted to probe whether a disturbance will escalate or decay over time. δk = k̂ exp[is1x + is2y + (λ− iω)t] and δv = v̂ exp[is1x + is2y + (λ− iω)t] (2.3) s1,2 are the spatial wave-numbers. these delimitate the wavelength along the longitudinal and lateral axis respectively, ω is the wave frequency, λ is the wave dumping, k̂ and v̂ are the amplitudes at some time t. further, equation (2.3) and its derivatives are substituted into (2.2). note that higher order terms are not considered in subsequent computations. the arrived simplification is k̂(λ− iω)m̃ + ivesik̂m̃ + ikes1v̂m̃ = 0 v̂(λ− iω)m̃ + (ve − c)is1v̂m̃− m̃ τ ( dve dk k̂ − v̂ ) + ds21v̂m̃ + is2v̂m̃ = 0 (2.4) where m̃ := exp[is1x + is2y + (λ− iω)t] 6= 0. equation (2.4) is then represented in its vector form as  λ̃ ikes1 − 1 τ dve dk λ̃ + 1 τ − ics1 + iφs2 + ds21  [k̂ v̂ ] = [ 0 0 ] (2.5) this (2.5) is of the typical form ax̂ = 0. the unknown vector x̂ to be determined are the amplitudes [k̂ v̂]′. λ̃ = λ− iω and ω̃ = ω −ves1 are abbreviations used to alleviate the computational process. the solution to equation (2.5) is non-trivial if the determinant of the matrix a is zero. the determinant of a produces the quadratic equation: viscous-diffusive traffic flow model 5 λ̃ + λ̃ ( 1 τ̆ − iη̆ ) + i τ ke dve dk s1 = 0 (2.6) where 1/τ̆ = 1/τ + ds21 > 0 and η̆ = cs1 −φs2 > 0. the solution to the characteristic polynomial (2.6) is: λ̃±(s) = 1 2 ( η̆ − 1 τ̆ ) ± √ 1 4 ( 1 τ̆2 − η̆2 ) + ( − 1 τ dve dk kes1 − η̆ 2τ̆ ) (2.7) as it can be seen, the square root term would yield a complex output, therefore the underlying equation (2.8) is used to simplify this root term. the variable r is used to denote the real part, while the imaginary part is denoted by i. from [19] √ r± ii = √ 1 2 (√ r2 + i2 + r ) ± i √ 1 2 (√ r2 + i2 −r ) (2.8) the discriminant for stability dwells on the real part (re) of the eigenvalues. deductively, the real part of the eigenvalues are re ( λ̃±(s) ) = − 1 2τ̆ ± √ 1 2 (√ r2 + i2 + r ) (2.9) a choice is made between re(λ̃−(s)) and re(λ̃+(s)) depending on which is more non-negative. observably, re(λ̃−(s)) < re(λ̃+(s)), which implies that any condition satisfying re(λ̃+(s)) will automatically satisfy re(λ̃−(s)). hence, the relevant eigenvalue to determine transitions from stationary traffic to unstable flow is re(λ̃+(s)). that is − 1 2τ̆ + √ 1 2 (√ r2 + i2 + r ) ≥ 0 simplified as i2 ≥ 1 4τ̆4 − r τ̆2 (2.10) the result of substituting the values of r = 1 4 ( 1 τ̆2 − η̆2 ) and ±|i| = 1 τ ∣∣∣∣dvedk ∣∣∣∣kes1 − η̆2τ̆ into (2.10) is( 1 τ ∣∣∣∣dvedk ∣∣∣∣kes1 − η̆2τ̆ )2 ≥ 1 4τ̆4 − 1 4 1 τ̆2 ( 1 τ̆2 − η̆2 ) (2.11) simplified as 1 τ ∣∣∣∣dvedk ∣∣∣∣kes1 · ( 1 τ ∣∣∣∣dvedk ∣∣∣∣kes1 − η̆τ̆ ) ≥ 0 finally, the instability condition is obtained as 1 τ ∣∣∣∣dvedk ∣∣∣∣kes1 ≥ η̆τ̆ = (cs1 −φs2) ( 1 τ + ds21 ) (2.12) the instability condition is gratified if the change in velocity as a result of change in density is quite larger. this scenario is evident within the synchronized regime (medium densities) of traffic flow. the convergence of individual vehicular velocity to the steady-state velocity is realized during either the free-flow regime or the congested regime. during these regimes, values for the velocity-density gradient term are quite smaller. hence, the instability criterion (2.12) will be violated. but in the absence of 6 g. o. fosu, a. adu-sackey, and j. ackora-prah the diffusion and viscosity rates, the threshold for equilibrium traffic corresponds to the value of the sonic speed c. this is given by the expression in equation (2.13) below.∣∣∣∣dvedk ∣∣∣∣ke ≥ c (2.13) 3. model analysis the discrete multilane version of the continuous viscous-diffusive model (1.9) is presented here by the introduction of the lane index l, that is ∂kl(x,t) ∂t + ∂ql(x,t) ∂x = 0 ∂vl(x,t) ∂t + vl(x,t) ∂vl(x,t) ∂x = c ∂vl(x,t) ∂x + ve(kl) −vl(x,t) τ + d ∂2vl(x,t) ∂x2 −φ ∂vl(x,t) ∂y (3.1) ql(x,t),kl(x,t) and vl(x,t) are the flow rate, density, and speed for the lth lane respectively. this equation (3.1) is presented to explain the inter-lane dependency of a multilane flow. existing models could only explain the association among a limited number of lanes, usually two lanes[12, 24]. here, an investigation for more than two lanes is presented. the model is solved numerically because of the computational difficulties associated with the analytical approach to obtaining a solution to this system. thus, the upwind finite difference scheme is employed to numerically solve this macroscopic system (3.1). the lane index continuity equation is discretized as klm(n + 1) = k l m(n) + φv l m(n) [ klm−1(n) −k l m(n) ] + φklm(n) [ vlm(n) −v l m+1(n) ] (3.2) the discretized acceleration equation is given by either of the following depending on the intensity of the traffic. when vlm(n) > c; being a lighter flow regime, then the acceleration equation is given as vlm(n + 1) = v l m(n) + φ [ c−vlm(n) ][ vlm(n) −v l m−1(n) ] − θ µ ·sd klm(n) ( vlm(n) −v l+1 m (n) ) + ∆t τ ( ve −vlm(n) ) + dω ( vlm−1(n) − 2v l m(n) + v l m+1(n) ) (3.3) in the case of heavy flow (vlm(n) < c), then the acceleration equation becomes vlm(n + 1) = v l m(n) + φ [ c−vlm(n) ][ vlm+1(n) −v l m(n) ] − θ µ2 ·sd klm(n) ( vlm(n) −v l+1 m (n) ) + ∆t τ ( ve −vlm(n) ) + dψ ( vlm−1(n) − 2v l m(n) + v l m+1(n) ) (3.4) where ∆t/∆y = θ, ∆t/∆x = φ, and ∆t/∆x2 = ψ. m,n,l are positive integers. xm = m∆x, yl = l∆y, and tn = n∆t. k l m(n) ≈ k(xm,yl, tn) is the density of vehicles at region m of lane l at time n. vlm(n) ≈ v(xm,yl, tn) is the velocity of vehicles at position m on lane l at time n. moreover, the steady-state velocity equation ve is defined as [10, 11, 15]: ve = vf { 1 − exp [ 1 − exp ( kw vf ( kf klm(n) − 1 ))]} (3.5) kw is the kinematic wave speed during a heavy traffic domain, vf and kf are respectively the maximum speed and density. for the initial condition, each lane has a specified density value with speed as derived. viscous-diffusive traffic flow model 7 kl(x, 0) =   k1, if l = 1 k2, if l = 2 ... ... ... kl, if l = l v l(x, 0) =   v(k1), if l = 1 v(k2), if l = 2 ... ... ... v(kl), if l = l 3.1. simulation result and analysis. a graphical solution of this lane-index macro-model is presented in this section. in addition, the information flow of the wave profiles under the perturbation of some key parameters are also presented. throughout the simulation, the total distance is taken as 3000m. a relatively shorter distance is chosen so that one could simultaneously observe both lateral and longitudinal happenings on a given multilane stretch. a simulation for five-lane infrastructure is presented with all vehicles moving along the same direction, with inter-lane interval ∆y = 1.5m. ∆x = 300 and ∆t = 4s are chosen to satisfy the courant-friedrichs-lewy numerical stability condition (3.6). max{vf − c,q′(k(x,t))} · ∆t ∆x ≤ 1 (3.6) the initial density profile for the five lanes is given as kl(x, 0) =   0.50000, if l = 1 0.38750, if l = 2 0.27500, if l = 3 0.16250, if l = 4 0.05000, if l = 5 the initial values for the first and last lanes are used as the boundary condition. the lanes are numbered as l = 1, 2, · · · , 5. lane one is the extreme outer lane, while five is the extreme inner lane. the initial density for lane one is denser due to parking on that lane and on the shoulders of the road; this following the no-slip condition of fluid [22, 33]. for the five lane analysis, lanes four and five are the high-speed lanes, while the outer lanes are the low-speed lanes. the following are details of the other parameters [16, 23, 38]: µ2 = 0.0041 d = 10 sd = 0.37 kf = 1 c = 11m/s vf = 20m/s τ = 10s kw = 11m/s figure 1 describes some traffic dynamics of a five lane carriageway. it could be observed that the density of the traffic becomes lighter moving from the outer lanes through to the inner lanes. lane five is less dense, and as such vehicles traversing on this lane have higher velocity compared to the other lanes. almost all vehicles on lane five could speed up to the maximum, but the situation is different on the outer lanes. this scenario typifies a realistic multilane traffic state in developing countries of which ghana is not an exception. here, passenger cars often transverse on the outer lanes, due to their intermittent stopping to pick and drop passengers, making these lanes denser. nonetheless, drivers change lanes oftentimes to the left as the density of their driving lane becomes compact. this is depicted through the simulation results with a speed drop and density rise in the higher speed lanes. deductively, a two-lane highway would easily get clumsy when there is an obstruction on the outer lane. but the situation is different on roadways with arbitrary many lanes. the case of a twenty-lane carriageway is shown in figure 2. the density of the traffic reduces moving towards the inner lanes because the model hinges on the no-slip condition. for this twenty-lane representation, maximum flow is achieved somewhere around lane fifteen through to twenty. 8 g. o. fosu, a. adu-sackey, and j. ackora-prah figure 1. speed and density profiles for a five-lane carriageway figure 2. multilane traffic profiles for a twenty-lane carriageway in figure 3, the sonic speed c is varied to determine it effect on flow. it was observed that predicting vehicular traffic using this macroscopic model would fail when the value of the sonic speed far exceeds vehicles speed. this is illustrated graphically with c = 150m/s. the absence of the anticipation term (c = 0m/s) produced quite shorter wave profiles vis-a-vis the benchmark value of c = 11m/s in figure 1. figure 3. multilane speed profiles for c = 0, left; c = 50m/s, middle; and c = 150m/s, right from the stability criterion (2.12), it was stated that the time taken for traffic to align with the velocity-density dependent value is key in determining the stationarity of the flow. thus, the adaptation time τ is varied to determine its effect on these multilane wave profiles. from figure 4, if drivers have enough time to adapt, their speed profiles are finer. that is to say drivers have a longer time to react without any repercussions. comparing the plot right of figure 4 to the other on the left, the speed profiles become unstable as τ converges to zero. viscous-diffusive traffic flow model 9 figure 4. multilane speed profiles for τ = 1s, left; τ = 5s, middle; and τ = 120s, right a similar perturbation of the lateral viscosity rate is represented in figure 5. the viscous rate ranges between zero and one; zero denotes the absence of any form of lateral resistance, and one denotes a highly viscous flow. from the simulation plots 5, the effect of viscosity is apparent as µ2 assumes values closer to the upper limit. it is seen that some amount of lateral obstruction has a consequential effect on the speed of vehicles. figure 5. multilane speed profiles under the perturbation of the viscous rate. top left: µ2 = 0, top right: µ2 = 0.041, bottom left: µ2 = 0.41, and bottom right: µ2 = 1.00 from figure 6, the prediction of traffic using this multilane model was found to be feasible whether diffusion was present or not. the diffusive term was barely substantiated in this simulation work. nonetheless, a highly enormous value showed inclinations of abnormal flow. 4. conclusion macroscopic traffic models of second-order consist of the continuity equation and the dynamic velocity equation. the dynamic velocity equation was introduced to rectify the shortcomings of the continuity equation. the components of the dynamic velocity equation are convection, anticipation, relaxation, diffusion, and viscosity. in this paper, a new macro-model that features all these dynamic speed quantities is presented. then after, the linear-stability condition of the new continuum second-order 10 g. o. fosu, a. adu-sackey, and j. ackora-prah figure 6. multilane speed profiles under the perturbation of the diffusion rate. top left: d = 0, top right: d = 50, bottom left: d = 500, and bottom right: d = 15000 model was determined through the analysis of wave profiles. the gradient of the velocity-density curve, the average density, and the sonic speed were found to be the determining variables for either stable or unstable flow. the proposed anisotropic model was again recast as a lane-indexed model and was solved numerically using the upwind finite difference scheme. this reformulation was done to remove the restriction on the proposed model as being single-piped. the viscosity rate, anticipation, diffusion, and relaxation time were perturbed to examine its effect on flow. by the simulation plots, it was observed that a smaller relaxation time, a larger anticipation rate, and a 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[42] s. zheng, r. jiang, b. jia, j. tian, and z. gao, impact of stochasticity on traffic flow dynamics in macroscopic continuum models, transp. res. rec. 2674 (2020), 690–704. corresponding author, department of mathematics, kwame nkrumah university of science and technology, kumasi, ghana email address: gabriel.of@knust.edu.gh department of applied mathematics, koforidua technical university, ghana email address: albert.adu-sackey@ktu.edu.gh department of mathematics, kwame nkrumah university of science and technology, kumasi, ghana email address: jaackora-prah.cos@knust.edu.gh 1. introduction 2. derivation of instability criterion 3. model analysis 3.1. simulation result and analysis 4. conclusion references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 3, number 1, march 2022, pp.24-49 https://doi.org/10.5206/mase/14332 dynamics of a stoichiometric producer-grazer model with maturation delay hua zhang, hao wang, and ben niu abstract. ecological stoichiometry provides a multi-scale approach to study macroscopic phenomena via microscopic lens. a stoichiometric producer-grazer model with maturation delay is proposed and studied in this paper. the interaction between stoichiometry and delay is novel and leads to more interesting insights beyond classical delay-driven periodic solutions. for example, the period doubling bifurcation route to chaos can occur as the minimal phosphorous:carbon ratio in producer decreases. mathematically, we establish the conditions for the existence and stability of positive equilibria, and study the occurrence of hopf bifurcation at positive equilibria. analytic results show that delay can change the number and stability of positive equilibria through transcritical bifurcation, saddle-node bifurcation and hopf bifurcation, and it further determines the grazer’s extinction. our model with a small delay behaves like lke model in terms of light intensity, and rosenzweig’s paradox of enrichment exists in a suitable light intensity. we plot bifurcation diagrams and show rich dynamics driven by delay, light intensity, phosphorous availability, and conversion efficiency, including that a large delay can drive the grazer to go extinct in an intermediate light intensity that is favorable for the survival of the grazer when there is no delay; a limit cycle can appear, then disappear as the delay increases; given the same initial condition, solutions with different delay values can tend to different attractors. 1. introduction there is increasing evidence that elemental imbalances between producer and grazer can significantly influence their growth, reproduction and survival. for example, the experiment studying zooplanktonphytoplankton interactions [32] showed zooplankton growth may suffer at high algal density instead of always being positively correlated with algal density. this inspires many researchers to explain the producer-grazer interactions from the stoichiometric point of view, which focuses on the relations between multiple key elements in organisms and the abiotic environment [22]. carbon (c, supplying energy) and phosphorus (p, measuring nutrient) are two vital elements for a cell that is the basic unit of living organisms. one classical stoichiometric producer-grazer model known as lke model [19] tracks how energy flow and nutrient cycling affect the grazer’s dynamics [17, 37]. the lke model allows p:c (phosphorous:carbon ratio) in producer to vary above a minimum structural p:c while to keep constant in grazer under the ”strict homeostasis” assumption [36, 35]. if p:c in producer is greater than that of grazer, then producer becomes low-quality food. thus, this model incorporates the effect of both producer’s quantity (light dependent) and quality (nutrient dependent) on the grazer’s dynamics. received by the editors 15 october 2021; accepted 2 february 2022; published online 6 february 2022. 2020 mathematics subject classification. 34k18, 37d45, 37n25, 92d40. key words and phrases. stoichiometric producer-grazer model, maturation delay, stability, bifurcation analysis, light intensity, phosphorus. hua zhang and ben niu were both supported by shandong provincial natural science foundation (zr2019qa020). hao wang was supported by natural sciences and engineering research council of canada (nos. rgpin-2020-03911 and rgpas-2020-00090). 24 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14332 stoichiometric producer-grazer model with maturation delay 25 the lke model reads dx dt = bx ( 1 − x min{k, (p −θy)/q} ) −f(x)y, dy dt = ê min { 1, p −θy θx } f(x)y −dy, (1.1) where x, y are the densities of producer and grazer, respectively. b is the intrinsic growth rate; k is the carrying capacity of producer, which is assumed to be determined positively by light intensity; ê is the maximal conversion efficiency of grazer; d is the loss rate of grazer involving metabolic losses and death; q is the minimal p:c in producer; θ is p:c in grazer keeping a constant; p is the total mass of phosphorus in the entire ecosystem system. all parameters are positive constants, and ê < 1, q < θ. f(x) reflects grazer ingestion rate, and it is a bounded smooth function with f(0) = 0, f ′ (0) < ∞, f ′ (x) > 0, for x ≥ 0. note that limx→0 f(x) x = f ′ (0) < ∞, which implies that system (1.1) is meaningful as x → 0 although there is the term min { 1, p−θy θx } . reasonable types of f(x) includes holling i, holling ii and holling iii functions. in [19], it was revealed that model (1.1) with holling ii type of f(x) admitted multiple positive equilibria, limit cycles and various bifurcation phenomena. the authors also observed the paradox of energy enrichment, where intense energy enrichment substantially elevated the producer density but decreased the grazer growth rate and may drive the grazer to extinction. an explanation on such phenomenon is that the grazer becomes poor-quality food in an extremely high light intensity due to amounts of carbon element is fixed by photosynthesis. complete analysis for model (1.1) on local and global stability of all equilibria and existence of limit cycles was provided by [17] and [37]. the authors in [17] dealt with two cases of f(x): holling type i and holling type ii. for the former case, theoretical analysis showed the unique internal equilibrium was always globally asymptotically stable if it existed. for the later case, limit cycles, bistability and several bifurcation types were exhibited when all parameters were fixed at realistic values except k varied. this work was enriched by [37], where authors gave a comprehensive dynamics analysis for holling ii type of f(x) with all flexible parameters. three new types of bistability comparing with [17] were found: between two positive equilibria, between one positive equilibrium and one boundary equilibria, between the limit cycle and one boundary equilibrium. for model (1.1), yuan et al. [39] accounted for the effect of environmental noises on the switch between two stochastic attractors in the bistable situation. wkl model in [34] mechanistically incorporates free nutrient in media. other extensions and applications of lke model can be found in [33, 24, 38, 16, 3] and the references therein. these studies make contributions to apply stoichiometric models for the improvements in the predictive power of population ecology and cancer treatment. notably, all existing stoichiometric models implicitly assume an instant process for grazer to be capable of preying on producer for its own growth. nevertheless, grazer often needs some time to become mature so that it has the ability to prey producer. inspired by [12], we assume grazer has two stage groups: immature grazer (yj) and mature grazer (y), and only mature grazer lives on producer. therefore, we propose a stoichiometric producer-grazer model with stage-structure for the grazer as 26 h. zhang, h. wang, and b. niu follows: dx(t) dt = bx(t) ( 1 − x(t) min{k, (p −θy(t))/q} ) −f(x(t))y(t), dy(t) dt = ê min { 1, p −θy(t− τ) θx(t− τ) } f(x(t− τ))y(t− τ)e−µτ −dy(t), dyj(t) dt = ê min { 1, p −θy(t) θx(t) } f(x(t))y(t) −µyj(t) − ê min { 1, p −θy(t− τ) θx(t− τ) } f(x(t− τ))y(t− τ)e−µτ, (1.2) where τ is the maturation delay, µ is the mortality rate of immature grazer, and e−µτ is the survival rate of immature grazer. system (1.2) can be derived from the standard age-structured population model, for example, see [28, 2]. here, we include it for the completeness. let y (t,a) be the density of grazer of age a at time t, τ is the maturation period, µ is the mortality rate of immature grazer, and d is the loss rate of mature grazer. assume that y (t,a) satisfies the following age-structured population model ∂y (t,a) ∂t + ∂y (t,a) ∂a = −µy (t,a), t > 0, 0 < a < τ, (1.3) and the mature grazer density y(t) := ∫∞ τ y (t,a)da safisfies dy(t) dt = y (t,τ) −dy(t), t > 0, (1.4) with y (t, 0) = y(t). fix s > 0 and let ws(t) := y (t,t−s) for s ≤ t ≤ s + τ. together with model (1.3), we have dws(t) dt = −µws(t), 0 ≤ s ≤ t ≤ s + τ, with ws(s) = y(s). this leads to ws(t) = e−µ(t−s)y(s). therefore, y (t,τ) = wt−τ (t) = e−µτy(t− τ). substituting y (t,τ) into eq. (1.4), and assuming that the prey activity of the mature grazer follows from that in system (1.1), we can obtain the second equation of system (1.2). it is reasonable to assumed that the density of immature grazer is small compared to that of mature grazer, so we only consider the dynamics of the first two equations in (1.2), that is, dx(t) dt = bx(t) ( 1 − x(t) min{k, (p −θy(t))/q} ) −f(x(t))y(t), dy(t) dt = ê min { 1, p −θy(t− τ) θx(t− τ) } f(x(t− τ))y(t− τ)e−µτ −dy(t). (1.5) the maturation delay in produce-grazer models has been widely studied, and it usually changes the stability/instability of equilibrium, for example, see [21, 4, 25, 18], but the interaction between stoichiometry and delay is novel. in this paper, we mainly study how delay affects the existence of positive equilibria and different types of bifurcation phenomena for system (1.5). as mentioned above, light intensity k and phosphorus availability p are two primary limiting factors for determining the persistence or extinction of the grazer. we also explore the change of their roles with the introduction of delay. the existence of delay and minimum function makes it challenging to give deeper theoretical analysis on the dynamics of the stoichiometric system, but numerical simulations provide another effective way to understand the dependence of underlying dynamics on delay and some key parameters in the stoichiometric system. some interesting results relative to stoichiometry and delay can be summarized as follows, and the implications of these results are provided in section 4. stoichiometric producer-grazer model with maturation delay 27 (1) there are two types of coexistence: a stable periodic solution and a locally asymptotically stable (las) positive equilibrium; two las positive equilibira, see fig 1. (2) our model with a small delay behaves like lke model in terms of light intensity, and rosenzweig‘s paradox of enrichment can occur for a suitable light intensity and small delay, but a large delay annihilates the oscillations, see figs 3 and 7. (3) the delay can not only produce a periodic solution (see appendix), but it can annihilate a periodic solution, see figs 4 and 5, where the amplitude of the periodic solution increases over delay, and an increasing delay drives the periodic solution to collide with the critical line x + y = p, which can make the solution change its convergent state. (4) there is a period doubling bifurcation route to chaos when both stoichiometry and delay are incorporated into the system, see figs 9 and 10. the rest of the paper is organized as follows. in section 2, we present the basic property of solutions to system (1.5) and the stability of boundary equilibria. with holling ii type functional response, the existence and stability of positive equilibria are discussed, and some conditions for the occurrence of hopf bifurcation are obtained. in section 3, using biologically meaningful parameter values, we depict some bifurcation diagrams to illustrate the conclusions obtained in section 2. further, we exhibit rich dynamics, and comprehensively show how the grazer’s dynamics depends on key parameters. in section 4, we relate our analytic results to some important biological phenomena. finally, we conclude the main results and suggest directions for future research. 2. basic analysis in this section, we first establish the nonnegativity and boundedness of solutions, then we consider the existence and stability of non-trivial equilibria. for simplicity, let p = p θ and s = q θ , and then system (1.5) becomes dx(t) dt = bx(t) ( 1 − x(t) min{k, (p−y(t))/s} ) −f(x(t))y(t), dy(t) dt = ê min { 1, p−y(t− τ) x(t− τ) } f(x(t− τ))y(t− τ)e−µτ −dy(t). (2.1) 2.1. nonnegativity and boundedness. in the biological perspective, the initial conditions are given as x(η) ≥ 0, q > y(η) ≥ 0,η ∈ [−τ, 0]. (2.2) denote k = min{k,p/q}, the basic property of solutions with initial values (2.2) is stated in the following theorem. theorem 2.1. let (x(t),y(t)) be any solution of system (2.1) subject to initial conditions (2.2). then, x(t) ≥ 0 for t ∈ (0,∞), and lim sup t→∞ x(t) ≤ k. moreover, if m := maxx∈[0,k] f(x) < dê , then 0 ≤ y(t) ≤ p for t ∈ (0,∞). proof. solving x(t) from the first equation of (2.1) gives x(t) = x(0)e ∫ t 0 [ b ( 1− x(u) min{k,(p−y(u))/s} ) −f(x(u)) x(u) y(u) ] du . thus, x(t) ≥ 0 for all t > 0, and x(t) > 0 if x(0) > 0. moreover, we see that dx(t) dt ≤ bx(t) ( 1 − x(t) min{k,p/s} ) = bx(t) ( 1 − x(t) k ) , from the standard comparison argument, we have lim sup t→∞ x(t) ≤ k. 28 h. zhang, h. wang, and b. niu using the variation-of-constant formula to the second equation in (2.1), we have y(t) = y(0)e−dt + ∫ t 0 ê min { 1, p−y(u− τ) x(u− τ) } f(x(u− τ))y(u− τ)e−µτe−d(t−u)du (2.3) which indicates y(t) ≥ y(0)e−dt for t ∈ [0,τ] provided that p > y(η) ≥ 0, η ∈ [−τ, 0]. moreover, if y(η) 6≡ 0 for η ∈ [−τ, 0], then y(t) > 0, t ∈ [0,τ]. for t ∈ [τ, 2τ], (i) if 0 ≤ y(t) < p, t ∈ [0,τ], then by a similar way, we have y(t) ≥ 0, and y(t) > 0 when y(η) 6≡ 0 for η ∈ [−τ, 0]. (ii) if there exists t1 ∈ [0,τ] such that y(t1) tends to p from below, and 0 ≤ y(t) < p for t ∈ [0, t1), we claim that under the condition m < d ê , it holds that 0 < y(t) ≤ p for t ∈ [τ, 2τ]. it follows from the second equation of (2.1) that dy(t1) dt = ê min { 1, p−y(t1 − τ) x(t1 − τ) } f(x(t1 − τ))y(t1 − τ)e−µτ −dy(t1) ≤ êmy(t1) −dy(t1). thus, dy(t1) dt < 0 when m < d ê , which implies that for any small ε > 0, y(t1 +ε) < y(t1). thus, y(t) ≤ p, t ∈ [0, t1 + ε]. we also see that y(t) ≤ p for t ∈ [t1,τ] with the similar argument. using (2.3), we have y(t) > y(0)e−dt ≥ 0 for t ∈ [t1 + ε,t1 + ε + τ]. as a consequence, 0 < y(t) ≤ p for t ∈ [0,τ]. similar to (i), we prove the claim. we can repeat the process for any [nτ, (n + 1)τ], n ≥ 2. � remark 2.1. due to the existence of a minimum function in the second equation of system (2.1), it is difficult to determine the global existence and the nonnegativity of the component y(t). when there is a strong restriction on f(x), theorem 2.1 gives the result. actually, when f(x) does not satisfy the restriction given in theorem 2.1, 0 ≤ y(t) < p can also hold (see section 3), but we can not prove it theoretically. such a minimum function can induce complex dynamics that is difficult to provide rigorous proofs, see subsection 3.3. 2.2. stability of boundary equilibria. system (2.1) always has equilibria: e0 = (0, 0) and e1 = (k, 0), and their local stability is obtained by the distribution of eigenvalues corresponding to the linearized system. theorem 2.2. (i) e0 is always unstable. (ii) if k < p, e1 is las (locally asymptotically stable) when êf(k) < d for all τ ≥ 0. (iii) if k > p, e1 is las when f(k) k êp < d for all τ ≥ 0. proof. the characteristic equation at e0 is (λ− b)(λ + d) = 0, (2.4) this suggests that two eigenvalues have different signs, so e0 is always unstable. at e1, if k < p, then the characteristic equation takes the following form (λ + b)[λ + d− êf(k)e−µτe−λτ ] = 0. (2.5) note that the stability of e1 depends on the real parts of the zeros of λ + d− êf(k)e−µτe−λτ . when τ = 0, λ = êf(k) −d. clearly, e1 is las if êf(k) < d and is unstable as êf(k) > d. when τ > 0, it is easy to see that λ = 0 is not a root of λ + d− êf(k)e−µτe−λτ = 0 provided êf(k)e−µτ 6= d. moreover, λ = ±iω (ω > 0) are a pair of purely imaginary roots of (2.5) if and only if ω satisfies cos ωτ = d êf(k)e−µτ , sin ωτ = −ω êf(k)e−µτ , and ω2 = [êf(k)e−µτ ]2 −d2. stoichiometric producer-grazer model with maturation delay 29 hence, (2.5) has no roots on the imaginary axis if êf(k)e−µτ < d. then we claim that all roots of (2.5) have negative real parts if êf(k) < d. for k > p, the characteristic equation is (λ + b) [ λ + d− f(k) k êpe−µτe−λτ ] = 0. (2.6) similarly, we obtain e1 is las when f(k) k êp < d. the proof is completed. � 2.3. existence and stability of positive equilibria. in this part, the existence of positive equilibria and their local stability are investigated with f(x) = cx a+x . motivated by [37], in the rest paper, we always assume (a) k ≤ { p q , θad q(cêe−µτ −d) } , ad + pd cp < êe−µτ. as a consequence, the positive equilibria are determined by{ 0 = b(1 − x k ) − cy a+x , 0 = cêx a+x e−µτ −d, or { 0 = b(1 − x k ) − cy a+x , 0 = cê(p−y) a+x e−µτ −d. denote h(x) = b c ( 1 − x k ) (a + x), l1 : x ∗ = ad cêe−µτ −d , l2 : y = ( p− da êc eµτ ) − d êc eµτx, x ∈ [ x∗, min { k, cêp d e−µτ −a }] . (2.7) when the parabola y = h(x) and the line l1 intersect in the region {(x,y) ∈ r2 : 0 < x < k, 0 < y,x + y < p}, there exists a positive equilibrium, denoted by e2. as the parabola y = h(x) enters the region {(x,y) ∈ r2 : 0 < x < k, 0 < y,x + y > p}, when it is tangent with the line l2, a newly positive equilibrium exists; when it intersects with the line l2 at two different points, system (2.1) has two newly positive equilibria, denoted by e3 and e4. thus, system (2.1) may have zero, one, two or three positive equilibria. to explore the existence of positive equilibria, it is convenient to define some critical curves, following the following steps: (i) let k1(τ) = ad/(cêe −µτ −d). (ii) if the line l1, the parabola y = h(x) and the critical line y = p−x intersect, then b c ( 1 − x∗ k ) (a + x∗) = p−x∗. denote k2(τ) = x ∗/[1 − c(p−x∗)/(b(a + x∗)], where x∗ is given in (2.7). (iii) if the parabola y = h(x) is tangent to the line l2 at (x̃, ỹ), then we have  ỹ = b c ( 1 − x̃ k ) (a + x̃), b ck (k −a− 2x̃) = − d êc eµτ, ỹ = ( p− da êc eµτ ) − d êc eµτx̃, x̃ ∈ [ x∗, min { k, cêp d e−µτ −a }] . vanishing x̃ and ỹ yields( b + deµτ ê )2 k2 + [ 2ab ( b + deµτ ê ) − 4bcp ] k + (ab)2 = 0. (2.8) therefore, if (2.8) has positive roots k+3 (τ) or k − 3 (τ) (it can hold that k + 3 (τ) = k − 3 (τ)), then the parabola y = h(x) is tangent to the line l2 in the region {(x,y) ∈ r2 : x ≥ x∗,x + y ≥ p} when k = k+3 (τ). 30 h. zhang, h. wang, and b. niu (iv) denoting the intersection point of the line l2 and x−coordinate as k4, we have k4(τ) = cêp d e−µτ − a. one can check that if τ satisfies assumption (a), then 0 < k1(τ) < p < k4(τ) for each τ. if cp < ab or cp > ab, êe−µτ < ad+pd cp−ab is further true, then 0 < k1(τ) < k2(τ) for each τ. k±3 (τ) exist if and only if cp ≥ a ( b + d êe−µτ ) , and k±3 (τ) = 2cpb−ab ( b + d êe−µτ ) ± 2 √ b2cp [ cp−a ( b + d êe−µτ )] ( b + d êe−µτ )2 . (2.9) note that if k2(τ) > 0 and k + 3 (τ) exists, then it must hold k + 3 (τ) < k2(τ). moreover, k+3 (τ) < k4(τ). based on [37, theorems 3.1-3.7], we state the existence of positive equilibria in the following theorem. theorem 2.3. assume that (a) is satisfied. (i) suppose that k ∈ (0,k1(τ)) or k ∈ (max{k2(τ),k4(τ)},∞). then system (2.1) has no positive equilibria. (ii) suppose that cp < ab or cp > ab but êe−µτ < ad+pd cp−ab holds. if k ∈ (k1(τ),k2(τ)), then system (2.1) has the positive equilibrium e2. (iii) suppose that cp > ab and ad cp−ab < êe −µτ hold. if k ∈ (k+3 (τ),k2(τ)), then system (2.1) has the positive equilibrium e3; if k ∈ (k+3 (τ),k4(τ)), then system (2.1) has the positive equilibrium e4. (iv) suppose that cp > ab, and ad cp−ab < êe −µτ < ad+pd cp−ab hold. if k belongs to sets (k1(τ),k2(τ)) ∩ (k+3 (τ),k2(τ)), (k1(τ),k2(τ)) ∩ (k + 3 (τ),k4(τ)), and (k + 3 (τ),k2(τ)) ∩ (k + 3 (τ),k4(τ)), respectively, then system (2.1) has two positive equilibria: e2 and e3, e2 and e4, or e3 and e4, respectively. if (k1(τ),k2(τ)) ∩ (k+3 (τ),k2(τ)) ∩ (k + 3 (τ),k4(τ)) 6= ∅, then in this interval, system (2.1) has three positive equilibria: e2, e3 and e4. remark 2.2. for the critical cases, we have (i) when k = k1(τ), that is, τ = 1 µ ln cêk d(a+k) , e2 collides e1 and disappears, system (2.1) undergoes a transcritical bifurcation. recalling theorem 2.2, we see that the boundary equilibrium e1 is las as e2 disappears, i.e., τ > 1 µ ln cêk d(a+k) . (ii) when k = k2(τ), e2 and e3 merge into one positive equilibrium. system (2.1) undergoes a saddle-node bifurcation, see fig 2(a). (iii) when k = k+3 (τ), e3 and e4 collide into one positive equilibrium. system (2.1) undergoes a saddle-node bifurcation, see fig 2(b). (iv) when k = k4(τ), namely, τ = 1 µ ln cêp d(a+k) , e4 collides e1 leaving one boundary equilibrium. system (2.1) undergoes a transcritical bifurcation. moreover, theorem 2.2 shows that e1 is las if τ > 1 µ ln cêp d(a+k) . the local stability of positives equilibria that are not on the critical line y = p−x can be analyzed using the method in [5], see appendix for details. we summarize the existence and stability of positive equilibria of system (2.1) in table 1, where i1, i2 and sn(τ) are defined in appendix. 3. numerical simulations in this section, numerical simulations are carried out to illustrate theoretical results and reveal some biological mechanisms intuitively. referring to [19], we take p = 0.025, ê = 0.6, b = 1.2, d = 0.25, θ = 0.03, q = 0.0038, c = 0.81, a = 0.25, µ = 0.003. (3.1) stoichiometric producer-grazer model with maturation delay 31 table 1. the existence and stability of positive equilibria for system (2.1) c êe−µτ k stability c < ab p k ∈ (k1(τ),k2(τ)) e2 hopf bifurcation occurs if sn(τ), n ∈ n0 has zeros in i1 c > ab p ad+pd cp < êe−µτ < ad cp−ab k ∈ (k1(τ),k2(τ)) e2 hopf bifurcation occurs if sn(τ), n ∈ n0 has zeros in i1 ad cp−ab < êe −µτ < ad+pd cp−ab k ∈ (k1(τ),k2(τ)) e2 hopf bifurcation occurs if sn(τ), n ∈ n0 has zeros in i1 k ∈ (k+3 (τ),k2(τ)) e3 unstable k ∈ (k+3 (τ),k4(τ)) e4 hopf bifurcation occurs if sn(τ), n ∈ n0 has zeros in i2 ad+pd cp−ab < êe −µτ k ∈ (k+3 (τ),k4(τ)) e4 hopf bifurcation occurs if sn(τ), n ∈ n0 has zeros in i2 3.1. biological mechanisms related to τ. it can be seen that c > ab p . due to assumption (a), we restrict 0 ≤ τ < 134.1278 and k ≤ min{6.5789, 2.0908}. 3.1.1. joint effect of τ and k on asymptotic dynamics. we present the existence of positive equilibria on τ − k plane, see fig 1(a). bifurcation diagrams provide direct understanding about theorem 2.3 and remark 2.2, which reflect the species persistence/extinction regulated by τ and k. the pair (τ,k) satisfying k > max{k4(τ),k2(τ)} is in region d1; (τ,k) satisfying k < k1(τ) is in region d7. theorem 2.3 indicates that system (2.1) has no positive equilibria in the two regions. the pair (τ,k) satisfying k1(τ) < k < k2(τ) is in regions d3, d4, d5 and d6, thus e2 exists in these regions. we further know that in regions d3 and d4, e2 is unstable induced by hopf bifurcation, and it remains las in regions d5 and d6. when k+3 (τ) < k < k2(τ), (τ,k) locates in regions d3 and d5, and e3 exists in these regions. when k+3 (τ) < k < k4(τ), (τ,k) belongs to regions d2, d3 and d5, e4 exists in these regions. all bifurcation curves described in remark 2.2 are also exhibited in fig 1(a). observe that three curves: k = k1(τ), k = k2(τ) and k = k4(τ) intersect at p3, at which two positive equilibria with one merged by e2 and e3 and the other e4 collide with e1 from its two sides, then all positive equilibria disappear and only one boundary equilibrium stays. in fact, the position of p3 on τ − k plane is (τ̂,k1(τ̂)), where τ̂ = 1 µ ln cêp d(a+p) . it is also seen that the curve k = k+3 (τ) is tangent with the curve k = k2(τ) at p2, thus e2, e3 and e4 collide into one positive equilibrium. meanwhile, it can be calculated that there is a zero eigenvalue for the corresponding characteristic equation. therefore, we assert p2 is a cusp bifurcation point. the hopf bifurcation curve near e2 intersects k-axis at (0, 0.7797) and k = k2(τ) at p1, respectively. at point p4, one can see that e3 and e4 collide into one equilibrium, e2 goes to a boundary equilibrium and disappears. as a result, system (2.1) has exactly one positive equilibrium. biologically, k = k1(τ) (the red curve) and k = k4(τ) (the magenta curve) are two critical curves determining the grazer’s persistence, which holds in the region between the two curves. we see that the survival region gradually declines as τ increases, which follows from the facts that k1(τ) is an increasing function and k4(τ) is a decreasing function of τ, respectively. therefore, beneficial light intensity for the growth of grazer negatively correlates with τ, and less light is needed for the grazer’s persistence as τ increases. it has been recognized that the producer is low quality food at high light level due to the conservation law of matter. the similar phenomenon is observed in fig 1(a) that system (2.1) has no positive equilibria for large k. of course, the extremely low light intensity leads to a very limited 32 h. zhang, h. wang, and b. niu τ k 0 50 100 150 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 assumption (a) is not satisfied e 2 e 4 p 1 e 2 d7 d2 d4 d1 p 2 p 3 d6 d3 e 2 ,e 3 ,e 4 d5 p 4 e 2 ,e 3 ,e 4 transcritical bifurcation curve no positive equilibrium saddle−node bifurcation curve hopf bifurcation curve transcritical bifurcation curve no positive equilibrium 0 50 100 150 200 0 0.5 1 1.5 2 τ k assumption (a) is not satisfied d1 d5 d6 d7 p 3 p 4 d3 p 2 d2 p 1 d4 p 0 no positive equilibriumtranscritical bifurcation curve saddle−node bifurcation curve hopf bifurcation curve transcritical bifurcation curve no positive equilibrium (a) (b) figure 1. the bifurcation diagrams on τ − k plane. the red curve is determined by k = k1(τ); the blue curve stands for k = k2(τ); the magenta curve is given by k = k4(τ); the green one is k = k + 3 (τ); the black one is hopf bifurcation curve at e2. (a) parameter values are given in (3.1). (b) ê = 0.655, a = 0.3 and other parameter values are the same as those in (3.1). quantity of the producer which drives the grazer to go extinct as well. moreover, the existence of p3 reflects that the grazer can go extinct at an intermediate light intensity when τ is too large. the above discussions show that system (2.1) can have sustainable oscillations without delay. does the appearance of limit cycles is substantially impacted by τ? the simulation in fig 1(b) illustrates it does, where ê = 0.655 and a = 0.3, and other parameters in (3.1) remain fixed. dynamics exhibited here are almost identical as in fig 1(a), except that hopf bifurcation curve intersects the curve k = k2(τ) at two distinct points, denoted by p0 and p1. in this case, both producer and grazer densities keep stable for τ = 0, and they change periodically only when τ is greater than a certain value. thus, we assert that τ plays a significant role in the oscillatory behavior of solutions. see appendix for an example on how τ affects the stability of the positive equilibrium e2. in view of fig 1(a) and (b), system (2.1) undergoes a saddle-node bifurcation when e2 and e3 (or e3 and e4) collide into one positive equilibrium. this phenomenon is presented in fig 2. 3.1.2. the bifurcation diagrams of the grazer over τ. to reveal how the grazer’s dynamics depend on τ under different light intensities, we choose four levels of solar energy: k = 0.3, k = 0.7, k = 0.87, k = 1.3, and sketch the change of existence and stability of non-trivial equilibria in fig 3. at low light (k = 0.3), the grazer declines at a stable equilibrium until it dies out. when light intensity is intermediate (k = 0.7), the variation of the grazer density is complex. the grazer density slowly reduces until τ = 1.278 where a supercritical hopf bifurcation occurs at e2 such that e2 loses its stability, and the grazer density changes periodically for 1.278 < τ < 21.308. at τ = 21.308, a periodic solution disappears and e2 regains stability, the grazer density continues to decline. two new equilibria merge through a saddle-node bifurcation at τ = 75.6630, then system has two stable equilibria simultaneously, both of which decrease as τ increases. as τ increases further, the grazer eventually becomes extinct. at high light (k = 0.87), whether the grazer density exhibits sustainable oscillations or declines at a stable equilibrium depends on the initial point. at τ = 0.0527, periodic solutions disappear through an infinite period bifurcation, after that the grazer density keeps at a stable equilibrium whatever its initial density is, and then gradually decreases when τ increases. when τ increases to τ = 123.0325, stoichiometric producer-grazer model with maturation delay 33 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.78 0.8 0.82 0.84 0.86 0.88 0.9 k x + y e 2 e 3 bifurcation point 0.8 1 1.2 1.4 1.6 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 k x + y e 4 e 3 bifurcation point (a) (b) figure 2. the saddle-node bifurcation portraits. solid and dot curves represent the existence and non-existence of equilibria, respectively. (a) the saddle-node bifurcation related to e2 and e3 at τ = 38.4432. (b) the saddle-node bifurcation related to e3 and e4 at τ = 10. the grazer becomes extinct. for extremely high light (k = 1.3), the trend of the grazer density is similar to that in the case of low light, except that the persistence interval of τ becomes small. of course, we have known that the extinction mechanisms are opposite in low light and in very high light. the above observations show that both moderate and high light enrichment can produce sustainable producer-grazer oscillations. 3.1.3. the solution behavior over different delays and initial states. note that unstable e2 and las e4 can coexist under certain conditions, such as in regions d3 and d4 in fig 1. in this case, system (2.1) presents a bistable phenomenon: a stable periodic solution and a stable positive equilibrium, thus the eventual steady state depends on the initial point. let k = 0.84 and other parameters be the same as those in (3.1). for initial points satisfying x + y < p, solutions converge to a periodic solution at τ = 0; as τ increases, the solution finally goes to e4, shown in fig 4(a) and (b). moreover, whatever the initial values are constant, periodic or monotone functions, when they satisfy x + y < p, solutions may converge to e4 if the periodic solutions collide with the critical line x + y = p. with initial values satisfying x + y > p, solutions directly converge to e4 for both τ = 0 and τ > 0 as shown in fig 4(c) and (d). this tendency is kept when the initial values are constant, periodic or monotone functions satisfying x + y > p. it is pointed out by [17, theorem 17] that under the following parameter values: p = 0.025, ê = 0.8, b = 1.2, d = 0.25, θ = 0.04, q = 0.004, c = 0.8, a = 0.25, µ = 0.003, (3.2) k = 0.585185 is an important threshold value, at which e2 is unstable and system (2.1) without delay has at least one limit cycle around e2, and e3 and e4 merge into a saddle-node equilibrium e3,4 on the critical line x + y = p. besides, solutions always tend to a periodic solution. we are particularly interested in the asymptotic behavior of solutions with different initial values if τ is incorporated into the system. simulation results imply that as τ increases, e3 and e4 leave the critical line x + y = p and separate, e3 is unstable and e4 gains stability. both solutions initiating from the regions x+y < p and x + y > p converge to the periodic solution when τ = 0, but as τ increases, all solutions converge 34 h. zhang, h. wang, and b. niu 0 20 40 60 80 100 120 0 0.02 0.04 0.06 0.08 0.1 τ y transcritical bifurcation 0 20 40 60 80 100 120 0 0.1 0.2 0.3 0.4 0.5 τ y s ad d le − n o d e b if u rc at io n p er io d ic s o lu ti o n 0 5 0 0.2 0.4 τ y h o p f b if u rc at io n transcritical bifurcation (a) (b) 0 20 40 60 80 100 120 0 0.1 0.2 0.3 0.4 0.5 τ y 0 0.05 0.45 0.55 τ y p er io d ic so lu ti o n 16 18 20 0.5337 0.534 τ y s ad d le − n o d e b if u rc at io n transcritical bifurcation 0 20 40 60 80 100 120 −2 0 2 4 6 8 10 12 14 16 x 10 −3 τ y transcritical bifurcation (c) (d) figure 3. the bifurcation diagrams of the grazer density over τ. all other parameter values are given in (3.1), except that k = 0.3, k = 0.7, k = 0.87 and k = 1.3 that correspond to (a)-(d), respectively. solid and dash curves stand for stable and unstable equilibria, respectively. black line represents a boundary equilibrium, magenta curve is e2, green and brown curves are e3 and e4, respectively. blue and red curves are the maxima and minima of amplitudes of periodic solutions, respectively. to e4, see fig 5. when τ = 0, the periodic solution attracts solutions starting from anywhere of the phase plane; introducing τ into the model can increase the maxima of periodic solutions, and once the periodic solution arrives at the critical line x + y = p from the region x + y < p, the solution may converge to e4. 3.1.4. joint effect of τ and p on asymptotic dynamics. by decreasing the ratio of p:c in the producer such that the chemical composition required and captured is unbalanced for the grazer, light enrichment harms the growth of the grazer and even leads to the extinction. naturally, we are curious whether we can change phosphorous to balance the adverse effect of light and significantly facilitate the grazer’s persistence. since p = p θ , where θ reflects the fixed p:c ratio in the grazer, the change of phosphorus availability can be described by varying p. we exhibit the dynamics of system (2.1) on τ −p plane in fig 6. stoichiometric producer-grazer model with maturation delay 35 0 100 200 300 400 500 600 0.4 0.45 0.5 0.55 0.6 t y (t ) τ=1 τ=0 0.2 0.3 0.4 0.5 0.6 0.3 0.35 0.4 0.45 0.5 0.55 x y e 4 (a) (b) 0 50 100 150 200 250 300 0.4 0.5 0.6 t y (t ) τ=1 τ=0 0.2 0.3 0.4 0.5 0.6 0.3 0.4 0.5 0.6 x y e 4 (c) (d) figure 4. initial values satisfy x + y < p in (a) and (b). (a) the solution starting from (0.2715, 0.5224) converges to a stable periodic solution when τ = 0 and tends to e4 when τ = 1. (b) when τ = 1, under distinct initial values (0.2665, 0.5224), (0.2665 − 0.2 sin(2t), 0.5224 − 0.2 sin(2t)), (0.2665 + 0.3t, 0.5224 + 0.3t) and (0.2665 − 0.5t, 0.5224 − t), all solutions converge to e4. the blue, magenta, green and black curves correspond to the above initial values, respectively. initial values are true for x + y > p in (c) and (d). (c) both solutions initiating from (0.3648, 0.6522) tend to e4 for τ = 0 and τ = 1. (d) when τ = 1 and initial values are taken as (0.3648, 0.6522), (0.3648 + 0.3 sin(2t), 0.6522 + 0.5 sin(2t)), (0.3648 + 0.3t, 0.6522 + 0.3t) and (0.3648 − 0.5t, 0.6522 − t), respectively, e4 keeps an attractor. the blue, magenta, green and black curves are solutions with the above initial values, respectively. it can be observed that system (2.1) admits the unique positive equilibrium e2 as τ and p vary simultaneously. moreover, when parameter values are in the region between two black vertical lines, e2 is unstable driven by hopf bifurcation. the increase of phosphorous availability in an ecosystem can increase the chance of grazer’s survival. we further claim that increasing phosphorous can weaken the negative effect of poor-food quality caused by increasing light intensity. this is based on the fact that the survival region remarkably reduces as k increases to extreme as shown in fig 1, however, 36 h. zhang, h. wang, and b. niu 0 100 200 300 400 500 600 0.4 0.42 0.44 0.46 0.48 t y (t ) τ=3 τ=0 0 100 200 300 400 0.35 0.4 0.45 0.5 0.55 t y (t ) τ=3 τ=0 (a) (b) figure 5. (a) the initial point satisfies x + y < p with (0.1636, 0.4469), the solution converges to a stable periodic solution for τ = 0 and tends to e4 for τ = 3. (b) the initial point satisfies x + y > p with (0.2626, 0.5469). for τ = 0, the solution converges to a stable periodic solution; for τ = 3, the solution eventually tends to e4. comparing (b) and (c) in fig 6, we observe this tendency stops when p rises. phosphorous availability has little effect on the oscillatory behavior which mainly depends on delay. 3.2. the bifurcation diagrams over k or ê. in this part, we study the relationship between dynamical behavior and two other parameters in system (2.1): light-dependent carrying capacity k and conversion efficiency ê. 3.2.1. the bifurcation diagrams of the grazer over k. the work of [19] and the above discussions have shown that the light-dependent carrying capacity k is a key factor on the grazer’s fate. we sketch the bifurcation diagrams of the grazer with respect to k in fig 7, where τ = 5 in (a) and τ = 100 in (b). in fig 7 (a), when 0 < k < 0.2732 and k > 1.3459, system (2.1) has no positive equilibria, which suggests the grazer cannot survive in such two scenarios. the extinction of grazer for 0 < k < 0.2732 is caused by the lack of producer, while the grazer can not persist for k > 1.3459 because of a mismatch between the nutrient content in the producer and the nutrient demand of the grazer. the rosenzweig’s paradox of enrichment [27] emerges in system (2.1) as k varies in the interval (0.2732, 0.8186): the grazer increases steadily for 0.2732 < k < 0.6543. at k = 0.6543, e2 loses its stability and a family of periodic solutions bifurcate from it, then the amplitude of the periodic solution gradually increases with k. as k increases to k = 0.8186, the periodic solution bursts into a heteroclinic orbit, and two new equilibria appear: unstable e3 and stable e4. as k further increases, the grazer density keeping on e4 declines. in other words, higher light intensity leads to lower grazer biomass, which is consistent with the results in [19]. finally, the grazer becomes extinct at k = 1.3459 caused by a transcritical bifurcation. it can also be seen that e2 and e3 disappear at k = 0.985 through a saddle-node bifurcation. according to fig 7 (b), we observe that the grazer survives only for 0.5680 < k < 0.9501. as k varies from 0.5680 to 0.6586, the grazer density rises at stable e2, then two new equilibria, e3 and e4, appear that coexist with e2 as 0.6586 < k < 0.7272, where the grazer density will increase if its initial value is small, but it will decrease if its initial value is large. at k = 0.7272, e2 collides with e3 and then disappears. the grazer continues to persist until k = 0.9501. stoichiometric producer-grazer model with maturation delay 37 τ p 0 50 100 150 0 0.2 0.4 0.6 0.8 1 1.2 coexistence region (at e 2 ) periodic solution exists τ p 0 50 100 150 200 0 0.5 1 1.5 2 2.5 periodic solution exists coexistence region (at e 2 ) (a) (b) τ p 0 50 100 150 200 250 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 periodic solution exists coexistence region (at e 2 ) (c) figure 6. the bifurcation diagrams on τ − p plane. the red curve is determined by τ = 1 µ ln cêk d(a+k) ; the blue curve stands for p(τ) = b c [1 − x ∗(τ) k ][a + x∗(τ)] + x∗(τ) with x∗(τ) = ad cêe−µτ−d; the green one is p(τ) = 1 4bck [ (b + de µτ ê )2k2 + 2ab(b + de µτ ê ) + (ab)2 ] ; the magenta curve represents p(τ) = d(k+a) cê eµτ . the black vertical line is hopf bifurcation line. there are periodic solutions bifurcating from e2 when parameter values are in the region between two vertical lines. parameter values are given in (3.1) except p varies and (a) k = 0.7, (b) k = 2, (c) k = 4. comparing with fig 4 in [19], we find that in the absence or presence of a small delay in system (2.1), the grazer density changes following a similar route, but the ranges of light intensity supporting the grazer’s persistence are different. the persistence window of light intensity seems narrowed by the maturation delay. from the mathematical point of view, delay indeed can induce richer dynamics, for example, the saddle-node bifurcation occurring at e2 and e3. according to fig 7, we observe that the increase of delay will reduce the survival chance of the grazer, and sustainable oscillations disappear for a very large delay. 3.2.2. the bifurcation diagrams over ê and k. mathematical models often assume the food assimilation of the grazer is a constant. in reality, the conversion efficiency depends on a variety of factors such as cell morphology and colony architecture [7]. some studies have pointed out that the paradox of enrichment 38 h. zhang, h. wang, and b. niu 0 0.4 0.8 1.2 1.6 2 0 0.2 0.4 k y h o p f b if u rc at io n p er io d ic s o lu ti o n s ad d le − n o d e b if u rc at io n saddle−node bifurcation transcritical bifurcation transcritical bifurcation 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 k y s ad d le − n o d e b if u rc at io n s ad d le − n o d e b if u rc at io n transcritical bifurcation transcritical bifurcation (a) (b) figure 7. the bifurcation diagrams of the grazer with respect to k. all other parameter values are given in (3.1) and τ is fixed at τ = 5 for (a) and τ = 100 for (b). all curves have the same interpretations as those in fig 3. shares such an assumption, see [1, 6, 11] and references therein. how does the conversion efficiency affect the dynamics of system (2.1)? motivated by this question, we draw a bifurcation diagram in fig 8(a) to illustrate the existence and stability of positive equilibria on ê−k plane. the coexistence region of producer and grazer is bounded by red and magenta curves. the coexistence window of light intensity becomes wider as ê increases. however, the grazer’s extinction still occurs when the light intensity is too high or too low. 0.45 0.5 0.55 0.6 0.65 0.7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 k ê no positive equilibrium no positive equilibrium e 4 e 2 , e 3 ,e 4 e 2 , e 3 ,e 4 transcritical bifurcation curve saddle−node bifurcation curve transcritical bifurcation curve e 2 (stable) hopf bifur− cation curve e2 (unstable) 0.4 0.45 0.5 0.55 0.6 0.65 0.7 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 y h o p f b if u rc at io n p er io d ic s o lu ti o n s ad d le − n o d e b if u rc at io n ê transcritical bifurcation (a) (b) figure 8. (a) the two-dimensional bifurcation diagram on ê−k plane. take τ = 5 and other parameter values given in (3.1), except that ê varies. the red curve is determined by k(ê) = x∗(ê); the blue curve stands for k(ê) = x∗(ê)/[1 − c(p − x∗(ê))/b(a + x∗(ê))]; the magenta curve is given by k(ê) = cêp d e−µτ − a; the green one is k = k+3 (ê); the black one is hopf bifurcation curve at e2. here, x ∗(ê) = ad/(cêe−µτ − d) and k+3 (ê) is defined in (2.9). (b) the bifurcation diagram of the grazer with respect to ê. the curves have the same meanings as those in fig 3. stoichiometric producer-grazer model with maturation delay 39 at medium light, increasing ê can generate or remove positive equilibria, destabilize the system. for example, a bifurcation diagram describing the grazer density over ê is depicted in fig 8(b) with k = 0.7. here, system (2.1) has a stable boundary equilibrium for ê < 0.4252. as ê varies from 0.4252 to 0.4861, system (2.1) has three positive equilibria: e2, e3 and e4, where two stable equilibria coexist. bistability implies that the grazer density converges to which one stable equilibrium depending on its initial value. at ê = 0.4861, unstable e3 and stable e4 disappear simultaneously through a saddle-node bifurcation. as ê further increases, the grazer density rises steadily until ê = 0.5726, where stable e2 loses its stability to a periodic solution. the amplitude of the periodic solution increases as ê increases. in conclusion, the grazer goes extinct if the conversion efficiency is less than 0.4252 and persists for 0.4252 < ê < 0.7. however, we want to claim that the persistence of the grazer can be threatened when the limit cycle has too large amplitude because the low point will be so small that a tiny perturbation/stochasticity will drive it to extinction. consistent with intuition, increasing the conversion efficiency of a grazer is generally beneficial for its survival, and our study further indicates subtle dependence of the grazer’s dynamics on the conversion efficiency. 3.3. period doubling route to chaos. we have presented some theoretical and numerical results under assumption (a) in the above discussions. actually, system (2.1) with f(x) = cx a+x can exhibit more complicate dynamics when ignoring the assumption, for example, the period doubling bifurcation route to chaos. we choose the following values of parameters to describe such phenomenon. p = 0.025, ê = 0.8, b = 1.2, d = 0.25, θ = 0.048, k = 2, µ = 0.004, a = 0.25, c = 0.81. (3.3) fixing τ = 10, we find that when 0.0125 < q < 0.1490, there is at least one positive equilibrium e5 = (x5,y5) with x5 = 0.1677, and y5 = 1 2 [ b c (a + x5) + p− √ m ] , m = [ b c (a + x5) + p ]2 − 4b c ( p− qx5 θ ) (a + x5). moreover, system (2.1) takes place the chaos routed by period doubling near e5 as q decreases in the interval [0.0238, 0.034], see fig 9(a). here we choose the component x of e5 as a poincáre section, and draw the change of the solution on the poincáre section over q. when q > 0.032, e5 is las. when 0.0288 < q < 0.032, e5 loses its stability to a stable periodic solution with period 1. when 0.0263 < q < 0.0288, the period-1 solution loses its stability to a stable periodic solution with period 2. as 0.0238 < q < 0.0263, the period-2 solution loses its stability to a stable periodic solution with period4, etc., and finally, e5 becomes chaotic. at q = 0.0238, the chaotic attractor suddenly disappears, which may be caused by the collision between the attractor and a periodic solution on the basin boundary of the attractor [23]. as 0.0207 < q < 0.0238, the solution converges to a stable period-1 solution. as q < 0.0207, the solution converges to a boundary equilibrium. the periodic solutions with period 1, 2 and 4 are shown in fig 10, respectively. when the solution always satisfies x + y < p, there is no period doubling route to chaos near e5, see fig 9(b). here, the maxima and minima of the solutions are drawn. it can be seen that e5 is las for q > 0.032, then it loses its stability to a period-1 solution that is similar to fig 10(a) or (b), we do not show it here. for system (2.1) without delay, the period doubling route to chaos is not observed in this paper. as far as we know, chaotic dynamics from lke model without or with delay is scarce, while it has been frequently displayed in communities consisting of two or more species, see [20, 31, 29, 30] for the lke model incorporating the maturation time of the grazer and the restriction of carbon and phosphorus elements in the producer, we see that the decrease of the minimal p:c in the producer can 40 h. zhang, h. wang, and b. niu 0.02 0.022 0.024 0.026 0.028 0.03 0.032 0.034 0.28 0.3 0.32 0.34 0.36 q y positive equilibrium 0.015 0.02 0.025 0.03 0.1 0.2 0.3 0.4 q y positive equilibrium (a) (b) figure 9. the bifurcation diagrams of the grazer with respect to q. (a) the solution on poincáre section of system (2.1) with delay and the minimum function in the second equation. (b) the solution of system (2.1) with delay and min { 1, p−y(t− τ) x(t− τ) } = 1. here the red line is e5 with the dash part being unstable and the solid part being las. cause chaotic oscillations of the producer-grazer population by many times binary decisions. this will make it impossible to predict the long-term population trajectories in time. 4. biological applications in the previous sections, we have observed richer dynamics in system (2.1) through theoretical and numerical analysis. in this section, we will describe the implications of some interesting qualitative dynamical behaviors in population prediction. time delays often destabilize an internal equilibrium and produce regular oscillations in population size for prey-predator systems, see [8, 9, 12]. however, fig 4(a) shows that, a properly large maturation delay can make the limit cycle vanish and drive the solution to converge to an equilibrium, which implies that a suitable delay can stabilize a prey-predator system. moreover, for a small delay, the rosenzweig’s paradox of enrichment induced by light intensity occurs in our model, which is similar to the existing studies on lke model in [17, 37]. while large delay impedes such phenomenon, and results in the coexistence of two different las positive equilibria, see fig 7. with an intermediate light input, a transition of population size from one coexistence state to another one may occur. it seems that the periodic oscillations are harmful for the ecological balance of a predator-prey system, but this dynamical behavior corresponds to predictable evolution in population, see [11] and the references therein for more applications. therefore, analyzing periodic behavior of system (2.1) is an important part in this paper. somehow surprisingly, in the presence of time delay, numerical analysis on the periodic solutions near the critical line x+y = p exhibits the period-doubling route to chaos, see subsection 3.3. the chaotic behaviour is related to the irregular fluctuations and variability in nature, which can be caused by many environmental factors. in this paper, we find that the stoichiometric producer-grazer system can be chaotic via period-doubling route. furthermore, such chaos can also be controlled as fig 9(a) shows that chaotic oscillations will disappear when the minimum p:c ratio in producer continues to decrease. this provides an appropriate interpretation regarding the irregular fluctuations in producer and grazer with stoichiometry. although there are numerous results on stoichiometric producer-grazer model with maturation delay 41 0.1 0.14 0.18 0.22 0.26 0.26 0.28 0.3 0.32 0.34 0.36 x y 0.1 0.15 0.2 0.25 0.3 0.26 0.28 0.3 0.32 0.34 0.36 x y (a) (b) 0.1 0.15 0.2 0.25 0.3 0.26 0.28 0.3 0.32 0.34 0.36 x y 0.1 0.15 0.2 0.25 0.3 0.28 0.3 0.32 0.34 0.36 x y (c) (d) figure 10. system (2.1) has a stable period-1 solution when (a) q = 0.0315, (b) q = 0.0295. (c) system (2.1) has a stable period-2 solution when q = 0.0275. (d) system (2.1) has a stable period-4 solution when q = 0.0259. the black line is x+y = p. the dynamics of population models with delays, the chaotic dynamics is uncommon, and the insights concerning the onset and control of chaos require further exploration. for our model with a small delay, two types of solutions coexist that converge to a limit cycle, or a las positive equilibrium when they have different initial values, see fig 11. moreover, the amplitude of the periodic solution increases over delay, and once the periodic solution collides with the critical line x + y = p, some unclear events take place such that the periodic solution vanishes and the solution finally goes to an equilibrium, see figs 4 and 5. the biological implication behind that may be as follows. the periodic behavior is sensitive to the initial population size, the increased maturation time can lead the change of population size. thus, with these population sizes being the new initial point, the solution finally converges to a stable constant state that is robust under tiny perturbation. in fact, our model may work as a theoretical interpretation for the switch of plankton abundances between a relatively stable state and oscillations in one year. there have been some reports reflecting such changes in view of different types of time series of plankton abundances. for instance, fig 5 in [26] shows that abundances of some microplankton like dinoflagellates, or some zooplankton like decapods and pteropods, may oscillate in several months and become relatively stationary in the remaining 42 h. zhang, h. wang, and b. niu period. when the plankton species coexist at a constant state, sudden change of some limiting factors can lead the population size to change, for instance, the increase of temperature or light intensity from april to may causes the significant increase of the abundances of synechococcus, dinoflagellates and copepods, see fig 5 in [26]. it is worth mentioning that the coexistence of a stable constant steady state and a stable limit cycle in plankton systems has been studied in [40], where that is induced by bautin bifurcation, while in this paper, it is the joint effect of delay and stoichiometry. 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.42 0.46 0.5 0.54 0.58 x y e 2 e 4 figure 11. the solutions of system (2.1) with different initial values: the blue one has an initial value as (0.2715, 0.4659), and the initial value of the red one is (0.2715, 0.45). here, k = 0.84, τ = 0.5 and other parameters are the same as those in (3.1). 5. discussion a series of newly emerging stoichiometric population models have captured biological features of lightand nutrient-dependent species growth, but they usually neglected the time taken for various physiological processes, such as the maturation time of the grazer. following lke model formulated in [19] and rigorous analysis of this model in [17, 37], we formulate a dde model and analyze the impact of the maturation delay on dynamical behavior. bifurcation analysis not only reproduces similar behavior as those in [19, 17, 37], but also generates more exciting dynamics beyond the ode results. for instance, delay drives a positive equilibrium to lose its stability to a stable limit cycle whose amplitude increases as delay increases. further increasing delay allows the limit cycle to collide with the critical line x + y = p, then the solution starting from a neighborhood of an unstable positive equilibrium converges to a different positive equilibrium instead of the stable limit cycle, see figs 4 and 5. due to delay and the liebig’s law of the minimum, there are period-2, period-4 solutions and the period doubling route to chaos, see figs 9(a) and 10. through rigorously mathematical analysis, we provide stability conditions for boundary equilibria and existence conditions for positive equilibria. various bifurcation phenomena are presented including transcritical bifurcation, saddle-node bifurcation and hopf bifurcation. we also obtain two types of coexistence: between two stable positive equilibria, between a stable positive equilibrium and a stable limit cycle. taking biologically relevant parameter values, we conduct simulations to demonstrate theoretical results. bifurcation analysis is further carried out to explain the joint effects of delay and light, delay and phosphorus on asymptotic dynamics. the increase of delay can decrease the survival stoichiometric producer-grazer model with maturation delay 43 region of the grazer on τ − k plane. when the system admits a small delay, rosenzweig’s paradox of enrichment holds for an intermediate light intensity, while such phenomenon disappears for a large delay. increasing phosphorus does not change sustainable oscillatory behavior, but it can elevate the survival chance of the grazer. moreover, the bifurcation diagrams of the grazer over maturation time, conversion efficiency and the minimal p:c ratio are sketched to illustrate their impacts on asymptotic dynamics. we further explain periodic solution, chaos and coexistence of a limit cycle and an positive equilibrium form the biological point of view in section 4. these behaviors in population models are not the first studied, and some results have been applied to control the harmful algal bloom [8], manage fisheries [15], and keep the population to evolve in order [30]. we expect that our results can be used in the development of ecological stoichiometry. there are some problems worthy of further study. the first is the non-negativity and boundedness of solutions. in this paper, we only obtain the conditional non-negativity of solutions and the boundedness of x component. due to the nonsmoothness induced by liebig’s law of the minimum, we need phase plane fragmentation and parameter space partitioning, but the delay makes phase space to be infinitedimensional, thus it is extremely challenging to achieve the goal. we pose this as an open mathematical question. another perspective is a full analysis of hopf bifurcation and all possible codimension-two bifurcations occurring at positive equilibria on the critical curve. note that a positive equilibrium on the curve has two different governing equations on two sides of the curve, thus the fréchet derivative of the functional differential equation at the equilibrium does not exist. hence, one cannot obtain the linear stability of the equilibrium by the distribution of corresponding eigenvalues. as pointed out in [17], bifurcation phenomena may widely exist near the equilibrium on the critical curve for the stoichiometric models, however, the minimum functions make eigenvalue analysis, normal form method and center manifold theory [14, 10, 13] inapplicable. therefore, novel mathematical approaches are needed to rigorously treat such special bifurcations that widely exist in many emerging non-smooth dynamical systems. acknowledgements we are grateful to two anonymous referees for valuable comments and suggestions which greatly improved the original manuscript. appendix here, we study the stability of positive equilibria that are not on the critical line y = p − x. to simplify the analysis, let e∗ = (x∗,y∗) be one positive equilibrium of system (2.1). linearizing system (2.1) at e∗ yields dx(t) dt = x∗fx(x ∗,y∗)x(t) + x∗fy(x ∗,y∗)y(t), dx(t) dt = −dy(t) + qxt(y ∗,x∗,y∗)x(t− τ) + qyt(y ∗,x∗,y∗)y(t− τ), (5.1) where one of the following cases is true: (i) fx = c a + x∗ h ′ (x∗), fy = − c a + x∗ , qxt = cêay∗ (a + x∗)2 e−µτ, qyt = cêx∗ a + x∗ e−µτ. (ii) fx = c a + x∗ h ′ (x∗), fy = − c a + x∗ , qxt = − cê(p−y∗)y∗ (a + x∗)2 e−µτ, qyt = cê(p− 2y∗) a + x∗ e−µτ, and h ′ (x∗) = b(k−a−2x∗) kc . therefore, the corresponding characteristic equation is λ2 + a1(τ)λ + a2(τ)λe −λτ + a3(τ) + a4(τ)e −λτ = 0, (5.2) 44 h. zhang, h. wang, and b. niu with a1(τ) = d−x∗fx(x∗,y∗), a2(τ) = −qyt(y ∗,x∗,y∗), a3(τ) = −dx∗fx(x∗,y∗), a4(τ) = x ∗[fx(x∗,y∗)qyt(y∗,x∗,y∗) −fy(x∗,y∗)qxt(y∗,x∗,y∗)]. when τ = 0, eq. (5.2) reads λ2 + [a1(0) + a2(0)]λ + a3(0) + a4(0) = 0. (5.3) obviously, e∗ is las provided that a1(0) + a2(0) > 0 and a3(0) + a4(0) > 0. system (2.1) undergoes a hopf bifurcation at e∗ as a1(0) + a2(0) = 0 and a3(0) + a4(0) > 0. in what follows, we assume that a1(0) + a2(0) > 0 and a3(0) + a4(0) > 0 always hold and investigate the stability change of positive equilibria induced by τ. one can check that λ = 0 is not a root of (5.2) for all τ > 0. assume that λ = iω (ω > 0) is a root of (5.2) for some τ > 0. substituting it into (5.2) and separating the real and imaginary parts, we have cos ωτ = − a1(τ)a2(τ)ω 2 + a4(τ)(a3(τ) −ω2) a24(τ) + a 2 2(τ)ω 2 , sin ωτ = a1(τ)a4(τ)ω −a2(τ)ω(a3(τ) −ω2) a24(τ) + a 2 2(τ)ω 2 . this leads to f(ω,τ) := ω4 + [a21(τ) −a 2 2(τ) − 2a3(τ)]ω 2 + [a23(τ) −a 2 4(τ)] = 0. (5.4) set ∆̃(τ) = [a21(τ) −a 2 2(τ) − 2a3(τ)] 2 − 4[a23(τ) −a 2 4(τ)]. it is simple to see that if (h1) a21(τ) −a 2 2(τ) − 2a3(τ) < 0, a 2 3(τ) > a 2 4(τ) and ∆̃(τ) > 0, is satisfied, then (5.4) has two differential positive roots given by ω±(τ) = 1 √ 2 [ a22(τ) + 2a3(τ) −a 2 1(τ) ± √ ∆̃(τ) ]1/2 . (5.5) moreover, if a21(τ) −a 2 2(τ) − 2a3(τ) < 0, a 2 3(τ) > a 2 4(τ) and ∆̃(τ) = 0, is true, then ω+(τ) = ω−(τ). if a 2 3(τ) < a 2 4(τ) is true, then only ω+(τ) can exist. otherwise, (5.4) has no positive roots. let the set i be i = {τ > 0 : (h1) or a23(τ) < a 2 4(τ) is satisfied}. when i is nonempty and ω(τ) > 0 solves (5.4) for some τ ∈ i, we can define θ ∈ [0, 2π) by cos θ(τ) = − a1(τ)a2(τ)ω 2(τ) + a4(τ)(a3(τ) −ω2(τ)) a24(τ) + a 2 2(τ)ω 2(τ) , sin θ(τ) = a1(τ)a4(τ)ω(τ) −a2(τ)ω(τ)(a3(τ) −ω2(τ)) a24(τ) + a 2 2(τ)ω 2(τ) . consequently, ω(τ)τ = θ(τ) + 2nπ and iω(τ∗) is a purely imaginary root of (5.2) if and only if τ∗ is the zero of functions sn(τ) with sn(τ) = τ − θ(τ) + 2nπ ω(τ) , τ ∈ i, n ∈ n0 = n∪{0}. it can be proved that sn(τ), n ∈ n0 are continuous and differentiable on i, see [5] for details. then the result established by [5] is followed. stoichiometric producer-grazer model with maturation delay 45 lemma 5.1. (i) assume sn(τ∗) = 0 for some τ∗ ∈ i and n ∈ n0. then eq. (5.2) has at least a pair of simple purely imaginary roots λ = ±iω(τ∗). (ii) let δ(τ∗) = sign { dreλ dτ ∣∣∣ λ=iω(τ∗) } = sign { ∂f(ω(τ∗),τ∗) ∂ω } sign { dsn(τ) dτ ∣∣∣ τ=τ∗ } . this pair of simple purely imaginary roots cross the imaginary axis from left to right if δ(τ∗) > 0 and from right to left if δ(τ∗) < 0. remark 5.1. note that sn(τ) > sn+1(τ) for all τ ∈ i, which implies that if s0(τ) = 0 has no root in i, then sn(τ) = 0 has no root in i for all n ∈ n0. therefore, the real parts of roots of (5.2) remain unchanged for τ ≥ 0. remark 5.2. a direct calculation yields that at τ = τ∗ ∈ i, ∂f(ω(τ∗),τ∗) ∂ω = ±2ω(τ∗) √ ∆̃(τ∗). therefore, if ω(τ∗) = ω+(τ∗), then δ(τ∗) = sign { dsn(τ) dτ ∣∣∣ τ=τ∗ } ; if ω(τ∗) = ω−(τ∗), then δ(τ∗) = −sign { dsn(τ) dτ ∣∣∣ τ=τ∗ } . applying the above results, we first deal with the stability of e2. through some calculations, we have e2 = (x ∗,h(x∗)), where h(x) = b c ( 1 − x k ) (a + x), x∗ = ad cêe−µτ −d . moreover, a1(τ) = d− bx∗(k −a− 2x∗) k(a + x∗) , a2(τ) = −d, a3(τ) = − dbx∗ k(a + x∗) (k −a− 2x∗), a4(τ) = dbx∗ k(a + x∗) [ k −a− 2x∗ + a x∗ (k −x∗) ] . it follows that a1(0) + a2(0) = − bx∗(0)(k −a− 2x∗(0)) k(a + x∗(0)) , a3(0) + a4(0) = abd(k −x∗(0)) k(a + x∗(0)) , where x∗(0) = ad cê−d > 0. the following proposition is immediate. proposition 5.2. at τ = 0, if ad cê−d < k < ad+acê cê−d , then e2 is las; if k > ad+acê cê−d , then e2 is a source-type equilibrium; if k < ad cê−d , then e2 is a saddle-type equilibrium. in particular, a hopf bifurcation occurs at e2 when k = ad+acê cê−d . for τ > 0, we have a21(τ) −a 2 2(τ) − 2a3(τ) = [ bx∗(k −a− 2x∗) k(a + x∗) ]2 > 0, a23(τ) −a 2 4(τ) = ab2d2 (a + x∗)2 ( 1 − x∗ k )[ 4x∗2 + 3ax∗ k − (a + 2x∗) ] . this means that (5.2) has exactly one pair of purely imaginary roots if (h2) k > 4x∗2 + 3ax∗ a + 2x∗ , is satisfied, and all roots of (5.2) have negative real parts if the inequality of (h2) is reversed. 46 h. zhang, h. wang, and b. niu now, we turn to e3 and e4. denote them as (x∗,h(x∗)), where h(x) = b c ( 1 − x k ) (a + x), and x∗ are two roots of êce−µτ [ p− b c ( 1 − x k ) (a + x) ] −d(a + x) = 0. we similarly obtain a1(τ) = d− bx∗(k −a− 2x∗) k(a + x∗) , a2(τ) = −d + êbe−µτ ( 1 − x∗ k ) , a3(τ) = −dbx∗(k −a− 2x∗) k(a + x∗) , a4(τ) = −bx∗ k(a + x∗) [ d(a + x∗) − êbe−µτ (k −a− 2x∗) ( 1 − x∗ k )] . then a1(0) + a2(0) = êb ( 1 − x∗(0) k ) − cx∗(0) a + x∗(0) h ′ (x∗(0)), a3(0) + a4(0) = − cx∗(0)h ′ (x∗(0)) a + x∗(0) cêy∗(0) a + x∗(0) − cx∗(0)h ′ (x∗(0)) a + x∗(0) cê(p−y∗(0))y∗(0) (a + x∗(0))2 , = c2êx∗(0)y∗(0) (a + x∗(0))2 [ −h ′ (x∗(0)) − p−y∗(0) a + x∗(0) ] . due to dx + cêe−µτy = cêe−µτp − ad on line l2, we see p−y∗(0) a+x∗(0) = d cê . at e3, the slope of parabola y = h(x) is larger than zero for all τ ≥ 0, thus, h ′ (x∗(0)) > 0, then a3(0) + a4(0) < 0. therefore, e3 is unstable when τ = 0. at e4, again using the slope of parabola y = h(x) and line l2, we find h ′ (x∗(0)) < 0 and −h ′ (x∗(0)) − dcê > 0, then a1(0) + a2(0) > 0 and a3(0) + a4(0) > 0 is confirmed. thus, e4 is always las when τ = 0. considering τ > 0, we have a21(τ) −a 2 2(τ) − 2a3(τ) > [bx∗(k −a− 2x∗) k(a + x∗) ]2 > 0, a23(τ) −a 2 4(τ) = − ( bx∗ a + x∗ )2( 1 − x∗ k )[( 1 − 2x∗ + a k ) êbe−µτ + d ] {( 1 − x∗ k )[( 1 − 2x∗ + a k ) êbe−µτ + d ] − 2d ( 1 − 2x∗ + a k )} . noticing that 2x∗ + a > dk+êbke−µτ êbe−µτ , we have( 1 − 2x∗ + a k ) êbe−µτ + d < 0, and a3(τ) 6= a4(τ). this shows λ = 0 is not an eigenvalue and e3 remains the saddle-type of equilibrium. in addition, (5.2) has a pair of purely imaginary roots if and only if (h3) ( 1 − x∗ k )[( 1 − 2x∗ + a k ) êbe−µτ + d ] − 2d ( 1 − 2x∗ + a k ) < 0. we define i1 = {τ ≥ 0 : e2 exists, (h2) is true}, i2 = {τ ≥ 0 : e4 exists, (h3) is true}, and conclude the stability of positive equilibria e2, e3 and e4 as follows. theorem 5.3. (i) assume ad cê−d < k < ad+acê cê−d . stoichiometric producer-grazer model with maturation delay 47 (i-1) if i1 is empty or s0(τ) = 0 has no positive root in i1 (is nonempty), then e2 is las for τ ≥ 0. (i-2) for n ∈ n0, denote jn = {τn : τn ∈ i1,sn(τn) = 0}. assume dsn(τ) dτ ∣∣∣ τ=τn 6= 0 and jn1 ∩jn2 = ∅ for n1 < n2. collect these roots in the set j = ⋃ n∈n0 jn = {τ0,τ1, ...,τm} with τi < τi+1, 0 ≤ i ≤ m− 1. e2 is las when τ ∈ ([0,τ0)∪(τm,∞))∩i1, and is unstable when τ ∈ (τ0,τm). when τ = τi ∈ j, system (2.1) takes place a hopf bifurcation at e2. (ii) e3 is unstable whenever it exists. (iii) if i2 is empty or s0(τ) = 0 has no positive root in i2 (is nonempty), then e4 is las for τ ≥ 0. moreover, the statement of (i2) is true for e4 when replacing i1 by i2. we take e2 as an example to show the stability change over τ. still use the parameter set (3.1), by calculation, ad cê−d = 0.2648 and ad+acê cê−d = 0.7797. the proposition 5.2 indicates that for τ = 0, e2 is las as 0.2648 < k < 0.7797 and system (2.1) possesses periodic solutions near e2 as k > 0.7797. as τ > 0, it can be checked that condition (h2) is satisfied at e2, and then the graph of sn on i1 is drawn in fig 12(a). we see that s0 has two zeros: τ + 0 = 1.2367 and τ − 0 = 21.3813, and sn, n ∈ n has no zeros, so j = j0 = {1.2367, 21.3813}. in addition, we have ds0(τ) dτ ∣∣∣ τ=τ + 0 = 0.9059 > 0, ds0(τ) dτ ∣∣∣ τ=τ − 0 = −14.1271 < 0. accordingly, when τ ∈ [0,τ+0 ), e2 is las; when τ ∈ (τ + 0 ,τ − 0 ), e2 is unstable; when τ ∈ [τ − 0 , τ̂), τ̂ is the zero of k− 4x ∗2(τ)+3ax∗(τ) a+2x∗(τ) , e2 regains its stability; when τ = τ ± 0 , system (2.1) occurs hopf bifurcations. the details are illustrated in fig 12(b), here, we show the change of amplitudes of solutions over τ. 0 5 10 15 20 25 −50 −40 −30 −20 −10 0 10 20 τ s n s 0 s 1 τ 0 + τ 0 − 0 5 10 15 20 25 30 0.35 0.4 0.45 0.5 0.55 τ y τ 0 + τ 0 − (a) (b) figure 12. 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[40] h. zhang and b. niu, dynamics in a plankton model with toxic substances and phytoplankton harvesting, internat. j. bifur. chaos appl. sci. engrg. 30 (2020), 2050035. hua zhang, department of mathematics, harbin institute of technology, weihai, shandong, 264209, p. r. china email address: zhanghua.math@foxmail.com hao wang, department of mathematical and statistical sciences, university of alberta, edmonton, alberta, t6g 2g1, canada email address: hao8@ualberta.ca ben niu, corresponding author, department of mathematics, harbin institute of technology, weihai, shandong, 264209, p. r. china email address: niu@hit.edu.cn 1. introduction 2. basic analysis 2.1. nonnegativity and boundedness 2.2. stability of boundary equilibria 2.3. existence and stability of positive equilibria 3. numerical simulations 3.1. biological mechanisms related to 3.2. the bifurcation diagrams over k or 3.3. period doubling route to chaos 4. biological applications 5. discussion acknowledgements appendix references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 4, number 1, march 2023, pp.1-11 https://doi.org/10.5206/mase/15397 on a new generalized tsallis relative operator entropy lahcen tarik, mohamed chergui, and bouazza el wahbi abstract. in this paper, we present a generalization of tsallis relative operator entropy defined for positive operators and we investigate some related properties. some inequalities involving the generalized tsallis relative operator entropy are pointed out as well. 1. introduction the relative entropy plays an important role in many areas. in the classical information theory, it serves as a notion to measure the difference between two probability distributions. for two discrete probability distributions p = (p1,p2, . . . ,pn) and q = (q1,q2, . . . ,qn), the relative entropy h(p |q) is defined as follows [12] h(p|q) = i=n∑ i=1 pi log pi qi . for q = ( 1 n , 1 n , . . . , 1 n ), we get h(p |q) = log n−ss(p), where ss(p) := − i=n∑ i=1 pi log pi stands for the famous shannon entropy. it represents a fundamental tool that caused an enormous change in studying many fields like physical quantum systems and modern communication. in [11], the authors provided a generalization for the entropy concept redefined by tsallis in [17]. namely, for a discrete probability distribution p of a random variable the tsallis entropy is defined as follows tq(p) ≡− i=n∑ i=1 p q i logq ( pi ) , where logq refers to the q-logarithmic function defined by the following formula logq(x) = x1−q − 1 1 −q , for any nonnegative real numbers x and q 6= 1. given the growing diffusion of the use of entropy, many studies have been interested in generalizing this notion to positive operators. to give an overview, let us start by recalling some notions and fixing some notations that will be used in the rest of this article. let h be a complex hilbert space endowed with an inner product 〈., .〉. b(h) will stand for the c∗algebra of all bounded linear operators acting on h. an operator a ∈ b(h) is called positive, in brief a ≥ 0, if a is selfadjoint and 〈ax,x〉 ≥ 0 for all x ∈ h. we denote by b+(h) the closed cone of all received by the editors 23 october 2022; accepted 10 february 2023; published online 20 march 202. 2010 mathematics subject classification. primary 54c70, 94a17, 47a63. key words and phrases. tsallis relative operator entropy, generalized tsallis relative operator entropy. 1 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15397 2 l. tarik, m. chergui, and b. el wahbi positive operators in b(h) and b+∗(h) the open cone of all positive invertible operators in b(h). for a,b ∈b(h) selfadjoint, we set a ≤ b to mean that b −a ∈b+(h). the relative operator entropy s(a|b) was introduced by fujii and kamei in [3] by the following expression s(a|b) = a 1 2 ( log a− 1 2 ba− 1 2 ) a 1 2 . yanagi et al. provided in [19] a parameterized generalization of the operator s(a|b). the authors introduced the tsallis relative operator entropy tp(a | b) for two operators a,b ∈ b+∗(h) and p ∈ (0, 1] as follows tp(a | b) = a]p b −a p . (1.1) the generalization is to be understood by the following result lim p→0 tp(a|b) = s(a|b). in (1.1), a]p b stands for the well known as p-power mean defined as follows a]pb := a 1 2 ( a− 1 2 ba− 1 2 )p a 1 2 . we recall also the operator means p-weighted arithmetic and p-weighted harmonic, defined for any a,b ∈b+∗(h) respectively by [2, 13, 16] a∇p b := (1 −p)a + pb and a!p b := { (1 −p)a−1 + pb−1 }−1 , where p ∈ [0, 1]. when p = 1 2 the subscript p will be omitted from the above notations. in quantum systems, tp(a|b) is an operator variant of the tsallis entropy [1, 9]. many nice properties of tp(a|b) can be found for example in [4, 5, 6, 7]. in [16], räıssouli et al provided an integral representation of tp(a|b) for p ∈ (0, 1] as follows tp(a|b) = sin pπ pπ ∫ 1 0 ( t 1 − t )p ( a!t b −a t ) dt. (1.2) the definition of p-weighted geometric operator mean a\p b for any p ∈ r by the following formula [10] a\pb := a 1 2 ( a− 1 2 ba− 1 2 )p a 1 2 , allows to generalize (1.1) for any p ∈ r∗ by setting, tp(a|b) = a\p b −a p . in [9], the authors proved that for any unitary operator u ∈b+∗(h), tp(u ∗au|u∗bu) = u∗tp(a|b)u. (1.3) fujii et kamei [3] and [20] provided the following upper and lower bounds of tsallis relative operator entropy, a−ab−1a ≤ t−p(a|b) ≤ s(a|b) ≤ tp(a|b) ≤ b −a. generalized tsallis relative operator entropy 3 by using the hermite-hadamard’s inequality, moradi et al. established in [15] for any p ∈ (0, 1] the following results, a 1 2 ( a− 1 2 ba− 1 2 + ih 2 )p−1 ( a− 1 2 ba− 1 2 − ih ) a 1 2 6 tp(a | b) 6 1 2 (a]pb −a\p−1b + b −a) . (1.4) from (1.4), the following inequalities can be deduced [9, 20] a−ab−1a ≤ tp(a|b) ≤ b −a. the present work is centered on generalizing the operator tp(a|b) and on investigating some related properties. some facets of our generalization are also highlighted. the rest of this paper is organized as follows. in section 2, we present a generalization for the tsallis relative operator entropy and we determine some of its properties. in section 3, after establishing some hermite-hadamard type inequalities, we provide some inequalities which involve the generalized tsallis relative operator. 2. generalized tsallis relative operator entropy we begin this section by providing the definition of a generalized tsallis relative operator entropy. then, we will deal with establishing some related properties. definition 2.1. let a,b ∈ b+∗(h), p ∈ r∗ and 0 ≤ ν,µ ≤ 1. we define the generalized tsallis relative operator entropy t(p,µ,ν)(a|b) by t(p,µ,ν)(a|b) = a\p(a∇µb) −a\p(a∇νb) p . (2.1) for the particular case µ = 1 and ν = 0, we get t(p,1,0)(a | b) = tp(a | b). this confirms that t(p,µ,ν)(a|b) represents effectively a generalization of tp(a|b). our first result concerning the generalized tsallis relative operator entropy t(p,µ,ν)(a|b) reads as follows. proposition 2.1. let a,b ∈b+∗(h), p ∈ r∗ and 0 ≤ ν,µ ≤ 1. we have lim p→0 t(p,µ,ν)(a|b) = s (a|a∇µ b) −s (a|a∇ν b) . proof. noticing that t(p,µ,ν)(a|b) = a\p(a∇µb) −a p − a\p(a∇νb) −a p , the desired result can be deduced. � proposition 2.2. let a,b ∈b+∗(h), p ∈ r∗ and 0 ≤ ν,µ ≤ 1. we have i) t(p,µ,ν)(u∗au|u∗bu) = u∗t(p,µ,ν)(a|b)u, for any unitary operator u ∈b+∗(h). ii) t(p,µ,ν) is homogenous, i.e. t(p,µ,ν)(αa|αb) = αt(p,µ,ν)(a|b) for any α > 0. proof. let us notice at first that t(p,µ,ν)(a|b) = tp(a|a∇µb) −tp(a|a∇νb). (2.2) 4 l. tarik, m. chergui, and b. el wahbi so, t(p,µ,ν)(u∗au|u∗bu) = tp(u∗au|(u∗au)∇µ(u∗bu)) −tp(u∗au|(u∗au)∇ν(u∗bu)). combining the formulas (2.2) and (1.3), we get t(p,µ,ν)(u∗au|u∗bu) = tp(u∗au|u∗(a∇µb)u) −tp(u∗au|u∗(a∇νb)u) = u∗ (tp(a|a∇µb) −tp(a|a∇νb)) u = u∗t(p,µ,ν)(a|b)u. using (2.2) and the fact that the operator mean ∇ and tp are homogenous, one can easily deduce the second assertion in the proposition. � the following result provides an integral representation for the generalized tsallis relative operator entropy. theorem 2.3. let a,b ∈b+∗(h). for any p ∈ r\{−1} and a,b ∈ [0, 1] with a < b, we have t(p+1,b,a)(a | b) = ∫ b a [a\p(a∇t b)] ( a−1b − ih ) dt. (2.3) proof. let us note that for all x ∈ r+, the following formula holds∫ b a (1 − t + tx)p(x− 1)dt = (1 − b + bx)p+1 − (1 −a + ax)p+1 p + 1 . so, by theory of functional calculus and substituting x by a −1 2 ba −1 2 , we get∫ b a ( (1 − t)ih + ta− 1 2 ba− 1 2 )p ( a− 1 2 ba− 1 2 − ih ) dt =( (1 − b)ih + ba− 1 2 ba− 1 2 )p+1 − ( (1 −a)ih + aa− 1 2 ba− 1 2 )p+1 p + 1 . (2.4) noticing that a− 1 2 ba− 1 2 − ih = a 1 2 ( a−1ba− 1 2 −a− 1 2 ) , the formula (2.4) is equivalent to ∫ b a ( a− 1 2 ((1 − t)a + tb)a− 1 2 )p a 1 2 ( a−1ba− 1 2 −a− 1 2 ) dt =( a− 1 2 ((1 − b)a + bb)a− 1 2 )p+1 − ( a− 1 2 ((1 −a)a + ab)a− 1 2 )p+1 p + 1 , which leads to (2.3) by multiplying on its both sides by a 1 2 . � remark 2.1. for a = 0,b = 1 and p ∈ r∗, we obtain tp(a | b) = ∫ 1 0 [a\p−1(a∇t b)] ( a−1b − ih ) dt. (2.5) it is worth mentioning that the formula (2.5) provides an integral representation for tp(a | b) more general than the one given by (1.2) stated by the authors in [16, definition 3.1] only for parameters p ∈ (0, 1) . the following results in the ongoing section deal with the monotonicity of t(p,µ,ν)(a|b) according to each of the parameters µ,ν and p. proposition 2.4. let a,b ∈b+∗(h), p 6= 0 and µ,ν ∈ [0, 1]. we have generalized tsallis relative operator entropy 5 i) if a ≤ b (a ≥ b) then t(p,µ1,ν)(a|b) ≤ (≥)t(p,µ2,ν)(a|b) for ν ≤ µ1 ≤ µ2 ≤ 1. ii) if a ≥ b (a ≤ b) then t(p,µ,ν1)(a|b) ≤ (≥)t(p,µ,ν2)(a|b) for 0 ≤ ν1 ≤ ν2 ≤ µ. proof. let µ,ν ∈ [0, 1] and x ≥ 0. if x ≥ 1 (0 < x ≤ 1), the function t 7→ (1 − t + tx)p − (1 −ν + νx)p p is increasing (decreasing) on [ν, 1]. thus, for ν ≤ µ1 ≤ µ2 ≤ 1 it holds (1 −µ1 + µ1x)p − (1 −ν + νx)p p ≤ (≥) (1 −µ2 + µ2x)p − (1 −ν + νx)p p . so, by theory of functional calculus, after replacing x by a− 1 2 ba− 1 2 and multiplying left and right by a 1 2 , we get t(p,µ1,ν)(a|b) ≤ (≥) t(p,µ2,ν)(a|b). the proof of the second statement can be done in a similar way to that of i). � to study he monotonicity of the map p 7−→ t(p,µ,ν)(a|b), we need the following lemma. lemma 2.5. let 0 ≤ ν ≤ µ ≤ 1, p ∈ r and x > 0. we have (p log(1 −µ + µx) − 1)(1 −µ + µx)p − (p log(1 −ν + νx) − 1)(1 −ν + νx)p ≥ 0. proof. we define on (0,∞) the real function by fp(t) = (p log(t)−1)tp and we put α = 1−µ + µx and β = 1 −ν + νx. if α = β, that is µ = ν or x = 1, the desired result is obvious. so, let us consider α 6= β. if 0 < x < 1 then by noticing that α−β = (µ−ν)(x− 1), we get 0 < α < β < 1. using the fact that fp is continuous on [α,β] and differentiable on (α,β), we can deduce by virtue of lagrange’s mean value theorem that there exists c ∈ (α,β) such that fp(α) −fp(β) = (α−β)f ′ p(c), or equivalently (p log(α) − 1)αp − (p log(β) − 1)βp = p2cp−1(α−β) log c. since c ∈ (α,β) then log c < 0 and consequently p2cp−1(α−β) log c ≥ 0. if x ≥ 1, one can follow similar steps used for the previous case. whence, the lemma is proved. � theorem 2.6. let a,b ∈b+∗(h), 0 ≤ ν ≤ µ ≤ 1 and p,q ∈ r∗ with p ≤ q. we have t(p,µ,ν)(a|b) ≤ t(q,µ,ν)(a|b). (2.6) proof. let us put for x > 0 and p ∈ r∗ α = 1 −µ + µx, β = 1 −ν + νx and φµ,ν,x(p) = αp −βp p . we have, d dp φµ,ν,x(p) = (p log(α) − 1)αp − (p log(β) − 1)βp p2 . by lemma 2.5, we have d dp φµ,ν,x(p) ≥ 0. that is p 7−→ φµ,ν,x(p) is increasing on r∗. so, if p ≤ q then φ(p,µ,ν,x) ≤ φ(q,µ,ν,x), 6 l. tarik, m. chergui, and b. el wahbi which implies, by virtue of theory of functional calculus and after substituting x by a− 1 2 ba− 1 2 , the following inequality (a− 1 2 (a∇µ b) a− 1 2 )p − (a− 1 2 (a∇ν b) a− 1 2 )p p ≤ (a− 1 2 (a∇µ b) a− 1 2 )q − (a− 1 2 (a∇ν b) a− 1 2 )q q . multiplying this last inequality by a 1 2 , we get (2.6). � remark 2.2. for µ = 1,ν = 0 and p,q ∈ [−1, 0)∪(0, 1] with p ≤ q, we get the well known inequality [8] tp(a|b) ≤ tq(a|b). this confirms, once more again, the generalization character of t(p,µ,ν)(a|b). 3. inequalities involving t(p,µ,ν)(a|b) in the current section, we aim to determine some estimations for t(p,µ,ν)(a|b). our first result is recited in the following proposition. proposition 3.1. let a,b ∈b(h)+∗, p > 0 and 0 ≤ ν,µ ≤ 1. we have[ a\p (a∇ν b) ] a−1 t(−p,µ,ν)(a|b) ≤ s(a|a∇µb) −s(a|a∇νb) ≤[ a\−p (a∇ν b) ] a−1 t(p,µ,ν)(a|b). (3.1) if p < 0, the inequalities (3.1) are reversed. proof. for p > 0 and y > 0, one can easily check by routine tools of real analysis that y−p − 1 −p ≤ log y ≤ yp − 1 p . so, by setting y = 1 −µ + µx 1 −ν + νx > 0 for x > 0, we obtain (1 −ν + νx)p (1 −µ + µx)−p − (1 −ν + νx)−p −p ≤ log(1 −µ + µx) − log(1 −ν + νx) ≤ (1 −ν + νx)−p (1 −µ + µx)p − (1 −ν + νx)p p . (3.2) thus, by theory of functional calculus, substitution of x by a− 1 2 ba− 1 2 and the following formula( 1 −ν + νa− 1 2 ba− 1 2 )p = a− 1 2 [a\p (a∇νb)] a− 1 2 , allow us to state ( 1 −ν + νa− 1 2 ba− 1 2 )p (1 −µ + µa−12 ba−12 )−p −(1 −ν + νa−12 ba−12 )−p −p ≤ log ( 1 −µ + µa− 1 2 ba− 1 2 ) − log ( 1 −ν + νa− 1 2 ba− 1 2 ) ≤ ( 1 −ν + νa− 1 2 ba− 1 2 )−p (1 −µ + µa−12 ba−12 )p −(1 −ν + νa−12 ba−12 )p p . (3.3) multiplying both sides of inequalities (3.3) by a 1 2 , we deduce the inequalities (3.1). if p < 0, we apply the inequalities (3.1) for −p > 0 to deduce the desired result. � generalized tsallis relative operator entropy 7 remark 3.1. it is worth mentioning that by taking µ = 1,ν = 0 and p ∈ (0, 1] in (3.1), we get particularly the inequalities t−p(a|b) ≤ s(a|b) ≤ tp(a|b), established by furuichi et al in [9, proposition 3.1]. proposition 3.2. let a,b ∈b+∗(h), q > 0, p ∈ [−q,q] and 0 ≤ ν ≤ µ ≤ 1. we have mp,q a− [ a\q(a∇ν b) ] a−1 [ a\−q(a∇µ b) ] ≤[ a\−p(a∇ν b) ] a−1t(p,µ,ν)(a|b) ≤[ a\−q(a∇ν b) ] a−1 [ a\q(a∇µ b) ] + np,q a, (3.4) with, mp,q := q p p+q −1 p + q −q p+q and np,q := q p p−q − 1 p −q q p−q . proof. for −q ≤ p ≤ q and y > 0, a simple study leads to the following inequalities q p p+q − 1 p + q −q p+q −y−q ≤ yp − 1 p ≤ yq + q p p−q − 1 p −q q p−q . whence, choosing y = 1 −µ + µx 1 −ν + νx > 0 for x > 0, we deduce q p p+q − 1 p + q −q p+q − (1 −ν + νx)q(1 −µ + µx)−q ≤ (1 −ν + νx)−p (1 −µ + µx)p − (1 −ν + νx)p p ≤ (1 −ν + νx)−q(1 −µ + µx)q + q p p−q − 1 p −q q p−q . (3.5) changing x by a− 1 2 ba− 1 2 and multiplying both sides of inequalities (3.5) by a 1 2 , we deduce the inequalities (3.4). � remark 3.2. proposition 3.2 provides a generalization of the proposition 3.4 stated in [9]. in fact, taking q = µ = 1,ν = 0 and p ∈ (0, 1] in (3.4), we get for all a,b ∈b+∗(h) a−ab−1a ≤ tp(a|b) ≤ b −a. for further results, the following hermite-hadamard type inequalities will be very useful. theorem 3.3. let f be a convex function on an open interval i ⊆ r. for all [x,y] ⊆ i and for each λ ∈ [0, 1], we have 2µf ((1 −λ)x + λy) ≤ ∫ λ+µ λ−µ f((1 − t)x + ty)dt ≤ 2µ [(1 −λ)f(x) + λf(y)] , (3.6) where µ = min{λ, 1 −λ}. if f is a concave function on i then the inequalities in (3.6) are reversed. proof. if f is a convex function on i, we get f ((1 −λ)x + λy) ≤ 1 2 [ f ((1 −λ− t)x + (λ + t)y) + f ((1 −λ + t)x + (λ− t)y) ] ≤ (1 −λ)f(x) + λf(y). 8 l. tarik, m. chergui, and b. el wahbi integrating this inequalities over t ∈ [0,µ], we obtain µf ((1 −λ)x + λy) ≤ 1 2 ∫ µ 0 f ((1 −λ− t)x + (λ + t)y) dt+ 1 2 ∫ µ 0 f ((1 −λ + t)x + (λ− t)y) dt ≤ µ ((1 −λ)f(x) + λf(y)) . using appropriately the changes of the variables u = λ + t and u = λ− t, it yields µf ((1 −λ)x + λy) ≤ 1 2 ∫ λ+µ λ−µ f ((1 − t)x + ty) dt ≤ µ ((1 −λ)f(x) + λf(y)) , which ends the proof. � remark 3.3. for µ = λ = 1 2 , we find the following well known hermite-hadamard inequalities f ( a + b 2 ) ≤ 1 b−a ∫ b a f(t)dt ≤ f(a) + f(b) 2 . (3.7) it is important to note that the inequalities (3.7) could also be deduced from the generalization pointed out separately in [14] and [18] which differs from the one stated in (3.6). an investigation on the extension of (3.6) to positive operators and the establishment of some applications will be an interesting topic for future work. the following theorem will be very useful to establish some inequalities involving generalized tsallis relative operator entropy. theorem 3.4. let a,b ∈b+∗(h) and λ ∈ [0, 1]. for all p < 0 with p 6= −1 or p > 1, we have 2µ (a\p(a∇λ b)) ≤ ∫ λ+µ λ−µ a\p (a∇t b)dt ≤ 2µ [a∇λ(a\p b)] , (3.8) where µ = min{λ, 1 −λ}. if p ∈ [0, 1], the inequalities in (3.8) are reversed. proof. consider on (0, +∞) the function defined by f(t) = tp, p ∈ (−∞, 0) ∪ (1, +∞). since f is a convex function on (0, +∞), taking x = 1 in (3.6) we get the following inequalities 2µ (1 −λ + λy)p ≤ ∫ λ+µ λ−µ (1 − t + ty)p dt ≤ 2µ (1 −λ + λyp) . (3.9) by theory of functional calculus and replacing y by a −1 2 ba −1 2 , we have 2µ ( (1 −λ)ih + λa −1 2 ba −1 2 )p ≤ ∫ λ+µ λ−µ ( (1 − t)ih + ta −1 2 ba −1 2 )p dt ≤ 2µ [ (1 −λ)ih + λ ( a −1 2 ba −1 2 )p] . (3.10) multiplying both sides of inequalities (3.10) by a 1 2 , we deduce the desired result. � in the following theorem, we will state another main result. theorem 3.5. let a,b ∈ b+∗(h) be two selfadjoint operators, with a ≤ b. for any λ ∈ (0, 1] and µ = min{λ, 1 −λ}, we have 2µ λ [a\p+1(a∇λ b) −a\p(a∇λ b)] 6 t(p+1,λ+µ,λ−µ)(a | b) 6 2µ [b∇λ(a\p+1 b) −a∇λ(a\p b)] , (3.11) generalized tsallis relative operator entropy 9 for all p < 0 with p 6= −1 or p > 1. if p ∈ [0, 1], the inequalities (3.11) are reversed. proof. according to the condition a ≤ b, we can set c = a −1 2 ( a −1 2 ba −1 2 − ih )1 2 a 1 2 . for p < 0 with p 6= −1 or p > 1 and by the use of the theorem 3.4, we get 2µ [a\p(a∇λ b)] 6 ∫ λ+µ λ−µ [a\p (a∇t b)] dt 6 2µ [a∇λ(a\p b)] . thus, 2µc∗ [a\p(a∇λ b)] c 6 ∫ λ+µ λ−µ c∗ [a\p (a∇t b)] cdt 6 2µc∗ [a∇λ(a\p b)] c. by setting d = a −1 2 ba −1 2 , we have c∗ [a\p(a∇λ b)] c = a 1 2 (d − ih) 1 2 ( a −1 2 (a∇λ b)a −1 2 )p (d − ih) 1 2 a 1 2 = a 1 2 (d − ih) 1 2 ((1 −λ)ih + d) p (d − ih) 1 2 a 1 2 = a 1 2 ((1 −λ)ih + d) p (d − ih) a 1 2 = a 1 2 ( a −1 2 (a∇λ b)a −1 2 )p ( a −1 2 ba −1 2 − ih ) a 1 2 = a 1 2 ( a −1 2 (a∇λ b)a −1 2 )p a 1 2 a −1 2 ( a −1 2 b −a 1 2 ) = [a\p(a∇λ b)] (a−1b − ih). furthermore, noticing that for any λ ∈ (0, 1], a−1b = 1 λ [ a −1 2 ( a −1 2 (a∇λ b)a −1 2 ) a 1 2 − (1 −λ)ih ] , it yields c∗ [a\p(a∇λ b)] c = 1 λ [a\p+1(a∇λ b) − (1 −λ)a\p(a∇λ b)] −a\p(a∇λ b) = 1 λ [a\p+1(a∇λ b) −a\p(a∇λ b)] . on the other hand, we have c∗ [a∇λ(a\p b)] c = a 1 2 (d − ih) 1 2 [(1 −λ)ih + λdp] (d − ih) 1 2 a 1 2 = a 1 2 (d − ih) 1 2 [(1 −λ)ih + λdp] (d − ih) 1 2 a 1 2 = a 1 2 [(1 −λ)ih + λdp] (d − ih) 1 2 (d − ih) 1 2 a 1 2 = a 1 2 [(1 −λ)ih + λdp] (d − ih) a 1 2 = [a∇λ(a\p b)] (a−1b − ih) = (1 −λ)b + λ(a\p b)a−1b −a∇λ(a\p b). noticing that a−1b = a −1 2 ( a −1 2 ba −1 2 ) a 1 2 , we obtain c∗ [a∇λ(a\p b)] c = (1 −λ)b + λ(a\p+1 b) −a∇λ(a\p b) = b∇λ(a\p+1 b) −a∇λ(a\p b). finally, by theorem 2.3 we have∫ λ+µ λ−µ c∗ [a\p (a∇t b)] cdt = ∫ λ+µ λ−µ [a\p (a∇t b)] (a−1b − ih)dt = t(p+1,λ+µ,λ−µ). 10 l. tarik, m. chergui, and b. el wahbi for p ∈ [0, 1], using the inverses of inequalities (3.8) and following the same steps used for the proof of (3.11), one can deduce the result. � remark 3.4. inequalities (3.11) provides a generalization for (1.4), in the sense that for p ∈ (0, 1] and λ = 1 2 in (3.11) we find (1.4). the following result provides also an extension for the inequalities (1.4). corollary 3.6. let a,b ∈b(h)+∗ with a ≤ b. for any p ∈ r∗\[1, 2], we have a 1 2  (ih + a−12 ba−12 2 )p − ( ih + a −1 2 ba −1 2 2 )p−1a12 6 2a 1 2  (ih + a−12 ba−12 2 )p − ( ih + a −1 2 ba −1 2 2 )p−1a12 6 tp(a | b) 6 1 2 (a\p b −a\p−1 b + b −a) . (3.12) proof. let p ∈ r∗\[1, 2]. by inequalities (3.11) when taking λ = 1 2 and replacing p by p− 1, we find 2 [a\p(a∇b) −a\p−1(a∇b)] ≤ t(p,1,0)(a | b) ≤ b∇(a\p b) −a∇(a\p−1 b), or equivalently 2a 1 2 [( a −1 2 ( b + a 2 ) a −1 2 )p − ( a −1 2 ( b + a 2 ) a −1 2 )p−1] a 1 2 6 t(p,1,0)(a | b) 6 1 2 (b + a\p b −a−a\p−1 b) . using the relation t(p,1,0)(a | b) = tp(a | b), we deduce the inequalities (3.12). � corollary 3.7. let a,b ∈ b+∗(h) be two selfadjoint operators with a ≤ b. for any λ ∈ ( 0, 1 2 ] and p ∈ r∗ \ [1, 2], we have 2[a\p(a∇λ b) −a\p−1(a∇λ b)] − 2 λ p (b −a) 6 tp(a | a∇2λb) 6 2λ [ b∇λ(a\p b) −a∇λ(a\p−1 b) ] − 2 λ p (b −a). (3.13) for p ∈ [1, 2], the inequalities reverse (3.13) are reversed. proof. as λ ∈ ( 0, 1 2 ] then µ = λ. so, applying the inequalities (3.11) combined with the following formula t(p,2λ,0) = tp (a|a∇2λb) + 2 λ p (b −a), we get (3.13). � acknowledgments the authors would like to thank the anonymous referee(s) for valuable comments and suggestions that have been implemented in the final version of the paper. generalized tsallis relative operator entropy 11 references [1] s. abe, monotonic decrease of the quantum non-additive divergence by projective measurements, phys. lett. a., 312(5-6) (2003), 336-338. [2] i. a. al-subaihi, m. räıssouli, further inequalities involving the weighted geometric operator mean and the heinz operator mean, linear and multilinear algebra, (2021), 1-23. [3] j. i. fujii, e. kamei, relative operator entropy in non-commutative information theory, math. japon., 34 (1989), 341-348. 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[20] l. zou, operator inequalities associated with tsallis relative operator entropy, math. inequal. appl., 18/2 (2015), 401-406. corresponding author, department of mathematics, science faculty, laga-lab ibn tofail university, kenitra, morocco email address: lahcen.tarik@uit.ac.ma department of mathematics, crmef-rsk, eream team, lareami-lab, kenitra, morocco email address: chergui m@yahoo.fr department of mathematics, science faculty, laga-lab ibn tofail university, kenitra, morocco email address: bouazza.elwahbi@uit.ac.ma 1. introduction 2. generalized tsallis relative operator entropy 3. inequalities involving t(p, , )(a|b) acknowledgments references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 4, number 1, 2023, pp.30-39 https://doi.org/10.5206/mase/15775 minimum dominating set for the prism graph family jebisha esther s and veninstine vivik j abstract. the dominating set of the graph g is a subset d of vertex set v , such that every vertex not in v −d is adjacent to at least one vertex in the vertex subset d. a dominating set d is a minimal dominating set if no proper subset of d is a dominating set. the number of elements in such set is called as domination number of graph and is denoted by γ(g). in this work the domination numbers are obtained for family of prism graphs such as prism cln, antiprism qn and crossed prism rn by identifying one of their minimum dominating set. 1. introduction domination in graphs is a wide research area in graph theory. the dominating set of the graph g is a subset d of a vertex set v such that every vertex not in v −d is adjacent to at least one vertex in the vertex subset d [6]. a dominating set d is a minimal dominating set if no proper subset of d is a dominating set. the number of elements in such a set is known as the domination number of graph [7]. the basic definition and details about the domination sets and domination numbers of graphs are discussed in [10]. although the mathematical study of domination in graphs began around 1960, there are some references to domination-related problems about 100 years prior. in 1862, de jaenisch attempted to determine the minimum number of queens required to cover an n×n chess board. with reference to [9] in many fields such as school bus routing, computer communication networks, radio stations, the locating radar station problem, modeling biological networks, facility location problems and coding theory, the domination is applied. in 2021, adel et al. [1] have successfully used the minimum dominating sets (mdsets) method to extract proteins that control protein-protein interaction (ppi) networks, revealing a link between structural analysis and biological functions. motivation many real-time situations can be modeled as graphs. for instance, each floor in a multi-storey building can be modelled as an n− prism graph [12] by considering the corners of the corridor positions in a floor to be vertices and the side brick walls on both sides as connecting edges. also, the vertices located in similar at the corner regions of the hallway are connected to each other. this reflects the structure of prism graph. consider the example of a 4−prism graph obtained from the following floor plan of a rectangular multi-storey building. to strengthen the security system of the building, surveillance cameras may be fixed at various positions. in order to minimize the cost of fixing cameras at various parts of the building, we need some leading positions to cover the whole area. such dominating points can be visualized using graphtheoretical concept of domination. the minimum number of cameras needed to be installed, which is cost-effective, can be achieved through a minimally dominant set. to handle these types of situations, received by the editors 20 january 2023; accepted 10 march 2023; published online 22 march 2023. 2020 mathematics subject classification. 05c38, 05c69. key words and phrases. minimal dominating set, domination number, prism, antiprism. 30 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15775 mds for prism family of graph 31 figure 1. floor plan of a multi-storey building v1 v2 v3v4 u1 u2 u3u4 figure 2: prism graph cl4 domination in graphs can be applied. consequently, it motivates us to study the minimal domination of the family of prism graphs. 2. preliminaries in this section, the literature review and some basic definitions related to our work are given. here we consider only the graphs that are undirected and cycle-structured. because of the richness in applications, huge collection of research works has been carried out in this area dealing with domination of graphs. recently [4] researchers behzad and et al. studied about an infinite family of regular graphs, the generalized petersen graphs. the authors [15] in the year 2019 examined the domination of fibonacci cubes and also they gave the pattern of minimum dominating sets of fibonacci cubes and in the same year [1] al-harere and breesam investigated the domination number of a new graph called as spinner graph. [3] alvarado et al. explored about the domination of tree and established the bound for minimal dominating set of forests. in 2021 [11] liu, chanjuan discussed about the upper and lower bounds of domination number for n maximal outer planar graph. [5] bermudo and others proposed some lower bounds on the domination number of a catacondensed hexagonal system using the number of hexagons and the number of branching hexagons[13]. sarah et al. found certain bounds for the domination number of latin square graphs. a dominating set [9] for a graph g = (v,e) is a subset d of v such that every vertex not in d is adjacent to at least one number of d. the domination number γ(g) is number of vertices in a smallest dominating set for g. 32 jebisha esther s and veninstine vivik j a prism graph [8], denoted by cln called also as circular ladder graph, is a graph corresponding to the skeleton of an n-prism graph had 2n vertices and 3n edges. an antiprism graph [16] is a graph that has one of the antiprism as its skeleton. an n-sided antiprism graph has 2n vertices and 4n vertices. they are regular, polyhedral. an antiprism graph is a special case of a circulant graph ci2n(1, 2). let us denote this graph by qn. let n be a positive even integer of at least 4. an n-crossed prism graph [14] is a graph obtained by taking two disjoint cycle graphs on n vertices, namely cn1 and c n 2 , where v (c n 1 ) = {x1x2, . . . ,xn} and v (cn2 ) = {w1,w2, . . . ,wn}, such that e(cn1 ) = (xixi+1,x1xn) where i = 1, 2, . . . ,n − 1 and e(cn2 ) = (wiwi+1,w1wn) such that i = 1, 2, . . . ,n−1 adding some edges wsxs+1 for s ∈{1, 3, . . . ,n−1} and wtxt−1 for t ∈{2, 4, . . . ,n}. due to many applications of domination in discrete mathematics, coding theory and networking, etc. it still remains an active research field for many decades. also it helps in optimal identification of minimum nodes to cover the entire graph. because of the cyclic structured nature of graphs in this work we have considered this family prism graphs and followed the method of mathematical induction to prove the bounds for domination number. in this paper the domination number for family of prism graphs namely prism graph, antiprism graph and crossed prism graph are obtained. 3. computation of domination numbers for the prism graph family theorem 3.1. the domination number of the prism graph cln, for n ≥ 4, is (1) γ(cln) = n 2 , if n ≡ 0 mod 4 (2) γ(cln) = dn2e, if n ≡ 1 mod 4 (3) γ(cln) = n+2 2 , if n ≡ 2 mod 4 (4) γ(cln) = dn2e, if n ≡ 3 mod 4 proof. let cln be the prism graph with vertex set v and edge set e. the vertex set is given by v = {vj}∪{uj} where j = 1, 2, . . . ,n. the number of vertices v in cln is |v | = 2n and the number of edges in cln is |e| = 3n. one of the minimum dominating set is identified from the following four cases on the various values of n. case 1. when n ≡ 0 mod 4 the minimum dominating set is given by dm = {v1,vi}∪{u3,uk} where i = 4p + 1,k = 4p + 3 and p = 1, 2, . . . , n− 4 4 . we used mathematical induction method [17], to prove that given above set is one of the minimal dominating sets of prism graph when n ≡ 0 mod 4. let n = 4q and q ≥ 1. when q = 1 it gives n = 4 and d1 = {v1,u3} = {vn−3,un−1} induction hypothesis on q: assume that the result is true for the case q = l and then prove that the result is true for q = l + 1. by induction hypothesis the given set dl = {v1,vi}∪{u3,uk} is the minimal dominating set. therefore p = 1, 2, . . . , l− 1, i = 5, 9, . . . , 4l− 3 and k = 7, 11, . . . , 4l− 1. =⇒ dl = {v1,v5,v9, . . . ,v4l−3}∪{u3, . . . ,u4l−1} is true. to prove: when q = l + 1 the result is true. let q = l + 1 =⇒ n = 4l + 4 =⇒ dl+1 = {v1,v5, . . . ,v4l+1}∪{u3, . . . ,u4l+3}. dl+1 = dl ∪d1 = {v1,v5,v9, . . . ,v4l−3}∪{u3, . . . ,u4l−1}∪{v1,u3}∪{v4l+1,u4l+3}. mds for prism family of graph 33 this proves the result is true for q = l + 1. =⇒ |dm| = n 2 therefore γ(cln) = n 2 . case 2. if n ≡ 1 mod 4 then dm = {v1,vi,vn}∪{u3,uk}where i = 4p + 1, k = 4p + 3 and p = 1, 2, . . . , n− 5 4 . let n = 4q + 1 and q ≥ 1. assume q = 1 =⇒ n = 5 =⇒ d1 = {v1,v5,u3} = {v1,vn,un−2} induction hypothesis on q: assume that the result is true for the case when q = l and then prove that the result is true for q = l + 1. by induction hypothesis the given set dl = {v1,vi,vn}∪{u3,uk} is the minimal dominating set. we have p = 1, 2, . . . , l− 1, i = 5, 9, . . . , 4l− 3 and k = 7, 11, . . . , 4l− 1 =⇒ dl = {v1,v5,v9, . . . ,v4l−3,v4l+1}∪{u3, . . . ,u4l−1} is true. to prove that when q = l + 1 the result is true. let q = l + 1 =⇒ n = 4l + 5 =⇒ dl+1 = {v1,v5, . . . ,v4l+1}∪{u3, . . . ,u4l+3}. dl+1 = dl ∪d1 = {v1,v5,v9, . . . ,v4l−3}∪{u3, . . . ,u4l−1}∪{v1,v4l+1,u4l+3} this proves that the result is true for q = l + 1. =⇒ |dm| = d n 2 e. hence γ(cln) = d n 2 e. case 3. if n ≡ 2 mod 4 then dm = {v1,vi}∪{u2,uk} where i = 4p, k = 4p + 2 and p = 1, 2, . . . , n− 2 4 . let n = 4q + 2 and q ≥ 1. when q = 1 =⇒ n = 6, d1 = {v1,v4,u2,u6} = {v1,vn−2,u2,un}. see fig.2. induction hypothesis on q: assume that the result is true for the case when q = l and we need to prove that the result is true for q = l + 1. by induction hypothesis the given set dl = {v1,vi}∪{u2,uk} is the minimal dominating set. therefore p = 1, 2, . . . , l, i = 4, 8, . . . , 4l and k = 6, 12, . . . , 4l + 2. =⇒ dl = {v1,v4,v8, . . . ,v4l}∪{u2, . . . ,u4l−2,u4l+2} is true. to prove: the result is true for q = l + 1. when q = l + 1 =⇒ n = 4l + 6 =⇒ dl+1 = {v1,v4, . . . ,v4l+4}∪{u2, . . . ,u4l+6}. dl+1 = dl ∪d1 = {v1,v4,v8, . . . ,v4l}∪{u2, . . . ,u4l−2,u4l+2}∪{v1,v4l+4,u4l+6} this proves that the result is true for q = l + 1. =⇒ |dm| = n + 2 2 therefore, it is obtained that γ(cln) = n + 2 2 . case 4. if n ≡ 3 mod 4 then dm = {v1,vi}∪{u3,uk} where i = 4p + 1, k = 4p + 3 and p = 1, 2, . . . , n− 3 4 . let n = 4q + 3 and q ≥ 1. when q = 1 =⇒ n = 7 =⇒ d1 = {v1,v5,u3,u7} = {v1,vn−2,u3,un}. induction hypothesis on q: assume that the result is true for the case when q = l and we need to prove 34 jebisha esther s and veninstine vivik j that the result is true for q = l + 1. by induction hypothesis the given set dl = {v1,vi}∪{u3,uk} is the minimal dominating set where, n = 4l + 3, p = 1, 2, . . . , l, i = 5, 9, . . . , 4l + 1 and k = 7, 11, . . . , 4l + 3 =⇒ dl = {v1,v5,v9, . . . ,v4l−3,v4l+1}∪{u3, . . . ,u4l−1,u4l+3} is true. to prove that the result is true when q = l + 1. let q = l + 1 =⇒ n = 4l + 7 =⇒ dl+1 = {v1,v5, . . . ,v4l+5}∪{u3, . . . ,u4l+7}. dl+1 = dl ∪d1 = {v1,v5,v9, . . . ,v4l−3,v4l+1}∪{u3, . . . ,u4l−1,u4l+3}∪{v1,v4l+5,u3,u4l+7}. this proves that the result is true for q = l + 1. hence |dm| = d n 2 e =⇒ γ(cln) = d n 2 e. � v1 v2 v3 v4v5 v6 u1 u2 u3 u4u5 u6 figure 1: prism graph cl6 v1 v2 v3 v4v5 v6 u1 u2 u3 u4u5 u6 figure 2: dominating vertices of cl6 remark:1 the domination numbers of some prism graphs cln of all the cases discussed are summarized in table 1. n v e γ n v e γ n v e γ 11 22 33 6 16 32 48 8 21 42 63 11 12 24 36 6 17 34 51 9 22 44 66 12 13 26 39 7 18 36 54 10 23 46 69 12 14 28 42 8 19 38 57 10 24 48 72 12 15 30 45 8 20 40 60 10 25 50 75 13 table:1 domination numbers (γ) of prism graphs cln with vertices v and edges e theorem 3.2. the domination number of the antiprism graph qn, where n ≥ 3, is γ(qn) = d2n5 e. proof. let qn be the antiprism graph with vertex set v and edge set e. the number of vertices in qn, |v | = 2n given by v = {vj}∪{uj} where j = 1, 2, . . . ,n and the number of edges |e| = 4n. see fig.3. case 1. if n ≡ 0 mod 5 then dm = {v1,vi}∪{u3,uk} where i = 5p + 1, k = 5p + 3 and p = 1, 2, 3, . . . , n−5 5 . the induction method is taken to prove that the above set is one of the minimal dominating set. let n = 5q and q ≥ 1. when q = 1 =⇒ n = 5 =⇒ d1 = {v1,u3} = {vn−4,un−2}. see fig.4. induction hypothesis on q: assume that the result is true for the case when q = l and then prove that the result is true when q = l + 1. by induction hypothesis the given set dl = {v1,vi}∪{u3,uk} is the minimal dominating set. therefore p = 1, 2, . . . , l− 1, i = 6, 11, . . . , 5l− 4 and k = 8, 13, . . . , 5l− 2. =⇒ dl = {v1,v6,v11, . . . ,v5l−4}∪{u3, . . . ,u5l−2} is true. mds for prism family of graph 35 we need to prove that when q = l + 1 the result is true. let q = l + 1 =⇒ n = 5l + 5 =⇒ dl+1 = {v1,v6, . . . ,v5l+1}∪{u3, . . . ,u5l+3}. dl+1 = dl ∪d1 = {v1,v6, . . . ,v5l−4}∪{u3, . . . ,u5l−2}∪{v5l+1,u5l+3} this prove that the result is true for q = l + 1. =⇒ |dm| = d 2n 5 e hence γ(qn) = d 2n 5 e. case 2. if n ≡ 1 mod 5 then dm = {v1,vi,vn−1}∪{u3,uk} where i = 5p+ 1, k = 5p+ 3 and p = 1, 2, 3, . . . , n−6 5 is one of the minimal dominating set. by induction method, let n = 5q+1 and q ≥ 1. when q = 1 =⇒ n = 6, d1 = {v1,v6,u3} = {v1,vn,un−4}. induction hypothesis on q: assume that the result is true for the case q = l and then prove that the result is true for q = l + 1. by induction hypothesis the given set dl = {v1,vi,vn}∪{u3,uk} is the minimal dominating set. therefore p = 1, 2, . . . , l− 1, i = 6, 11, . . . , 5l− 4 and k = 8, 13, . . . , 5l− 2. =⇒ dl = {v1,v6,v11, . . . ,v5l−4,v5l+1}∪{u3, . . . ,u5l−2} is true. to prove: when q = l + 1 the result is true. let q = l + 1 =⇒ n = 5l + 6 =⇒ dl+1 = {v1,v6, . . . ,v5l+6}∪{u3, . . . ,u5l+3} dl+1 = dl ∪d1 = {v1,v6, . . . ,v5l−4,v5l+1}∪{u3, . . . ,u5l−2}∪{v5l+6,u5l+3} hence the result is true when q = l + 1. =⇒ |dm| = d 2n 5 e hence γ(qn) = d 2n 5 e. case 3. if n ≡ 2 mod 5 then dm = {v1,vi,vn−1}∪{u3,uk} where i = 5p + 1, k = 5p + 3 and p = 1, 2, 3, . . . , n−7 5 . prove by induction that the above set is one of the minimal dominating sets. let n = 5q + 2 and q ≥ 1. when q = 1 =⇒ n = 7 and d1 = {v1,v6,u3} = {v1,vn−1,un−4}. induction hypothesis on q: assume that the result is true when q = l and then prove that the result is true when q = l + 1. by induction hypothesis the given set dl = {v1,vi,vn−1}∪{u3,uk} is the minimal dominating set. so that p = 1, 2, . . . , l− 1, i = 6, 11, . . . , 5l− 4 and k = 8, 13, . . . , 5l− 2. =⇒ dl = {v1,v6,v11, . . . ,v5l−4,v5l+1}∪{u3, . . . ,u5l−2} is true. to prove that if q = l + 1 then the result is true. let q = l + 1 =⇒ n = 5l + 7. =⇒ dl+1 = {v1,v6, . . . ,v5l+6}∪{u3, . . . ,u5l+3} dl+1 = dl ∪d1 = {v1,v6,v11, . . . ,v5l−4,v5l+1}∪{u3, . . . ,u5l−2}∪{v5l+6,u5l+3} hence result is true for q = l + 1. =⇒ |dm| = d 2n 5 e thus γ(qn) = d 2n 5 e. case 4. if n ≡ 3 mod 5 then dm = {v1,vi}∪{u3,uk} 36 jebisha esther s and veninstine vivik j where i = 5p + 1, k = 5p + 3 and p = 1, 2, 3, . . . , n−3 5 . prove by induction that the above set is one of the minimal dominating sets. let n = 5q + 3 and q ≥ 0. assume q = 0 =⇒ n = 3 =⇒ d1 = {v1,u3} = {vn−2,un} induction hypothesis on q: assume that the result is true for the case when q = l and then prove that the result is true for q = l + 1. by induction hypothesis dl = {v1,vi}∪{u3,uk} is the minimal dominating set. we have p = 1, 2, . . . , l, i = 6, 11, . . . , 5l − 4, 5l + 1 and k = 8, 13, . . . , 5l − 2, 5l + 3. =⇒ dl = {v1,v6,v11, . . . ,v5l+1}∪{u3, . . . ,u5l+3} is true. to prove that when q = l + 1 the result is true. let q = l + 1 =⇒ n = 5l + 8. =⇒ dl+1 = {v1,v6, . . . ,v5l+6}∪{u3, . . . ,u5l+8} dl+1 = dl ∪d1 = {v1,v6, . . . ,v5l+1}∪{u3, . . . ,u5l+3}∪{v5l+6,u5l+8} this implies the result is true when q = l + 1. =⇒ |dm| = d 2n 5 e therefore γ(qn) = d 2n 5 e. case 5. if n ≡ 4 mod 5 then the minimum dominating set dm = {v1,vi}∪{u3,uk} where i = 5p + 1, k = 5p + 3 and p = 1, 2, 3, . . . , n−4 5 . prove by induction that the above set is one of the minimal dominating sets. let n = 5q + 4 and q ≥ 0. assume the case q = 0 =⇒ n = 4 =⇒ d1 = {v1,u3} = {vn−3,un−1} induction hypothesis on q: assume that the result is true for q = l and then prove that the result is true when q = l + 1. by induction hypothesis the given set dl = {v1,vi}∪{u3,uk} is the minimal dominating set. here p = 1, 2, . . . , l, i = 6, 11, . . . , 5l− 4, 5l + 1 and k = 8, 13, . . . , 5l− 2, 5l + 3 =⇒ dl = {v1,v6,v11, . . . ,v5l+1}∪{u3, . . . ,u5l+3} is true. we have to prove that when q = l + 1 the result is true. let q = l + 1 =⇒ n = 5l + 9 =⇒ dl+1 = {v1,v6, . . . ,v5l+6}∪{u3, . . . ,u5l+8}. dl+1 = dl ∪d1 = {v1,v6,v11, . . . ,v5l+1}∪{u3, . . . ,u5l+3}∪{v5l+6,u5l+8} this prove that the result is true when q = l + 1. =⇒ |dm| = d 2n 5 e thus γ(qn) = d 2n 5 e. � mds for prism family of graph 37 figure 3: antiprism graph q5 v1 v2 v3 v4 v5 u1 u2 u30u4 u5 v1 v2 v3 v4 v5 u1 u2 u3u4 u5 figure 4: dominating vertices of q5 remark 2: table 2 summarises the domination numbers (γ ) of few antiprism graphs qn with vertices v and edges e. n v e γ n v e γ n v e γ 11 22 44 5 16 32 64 7 21 42 84 9 12 24 48 5 17 34 68 7 22 44 88 9 13 26 52 6 18 36 72 8 23 46 92 10 14 28 56 6 19 38 76 8 24 48 96 10 15 30 60 6 20 40 80 8 25 50 100 10 table 2: domination numbers (γ ) of crossed prism graph qn theorem 3.3. the domination number of the crossed prism graph rn, where n ≥ 4 is (1) γ(rn) = n 2 , if n ≡ 0 mod 4 (2) γ(rn) = n+2 2 , if n ≡ 2 mod 4. proof. let rn be the crossed prism graph with vertex set v and edge set e. the number of vertices in rn is |v | = 2n and the number of edges in rn, |e| = 3n. see fig.5. for n ≥ 4 the form of minimum dominating set is case 1. if n ≡ 0 mod 4 then dm = {v1,vi}∪{u4,uk} where i = 4p + 1, k = 4p + 4 and p = 1, 2, . . . , n− 4 4 . prove by induction that the above set is one of the minimal dominating sets. let n = 4q and q ≥ 1. when q = 1 =⇒ n = 4 =⇒ d1 = {v1,u4} = {vn−3,un}. induction hypothesis on q: assume that the result is true when q = l and then prove that the result is true for q = l + 1. by induction hypothesis dl = {v1,vi}∪{u4,uk} is the minimal dominating set, therefore p = 1, 2, . . . , l− 1, i = 5, 9, . . . , 4l− 3 and k = 4, 8, . . . , 4l. =⇒ dl = {v1,v5,v9, . . . ,v4l−3}∪{u4, . . . ,u4l−4} is true. we have to prove that when q = l + 1 the result is true. let q = l + 1 =⇒ n = 4l + 4 =⇒ dl+1 = {v1,v5, . . . ,v4l+1}∪{u3, . . . ,u4l+4} dl+1 = dl ∪d1 = {v1,v5,v9, . . . ,v4l−3}∪{u4, . . . ,u4l}∪{v4l+1,u4l+4} 38 jebisha esther s and veninstine vivik j this proves that the result is true for q = l + 1. hence |dm| = n 2 =⇒ γ(rn) = n 2 . case 2. if n ≡ 2 mod 4 then the one of the minimum dominating set dm = {v1,vi}∪{u1,uk} where i = 4p, k = 4p + 1 and p = 1, 2, . . . , n− 2 4 . the proof is given by mathematical induction method. let n = 4q + 2 and q ≥ 1. assume that q = 1 =⇒ n = 6 and d1 = {v1,v4,u1,u5} = {v1,vn−2,u1,un−1}. see fig.6. induction hypothesis on q: assume that the result is true for the case, when q = l and then prove that the result is true for q = l + 1. by induction hypothesis dl = {v1,vi}∪{u4,uk} is the minimal dominating set. therefore p = 1, 2, . . . , l, i = 4, 8, . . . , 4l and k = 5, 9, . . . , 4l + 1. =⇒ dl = {v1,v4,v8, . . . ,v4l}∪{u5, . . . ,u4l+1} is true. to prove: when q = l + 1 the result is true. let q = l + 1 =⇒ n = 4l + 6 =⇒ dl+1 = {v1,v5, . . . ,v4l+4}∪{u3, . . . ,u4l+5}. dl+1 = dl ∪d1 = {v1,v5,v9, . . . ,v4l}∪{u4, . . . ,u4l+1}∪{v4l+4,u4l+5} this proves that the result is true when q = l + 1. =⇒ |dm| = n + 2 2 . thus γ(rn) = n + 2 2 . � v1 v2 v3 v4v5 v6 u1 u2 u3 u4u5 u6 figure 5: crossed prism graph r6 v1 v2 v3 v4v5 v6 u1 u2 u3 u4u5 u6 figure 6: dominating vertices of r6 remark:3 the domination numbers of various crossed prism graphs rn for some of the cases n ≡ 0 mod 4 and n ≡ 2 mod 4 are summarized in table 3. n v e γ n v e γ n v e γ 10 20 30 6 20 40 60 10 30 60 90 16 12 24 36 6 22 44 66 12 32 64 96 16 14 28 42 8 24 48 72 12 34 68 102 18 16 32 48 8 26 52 78 14 36 72 108 18 18 36 54 10 28 56 84 14 38 76 114 20 table 3: domination number (γ ) for crossed prism graph rn with vertices v and edges e mds for prism family of graph 39 4. conclusion the idea of computing bounds of domination number for graphs remains to be an active area of research for decades. this led to the focus of investigating the domination number of the family of prism graphs. in this work we have generalized the minimum dominating set for each case of considered graphs and proved using mathematical induction method. the upper bound of domination numbers for prism, antiprism and crossed prism graphs are determined. future work could establish bounds for domination in various structured graphs that can be explored. references [1] m. 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[17] r. j. wilson, introduction to graph theory, pearson education india, 1979. jebisha esther s, department of mathematics, karunya institute of technology and sciences, coimbatore641 114, tamil nadu, india. email address: jebishaesther@gmail.com veninstine vivik j, corresponding author, department of mathematics, karunya institute of technology and sciences, coimbatore-641 114, tamil nadu, india. email address: vivikjose@gmail.com 1. introduction motivation 2. preliminaries 3. computation of domination numbers for the prism graph family 4. conclusion references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase online first, pp.1-11 https://doi.org/10.5206/mase/15477 fisher information approach to understand the gompertz model avan al-saffar and eun-jin kim abstract. as a measure of sustainability, fisher information is employed in the gompertz growth model. this article examines the gompertz growth equation which describes the time evolution of large-scale variables such as the human body, with control parameters incorporating the effect of therapies on small-scale variables. control parameters are typically assumed to be constants, but in reality, they may be subject to fluctuations. the effect of different oscillatory modulations is examined on the system’s evolution and probability density function (pdf). for a sufficiently large frequency of periodic fluctuations occurring in both positive and negative feedbacks, the system maintains its initial conditions. a similar pdf is shown regardless of the initial values when there are periodic fluctuations in positive feedback. by periodic fluctuations in negative feedback, the gompertz model can lose its self-organization. finally, despite the fact that the gompertz and logistic systems evolve differently over time, the results show that they are exceptionally similar in terms of information and sustainability. 1. introduction the gompertz model has been evoked as a growth curve by numerous researchers for both biological and economic phenomena. specifically, the gompertz model (1.1) was employed significantly in different areas, in order to demonstrate population dynamics and to track their properties such as: specifying a mortality law in actuarial science, tumor growth and suppression modeling in medicine, depicting the outgrowth of organisms in biology, in ecology, etc. (see [12], [4] and references there in). in common, the gompertz model is written as follows: x(t) = ea(1−e −bt), (1.1) where a and b are positive quantities [4, 15, 17]. from eq. (1.1), x approaches zero when t display a tendency towards negative infinity; whereas x gets close to the equilibrium point x∗ = ea as t tends towards positive infinity, which is asymptotically stable and attracts all the initial states [10]. eq. (1.1) presents a sigmoidal curve for x(t) and can model human’s height and mass, a fish community, a rabbit (see [18], [14] and references therein), diverse malicious tumors [7, 8]. in particular, the gompertz equation was successfully fitted to data of tumor growth for the first time in the 1960s by laird [7], where tumors are cellular populations increasing in a restricted space with a limited availability of nutrients. furthermore, waliszewski [15] showed that the derivative of the gompertz function is a pdf, and the probability distribution of it is a solution of non-gaussian form. the verhulst, richards, comperzt, and modified malthus model were used to to understand the dynamics of water volume growth in a reservoirs [5]. received by the editors 2 november 2022; accepted 14 december 2022; published online 26 december 2022. 2010 mathematics subject classification. primary 54c40, 14e20; secondary 46e25, 20c20. key words and phrases. dynamical system, sustainability, fisher information, periodic fluctuations, probability density function, equilibrium points. 1 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15477 2 a. al-saffar and e.-j. kim a periodic gompertz model with stochastic impulsive coefficients is investigated by wang and liu [16]. a modulation in the model parameters is studied using fisher information. fisher information indexes for dynamic systems in periodic steady states were developed and applied to a simple two-species lotkavolterra predator-prey model [9]. further comparison is conducted based on changes in information during growth between the logistic growth model and gompertz growth model [13]. in order to achieve regime change, an ecological system must be able to change its properties to a great extent [6]. the purpose of this article is to study the effects of oscillatory modulations on the parameters and examine sustainability by using fisher information. also, the similarity and differences between the gompertz model and the logistic function of growth are presented. the reminder of the article is organized as follows. section §2 introduces the model. in section §3, the gompertz model with fluctuations in both terms is presented, and present pdfs, fisher information. also, it is compared with the logistic model. in sections §4 and §5, we summarize the results for a periodic modulation of the model parameters in the negative feedback or positive feedback, respectively. conclusions are provided in section §6. 2. the model the model is given by: dx dt = cx− bx log x. (2.1) where, c and b are experimental coefficients specifying the curve’s slope (e.g [15] and references there in); c is the rate of population growth, and b is the average of population death, t is time, x is the population of any type, and x0 = x(t = 0) is the population initial size. eq.(2.1) presents a population asymptotically reaching the maximum value x∗ = e(c/b) which is the stable equilibrium point. the first term cx (linear) with c > 0 represents positive feedback whereas the second term bx log (x) (nonlinear) with b¿0 represents negative feedback. in the case of a periodic fluctuation d0 sin(ωt), is included in the different model parameters, where d0 and ω are the amplitude and frequency of the modulation, respectively. as in the logistic equation in [1], the value of the amplitude is fixed to be d0 = 7, to explore the influence of different values of ω and x0 on the response of the gompertz model. specifically, it may be beneficial to study three different cases of the gompertz model. each case will be presented individually in the next sections iii-v. numerical simulations and analytical analyses are performed. case-1: dx dt = [d0 sin(ωt)] x (1 − log x). case-2: dx dt = [d0 sin(ωt)] x− b x log x. case-3: dx dt = cx− [1 + d0 sin(ωt)] x log x. each case will be discussed separately in the following sections 3-5. we perform both analytical and numerical analyses. 3. case-1: similar fluctuation in the positive and negative feedback the following model is considered: dx dt = [d0 sin(ωt)] x(1 − log x). (3.1) fisher information approach to understand the gompertz model 3 time 0 10 20 30 40 50 x (t ) 0 1 2 3 (a) x 0 = 0.1, ω= 1 time 0 2 4 6 8 10 x (t ) 0 0.5 1 1.5 (b) x 0 =0.1, ω=10 time 0 10 20 30 40 50 x (t ) 2 2.2 2.4 2.6 2.8 3 (c) x 0 =2, ω=1 time 0 2 4 6 8 10 x (t ) 2 2.2 2.4 2.6 (d) x 0 =2, ω=10 figure 1. time trace of x(t) for x0 = 0.1, 2 and ω = 1, 10. the analytical solution to eq. (3.1) can be found in the following form: x(t) = e ( 1− [( 1−log (x0) ) e { − d0 ω ( 1−cos(ωt) )}]) , (3.2) where x0 is the initial value of x at t = 0. we show the typical time history of x(t) for different values of ω and x0 in fig. 1. in fig. 1, the time trace of x for two different values of ω, ω = 1, 10 and two initial values x0 = 0.1, 2 are shown. from fig. 1, for small ω, x(t) tends to reach the equilibrium point (x∗ = e) without paying attention to the value of x0 where the perturbation’s time-scale becomes much larger than the system’s response time. in contrast, for large ω, the time-scale is much shorter than the system’s response time and so that x(t) preserves its initial values and on no occasion can outreach the equilibrium point (see fig. 1). in fig. 2, the maximum and minimum of x in blue solid line and red dashed lines, respectively against ω for different values of x0 is presented. it is obvious that a difference between column 1 and 2 which show the results for different x0. for x0 = 0.1 the maximum value of x decreases extremely faster as ω increases than for x0 = 2. in comparison, for x0 = 2, x(t) does not deviate from its initial conditions similar to fig. (1) (b), (d). to show this clearly, the measure of relative variation of the maximum values of x is displayed to quantify the conservation of x0. . this is shown in fig. 2 (e)-(f). we compute the ratio of the change in the maximum of x to determine the maintenance of initial conditions. initial value−maximum ofx maximum ofx 100% (3.3) 4 a. al-saffar and e.-j. kim 0 100 200 300 400 500 0 1 2 3 m a x & m in o f x (a) x 0 = 0.1 max(x) min(x) 0 50 100 0 2 4 100 200 300 400 500 1.5 2 2.5 3 m a x & m in o f x (b) x 0 = 2 max(x) min(x) 10 0 10 1 10 2 10 -1 10 0 10 1 m a x & m in o f x (c) x 0 = 0.1 max(x) min(x) 10 0 10 1 10 2 2 2.5 3 m a x & m in o f x (d) x 0 = 2 max(x) min(x) 10 0 10 1 10 2 10 1 10 2 % (e) x 0 = 0.1 max(x) 10 0 10 1 10 2 10 -1 10 0 10 1 10 2 % (f) x 0 = 2 max(x) figure 2. we fixed d0 to be d0 = 7, x0 = 0.1 in panels (a), (c) and (e) and x0 = 2 in panels (b), (d) and (f). (c), (d), (e) and (f) are shown in log-log scales. it has been found that % is almost 100 in fig. 2 (e)-(f) for ω ≤ 10. for sufficiently large ω ≥ 10, we observe nearly straight lines which indicates that the percentage change decreases with ω as a power-law in both cases where panels (e) and (f) are presented in log-log scale. 3.1. probability density function. in this section, we compute the probability density function (pdf) of x and examine the effect of ω and x0 on pdf. to this end, we link the time spent by the system state at x to the probability of observing the system at a particular value of x [2, 3, 11]. p[x] dx = p[t] dt. (3.4) since t is a continuous variable with a uniform probability density: p[t] = constant = a, (3.5) we combine eqs. (3.4) and (3.5) and obtain a pdf of x as: p[x] = p[t] ∣∣∣∣ dtdx ∣∣∣∣ = a ∣∣∣∣ dtdx ∣∣∣∣ = au , (3.6) where u = dx dt . (3.7) now we can obtain pdf of x as follow: p[x] = a d0 sin (ωt) x (1 − log (x)) . (3.8) we use eq. (3.2) to replace sin(ωt) in eq. (3.8) by a function which only depends on x [see eq. (3.9)] through using the identity ( sin (ωt) = √ 1 − cos2 (ωt) ) as follows: cos (ωt) = 1 + ( ω d0 log [ 1 − log (x) 1 − log (x0) ]) . (3.9) fisher information approach to understand the gompertz model 5 0 1 2 3 x 10 -1 10 1 10 3 10 5 10 7 p d f =0.5,x 0 =0.01 0 1 2 3 10 -1 10 1 10 3 10 5 10 7 x 0 =0.1 1 1.5 2 2.5 3 10 -1 10 1 10 3 10 5 10 7 x 0 =1 2 2.2 2.4 2.6 2.8 10 1 10 3 10 5 10 7 x 0 =2 2.7 2.705 2.71 2.715 2.72 10 3 10 5 10 7 x 0 =2.7 2.717 2.7175 2.718 2.7185 10 3 10 5 10 7 x 0 =2.717 2.718 2.7185 2.719 2.7195 2.72 10 3 10 5 10 7 x 0 =2.72 2.6 2.8 3 3.2 3.4 10 1 10 3 10 5 10 7 x 0 =3.5 3 4 5 10 -1 10 1 10 3 10 5 10 7 x 0 =5 figure 3. pdf of x(t) for d0 = 7 and ω = 0.5 by using different values of x0. all the sub-figures have the same axis labels. it is mentioned that the pdf relies on x0 forω = 0.5 in fig. 3. a bimodal pdf is noticed in every case with various distance between the two ends. the first peak appears at x = x0 whereas the second peak can be seen at x∗. for x0 < x ∗, the left peak is always lower than the right peak at equilibrium while for x0 > x ∗; the left peak becomes higher than the right peak as x0 is larger than the equilibrium point. a very identical behavior is observed for differentω (e.g., ω = 2 [figures are not shown]. for small x0, the system exhibits a wide pdf but for large values of x0, the pdf starts to shrink as can be seen in fig. 1; the system can never reach the equilibrium point and maintains its initial conditions. in the following section we will relate these findings to fisher information. 3.2. fisher information. fisher information is utilized to study the effects of modulation in model parameters. fisher information as a function of the system’s variability is evoked to investigate whether the gompertz function is sustainable or not (the fisher information changes as the system’s state trajectory tracks changes in its dynamic regime). note that a pdf biased to particular x values has higher fisher information. by following the results in [1], for a single measurable variable x, we find the fisher information fi calculated from the pdf of x (p(x, t)) as follows: fi = ∫ 1 p(x) ( dp(x) ∂x )2 dx. (3.10) since ∂p[x] ∂t = − a u2 du dt , (3.11) 6 a. al-saffar and e.-j. kim 0 1 2 3 4 5 6 7 8 9 10 x 0 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 lo g (f t ) d 0 =7 2.71 2.715 2.72 2.725 10 12 10 13 10 14 figure 4. asymptotic value of ft against x0 for ω = 0.5. a local maximum is observed at x0 ' x∗ where x∗ = exp(1) we compute the time averaged fisher information (ft ): ft = a t ∫ t 0 1 (u(t))4 ( du dt )2 dt = 1 t ∫ t 0 1 a ( ∂p(x) ∂t )2 dt. (3.12) from eqs. (3.1), (3.11) and (3.12), we obtain: ft = a t ∫ t 0 [ ω cos(ωt) sin(ωt) − log(x)d0 sin(ωt)]2 [d0 sin(ωt) (x−x log(x))]2 dt. (3.13) ft is the fisher information averaged over the total time duration t ; a is a normalization constant. in this section, the sustainability of our system is checked by computing ft for different initial conditions. ft is displayed for (d0 = 7), different x0 and ω = 0.5. in fig. 4, fisher information is increased as x0 increases close to x0 ' x∗ while ft decreases beyond x0 ' x∗. this means that a local maximum is found at x0 close to x∗. also, for x0 > x∗, it is observed that the curve monotonically decreases as x0 increases because of the presence of high variability, as the system loses its functionality, thus, lower fisher information is noticed (see fig. 4. it is found that similar behavior for ω = 2 [figures are not shown]. in this case, the high fisher information indicates that the system is sustainable (less disorder), whereas the low fisher information indicates that the gompertz equation is unsustainable, which means that the population cannot survive under these conditions (specific parameter values). 3.3. comparison with the logistic model. the above results are compared with the logistic model governed by: dx dt = n0 sin(ωt) x ( 1 − x k ) , (3.14) where n0 and ω are amplitude and frequency of the modulation, respectively, and k is the carrying capacity of the logistic system. a pdf of x at different initial conditions in is displayed in fig. 5. in fig. 5, pdf of x(t) is displayed for fixed value of ω, ω = 0.5 but for different initial conditions. it is observed similar results as in fig. 3. this bimodal distribution produces from the maintenance of fisher information approach to understand the gompertz model 7 0 5 10 x 10 -2 10 0 10 2 10 4 10 6 p d f =0.5,x 0 =0.1 0 5 10 10 -2 10 0 10 2 10 4 10 6 x 0 =1 6 8 10 10 0 10 2 10 4 10 6 x 0 =5 7 8 9 10 10 0 10 2 10 4 10 6 x 0 =7.5 9.6 9.8 10 10 0 10 2 10 4 10 6 x 0 =9.5 9.96 9.98 10 10 2 10 4 10 6 x 0 =9.95 10 10.05 10.1 10 2 10 4 10 6 x 0 =10.1 10 11 12 13 14 10 -2 10 0 10 2 10 4 10 6 x 0 =13.5 10 12 14 10 -2 10 0 10 2 10 4 10 6 x 0 =15 figure 5. pdf for n0 = 5, k = 10 and ω = 0.5 and different values of x0. all the sub-figures have the same axis labels. 0 2 4 6 8 10 12 14 16 18 20 10 24 10 25 10 26 10 27 10 28 10 29 10 30 10 31 10 32 10 33 n 0 =5, =0.5 figure 6. ft for n0 = 5, k = 10 and ω = 0.5 and different x0. x0 against the tendency of x approaching (k = 10). analytically, it is show ft as follows: ft = a t ∫ t 0 [ ω cos(ωt) sin(ωt) + ( (1 − 2x k )n0 sin(ωt) )]2 [ n0 sin(ωt) (x− x 2 k ) ]2 dt. (3.15) 8 a. al-saffar and e.-j. kim 0 10 20 30 40 50 t 0 1 2 3 x d 0 =b=7, =1, x 0 =0.1 0 0.5 1 1.5 2 2.5 3 x 0 5 10 15 p (x ) 10 5 d0 =b=7, =1, x 0 =0.1 0 10 20 30 40 50 t 0 2 4 6 x x 0 =5 0 0.5 1 1.5 2 2.5 3 x 0 1 2 3 p (x ) 10 6 x0 =5 0 10 20 30 40 50 t 0 5 10 x x 0 =10 0 0.5 1 1.5 2 2.5 3 x 0 1 2 3 p (x ) 10 6 x0 =10 figure 7. time trace of x(t) and pdf of x for different values of x0 = 0.1, 5, 10 and ω = 1. ft in fig. 6 where it can see the maximum ft at x0 ' x∗ similarly to the gompertz model. therefore, both the gompertz case-1 and logistic case-1 (see [9]) show less variability (more sustainability) close to the stable equilibrium point. 4. case-2: perturbation in the positive feedback (linear term) a periodic modulation for the linear term is added while keeping (b=constant). dx dt = ( d0 sin(ωt) x ) − b x log (x), (4.1) where the value of b is kept constant. the analytical solution to eq. (4.1) can be found as: x = ( d0(b sin(ωt) −ω cos(ωt)) ) + (d0ωd) + ((b 2 + ω2)d log (x0)) (b2 + ω2) , (4.2) where d = exp(−bt). by using this analytical solution, x(t) is presented for different x0 in the first column and pdf of x in the second column (see fig. 7). specifically in fig. 7, the impact of different x0 on pdfs for b = d0 = 7, ω = 1 is observed. the pdfs show the same bimodal pdf for all initial conditions. thus, case-2 reaches the same equilibrium pdf independent of x0 similar to the results of the logistic model. that is, fisher information ft is also independent of x0. fisher information approach to understand the gompertz model 9 0 200 400 600 800 1000 t 10 -2 10 0 10 2 10 4 10 6 x (a) d 0 =1, = 0.1 0 10 20 30 40 50 t 0 0.5 1 1.5 2 2.5 3 x (b) d 0 =1, = 10 0 200 400 600 800 1000 t 10 0 10 20 10 40 x (b) d 0 =10, = 0.1 0 10 20 30 40 50 t 0 2 4 6 8 x 10 4 (b) d 0 =10, = 10 figure 8. x(t) against t for d0 = 1, 10 and ω = 0.1, 10. for all cases x0 = 0.1, b = 1, c = 1. 5. case-3: perturbation in the negative feedback (nonlinear term) in this case, the negative feedback contains modulation: dx dt = cx− (1 + d0 sin(ωt))x log (x). (5.1) the nonlinear term contained a periodic modulation effectively minimizes the influence of the death rate and promotes exponential growth (see fig. (8)). for large value of d0 and small ω in fig. 8, it is observed that the solution grows exponentially. in fig. 9, pdfs for d0 = 0.5 emerged in the first row and d0 = 1 in the second row, respectively. for d0 = 1, a bimodal pdf is noticed for all ω, the pdfs become broader as ω decrease. the intermittency burst demonstrated by the high-amplitude peaks leads to the broadening of pdfs as ω decrease. 6. conclusions the effects of different initial conditions and different modulations in the model parameters are considered for both linear and/or nonlinear terms. specifically, identical periodic modulation of the model parameters for the positive and negative feedback is included in the first case, where it is observed that the gompertz model does not forget its initial values as the system’s response time is quite large than the disturbance’s time-scale. that is, initial values far from equilibrium x∗ can never reach the equilibrium point. in view of sustainability, a maximum fisher information is noticed for x0 ' x∗; fisher information increases with x0 up to x0 ' x∗ and decreases beyond x0 ' x∗. this behavior is due to the high variability in the model parameters beyond x0 ' x∗; low variability (more sustainable) yields high fisher information whereas high variability leads to low fisher information. 10 a. al-saffar and e.-j. kim figure 9. pdfs of x for two different d0 = 0.5 and 1 in the upper and lower panels, respectively. for all cases, x0 = 0.1, b = 1, and c = 1, pdfs are bimodal in all cases. in the cases tested, the logistic and gompertz models behave very similar in terms of information and sustainability although they evolve differently in time. it will be interesting to investigate different models using similar methods in the future. references [1] a. al-saffar and e. kim, sustainable theory of a logistic model-fisher information approach, math. biosciences, 285(2017), 81–91. [2] h. cabezac, and b. d. fath, towards a theory of sustainable systems, fluid phase equilibria, 194(2002), 3–14. [3] b. d. fath, h. cabezas, and ch. w. pawlowski, regime changes in ecological systems: an information theory approach, j. theor. bio., 222(2003), 517–530. [4] d. jukic, g. kralik, and r. scitovski, least-squares fitting gompertz curve, j. computational and applied math., 169(2004), 359–375. [5] k. kartono, p. purwanto, and s. suripin, ecological implications of the dynamics of water volume growth in a reservoir, ecological engineering & environmental technology, 22(2021), 22–29. [6] e. konig, cabezas, h. and a. l. mayer, detecting dynamic system regime boundaries with fisher information: the case of ecosystems, clean techn environ policy, 21(2019), 1471–1483. [7] a. k. laird, dynamics of tumour growth, br. j. cancer, 18(1964), 490-502. 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[13] l.m. tenkes, r. hollerbach, and e. kim, time-dependent probability density functions and information geometry in stochastic logistic and gompertz models, j. stat. mech.: theory and experiment 12(2017): 123201. [14] p. waliszewski, and j. konarski, the gompertzian curve reveals fractal properties of tumor growth, chaos, solitons and fractals 16(2003), 665–674. [15] p. waliszewski, a principle of fractal-stochastic dualism and gompertzian dynamics of growth and self-organization, biosystems 82(2005), 61–73. [16] z. wang, and m. liu, optimal impulsive harvesting strategy of a stochastic gompertz model in periodic environments, app. math. letters 125(2022): 107733. [17] ch. p. winsor, the gompertz curve as a growth curve, proceedings of the national academy of sciences 18(1932), 1–8. [18] m. h. zwietering, i. jongenburger, f. m. rombouts and k. van’t riet, modeling of the bacterial growth curve, app. and enviro. microbiology, 56(1990), 1875–1881. corresponding author, department of statistics, university of duhok, duhok, iraq email address: avan.elias@uod.ac fluid and complex systems center, coventry university, coventry, united kingdom email address: e.kim@shef.ac.uk 1. introduction 2. the model 3. case-1: similar fluctuation in the positive and negative feedback 3.1. probability density function 3.2. fisher information 3.3. comparison with the logistic model 4. case-2: perturbation in the positive feedback (linear term) 5. case-3: perturbation in the negative feedback (nonlinear term) 6. conclusions references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 4, number 2, june 2023, pp.144-153 https://doi.org/10.5206/mase/16387 analysis of thermal stresses to 2d plane thermoelastic inhomogeneous strip abhijeet b. adhe, kirtiwant p. ghadle, and uday s. thool abstract. this paper deals with study of the plane elasticity of thermoelastic problems for inhomogenous strip. here, the original problems are reduced to set the governing equations in the volterra integral equations by making the use of direct integration method. further using the iteration technique the numerical calculations have been performed. the stress distribution obtained and calculated numerically and shown graphically. 1. introduction a large development of the subject, thermoelasticity is motivated by various fields of engineering sciences, during the last few decades. the main physical drawback in the theory of uncoupled thermoelasticity is that an elastic body has no effect on the temperature and vice versa. the interest of researchers to study elasticity and thermo-elasticity problems has grown very fast due to wide applications to real world. biot [3] derived the equation of thermal conductivity by including the coupling between thermal fields and strain fields. a novel work done by lord et.al. [10] introduced two generalizations to the coupled theory of thermoelasticity and given successful alternate to fourier’s law in heat conduction. tokovyy et.al. in [15] emphasized on analytical treatment of the one dimensional and two dimensional elasticity and thermoelasticity problems using direct integration method, for a long hollow cylinder and a long annular radially non-homogeneous cylinder respectively. babich et al. [2] solved the plane problem of a horizontal concentrated load by using the linearized elasticity theory from an infinite inhomogeneous stringer to an elastic infinite strip with initial stresses clamped at one edge. the problem is reduced to system of integro-differential equations which then solved by means of fourier transform. manthena et. al. [12] analysed the same problems for a mixture of metals like copper and zinc. jafari et al. [6] discussed the stress analysis in an orthotropic infinite plate with a circular hole using complex variable technique to the two dimensional thermoelastic problem. kalynyak et al.[7] focused on development by prof. vihak in the field of direct and inverse problems of heat conduction and thermomechanics which are important in investigating problems of thermal power engineering. by considering an inverse thermoelastic problem in [8] prof. kalyanyak discussed the presence of a stationary temperature field for a long rectangular beam of inhomogeneous nature. mahakalkar et. al. [11] studied thermoelastic transient heat conduction problem with internal heat by using classical method. they investigated results on temperature distibution, thermal deflection and stresses by integral transform. iqbal kaur et.al.[5] studied recent thermoelastic theories and models related to micro-nano beams and bars, their uses and limitations. received by the editors 3 april 2023; accepted 21 june 2023; published online 26 june 2023. 2020 mathematics subject classification. primary 74a10 ; secondary 74a15, 74e05. key words and phrases. 2d plane thermoelastic problems, inhomogeneous strip, thermal stresses, volterra integral equation, analytical solution. 144 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/16387 analysis of thermal stresses to inhomogeneous strip 145 tokovyy [16, 17] discussed an analytical solution of plane themroelasticity inhomogeneous problem for planes, half-planes and strips with the aid of direct integration method, here, the governing equations reduced to an integral equations which are then solved by using iteration method which produces a solution of the problem in explicit form. and extended their work in [18] in terms of stresses for a infinite strip for a case of inhomogeneous isotropic material. solution is found using fourier transforms and iteration method. kushnir [9] used direct integration method for generalization of the original equations of solution of 2d problems of thermoelasticity for solids with corner points and they are reduced to a governing integrodifferential equations for a key function, an explicite form solution is found. tianhu [19] investigated the magneto-thermoelastic response of a homogeneous and isotropic finite thin slim strip subjected to a moving heat source by using lord-shulman theory and laplace transform. vigak [20, 21, 22] has been developed a method to find solution of the elasticity problems in a semi-plane using the method of direct integration of equilibrium equation. equilibrium conditions for tractions and compatibility equations for the displacements has been found correct. in [23] vigak et. al. invented a new analytic method for solving quasi-static thermoelastic problem for stresses in rectangular region, the initial problem is reduced to governing integral differential equations for stress components. the solution is obtained as the series expansion according to saint-venant’s principle. youssef et al.[24] developed a new model of three dimensional generalized thermoelasticity problem by using classic ls model. the double fourier transform and laplace technique had been applied to the governing equations subjected to rectangular traction free surface, with the study of the temperature analysis, stresses, strain and displacement in a three dimensional half-space. zhihe et. al. [25] emphasizes on characterization of fgm strip using thermoelastic problem. in the thermoelasticity one can determine the stresses produced due to the temperature field and moreover to find the temperature distribution by internal forces which vary with time. our intent of this paper is to extend our own work [1, 4] for obtaining an analytical solutions to the thermoelastic problems which occures in isotropic and inhomogeneous strip under some thermal condition applied. 2. problem formulation consider, 2d plane thermoelastic problem in the strip of inhomogeneous isotropic material with infinite width r = {(x,y) ∈ (−∞,∞) × (−a,a)}, where a > 0 is dimensionless parameter. thermoelastic equillibrium of plane r is ruled by the equillibrium equations,  ∂σxx ∂x + ∂σxy ∂y + fx = 0, ∂σxy ∂x + ∂σyy ∂y + fy = 0, (2.1) strain-compability equations, ∂2�yy ∂x2 + ∂2�xx ∂y2 = ∂2�xy ∂x∂y , (2.2) stress strain relations,   �xx = σxx e1(x) − v1(x)σyy e1(x) + α(x)t(x,y), �yy = σyy e1(x) − v1(x)σx e1(x) + α(x)t(x,y), �xy = σxy g(x) , g1(x) = e1(x) 2(1 + v1(x)) . (2.3) 146 a. b. adhe, k. p. ghadle, and u. s. thool figure 1. schematic diagram of strip under consideration. here, σxx,σyy,σxy are stress-tensor components, �xx,�yy,�xy denotes strain components, fx,fy are stress dimensional projections of forces in dimensionless components respectively, and e1,g1,α,v1 denotes young’s modulus, shear modulus, coefficient of thermal expansion and poisson’s ratio. due to temperature distribution, the normal and shearing stresses arise on the boundarres y = ±a in the strip r, σyy(x,−a) = −p1(x), σyy(x,a) = p2(x), (2.4) σxy(x,−a) = −q1(x), σxy(x,a) = q2(x). the two dimensional steady-state temperature t(x,y) can be found from the heat conduction equation [13] ∂2t ∂x2 + ∂2t ∂y2 = −w(x,y), (2.5) under conditions imposed on the boundary in the region −∞≤ x ≤∞  ∂t ∂x = 0 at x = ±∞, t(x,y) + k1 ∂t ∂y = 0 at y = a, t(x,y) −k2 ∂t ∂y = 0 at y = −a, (2.6) where, w(x,y) = q(x,y) k and q(x,y) denoting the heat generated due to internal heat generated and k1,k2 are coefficient of thermal conductivity. using equilibrium condition (2.1) we have ∂2σxx ∂x2 − ∂2σyy ∂y2 + ∂fx ∂x − ∂fy ∂y = 0. differentiating first equation in (2.1) with respect to x and second equation with respect to y and subtracting, we get ∂2σxx ∂x2 + ∂fx ∂x = ∂2σyy ∂y2 + ∂fy ∂y . adding ∂2σyy ∂x2 on both sides which yields ∆σyy = ∂2σ ∂x2 + ∂fx ∂x − ∂fy ∂y (2.7) where ∆ = ∂2 ∂x2 + ∂2 ∂y2 analysis of thermal stresses to inhomogeneous strip 147 denotes the two-dimensional laplace differential operator. putting stress-strain relations (2.3) and equillibrium conditions (2.1), equation (2.2) can be written as ∆ [ (1 −v1) 2g σ + α(1 + v1)t ] = σyy 2 d2 dy2 ( 1 g ) −fy d dy ( 1 g ) − 1 2g ( ∂fx ∂x + ∂fy ∂y ) . (2.8) the equations (2.7) and (2.8) are bounded by two boundary conditions (2.4) for σyy and for their derivatives which satisfies equilibrium condition (2.1) at y = ±a:  σyy ∂y = − ∂q1 ∂x −fy(x) at y = −a, σyy ∂y = − ∂q2 ∂x −fy(x) at y = a. (2.9) the shear stress is found by integrating the equilibrium conditions which gives 2σxy = q1 + q2 − ∫ a −a ( ∂σxx ∂x + fx)sgn(y − ξ)dξ − ∫ a −a ( ∂σyy ∂y + fy)sgn(x−η)dη (2.10) where, sgn =   1, for x > 0, 0, for x = 0, −1, for x < 0. 3. solution of thermoelastic problem to find the solution of the formulated problem, we apply the fourier transform [14] with respect to x defined by f̄(y; ω) = ∫ ∞ −∞ f(x,y) exp(−iωx)dx (3.1) where f(x,y) is an arbitrary function, i2 = −1; ω is a parameter. we choose σyy and σ to be the governing functions. to calculate the key stresses, we apply the fourier transform (3.1) to equation (2.7) to get( d2 dy2 −ω2 ) σ̄yy = −ω2σ̄ + iωf̄x − d dy f̄y. (3.2) applying fourier integral transform (3.1) to equation (2.8) with the conditions in (2.4), we obtain( d2 dy2 −ω2 )[ (1 −v1) 2g σ̄ + α(1 + v1)t̄ ] = σ̄yy 2 d2 dy2 ( 1 g ) − f̄y d dy ( 1 g ) − 1 2g ( iωf̄x ∂x + ∂f̄y ∂y ) , (3.3) σ̄yy(x,−a) = −p̄1, σ̄yy(x,a) = p̄2 (3.4) ∂σ̄yy ∂y (x,−a) = −iωq̄1 − f̄y(x,−a), ∂σ̄yy ∂y (x,a) = iωq̄2 + f̄y(x,a). the solution of differential equation (3.2) is σ̄yy = c1coshωy + c2sinhωy + 1 ω ∫ y −a ( iωf̄x − df̄y dy −ω2σ̄ ) sinh(ω(y − ξ))dξ (3.5) 148 a. b. adhe, k. p. ghadle, and u. s. thool where c1 and c2 are the constants of integration. using the first two boundary conditions in (2.4), the solution can be expressed as σ̄yy = −p̄2cosh(ω(y + a)) − ( iq̄2 + f̄x(x,−a) ω ) sinh(ω(y + a)) + 1 ω ∫ y −a ( iωf̄x − df̄y dy −ω2σ̄ ) sinh(ω(y − ξ))dξ. (3.6) it satisfies two integral conditions:∫ a −a σ̄sinhωξdξ = i(q̄1 + q̄2) sinhωa ω + (p̄2 − p̄1) coshωa ω + ( f̄y(x,a) + f̄y(x,−a) ) sinhωa ω2 + 1 ω ∫ a −a ( if̄x − 1 ω df̄y dξ ) sinhωξdξ, (3.7) and ∫ a −a σ̄coshωξdξ = i(q̄1 − q̄2) coshωa ω − (p̄2 + p̄1) sinhωa ω + ( f̄y(x,a) − f̄y(x,−a) ) coshωa ω2 + 1 ω ∫ a −a ( if̄x − 1 ω df̄y dξ ) coshωξdξ. (3.8) hence, the solution of equation (3.3) can be found as σ̄ = 2g1 1 −v1 [ c1coshωy + c2sinhω −α(1 + v1)t̄ + θ(y) + φ(y) + q(y) + ψ(y) + ∫ y −a σ̄(η)k(y,η)dη ] (3.9) where, θ(y) = − p̄2 2ω ∫ y −a d2 dξ2 ( 1 g(ξ) ) cosh(w(a + ξ))sinh(w(y − ξ))dξ, φ(y) = − iq̄2 2ω ∫ y −a d2 dξ2 ( 1 g(ξ) ) sinh(w(a + ξ))sinh(w(y − ξ))dξ, q(y) = 1 2ω ∫ y −a d2 dξ2 ( 1 g(ξ) ) sinh(ω(y − ξ)) ∫ ξ −a ( if̄x − 1 ω df̄y dξ ) sinh(ω(ξ −η))dηdξ, ψ(y) = − 1 ω ∫ y −a ( f̄y(ξ) ( 1 g(ξ) ) + 1 2g1(ξ) (isf̄x + df̄y dξ ) ) sinh(ω(y − ξ))dξ − f̄y(−a) q̄2 2ω2 ∫ y −a d2 dξ2 ( 1 g(ξ) ) sinh(w(a + ξ))sinh(w(y − ξ))dξ, k(y,η) = ∫ y −η d2 dξ2 ( 1 g(ξ) ) sinh(w(y − ξ))sinh(w(ξ −η))dξ. different types of techniques can be used to determine the solution of equation (3.9). here we use the method of resolvent kernel σ̄ = 2g1 1−v1 [ c1ncoshωy + c2nsinhω −α(1 + v1)t̄ + θ(y) + φ(y) + q(y) + ψ(y) + ∫y −a σ̄n(η)k(y,η)dη ] . the resolvent kernel is calculated as <(y,η) = ∞∑ n=0 kn+1(y,η) (3.10) analysis of thermal stresses to inhomogeneous strip 149 where   k1(y,ξ) = k(y,ξ), kn+1 = ∫ y −a k(y,ξ)kn(ξ,η)dη, n = 1, 2, · · · . we have a fact that the recurring kernels kn+1 −→ 0 as n −→∞ which shows the initial condition for convergence holds . consequently, for a natural number n, <(y,ξ) ≈<n (y,ξ) = n∑ n=0 kn+1(x,ξ) (3.11) we construct a solution of equation (3.9) by using resolvent-kernel technique, yields, σ̄ = 2g1 1 −v1 [ cs1(y) + ds2(y) + τ(y) −α(1 + v1)t̄ ] (3.12) where, s1(y) = cosh ωy + ∫ y −a cosh ωξ<(y,ξ)dξ, s2(y) = sinh ωy + ∫ y −a sinh ωξ<(y,ξ)dξ, τ(y) = θ(y) + φ(y) + q(y) + ψ(y) + ∫ y −a (θ(ξ) + φ(|xi) + q(ξ) + ψ(ξ))<(y,ξ)dξ, c = d2r1 −d1r2 d2d3 −d21 , d = d3r2 −d1r1 d2d3 −d21 , r1 = 1 2 ∫ a −a σ̄ cosh ωξ<(y,ξ)dξ, r2 = 1 2 ∫ a −a σ̄ sinh ωξ<(y,ξ)dξ, d1 = ∫ a −a g(ξ) 1 −v1(ξ) sinh ωξ cosh ωξdξ, d2 = ∫ a −a g(ξ) 1 −v1(ξ) sinh2 ωξdξ, d3 = ∫ a −a g(ξ) 1 −v1(ξ) cosh2 ωξdξ. integral conditions of σ̄ can be computed using equation (3.7)-(3.8) as σ̄yy = −p̄2cosh(ω(y + a)) − ( iq̄2 + f̄x(x,−a) ω ) sinh(ω(y + a)) + ω ∫ y −a 1 1 −v1 [2g (c cosh ωξ + d sinh ωξ − τ(ξ)) + α(ξ)e1(ξ)] sinh(ω(y − ξ))dξ. (3.13) after the total plane stress is found in the form of equation (3.12), the normal stress σ̄yy can be found by using equation (3.6). then the stress σ̄xx is computed by σ̄xx = σ̄ − σ̄yy. (3.14) 150 a. b. adhe, k. p. ghadle, and u. s. thool if e1,g1 and v1 are constant, then equation (3.12) gives the same expressions for σ and σyy. then, equation (3.9) takes the form σ̄ = e1 [ ccoshωy + dsinhω −α(1 + v1)t̄ + θ(y) + φ(y) + q(y) + ψ(y) + ∫ y −a σ̄(η)k(y,η)dη ] (3.15) which is an analytic solutions to the given thermoelastic problem in r. the shear stress can be determined from equation (2.1) as σ̄xy = i ω ( dσ̄yy dy + f̄y ) , which yields σ̄xy = −i [ p̄2 sinh(ω(y + a)) − i ( iq̄2 + f̄x(x,−a) ω ) cosh(ω(y + a)) +ω ∫ y −a 1 1 −v1 [2g (c cosh ωξ + d sinh ωξ − τ(ξ)) + α(ξ)e1(ξ)] cosh(ω(y − ξ))dξ + f̄y ] . (3.16) we can see that, the stresses σ̄ and σ̄xy are depending only on poisson’s ratio which is deviating in y-coordinate. the stress-tensor components (3.13) − (3.16) can be found by means of the inversion formula f(x,y) = 1 √ 2π ∫ ∞ −∞ f̄(y,ω) exp(iωx)dω 4. numerical results consider a strip loaded by p1 = p2 = aµ(x) , µ(x) = exp(−bx2) (4.1) where, q1 = q2 = 0 at fx = fy = 0 for t = 0. here, a,b are constants and a > 0. numerical computations have been done using python programming language. let, g = g0 exp(ky), v1 = constant, where, k is constant and g0 = e0/2(1 + v0). distribution of a function µ(x) for b = 2 is depicted in figure 2. it shows equation (4.1) gives smooth curve and highest value for x → 0 and vanishes for x →±∞ rapidly, which makes the equation more useful to verify analytic solution. introduce the parameter v0 = 1 + 1 1 − by figure 3 demonstrates the distribution of dimensionless stresses in r for different valus of b. the solid line curve shows the case of the homogeneous material properties i.e. b=0. the case b = 0.5 and b = 1 corresponds to dotted and dashed lines respectively on stress distribution. as the expection the curves are symmetric about y = 0 which clearly shows an effect of inhomogeneity. thus, transversal stress σ̄yy should have maximum value at x = 0 which is shifted towards the direction of inhomogeneity increase. 5. conclusions this article develops an approach for solving an analytic solution of the plane two dimensional thermoelastic problems in terms of inhomogeneous isotropic strip. in this study we arrive at following conclusions. • the original thermoelastic problem is reduced to that of solution of integral equations using direct integration method. analysis of thermal stresses to inhomogeneous strip 151 figure 2. distribution of µ(x) figure 3. distribution of σ̄yy for b=0, 0.5, 1 • it provides the solution of volterra integral equation of second kind which is then solved by resolvent kernel method which provides an efficient technique for analysis of inhomogeneous thermoelastic problems in terms of stress components in the strip r. • the presesnted technique can be applied without any restrictions for material properties . • one can solve corresponding inverse thermoelastic problem in displacements using constructed solutions. • in the present article, analytical solution of thermal stresses is constructed by assuming the fact that, stresses are vanishing at infinity. we can see that, same technique can be used for problems with different loading conditions, instead of fourier transform. 152 a. b. adhe, k. p. ghadle, and u. s. thool references [1] a. adhe and k. ghadle , thermal stress analysis of inhomogeneous infinite solid to 2d elasticity of thermoelastic problems, springer proceedings in mathematics and statistics (2022), 509–521. https://doi.org/10.1007/ 978-981-19-9307-7_41 [2] s.babich and n. dikhtyaruk, load transfer from an infinite inhomogeneous stringer to a prestressed elastic strip clamped at one edge, international applied mechanics, 56(6)(2020), 708-716. https://doi.org/10.1007/ s10778-021-01047-9. 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[24] h. youssef and a. al-lehaibi,the boundary value problem of a three dimensional generalized thermoelastic half-space subjected to moving rechangular heat source, boundary value problems, 8(2019), 1-15. https://doi. org/10.1186/s13661-019-1119-y. [25] j. zhihe and r. batra,thermal shock resistance of functionally graded materials, encyclopedia of thermal stresses, (2014), 5135-5146. https://doi.org/10.1007/978-94-007-2739-7. a. b. adhe, corresponding author, csmss chh. shahu college of engineering, aurangabad 431010 india. email address: adhe.abhijeet@gmail.com k. p. ghadle, department of mathematics, dr. babasaheb ambedkar marathwada university, aurangabad 431004 india. email address: drkp.ghadle@gmail.com u. s. thool, department of mathematics, institute of science, nagpur 440001 india. email address: maths.thool@iscnagpur.ac.in https://doi.org/10.1186/s13661-019-1119-y https://doi.org/10.1186/s13661-019-1119-y https://doi.org/10.1007/978-94-007-2739-7 1. introduction 2. problem formulation 3. solution of thermoelastic problem 4. numerical results 5. conclusions references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 3, number 1, march 2022, pp.60-85 https://doi.org/10.5206/mase/14537 how does the latency period impact the modeling of covid-19 transmission dynamics? ben patterson and jin wang abstract. we introduce two mathematical models based on systems of differential equations to investigate the relationship between the latency period and the transmission dynamics of covid-19. we analyze the equilibrium and stability properties of these models, and perform an asymptotic study in terms of small and large latency periods. we fit the models to the covid-19 data in the u.s. state of tennessee. our numerical results demonstrate the impact of the latency period on the dynamical behaviors of the solutions, on the value of the basic reproduction numbers, and on the accuracy of the model predictions. 1. introduction the coronavirus disease 2019 (covid-19), caused by severe acute respiratory syndrome coronavirus 2 (sars-cov-2), has been a global pandemic for almost two years. despite tremendous efforts in disease control and management, particularly the development and deployment of efficacious vaccines, most countries and territories continue to struggle with the pandemic. as of december 2021, more than 270 million cases were reported throughout the world [43]. in the u.s. alone, covid-19 already led to nearly 50 million cases and over 800 thousand deaths [38]. mathematical and computational models, which have long been used in epidemiological research [4, 6, 13, 30], can help us better understand the transmission dynamics of covid-19 so as to design more effective intervention strategies. a number of mathematical models for covid-19 have been published (see, e.g., [16, 17, 19, 20, 27, 29, 33, 34, 36, 37]). most of these models are based on the susceptible-infectiousrecovered (sir) or susceptible-exposed-infectious-recovered (seir) compartmental framework. sir and seir (and their variants), as two basic types of epidemic models, have been studied extensively and applied to numerous infectious diseases [13, 23]. a distinctive feature of covid-19 not reflected by sir and seir models, however, is that asymptomatic and pre-symptomatic infection is common; i.e., infected individuals could be contagious without showing any symptoms [10, 26, 28]. individuals during this time interval, referred to as an incubation period, may not be aware of their infection at all (since they do not exhibit symptoms) and may easily transmit the disease to other people through movement and contact. this has been one of the important factors that lead to the fast spread of covid-19. it is widely believed that the incubation period for covid-19 could be as long as 14 days [38]. quite a few publications have also estimated the mean incubation period for covid-19. for example, a study for covid-19 patients in china found that the median incubation period was 4.4 days before january 25, 2020 and 11.5 days after january 31, 2020 [21]. a meta-analysis in [9] showed that the mean incubation period of covid-19 was 6.0 days globally from 2010 mathematics subject classification. primary 92d30, 92d25; secondary 37n25, 34a34. key words and phrases. covid-19, mathematical modeling, latency period, data fitting. this work was partially supported by the national institutes of health under grant number 1r15gm131315. 60 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14537 latency period and covid-19 transmission dynamics 61 december 2019 to may 2021. another pooled estimate of the covid-19 incubation period yielded 5.74 days [25]. a concept closely related to the incubation period is the latency period. while the incubation period measures the time between infection and onset of symptoms, the latency period represents the time from infection to infectiousness. the presence of asymptomatic infection for covid-19 indicates that the incubation of the disease typically lasts longer than the latency [12, 24]. this is supported by clinical observations of covid-19 patients that their mean incubation period was longer than their mean latency period [35]. a meta-analysis of reported data from seven countries found that the mean latency period was about 2.52 days [22]. another study found that on average the peak infectivity of covid-19 occurred about 1 day before symptom onset [21]. the specific length of the latency period, however, varies from individual to individual, which contributes to a highly heterogeneous pattern in the course of covid-19 infection. the main goal of this paper is to investigate the impact of the latency period on the transmission dynamics modeling of covid-19. among the large number of mathematical and computational models published thus far for covid-19, very few have been devoted to this point. sadun [27] considered the effects of latency on the basic reproduction number of covid-19 and estimated the fraction of the population that would become infected in the long run. liu et al. [20] incorporated the latency period into two differential equation-based models, including one with a time delay, and fitted the models to covid19 data in china. despite these studies, our understanding of the relationship between the latency period and disease transmission and spread remains inadequate, and our current knowledge is limited regarding the interplay between the asymptomatic infection and the latency in shaping the transmission dynamics of covid-19. to address this issue, we will introduce two simple models in this work, denoted as sair and seair, respectively. the first model consists of the susceptible (s), asymptomatic infectious (a), symptomatic infectious (i), and recovered (r) individuals. the second model includes an additional compartment e for the exposed, or latent, individuals. we will then compare the dynamics of these two models, with a focus on the different dynamical behaviors introduced by the presence of the e compartment. in particular, we will use perturbation theory [3, 14, 15] to analyze the relationship between the two models. additionally, we will apply both models to the covid-19 cases in the u.s. state of tennessee, using data from the tennessee department of health [41]. through data fitting and numerical simulation, we will demonstrate the connection between the sair and seair models and their different dynamical behaviors in a real-world application. the remainder of this article is organized as follows. in section 2, the sair and seair models are formulated, and the main results of their equilibrium analysis are summarized (with details provided in the appendices). in section 3, an asymptotic analysis is conducted for the seair model in terms of small and large latency periods. in section 4, data fitting and numerical simulation results for both models are presented. finally, some discussion is made in section 5 to conclude the paper. 2. model formulation and equilibrium dynamics 2.1. sair model. to study the transmission behavior of covid-19, we consider the following sair system of differential equations: 62 b. patterson and j. wang ds dt = λ −βasa−βisi −µs, da dt = βasa + βisi − (α + γ1 + µ)a, di dt = αa− (w + γ2 + µ)i, dr dt = γ1a + γ2i −µr, (2.1) where s, a, i, and r are the compartments for susceptible, asymptomatic infectious, symptomatic infectious, and recovered individuals, respectively. susceptible individuals become infected by contacting infectious individuals. infected individuals first become asymptomatic infectious; some of them will develop symptoms later, while others will remain asymptomatic over the entire course of their infection. the parameter λ is the population influx rate, βa and βi are the rates of transmission from asymptomatic and symptomatic infections, respectively, µ is the natural death rate, α−1 is the incubation period (so α is the rate of transfer from a to i), γ1 is the rate at which individuals recover from asymptomatic infection, γ2 is the rate at which individuals recover from symptomatic infection, and w is the disease-induced death rate. it is assumed that all of these parameters are positive constants. using the standard next-generation matrix analysis (with details provided in appendix a), we obtain the basic reproduction number for the sair model (2.1) as rsair0 = βaλ µ(α + γ1 + µ) + αβiλ µ(α + γ1 + µ)(w + γ2 + µ) := rsair1 + r sair 2 , (2.2) where rsair1 and rsair2 quantify the contributions from asymptomatic and symptomatic infections, respectively, to the disease transmission risk. meanwhile, from an equilibrium analysis (appendix b), we can establish the following results for the sair model: when rsair0 < 1, there is a unique disease-free equilibrium (dfe): (λ/µ, 0, 0, 0), which is locally asymptotically stable; when rsair0 > 1, the dfe is unstable, and there is a unique endemic equilibrium which is locally asymptotically stable. 2.2. seair model. next, we consider the following seair system for the transmission dynamics of covid-19: ds dt = λ −βasa−βisi −µs, de dt = βasa + βisi − (v + µ)e, da dt = ve − (α + γ1 + µ)a, di dt = αa− (w + γ2 + µ)i dr dt = γ1a + γ2i −µr, (2.3) where the additional compartment e represents exposed individuals who are infected but not yet infectious. this model differs from our previous sair model in that infected individuals will not begin infecting other people immediately, instead experiencing a latency period where they neither show symptoms nor infect others. the parameter v denotes the rate of transfer from e to a; i.e., v−1 represents the latency period. other parameters have the same meanings as those in the sair model. latency period and covid-19 transmission dynamics 63 using the next-generation matrix analysis again (appendix a), we obtain the basic reproduction number for the seair model (2.3) as rseair0 = vβaλ µ(v + µ)(α + γ1 + µ) + vαβiλ µ(v + µ)(α + γ1 + µ)(w + γ2 + µ) := rseair1 + r seair 2 , (2.4) where rseair1 and rseair2 measure the contributions from asymptomatic and symptomatic infections, respectively, to the disease risk in the current seair model. similarly, an equilibrium analysis (appendix c) yields the following results for the seair model: when rseair0 < 1, there is a unique disease-free equilibrium (λ/µ, 0, 0, 0, 0), which is locally asymptotically stable; when rseair0 > 1, the dfe is unstable, and there is a unique endemic equilibrium which is locally asymptotically stable. 3. asymptotic analysis from equations (2.2) and (2.4), it is straightforward to observe that when v → 0, rseair0 → 0; when v → ∞, rseair0 → rsair0 . these suggest special dynamical behaviors with respect to small and large values of v. to explore such dynamical properties, we conduct an asymptotic analysis for the seair model in what follows. when v = 0, the seair system (2.3) becomes ds dt = λ −βasa−βisi −µs, de dt = βasa + βisi −µe, da dt = −(α + γ1 + µ)a, di dt = αa− (w + γ2 + µ)i. (3.1) we have dropped the equation for r, since it is not needed in the analysis. given an initial condition (s,e,a,i) = ( s(0), e(0), a(0), i(0) ) at t = 0, it can be easily verified that the solution of system (3.1) satisfies a(t) = a(0)e−(α+γ1+µ)t, i(t) = αa(0)e−(α+γ1+µ)t w + γ2 −α−γ1 + [ i(0) − αa(0) w + γ2 −α−γ1 ] e−(w+γ2+µ)t and s(t) + e(t) = λ µ + [ s(0) + e(0) − λ µ ] e−µt. when t → ∞, a(t) → 0 and i(t) → 0, which implies that e(t) → 0 based on the second equation of system (3.1). hence, when t → ∞, any solution of system (3.1) approaches the disease-free equilibrium( λ µ , 0, 0, 0 ) . by the continuous dependence on parameters for the solutions of the differential equations, we obtain that when v → 0, all solutions of the original seair system will approach the dfe. in other words, the case with a very large latency period (i.e., v → 0) can be treated as a regular perturbation [3] to the differential system. the result can be easily expected from a biological perspective: when the latency period is sufficiently long, the infection will not spread out and will be eradicated. this is consistent with our observation that rseair0 → 0 as v → 0. 64 b. patterson and j. wang the case with a very small latency period (i.e., v →∞), however, represents a singular perturbation. to analyze this scenario, we let ε = 1 v be the length of the latency period, and examine the solution of the seair system for a small ε > 0 using techniques of singular perturbations [5, 14, 15]. we first re-write system (2.3) as ds dt = λ −βasa−βisi −µs, di dt = αa− (w + γ2 + µ)i, d dt (e + a) = βasa + βisi −µe − (α + γ1 + µ)a, ε de dt = ε [ βasa + βisi −µe ] −e, (3.2) where we have replaced the equation for a with an equation for e +a, obtained by adding the second and third equations in the original system, and where we have multiplied ε on both sides of the e equation. the first three equations in system (3.2) are on a ‘slow’ time scale, compared to the last equation which is on a ‘fast’ scale due to the multiplication of a small ε to the time derivative. to proceed, we introduce a fast time variable τ by τ = t ε . then system (3.2) can be transformed to ds dτ = ε ( λ −βasa−βisi −µs ) , di dτ = ε [ αa− (w + γ2 + µ)i ] , d dτ (e + a) = ε [ βasa + βisi −µe − (α + γ1 + µ)a ] , de dτ = ε [ βasa + βisi −µe ] −e. (3.3) since system (3.2) is formulated by the slow time variable t, we refer to it as the slow system. in contrast, system (3.3) is in terms of the fast time variable τ, and we refer to it as the fast system. these two systems have the same phase portraits for ε > 0, but they have different limit behaviors at ε = 0: the limit of the slow system describes the dynamics on a large time interval away from 0, whereas the limit of the fast system describes the dynamics for a small neighborhood of time 0. details are provided below. setting ε = 0 in system (3.2), we obtain the reduced slow system (or, the slow limit system) ds dt = λ −βasa−βisi −µs, di dt = αa− (w + γ2 + µ)i, d dt (e + a) = βasa + βisi −µe − (α + γ1 + µ)a, e = 0. (3.4) substitution of the last equation (e = 0) into the third equation of system (3.4) yields latency period and covid-19 transmission dynamics 65 ds dt = λ −βasa−βisi −µs, di dt = αa− (w + γ2 + µ)i, da dt = βasa + βisi − (α + γ1 + µ)a. (3.5) system (3.5) is identical to the sair model (2.1), where the equation for r does not need to be included. hence, when ε → 0, the dynamical behavior of the seair model will approach that of the sair model as long as the time t is not very close to 0. this scenario is relevant to our research interest, since the focuses in most epidemiological applications are how an epidemic would spread after its onset and what would be its long-term progression, and those are concerned with relatively large time. this result is consistent with our previous observation that rseair0 → rsair0 when v → ∞. the biological interpretation is that when the latency period is sufficiently short, the impact of the latency could be disregarded. consequently, the dynamics of the disease transmission could be described by the sair model, without incorporating the e compartment. on the other hand, when ε → 0, the dynamical behavior of the seair model very close to the time at 0 would be described by the limit of the fast system (3.3). setting ε = 0 in (3.3) leads to the reduced fast system (or, the fast limit system) ds dτ = 0, di dτ = 0, d dτ (e + a) = 0, de dτ = −e. (3.6) system (3.6) yields s(τ) = s(0), i(τ) = i(0), a(τ) = a(0) + e(0) −e(0)e−τ, e(τ) = e(0)e−τ. in particular, e(τ) will quickly decrease to 0 when τ is increasing. hence, in a small neighborhood of time 0, the solution for e(τ) exhibits a rapid change. this is an analogue to a boundary layer solution in fluid dynamics [2, 32]. we will use numerical simulation in the next section to verify some of these asymptotic predictions. in particular, we will quantify the impact of the latency period on the dynamics of the seair model using real data. 4. numerical simulation 4.1. fitting of data. we apply our models to the covid-19 data in the u.s. state of tennessee in the period between january 1st, 2021 and june 30th, 2021. the most recent estimate from the u.s. census bureau puts the total population of tennessee at n = 6,829,174 [42]. since the time period we consider is short, we assume that immigration and emigration are equal and that the natural birth rate is the same as the natural death rate µ. the natural death rate is defined as the reciprocal of the average life expectancy in the state, which is 75.5 years [1]. we then define the population influx rate as the product of the natural birth rate and total population, λ = µn. meanwhile, we calculate the recovery rate of asymptomatic infection (γ1), the recovery rate of symptomatic infection (γ2), and the incubation rate (α) by the reciprocal of the asymptomatic infection period, the symptomatic infection period, and 66 b. patterson and j. wang the incubation period, respectively, reported in [8]. similarly, we calculate the latency rate (v) using the reciprocal of the average latency period reported in [22]. all these parameter values used in this section are provided in table 1. the remaining parameters are the transmission rates βa and βi, and the disease-induced death rate w. these three parameters may vary significantly from place to place. particularly, a sensitivity analysis (appendix d) indicates that βa and βi are among the most sensitive parameters for the basic reproduction numbers of both models. hence, we will estimate the values of these three parameters by fitting our models to the infection data reported from tennessee department of health [40]. table 1. values of parameters parameter definition value source λ influx rate 247.815 persons per day [1, 42] µ natural death rate 3.629 × 10−5 per day [1] α−1 incubation period 4 days [8] γ−11 asymptomatic recovery period 9.5 days [8] γ−12 symptomatic recovery period 18.07 days [8] v−1 latency period 2.52 days [22] βa asymptomatic transmission rate found by data fitting βi symptomatic transmission rate found by data fitting w disease-induced death rate found by data fitting we first conduct data fitting for the sair model (2.1) based on the reported data from january 1st, 2021 to june 30th, 2021. we fit the number of cumulative confirmed cases using the standard least squares method. we assume that cases are not detected prior to symptom onset, and that fully asymptomatic cases are not taken into account in the recorded number of active cases. the numbers for the active cases and the susceptible and recovered people on january 1st, 2021 are available [40] and they are used in our initial condition. to obtain the initial condition for the number of asymptomatic cases, we use the estimate given by cdc that asymptomatic cases make up about 30% of all active cases [39]; that is, a(0) = 3 7 ×i(0). thus, our initial condition is given by (s(0),a(0),i(0),r(0)) = (6218190, 25642, 59831, 525511). from the numerical solution, the number of cumulative confirmed cases at time t is approximated by i(t) + r(t) + ∫ t 0 wi(τ) dτ , where the first term represents the number of active infections, the second term represents the number of infected individuals that have recovered, and the third term represents the number of disease-induced deaths, as of time t. table 2 contains the results from fitting the cumulative cases by the sair model for βa, βi and w, as well as their 95% confidence intervals. the normalized mean square error for this fitting is found to be 0.00023. table 2. parameter estimates for the sair model parameter resultant value 95% confidence interval βa 3.01 × 10−8/person/day (2.78 × 10−8, 3.23 × 10−8) βi 4.59 × 10−9/person/day (4.53 × 10−9, 4.66 × 10−9) w 0.012/day (0.000, 0.034) using these parameters, we calculate the basic reproduction number of the sair model and obtain rsair0 = 0.907, where rsair0 is defined in equation (2.2). broken down into two parts, we have rsair1 = latency period and covid-19 transmission dynamics 67 0.578, and rsair2 = 0.329. each of the two values quantifies a different route of disease transmission, with rsair1 representing the risk of transmission from asymptomatic individuals and rsair2 representing the risk of transmission from symptomatic individuals. from these values, we immediately observe the following: (1) the basic reproduction number rsair0 is lower than unity, indicating that the epidemic was decaying during the time period of our consideration. this is evidenced by the fact that the reported number of active cases in tennessee decreased from almost 60,000 on january 1st, 2021 to about 1,200 on june 30th, 2021 [40]. the significant reduction of cases was most likely resulting from the vaccination campaign which started in the u.s. from mid-december 2020. (2) between the two components of the basic reproduction number, rsair1 > rsair2 , indicating a higher disease risk from the group of asymptomatic infectious individuals. this is due to the fact that fewer intervention steps are taken to reduce the chance of virus spread, including quarantine or hospitalization, when an individual is not currently experiencing symptoms. figure 1 visualizes the fitting results for the number of cumulative cases based on the sair model and the parameter values provided in tables 1 and 2. figure 2 shows the simulation results for the asymptomatic and symptomatic infectious individuals in tennessee during this time period that starts from january 1st, 2021. 20 40 60 80 100 120 140 160 180 days from january 1st, 2021 5 5.5 6 6.5 7 7.5 8 8.5 c u m u la tiv e c o n fir m e d c a se s 10 5 model data actual data figure 1. a comparison of the cumulative confirmed cases in tennessee from january 1st, 2021 to june 30th, 2021 and the results of the sair model simulation using the known and fitted parameter values from tables 1 and 2. 68 b. patterson and j. wang 20 40 60 80 100 120 140 160 180 200 days from january 1st, 2021 0 1 2 3 4 5 6 a ct iv e c a se s 10 4 asymptomatic symptomatic figure 2. sair model simulation results showing the number of individuals in the asymptomatic and symptomatic infectious compartments for the time period starting from january 1st, 2021, based on the parameter values from tables 1 and 2. next, we conduct data fitting to the seair mode (2.3) and estimate the values of βa, βi, and w using the tennessee covid-19 data for the same 6-month period (from january 1st, 2021 to june 30th, 2021). the initial condition is given by (s(0),e(0),a(0),i(0),r(0)) = (6198190, 20000, 25642, 59831, 525511), where we estimate that the number of exposed individuals to be 20,000 initially. table 3 displays the results from fitting for βa, βi, and w, as well as their 95% confidence intervals. the normalized mean square error in this case is found to be 0.00089. table 3. parameter estimates for the seair model parameter resultant value 95% confidence interval βa 2.98 × 10−8/person/day (2.91 × 10−8, 3.05 × 10−8) βi 4.56 × 10−9/person/day (4.43 × 10−9, 4.70 × 10−9) w 0.013/day (0.009, 0.016) using these parameters, we calculate the basic reproduction number of the seair model and obtain rseair0 = 0.894, with rseair1 = 0.573 and rseair1 = 0.321. we observe a similar patten as that of the sair model; i.e., rseair0 < 1 and rseair1 > rseair2 . meanwhile, figure 3 illustrates the fitting results for the number of cumulative cases based on the seair model, and figure 4 displays the simulation results for the exposed (or, latent), asymptomatic, and symptomatic individuals during this time period starting from january 1st, 2021. we see that the curves for the asymptomatic and symptomatic individuals in figure 4 show a similar behavior as that in figure 2. we will make a careful comparison between the two models in the next section. latency period and covid-19 transmission dynamics 69 20 40 60 80 100 120 140 160 180 days from january 1st, 2021 5 5.5 6 6.5 7 7.5 8 8.5 c um ul at iv e c on fir m ed c as es 10 5 model data actual data figure 3. a comparison of the cumulative confirmed cases in tennessee from january 1st, 2021 to june 30th, 2021 and the results of the seair model simulation using the known and fitted parameter values from tables 1 and 3. 20 40 60 80 100 120 140 160 180 200 days from january 1st, 2021 0 1 2 3 4 5 6 a ct iv e c as es 10 4 exposed asymptomatic symptomatic figure 4. seair model simulation results showing the number of individuals in the exposed, asymptomatic, and symptomatic compartments for the time period starting from january 1st, 2021, based on parameter values from tables 1 and 3. 4.2. comparison of models. now we focus our attention on the comparison of the simulation results generated by the sair and seair models. in particular, we examine how the value of the latency period impacts the numerical solution of the seair model in reference to that of the sair model. to that end, we pick several different values of the latency period, with v−1 = 0.1 days, 1 day, 3 days, and 5 days, for the seair model. we then run the sair model and the seair model (for each given value of v) for the same time period from january 1st, 2021 to june 30th, 2021. except for the parameter v, all the other parameters are fixed and their values are provided in tables 1 and 3. figure 5 shows a comparison of the simulation results for the active asymptomatic cases generated by the sair model and the seair models with different latency periods. meanwhile, figure 6 compares the the simulation results for the active symptomatic cases. we observe a clear pattern from these two figures: the shorter the latency period is, the closer the corresponding seair curve is to the sair curve. this indicates that as the latency period approaches 0, the seair model acts more similarly to the sair 70 b. patterson and j. wang model. an additional evidence is provided in figure 7 where we plot the relative error of the seair model with each given latency period for the total active cases, compared to those generated by the sair model. we clearly see that the relative error decreases when the latency period is reduced. when the latency period is 0.1 days, the relative error is already very close to 0. these findings provide a numerical demonstration of the asymptotic results derived in section 3. 20 40 60 80 100 120 140 160 180 days from january 1st, 2021 0.5 1 1.5 2 2.5 a ct iv e a sy m pt om at ic c as es 10 4 sair data seair with v = 1/0.1 seair with v = 1 seair with v = 1/3 seair with v = 1/5 figure 5. simulation results for the active asymptomatic cases generated by the sair model and the seair models with latency periods of 0.1 days, 1 day, 3 days, and 5 days. 20 40 60 80 100 120 140 160 180 days from january 1st, 2021 1 2 3 4 5 6 a ct iv e s ym pt om at ic c as es 10 4 sair data seair with v = 1/0.1 seair with v = 1 seair with v = 1/3 seair with v = 1/5 figure 6. simulation results for the active symptomatic cases generated by the sair model and the seair models with latency periods of 0.1 days, 1 day, 3 days, and 5 days. latency period and covid-19 transmission dynamics 71 20 40 60 80 100 120 140 160 180 days from january 1st, 2021 -20% 0% 20% 40% 60% 80% 100% 120% 140% r el at iv e e rr or seair with v = 1/0.1 seair with v = 1 seair with v = 1/3 seair with v = 1/5 figure 7. relative error of total active cases generated by the seair models with latency periods of 0.1 days, 1 day, 3 days, and 5 days, compared to those generated by the sair model. in addition, we calculate the basic reproduction number for each model using these parameters and present the results in table 4. the results numerically confirm that when the latency period approaches 0 (i.e., v →∞), rseair0 →rsair0 , as can be seen from the analytical expressions in equations (2.2) and (2.4). another observation is that in all these cases, the values of the basic reproduction numbers are very close to each other. in fact, with a latency period of 5 days, rseair0 and rsair0 only differ in the order of 10−4, and with a latency period of 0.1 days, rseair0 and rsair0 match each other up to 5 decimal digits. this is due to the fact that the basic reproduction number has a low sensitivity on the parameter v (see the sensitivity analysis results in appendix d). an implication is that, for the evaluation of the basic reproduction number for covid-19, using the sair model or seair model will not make a significant difference. table 4. basic reproduction numbers for the sair model and the seair model with latency periods of 0.1 days, 1 day, 3 days, and 5 days model basic reproduction number sair 0.89158 seair with v = 1/0.1 per day 0.89158 seair with v = 1/1 per day 0.89155 seair with v = 1/3 per day 0.89148 seair with v = 1/5 per day 0.89142 4.3. accuracy of predictions. having compared the dynamical properties of the sair and seair models, we now perform a more detailed study on the model output in terms of different latency periods, and assess the accuracy of these models and their abilities to predict future trends of covid-19. for this purpose, we divide the 6-month time frame of our concern into two periods: (1) from january 1st, 2021 to may 31st, 2021, as the fitting period; and (2) from june 1st, 2021 to june 30th, 2021, as the testing period. we run each model with a different latency period to estimate the parameters βa, βi, and w using the reported infection data in the 5-month fitting period. based on the fitted parameter values, a prediction is then generated from the model and compared to the reported data in the 1-month testing period. table 5 displays the parameter estimates in the fitting period using several values for the latency period: 0.1 days, 1 day, 2 days, 2.52 days, 3 days, and 4 days. we note that an average latency period 72 b. patterson and j. wang table 5. parameter estimates for different values of the latency period latency period parameter resultant value 95% confidence interval βa 2.85 × 10−8/person/day (2.60 × 10−8, 3.11 × 10−8) 0.1 days βi 4.60 × 10−9/person/day (4.31 × 10−9, 4.89 × 10−9) w 0.013/day (0.004, 0.026) βa 2.89 × 10−8/person/day (2.71 × 10−8, 3.06 × 10−8) 1 day βi 4.60 × 10−9/person/day (4.35 × 10−9, 4.84 × 10−9) w 0.013/day (0.006, 0.020) βa 2.93 × 10−8/person/day (2.81 × 10−8, 3.06 × 10−8) 2 days βi 4.60 × 10−9/person/day (4.40 × 10−9, 4.80 × 10−9) w 0.013/day (0.009, 0.018) βa 2.94 × 10−8/person/day (2.83 × 10−8, 3.06 × 10−8) 2.52 days βi 4.62 × 10−9/person/day (4.45 × 10−9, 4.64 × 10−9) w 0.013/day (0.010, 0.017) βa 2.95 × 10−8/person/day (2.85 × 10−8, 3.05 × 10−8) 3 days βi 4.66 × 10−9/person/day (4.50 × 10−9, 4.81 × 10−9) w 0.013/day (0.011, 0.016) βa 2.96 × 10−8/person/day (2.87 × 10−8, 3.04 × 10−8) 4 days βi 4.70 × 10−9/person/day (4.56 × 10−9, 4.83 × 10−9) w 0.013/day (0.011, 0.015) of 2.52 days for covid-19 was reported in [22]. among these different latency periods, we pick v−1 = 1 day, 2 days, 2.52 days, and 4 days, and present the numerical results for these four latency periods in figure 8. on the left panel of figure 8, we show the comparison between the actual data and the simulation results for the cumulative cases on both the fitting and testing periods, with each choice of the latency period. on the right panel of figure 8, we plot the relative errors of the simulation results on the testing period, in reference to the actual data. from figure 8 (c) and (e), we observe that when the latency period is 2 days or 2.52 days, the simulation results are very close to the actual data in the testing period. this is confirmed by the plot of the relative errors on the right panel, (d) and (f), each of which shows an error close to 0. furthermore, a careful examination of the right panel indicates that when the latency period is less than or equal to 2 days, the relative error remains negative (a pattern of undershooting in the testing period), and the magnitude of the error decreases when the latency period increases. on the other hand, when the latency period is larger than or equal to 2.52 days, the relative error becomes positive (a pattern of overshooting in the testing period), and the magnitude of the error increases when the latency period increases. hence, we expect that when the latency period is between 2 and 2.52 days, the relative error will change its sign with a further reduced magnitude, and there is possibly a critical value for the latency period that minimizes the error. to verify this, we have chosen a number of additional values between 2 and 2.52 days for the latency period, and repeatedly run the fitting and testing steps for each case. figure 9 shows the relative errors in the testing period for a few typical scenarios, with latency periods of 2.30, 2.35, 2.40 and 2.45 days. as expected, the error in each of these scenarios has a very small magnitude and crosses the horizontal axis. to quantify the difference, we have also computed the l1 norm (i.e., the absolute sum) of the error over the 30-day testing period for each case, and found that the l1 norm of the error is minimized when the latency period is 2.40 days (see figure 9). furthermore, as a comparison, we add the parameter v into the set of parameters to be fitted by data, and conduct the fitting and testing procedure again for the seair model. now we need to estimate four latency period and covid-19 transmission dynamics 73 parameters: βa, βi, w, and v, in the fitting period, and the results are provided in table 6. in this case, we see that the latency period directly estimated from data is v−1 ≈ 5 days. the comparison between the simulation results and the actual data and the calculation of the relative error in the testing period are presented in figure 10. we clearly observe that the accuracy of the prediction in figure 10 is lower than that in figure 9. the implication is that this approach of directly fitting the latency period from data, though efficient, may not be as accurate in the predictions as the previous approach of indirectly (and repeatedly) calibrating the latency period. 20 40 60 80 100 120 140 160 180 days from january 1st, 2021 5 5.5 6 6.5 7 7.5 8 8.5 cu m ul at ive c on fir m ed c as es 10 5 en d of d at a fi tti ng model data actual data (a) latency period = 1 day 155 160 165 170 175 180 days from january 1st, 2021 -1% -0.5% 0% 0.5% 1% 1.5% 2% 2.5% 3% 3.5% 4% re la tiv e er ro r o f c um ul at ive c as es (b) latency period = 1 day 20 40 60 80 100 120 140 160 180 days from january 1st, 2021 5 5.5 6 6.5 7 7.5 8 8.5 cu m ul at ive c on fir m ed c as es 10 5 en d of d at a fi tti ng model data actual data (c) latency period = 2 days 155 160 165 170 175 180 days from january 1st, 2021 0% 0.5% 1% 1.5% 2% 2.5% 3% 3.5% re la tiv e er ro r o f c um ul at ive c as es (d) latency period = 2 days 20 40 60 80 100 120 140 160 180 days from january 1st, 2021 5 5.5 6 6.5 7 7.5 8 8.5 cu m ul at ive c on fir m ed c as es 10 5 en d of d at a fi tti ng model data actual data (e) latency period = 2.52 days 155 160 165 170 175 180 days from january 1st, 2021 0% 0.5% 1% 1.5% 2% 2.5% 3% 3.5% re la tiv e er ro r o f c um ul at ive c as es (f) latency period = 2.52 days 20 40 60 80 100 120 140 160 180 days from january 1st, 2021 5 5.5 6 6.5 7 7.5 8 8.5 cu m ul at ive c on fir m ed c as es 10 5 en d of d at a fi tti ng model data actual data (g) latency period = 4 days 155 160 165 170 175 180 days from january 1st, 2021 -1% -0.5% 0% 0.5% 1% 1.5% 2% 2.5% 3% re la tiv e er ro r o f c um ul at ive c as es (h) latency period = 4 days figure 8. actual data and simulation results for the cumulative cases using latency periods of 1, 2, 2.52 and 4 days. the left panel shows the comparisons in both the fitting and testing periods, and the right panel shows the relative errors in the testing period. 74 b. patterson and j. wang table 6. parameter estimates involving latency parameter resultant value 95% confidence interval βa 2.96 × 10−8/person/day (2.42 × 10−8, 3.08 × 10−8) βi 4.72 × 10−9/person/day (4.55 × 10−9, 4.90 × 10−9) w 0.012/day (0.004, 0.020) v 0.201/day (0.148, 0.255) 155 160 165 170 175 180 days from january 1st, 2021 0% 0.5% 1% 1.5% 2% 2.5% 3% 3.5% r e la tiv e e rr o r o f c u m u la tiv e c a se s (a) latency period = 2.30 days 155 160 165 170 175 180 days from january 1st, 2021 0% 0.5% 1% 1.5% 2% 2.5% 3% 3.5% r e la tiv e e rr o r o f c u m u la tiv e c a se s (b) latency period = 2.35 days 155 160 165 170 175 180 days from january 1st, 2021 0% 0.5% 1% 1.5% 2% 2.5% 3% 3.5% r e la tiv e e rr o r o f c u m u la tiv e c a se s (c) latency period = 2.40 days 155 160 165 170 175 180 days from january 1st, 2021 0% 0.5% 1% 1.5% 2% 2.5% 3% 3.5% r e la tiv e e rr o r o f c u m u la tiv e c a se s (d) latency period = 2.45 days figure 9. relative errors between actual data and simulation results in the testing period for the cumulative cases: (a) latency period = 2.30 days, l1 norm of error = 0.02252; (b) latency period = 2.35 days, l1 norm of error = 0.01725; (c) latency period = 2.40 days, l1 norm of error = 0.01555; (d) latency period = 2.45 days, l1 norm of error = 0.01780. 20 40 60 80 100 120 140 160 180 days from january 1st, 2021 5 5.5 6 6.5 7 7.5 8 8.5 c um ul at iv e c on fir m ed c as es 10 5 e nd o f d at a fi tti ng model data actual data (a) latency period = 1/0.201 days 155 160 165 170 175 180 days from january 1st, 2021 -1.5% -1% -0.5% 0% 0.5% 1% 1.5% 2% 2.5% r el at iv e e rr or o f c um ul at iv e c as es (b) latency period = 1/0.201 days figure 10. actual data and simulation results for the cumulative cases with the latency period directly fitted from data: (a) comparisons in both the fitting and testing periods; and (b) relative error in the testing period. latency period and covid-19 transmission dynamics 75 additionally, we list in table 7 the values of the basic reproduction number (rseair0 ) and its two components (rseair1 and rseair2 ) for each different choice of the latency period discussed in this section. note that each latency period here corresponds to a different set of fitted parameters βa, βi, and w (see table 5). hence, we see a larger variation of the basic reproduction numbers than that presented in table 4. table 7. basic reproduction numbers for different fitting scenarios latency period rseair1 rseair2 rseair0 0.1 days 0.549 0.321 0.870 1 day 0.555 0.322 0.877 2 days 0.564 0.321 0.885 2.30 days 0.566 0.322 0.888 2.35 days 0.566 0.322 0.888 2.40 days 0.566 0.322 0.888 2.45 days 0.566 0.323 0.889 2.52 days 0.566 0.323 0.889 3 days 0.566 0.326 0.892 4 days 0.569 0.330 0.899 5. discussion we have performed a modeling study for the impact of the latency period on the transmission dynamics of covid-19, by considering an sair model and an seair model and comparing their dynamical behaviors. each model incorporates the dual (asymptomatic and symptomatic) transmission routes of covid-19, but the second model (seair) additionally includes a latent compartment that is absent from the first one (sair). for model comparison, we have combined equilibrium analysis, asymptotic study, and numerical simulation. using the covid-19 data in the state of tennessee, we have examined various scenarios associated with different latency periods in this work. in particular, our findings show that within the biologically meaningful regime, the dynamics of the seair model approach those of the sair model when the length of the latency period tends to 0. meanwhile, we have numerically tested several values of the latency period and found that when the latency period equals 2.40 days, the predictions generated by the seair model achieve the best performance in the sense that the relative error is minimized over the testing period. this numerically found ‘optimal’ value is close to the mean latency period of 2.52 days estimated in the meta-analysis of covid-19 data from seven countries [22]. among our numerical findings, we highlight the following: (1) the value of the latency period has a substantial impact on the state variables, particularly the numbers of the asymptomatic and symptomatic infectious cases generated by the models. (2) the latency period has a less significant impact on the basic reproduction number. in this regard, using the sair model or the seair model with different latency periods to evaluate the basic reproduction number would lead to similar results. (3) the accuracy of the predictions from the seair model strongly depends on the latency period, and a carefully calibrated value of the latency period could minimize the error of the prediction. these results could provide useful guidelines in the selection of mathematical models and calibration of parameters to study the transmission and spread of covid-19. although some of the findings in this work are specific to the covid-19 data in tennessee, the methodology can be generalized toward broader applications. as is common in data fitting studies, 76 b. patterson and j. wang potential inaccuracy of the data could impact the quantitative outcome of our models, though we expect that our qualitative predictions would still hold. by presenting two very basic and simple models, we have adopted the point of view that “useful models are simple and extendable” [11]. the two models (sair and seair) are analytically tractable and their dynamics can be compared through both asymptotic and numerical studies, which serve our main purpose of investigating the interplay between the asymptomatic and symptomatic transmission routes and the latency period in covid-19 transmission dynamics. on the other hand, we have only discussed the local asymptotic stability of the equilibrium solutions (appendices b and c) and have not studied the global stability properties, since the stability of the dynamical systems is not our focus in this work. nevertheless, standard techniques, such as the geometric approach [18] and lyapunov functions, can be employed to conduct the global stability analysis when needed. moreover, our modeling framework can be extended in a number of ways by incorporating factors such as disease prevention and intervention measures, sars-cov-2 variants, and hospitalization. we expect, however, that the qualitative relationship between a model incorporating the latency period and one without considering latency would remain the same. an additional remark is that we have not explicitly considered vaccination in our current models. instead, the impact of vaccination is implicitly incorporated into the models through the diseases transmission rates whose values fitted from real data could (at least) partially reflect the reduced infection risk due to vaccine deployment. appendix a: basic reproduction numbers we first consider the sair model (2.1). it is straightforward to determine that the disease-free equilibrium (dfe) of the system is x0 = (λ/µ, 0, 0, 0). to find the basic reproduction number of the system, rsair0 , we use the next-generation matrix method described by driessche and watmough [31]. to do this, we define two vectors, f and v , where f represents the rate at which new infection is introduced in the asymptomatic and symptomatic compartments and v represents the rate of transfer into and out of these compartments. thus we obtain[ da dt di dt ] = f − v = [ βasa + βisi 0 ] − [ (α + γ1 + µ)a (w + γ2 + µ)i −αa ] . now we find the jacobians of f and v at the dfe, f = [ ∂f ∂x (x0) ] = [ βaλ/µ βiλ/µ 0 0 ] , v = [ ∂v ∂x (x0) ] = [ α + γ1 + µ 0 −α w + γ2 + µ ] . this gives us the next-generation matrix fv −1 = [ βaλ µ(α+γ1+µ) + αβi λ µ(α+γ1+µ)(w+γ2+µ) βi λ µ(w+γ2+µ) 0 0 ] . the basic reproduction number is then defined as the spectral radius of the next-generation matrix fv −1. so rsair0 = βaλ µ(α + γ1 + µ) + αβiλ µ(α + γ1 + µ)(w + γ2 + µ) . (a.1) latency period and covid-19 transmission dynamics 77 next, we consider the seair model (2.3). the dfe of the system is given by x0 = (λ/µ, 0, 0, 0, 0). to find the basic reproduction number, rseair0 , we similarly separates the system into two parts: dedtda dt di dt   = f − v =  βasa + βisi0 0  −   (v + µ)e(α + γ1 + µ)a−ve (w + γ2 + µ)i −αa   . the jacobians of f and v evaluated at the dfe are f = [ ∂f ∂x (x0) ] =  0 βaλ/µ βiλ/µ0 0 0 0 0 0   , v = [ ∂v ∂x (x0) ] =  v + µ 0 0−v (α + γ1 + µ) 0 0 −α (w + γ2 + µ)   . thus the next-generation matrix is fv −1 =   vβaλ µ(v+µ)(α+γ1+µ) + vαβi λ µ(v+µ)(α+γ1+µ)(w+γ2+µ) βaλ µ(α+γ1+µ) + αβi λ µ(α+γ1+µ)(w+γ2+µ) βi λ µ(w+γ2+µ) 0 0 0 0 0 0   . consequently, the spectral radius of the matrix fv −1 yields rseair0 = vβaλ µ(v + µ)(α + γ1 + µ) + vαβiλ µ(v + µ)(α + γ1 + µ)(w + γ2 + µ) . (a.2) appendix b: equilibrium analysis for the sair model an equilibrium point of the sair system (2.1) implies that ds dt = da dt = di dt = dr dt = 0 and so should satisfy the following equations λ = βasa + βisi + µs, (b.1) (α + γ1 + µ)a = βasa + βisi, (b.2) 0 = αa− (w + γ2 + µ)i, (b.3) 0 = γ1a + γ2i −µr. (b.4) we show that when rsair0 < 1 the only equilibrium point is the dfe, and when rsair0 > 1 there exists a positive endemic equilibrium. firstly, we rewrite equation (b.3) so that a is expressed as a function of i: a(i) = (w + γ2 + µ)i α (b.5) which is increasing for all i > 0. next, we substitute the right-hand side of equation (b.2) into equation (b.1) to get λ = (α + γ1 + µ)a(i) + µs and from this we derive s(i) = λ − (α + γ1 + µ)a(i) µ , (b.6) which is decreasing since a(i) is increasing. by substituting equation (b.5) into equation (b.2), we obtain (α + γ1 + µ)(w + γ2 + µ)i α = βas(i)(w + γ2 + µ)i α + βis(i)i. 78 b. patterson and j. wang when i 6= 0, this simplifies into 1 = s(i) ( βa α+γ1+µ + βiα (α+γ1+µ)(w+γ2+µ) ) . (b.7) we denote the right-hand side of equation (b.7) by f(i), which is decreasing since s(i) is decreasing. clearly, f(i) < 1 for sufficiently large i. therefore, to guarantee that a unique endemic equilibrium exists, f(0) > 1 must be true, where f(0) = s(0) ( βa α+γ1+µ + βiα (α+γ1+µ)(w+γ2+µ) ) = λ µ ( βa α+γ1+µ + βiα (α+γ1+µ)(w+γ2+µ) ) . (b.8) by equation (a.1), we have f(0) = rsair0 . this implies that when rsair0 > 1, there is a unique endemic equilibrium, and when rsair0 < 1, the only possible equilibrium is the dfe. based on the standard result in [31], the unique dfe is locally asymptotically stable when rsair0 < 1 and unstable when rsair0 > 1. next, we use the routh-hurwitz criteria to show that the unique endemic equilibrium is locally asymptotically stable when rsair0 > 1. firstly, we let x̂ = (ŝ,â, î, r̂) be the unique endemic equilibrium, and construct the jacobian of the system at the endemic equilibrium: j =   −βaâ−βiî −µ −βaŝ −βiŝ 0 βaâ + βiî βaŝ −α−γ1 −µ βiŝ 0 0 α −(w + γ2 + µ) 0 0 0 γ1 + γ2 −µ   . next we find the eigenvalues using the characteristic polynomial defined by det(λi −j) = (λ + µ) ∣∣∣∣∣∣∣ λ + (βaâ + βiî + µ) βaŝ βiŝ −βaâ−βiî λ− (βaŝ −α−γ1 −µ) −βiŝ 0 −α λ + (w + γ2 + µ) ∣∣∣∣∣∣∣ which, for simplicity, we denote as (λ + µ) ∣∣∣∣∣∣ λ + a11 a12 a13 a21 λ + a22 a23 0 −α λ + a33 ∣∣∣∣∣∣ . this gives us the characteristic equation (λ + µ)(λ3 + xλ2 + yλ + z) (b.9) where x = a11 + a22 + a33, (b.10) y = a23α + a11(a22 + a33) + a22a33 −a21a12, (b.11) z = a11a22a33 −a21a12a33 + α(a11a23 −a21a13). (b.12) to satisfy the routh-hurwitz criteria and prove local asymptotic stability, we need to show that x > 0, y > 0, z > 0, and xy > z. we start off by showing that x > 0. it is simple to confirm that each component of x is positive: a11 = βaâ + βiî + µ, (b.13) which is clearly positive, and a33 = w + γ2 + µ, (b.14) latency period and covid-19 transmission dynamics 79 which is also positive. using equation (b.2), (βaŝ −α−µ) can be rewritten as −βiŝî â , therefore a22 = βiŝî â , (b.15) so a22 is positive as well. since each component of x is positive, we conclude that x > 0. next we show that y > 0. it is clear that since a11, a22 and a33 are all positive, a11(a22 + a33) is positive as well. now, −a21a12 > 0 since −a21a12 = −(−βaâ−βiî)(βaŝ) = (βaâ + βiî)(βaŝ). finally, we look at a23α + a22a33. using equation (b.3), a33 can be rewritten as a33 = (w + γ2 + µ) = αâ î . (b.16) therefore, using the value of a22 given by equation (b.15), a23α + a22a33 = −βiŝα + βiŝî â ( αâ î ) = 0, giving us a new equation for y, y = a11(a22a33) −a21a12, (b.17) which allows us to conclude y > 0. now we show that z > 0. to begin, we look at α(a11a23 −a21a13): α(a11a23 −a21a13) = α((βaâ + βiî + µ)(−βiŝ) − (−βaâ−βiî)(βiŝ)) = α(−(βaâ + βiî)(βiŝ) + (βaâ + βiî)(βiŝ) −µβiŝ) = −µβiŝα. (b.18) using this result and the product of a11a22a33, and using the values of a22 and a33 from equations (b.15) and (b.16), respectively, we can see that a11a22a33 + α(a11a23 −a21a13) = (βaâ + βiî)(βiŝα), which is clearly positive. now, −a21a12a33 = −(−βaâ−βiî)(βaŝ)( αâ î ) = (βaâ + βiî)(βaŝ)( αâ î ) , which is also positive. therefore, since each component of z is positive, we can conclude that z > 0. finally, we need to verify that xy > z. to do this, we need to look at a33y : a33y = ( αâ î )((βaâ + βiî + µ)( βiŝî â + αâ î ) + (βaâ + βiî)(βaŝ)) = (βaâ + βiî)(βiŝα) + (βaâ + βiî)(βaŝ)( αâ î ) + (βaâ + βiî)( αâ î )2 + µ(βiŝα + ( αâ î )2). this is greater than z because z = (βaâ + βiî)(βiŝα) + (βaâ + βiî)(βaŝ)( αâ î ), and since x > a33, we conclude that xy > z. thus, all of the routh-hurwitz criteria are satisfied, and we conclude that the unique endemic equilibrium is locally asymptotically stable when it exists; i.e., when rsair0 > 1. 80 b. patterson and j. wang appendix c: equilibrium analysis for the seair model an equilibrium of the seair system (2.3) should satisfy λ = βasa + βisi + µs, (c.1) (v + µ)e = βasa + βisi, (c.2) 0 = ve − (α + γ1 + µ)a, (c.3) 0 = αa− (w + γ2 + µ)i, (c.4) 0 = γ1a + γ2i −µr. (c.5) using equation (c.4) we get a(i) = (w + γ2 + µ)i α , (c.6) and by equation (c.3) we get e(i) = (α + γ1 + µ)(w + γ2 + µ)i vα , (c.7) both of which are increasing for all i > 0. substituting equation (c.6) into equation (c.1), we have s(i) = λ βa(w+γ2+µ)i α + βii + µ , (c.8) which is decreasing for all i > 0. next, we substitute equations (c.6) and (c.7) into equation (c.2) to obtain 1 = s(i) ( βav (v+µ)(α+γ1+µ) + βivα (v+µ)(α+γ1+µ)(w+γ2+µ) ) , the right-hand side of which is denoted by f(i). now, f(i) is decreasing since s(i) is decreasing for all i > 0 and f(i) < 1 for sufficiently large i. thus, to guarantee a unique endemic equilibrium, f(0) > 1 must be true, where f(0) = λ µ ( βav (v+µ)(α+γ1+µ) + βivα (v+µ)(α+γ1+µ)(w+γ2+µ) ) , (c.9) which is identical to rseair0 . thus, we conclude that when rseair0 < 1 the only equilibrium point is the dfe, and when rseair0 > 1, there exists a unique positive endemic equilibrium. next, as in the sair model, we use the routh-hurwitz criteria to show that the endemic equilibrium of the seair model is locally asymptotically stable when rseair0 > 1. let x̂ = (ŝ, ê,â, î, r̂) be the unique endemic equilibrium for the system. we construct the jacobian, j, of the system at the endemic equilibrium: j =   −βaâ−βiî −µ 0 −βaŝ −βiŝ 0 βaâ + βiî −(v + µ) βaŝ βiŝ 0 0 v −(α + γ1 + µ) 0 0 0 0 α −(w + γ2 + µ) 0 0 0 γ1 γ2 −µ   . to determine the characteristic polynomial, we compute det(iλ−j) = (λ + µ) ∣∣∣∣∣∣∣∣∣ λ + βaâ + βiî + µ 0 βaŝ βiŝ −βaâ−βiî λ + (v + µ) −βaŝ −βiŝ 0 −v λ + (α + γ1 + µ) 0 0 0 −α λ + (w + γ2 + µ) ∣∣∣∣∣∣∣∣∣ , latency period and covid-19 transmission dynamics 81 which for simplicity we denote as (λ + µ) ∣∣∣∣∣∣∣∣ λ + a11 0 a13 a14 a21 λ + a22 a23 a24 0 −v λ + a33 0 0 0 −α λ + a44 ∣∣∣∣∣∣∣∣ . this gives us the characteristic polynomial (λ + µ)(λ4 + xλ3 + yλ2 + zλ + c), (c.10) where x = a11 + a22 + a33 + a44, (c.11) y = va23 + a11a22 + a11a33 + a11a44 + a22a33 + a22a44 + a33a44, (c.12) z = vαa24 + va23a44 + va11a23 −a21a13 + a11a22a33 + a22a33a44 +a11a33a44 + a11a22a44, (c.13) c = a11(va44a23 + vαa24) −vαa21a14 −va44a21a13 + a11a22a33a44. (c.14) to verify the routh-hurwitz criteria for stability, we need to show x > 0, y > 0, z > 0, c > 0, and xyz > z2 + cx2. to begin, we already know x = a11 +a22 +a33 +a44 is positive since each of our parameters is assumed to be positive. now, to show that y is positive we need to look at va23 : va23 = −vβaŝ. substituting equations (c.6) and (c.7) into equation (c.2), we can get −vβaŝ = vαβiŝ (w + γ2 + µ) − (v + µ)(α + γ1 + µ). (c.15) the latter term of this, −(v + µ)(α + γ1 + µ), is equal to −a22a33 so the terms will cancel out. we are left with y = vαβiŝ (w + γ2 + µ) + a11a22 + a11a33 + a11a44 + a22a44 + a33a44, (c.16) which is clearly positive. to show that z > 0, we first need to look at the terms vαa24 + va23a44 : vαa24 + va23a44 = −vαβiŝ −vβaŝ(w + γ2 + µ) substituting equations (c.6) and (c.7) into equation (c.2) yields −vαβiŝ = vβaŝ(w + γ2 + µ) − (v + µ)(α + γ1 + µ)(w + γ2 + µ). (c.17) therefore, vαa24 + va23a44 = −(v + µ)(α + γ1 + µ)(w + γ2 + µ), which will cancel with the term a22a33a44. next, we have va11a23 = −vβaŝ(βaâ + βiî + µ), which can be rewritten using equation (c.15) to be va11a23 = vαβiŝ (w + γ2 + µ) (βaâ + βiî + µ) − (v + µ)(α + γ1 + µ)(βaâ + βiî + µ) 82 b. patterson and j. wang which cancels out the term a11a22a33, and we are left with vαβiŝ (w+γ2+µ) (βaâ + βiî + µ), which is clearly positive. finally, the last term in question is −a21a13 = (βaâ + βiî)(βaŝ), which is also clearly positive. after all the canceling terms we are left with z = vαβiŝ (w + γ2 + µ) a11 −a21a13 + a11a33a44 + a11a22a44, (c.18) every term of which is positive, and so we conclude that z > 0. next, we need to show that c > 0. firstly, we know the terms −vαa21a14 − va44a21a14 are positive since −vαa21a14 −va44a21a14 = vα(βaâ + βiî)(βiŝ) + v(βaâ + βiî)(βaŝ)(w + γ2 + µ). secondly, we have a11(vαa24 + va23a44), which, using equation (c.17), can be written as a11(vαa24 + va23a44) = −(βaâ + βiî + µ)(v + µ)(α + γ1 + µ)(w + γ2 + µ), which will cancel out with a11a22a33a44. thus, we are left with c = −vαa21a14 −va44a21a13, (c.19) and we conclude that c > 0. finally, we may conduct similar calculations to show xyz > z2 + cx2, though the algebraic manipulations become very tedious and the details are not provided here. verification of all these routh-hurwitz criteria will lead to our conclusion that the endemic equilibrium of the seair model is locally asymptotically stable when it exists; i.e., when rseair0 > 1. appendix d: sensitivity of the basic reproduction number to quantify the influence of model parameters in shaping the disease risk, we conduct a sensitivity analysis for the basic reproduction numbers of the sair and seair models. this is usually achieved by evaluating the relative sensitivity. to compute the relative sensitivity of the basic reproduction number, r0, in terms of a parameter, h, we take the partial derivative of r0 with respect to h, and then normalize the result by dividing it with the quotient r0/h. that is, ∂r0 ∂h · h r0 , and we substitute the parameter values from data fitting (section 4) to complete the evaluation of the relative sensitivity. this procedure is applied to each parameter that the basic reproduction number depends on. figures 11 and 12 show the sensitivity analysis results for the sair mode and the seair model, respectively. latency period and covid-19 transmission dynamics 83 2 1 w i a parameter -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 r e la ti v e s e n s it iv it y figure 11. sensitivity analysis results for the basic reproduction number of the sair model. 1 2 w v i a parameter -0.3 -0.2 -0.1 0 0.1 0.2 0.3 r e la ti v e s e n s it iv it y figure 12. sensitivity analysis results for the basic reproduction number of the seair model. references [1] e. arias, b. bastian, j. xu, and b. tejada-vera, u.s. state life tables, 2018, national vital statistics reports, 70(2021). 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[36] c. yang and j. wang, a mathematical model for the novel coronavirus epidemic in wuhan, china, mathematical biosciences and engineering, 17(2020), 2708-2724. https://www.imperial.ac.uk/mrc-global-infectious-disease-analysis/news--wuhan-coronavirus/ https://www.imperial.ac.uk/mrc-global-infectious-disease-analysis/news--wuhan-coronavirus/ latency period and covid-19 transmission dynamics 85 [37] c. yang and j. wang, modeling the transmission of covid-19 in the us – a case study, infectious disease modelling, 6(2021), 195-211. [38] centers for disease control and prevention: coronavirus disease 2019 (covid-19). available at https://www.cdc. gov/coronavirus/2019-ncov [39] centers for disease control and prevention: covid-19 pandemic planning scenarios. available at https://www.cdc. gov/coronavirus/2019-ncov/hcp/planning-scenarios.html [40] datasets from tennessee department of health. available at https://www.tn.gov/health/cedep/ncov/data/ downloadable-datasets.html [41] tennessee department of health. available at https://www.tn.gov/health.html [42] u.s. census bureau quick facts: tennessee. available at https://www.census.gov/quickfacts/fact/table/tn [43] who coronavirus disease (covid-19) weekly epidemiological update and weekly operational update. available at https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports department of mathematics, university of tennessee at chattanooga, chattanooga, tn 37403, usa email address: gvj998@mocs.utc.edu corresponding author, department of mathematics, university of tennessee at chattanooga, chattanooga, tn 37403, usa email address: jin-wang02@utc.edu https://www.cdc.gov/coronavirus/2019-ncov https://www.cdc.gov/coronavirus/2019-ncov https://www.cdc.gov/coronavirus/2019-ncov/hcp/planning-scenarios.html https://www.cdc.gov/coronavirus/2019-ncov/hcp/planning-scenarios.html https://www.tn.gov/health/cedep/ncov/data/downloadable-datasets.html https://www.tn.gov/health/cedep/ncov/data/downloadable-datasets.html https://www.tn.gov/health.html https://www.census.gov/quickfacts/fact/table/tn https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports 1. introduction 2. model formulation and equilibrium dynamics 2.1. sair model 2.2. seair model 3. asymptotic analysis 4. numerical simulation 4.1. fitting of data 4.2. comparison of models 4.3. accuracy of predictions 5. discussion appendix a: basic reproduction numbers appendix b: equilibrium analysis for the sair model appendix c: equilibrium analysis for the seair model appendix d: sensitivity of the basic reproduction number references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 4, number 2, june 2023, pp.115-127 https://doi.org/10.5206/mase/15591 a conceptual model of mixed integer linear programming water distribution system habiba babangida awwalu, nasiru abdullahi, and muktar hussaini abstract. water is a basic part of our daily lives, as such effective water supply is of paramount importance. thus, as a result of the rise in population size and water shortage there is the need for proper, suitable and optimal utilization of water resources to efficiently be distributed among the populace. the proper allocation and distribution of water in the field of network planning need to be modelled through mathematical parameters for objective of water distribution system. this mathematical approach requires of solving an optimization problem based on multi-objective function subjected to certain constraints of mixed integer linear programming objective function which is proportional to the cost of the water distribution network. this paper present a conceptual model of multi-objective optimization proposed for determination of design parameters of water distribution system by considering the significant number of constraints, decision variables, cost and reliability objective functions. the model was proposed to solve the reliability problem of water production and reduce the design and operational costs. keywords: conceptual model, cost function, differential evolution, reliability function, multi-objective function, mixed-integer linear programming 1. introduction water distribution system (wds) is a complex structure to be managed by each city. therefore, the design of the water distribution network must be effective, in order to ensure adequate, quality and constant water supply to the population. in addtion, it is of paramount importance for wdss to be designed effectively in order to sufficiently supply water to the end users. a water distribution system consists of pipes, reservoirs, pumps, and valves of different types, which are connected to each other to provide water to consumers. providing water at adequate pressure, quantity and quality to the consumers at minimized cost is the fundamental goal of water distribution system. hence, improper design of the water distribution system can lead to deterioration of the system and consequently affects the operation and maintenance cost. the entire cost associated with the design of the wdss is a function of the pipe design diameters and material, and the diameters in turn affect the network pressures and leading to decline in network reliability, inadequate water supply and pressure. this research is into providing a multi-objective model for optimal water distribution system to promote a sustainable improvement of water pipe-lining at minimal cost of design and maintenance, also to improve the reliability of water production taking into consideration of model constraints. received by the editors 9 september 2022; accepted 9 april 2023; published online 22 may 2023. 2020 mathematics subject classification. primary 90c29. key words and phrases. differential evolution (de), water distribution system (wds), pipe network, mixed integer linear programming (milp), pareto fronts, multi-objective functions. 115 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15591 116 h. b. awwalu, n. abdullahi, and m. hussaini moreover, in te study of comparative analysis with some wdss case studies around the world, the author in [5] performances in the optimization design of wdss are compared among the six well-known benchmark wdss were used to verify the effectiveness of their proposed optimization approach, including the two-loop network, the goyang water distribution network, the bakryun water distribution network, the new york tunnels problem(nytp), the hanoi problem (hp) and balerma network (bn). previous researches were conducted to determine an optimal “water distribution system”, wds or network [14], [15], [8].however, in [5],[15], [13] were studies on single objective optimization problem for wds. moreover, in [8] multi-objective optimization problem was considered, but minimization of cost for water system network design and operational cost were not considered. therefore, in this research endeavor, three objectives function that were subjected to five number of constraints is considered as the proposed mixed-integer linear programming(milp) model for wds and used differential evolutionary (de)algorithm for optimal wds. in this milestone, the formulated milp mathematical model for multi-objective optimization problem to: (1) minimize total cost of pipe-line network, pump and fuel cost, and (2) maximize the network reliability for better production and (3) minimize the water head loss and water loss due to leakages to satisfy the consumer demand would be verified and validated. to understand the concept of multi-objective optimization problem (moop) and its attainment, a definition of single-objective optimization problem is required. a single-objective optimization problem (soop) can be defined as: definition1: general single objective optimization problem (gsoop). chiandussi et al (2012),a general single-objective optimization problem (gsoop) is defined as the minimization (or maximization) of a scalar objective function f(x) subject to inequality constraints gi(x) ≤ 0, i = (1, ...,m) and equality constraints hi(x) = 0,j = (1, ...,p) where x is an n-dimensional decision variable vector x = (x1, ...,xn) from some universe ω. ω contains all possible x that can be used to satisfy an evaluation of f(x) and its constraints. the method of finding the global optimum of any function (may not be unique) is referred to as global optimization. generally, the global minimum of a given single-objective problem is : definition2: (single-objective global minimum optimization) [4], given a function f : ω ⊆ rn → , ω 6= φ, for x ∈ ω, the value f∗ , f(x∗) ≥ −∞ is called a global minimum if and only if ∀x ∈ ω : f(x∗) ≤ f(x) where x∗ is by definition the global minimum solution, f is the objective function and the set ω is the feasible region of x . the goal of determining the global minimum solution is called the global optimization problem for a single-objective problem, [4]. multi-objective problems are those problems where the goal is to optimize simultaneously k objective functions designated as f1(x),f2(x), ...,fk(x), and forming a vector function f(x) = (f1(x),f2(x), · · · ,fk(x)). although soops may have a unique solution, multi-objective problems (as a rule) presents a possibly uncountable set of solutions. two n-dimensional euclidean spaces rn are considered in multi-objective problems (fig. 1.1) [4]. • the n-dimensional space of the decision variables in which each coordinate axis corresponds to a component of vector x mixed integer linear programming water distribution system 117 • the k-dimensional space of the objective functions in which each coordinate axis correspond to a component vector fk(x) the evaluation function of a multi-objective problem, f : ω → λ maps the decision variables x = (x1, ...,xn) to vectors (y = a1, ...,an). the set of solutions is found through the use of the pareto optimality theory, [6]. this mapping may or may not be onto some region of the objective function space depending on the functions and the constraints defining the multi-objective problem. most of the previous researchers were aimed to solve the efficiency of water distribution system by proposing a single objective model such as minimization of cost, minimization of water head loss, or maximization of water production and others. therefore, this paper aims to proposed a multiobjective model for wds, that uses milp concept, also presents a survey and intensive review of water distribution system, differential evolutionary algorithm challenges in relation to both single and multi-objective optimization problems. the rest of the paper is organized into six sections, starting from the introductory, optimization of pipe network, optimization algorithm, proposed milp model for wds, then verification and validation process of the model. the paper is concluded with the proposal of full experimentation and benchmarking of the proposed model in the near future. 2. related works in order to minimize the costs associated with operating a wds according to [3], [2] water loss and pump energy consumption have to be reduced through optimal pressure set point for pressure reducing valves (prvs) and pump speed as reviewed in [1]. the energy consumption, titled as energy objective function (ofe) for a full day (24hrs) period is written as ofe = 24∑ t=1 np∑ p=1 γ ·qp(α) ·hp(α) · t ·ct η(α) (2.1) from (2.1), t denotes the time step as pump (α) supplies a flow rate qp(α), operating based on the relative speed (α) for hydraulic head hp with efficiency η. the energy cost is linked with the particular time of the day and is represented by ct in a network with np pumps the cost for water loss is estimated through the estimation of any water leaks and the associated water cost. however, based on the rate of energy consumption, the water loss cost can be minimal, hence hindering the optimization process. an alternative approach of pressure management without estimation of water losses cost is the relative 118 h. b. awwalu, n. abdullahi, and m. hussaini difference between the minimum operating pressure (pmin) that is required by the system and actual operating pressure. this is termed as pressure objective function (ofp), as shown below ofp = 24∑ t=1 nn∑ n=1 |pt,n −pmin| pmin (2.2) from (2.2), p(t,n) represents the pressure at the node n in time step t within the network of nn nodes. it can be clearly seen that (2.2) is a non-dimensional function, and (2.1) is the energy cost, in monetary units. in order to make both functions in the common form of non-dimensional representation, equation (2.1) is divided by the maximum value of energy cost which is found using the nominal speed of the pump denoted as α = 1. thus, the objective function is written as of = 24∑ t=1 nn∑ n=1 |pt,n −pmin| pmin + 24∑ t=1 np∑ p=1 γ ·qp(α) ·hp(α) · t ·ct η(α) (2.3) based on operation constraints, the standard pressures (pminandpmax) and tank levels (l(t,min) and (l(t,max)) are taken into consideration in finding the operational schedule that safely delivers the demand that was forecast. according to [21] in order to avoid maintenance problems and noise at pumps there is need to define minimal speed. thus, the optimization problem is represented as: pmin ≤ p(t,n) ≤ pmax (2.4) lt,min ≤ lt,t ≤ lt,max (2.5) α ≥ 0.5 (2.6) hence, the penalty function is presented as: pen = nc∑ t=1 βi · |xsi −x lim i | k (2.7) from (2.7), pen denotes the total penalty for the solution of the problem nc constrains which is represented by xlimi . the adjusted parameters to aid convergence are denoted as βi and k. finally, the compound objective function (cof) is presented by cof = of + pen. (2.8) 3. conceptual model of proposed milp wds traditionally, ideas can be developed into mathematical formulation to represents a model as proposed in [11],[12], [10] which is called a conceptual model. moreover, as previous research used milp to model wds and combined it with de to obtain the optimal solution, though they consider pipe size and inlet pressure as objective function [14].in this study, the milp of wds is formulated as a multiobjective optimization (moo) problem to minimize pipe-line cost and the total head loss of the system, also maximize reliability of water production. in particular, the proposed model aims to proposed the optimal wds incorporated with pipe length, diameter of all arcs in the network, pump capacity, peak and fire flow demand,cost of pump and fuel, subject to the constraints of mass conservation, energy conservation, minimum pressure, flow requirement and pipe size available while the pipe layout and its connectivity,nodal demand, and minimum head requirements would be imposed. in addition, this study proposes a new term of flow conservation constraint in which the total quantity of water that flow through any pipe is equal to the demand of destination node plus with all flow remaining flow along the pipe path. the parameters parameters were mathematically proposed for the formulation of the milp wds model with three objective functions and five constraints: aij = pipe for start node i to destination node j, where i,j = {1, 2, 3...np}; mixed integer linear programming water distribution system 119 np = number of pipe; npu = number of pumps in the network; ci = cost of pipe per unit length; di = diameter of pipe aij; li = length of pipe aij; cpi = cost of pump per unit total power; pi = power of i th pump; cfu =cost of fuel; t = time to run a pump or generator per month ∀t ∈ t = {1, 2, 3, ...t}; zt = binary variable indicating the water demand in month t is supplied or not; djpeak =peak demand of destination node j; dff = fire flow demand; hi = head of start node i; hj = head of destination node j; hlaij = head loss along pipe aij; flowaij= total quantity of water flow through pipe aij; rk = the resistance coefficient of the kth pipe with flow rate; qk = flow rate of kth pipe and; n = 1.852 a constant number depending on the head loss equation. 4. proposed milp wds objective functions an objective function defines the quantity to be optimized, and the goal of linear programming is to fine the values of the variables that minimize or maximize the objective function [8], [18]. 4.1. cost objective function. the proposed milp cost objective function is for the minimization of total piping cost with the selection of pipe size diameters as the decision variables for the first objective function (f1) as shown in equation (4.1). also, the cost of pump with the selection of pump’s power as the decision variables, while pipe diameter from the layout and minimum pressure are imposed as constraints as shown in (5.3) and (5.4): minimize cost f1 = np∑ i=1 ci di li + np u∑ i=1 cpi pi + cfu (4.1) 4.2. reliability of production objective function. the milp optimization model of the second objective function (f2) is stated mathematically as the maximization of reliability of the water production of the total water demand per unit time or month t shown in equation (4.2). in addition, with the selection of length binary variable indicating the water demand in month t is supplied or not as decision variable, subjected to the peak demand must not be more that the fire flow demand constraint presented in equation (5.2). maximize realiability f2 = t∑ t=1 zt t djpeak (4.2) 4.3. head loss function. the milp optimization model of the third objective function (f3) is proposed as the minimization of water head loss. as shown in equation (4.3), the model is to minimize the total head loss around a closed pipe loop should be equal to the head loss along a loop for each 120 h. b. awwalu, n. abdullahi, and m. hussaini length of the pipe and the diameter of the pipe as decision variable. the third objective function is subjected to energy conservation in the loops of wds as presented below: minimize headloss f3 = np∑ aij=1 hlaij di li (4.3) 5. constraints the restrictions or limitations on the total amount of a particular resource required to carry out the activities that would decide the level of achievement in the decision variables is called constraints. linear programming problem (lpp) in standard form requires all constraints to be in equations form [5], [19]. in a nut shell, a constraint is an inequality that defines how the values of the variables in a problem are limited. in order for linear programming techniques to work, all constraints should be linear inequalities. in this study the energy conservation, minimum node pressure, available pipe size and continuity flow of water through the pipes are considered as constraints. (i) energy conservation in loops. the total head loss around a closed pipe loop should be equal to zero, or the head loss along a loop between two fixed head reservoirs should be equal to the difference in water level of the reservoirs presented by hlaij = ∆haij = hi −hj = rkqnk (5.1) where hi and hj = the nodal heads at the start and end at the node of the pipe (m), respectively; rk = the resistance coefficient of the k th pipe with flow rate qk(s/m 2); and n = constant number depending on the head loss equation, and is 1.852 for the most common expression for head loss, from the hazen-williams head loss formulation [15]. (ii) demand satisfaction. the product of a binary variable indicating the water demand in month t is supplied or not based on customer peak demand must be less than or equal to the fire flow demand as shown below: t∑ t=11 zt t ·djpeak ≤ dff (5.2) (iii) minimum pressure at nodes. this constraint determines the minimum pressure for each node in the network. for each junction node in the network, the pressure head should be greater than the prescribed minimum pressure head as described below: hj ≥ hminj ∀j ∈ nn, (5.3) where hj = pressure head at node j (m); nn = number of nodes; and h min j = minimum required pressure head (m). (iv) pipe size availability. the diameter of the pipes should be selected from a set of commercially available sizes, and are thus discrete as shown below: dk = {d1,d2,d3, ...dns} ∀k ∈ d, (5.4) where ns = number of candidate diameters and d is the set of pipe diameters. (v) conservation of flow. for each node, the flow must satisfy the following equation: flowaij = dj peak + np∑ ajk=1 flowaik. (5.5) mixed integer linear programming water distribution system 121 the conservation of flow constraint is derived in which the total quantity of water that flow through any pipe is equal to the demand of destination node j plus all remaining flow along the pipe path. 5.1. decision variables. decision variables are set of quantities that need to be determine in order to solve the problem. each decision variable in any lp model must be positive irrespective of whether the objective function is to minimize or maximize the net present value of an activity. therefore, calculating the objective functions needs an approach to determine timely release of water from the reservoir as demanded by the customers. hence, zt is a binary decision variable with 1 or 0 indicating the water demand in month t is supplied or not and di is the required pipe diameter that can satisfied the demand. also, the cost of pump with the selection of pump’s power as the decision variable. 5.2. differential evolutionary (de) algorithm. differential evolution (de) algorithm is a branch of evolutionary programming developed by rainer storn and kenneth price for optimization problems over continuous domains [17]. in de, each variable’s value is represented by a real number. the advantages of de are its simple structure, ease of use, speed and robustness. de is one of the best genetic type algorithms for solving problems with the real valued variables. differential evolution is a design tool of great utility that is immediately accessible for practical applications. de has been used in several science and engineering applications to discover effective solutions to nearly intractable problems without appealing to expert knowledge or complex design algorithms. differential evolution uses mutation as a search mechanism and selection to direct the search toward the prospective regions in the feasible region. genetic algorithms generate a sequence of populations by using selection mechanisms. genetic algorithms use crossover and mutation as search mechanisms. the principal difference between genetic algorithms and differential evolution is that genetic algorithms rely on crossover, a mechanism of probabilistic and useful exchange of information among solutions to locate better solutions, while evolutionary strategies use mutation as the primary search mechanism [20]. differential evolution (de) algorithm has strong ability for solving single-objective problem, which is extended into a multi-objective evolution algorithm for deed in present study. the proposed emode algorithm combined two selection strategies based on de, and the constraint processing technique of each strategy is different, which could make full use of the advantages of these two strategies. differential evolution (de) algorithm has strong ability for solving single-objective problem, which is extended into a multi-objective evolution algorithm for deed in present study. the proposed emode algorithm combined two selection strategies based on de, and the constraint processing technique of each strategy is different, which could make full use of the advantages of these two strategies. differential evolution (de) algorithm has strong ability for solving single-objective problem, which is extended into a multiobjective evolution algorithm for deed in present study. the proposed emode algorithm combined two selection strategies based on de, and the constraint processing technique of each strategy is different, which could make full use of the advantages of these two strategies differential evolution (de) algorithm has strong ability for solving single-objective problem, which is extended into a multiobjective evolution algorithm for mixed-integer linear programing problem of wds as in this study. the proposed milp-de algorithm combined the selection strategies based on de, and the constraint processing technique of each objective function is different, which could make full use of the advantages of both milp and de for optimal solution. 5.2.1. initial population. differential evolution (de) being a parallel direct search method utilizes the number of the population np (size of the population) and d-dimensional parameter vectors. de starts with an initial population generated randomly as xi,g, and the population size is np. in this study, the 122 h. b. awwalu, n. abdullahi, and m. hussaini population are generated as a set of optimal value from the objective functions of milp. the initial population is generated as: xi,g, i = 1, 2, 3, ...,np. (5.6) as a population for each generation g the np does not change during the minimization or maximization process. the initial vector population is chosen randomly and should cover the entire parameter space. as a rule, there is assumption of a uniform probability distribution for all random decisions unless otherwise stated. in case a preliminary solution is available, the initial population might be generated by adding normally distributed random deviations to the nominal solution xnom,0. de generates new parameter vectors by adding the weighted difference between two population vectors to a third vector by a process be called mutation. 5.2.2. mutation. the mutated vector’s parameters are then mixed with the parameters of another predetermined vector, the target vector, to yield the so-called trial vector. if the trial vector yields a lower function value than the target vector, the trial vector replaces the target vector in the following generation in the case of cost and head loss minimization, and the upper function value in the case of reliability maximization. more specifically the de’s basic mutation strategy can be described as follows: for each target vector xi,g, i = 1, 2, 3, · · · ,np, a mutant vector is generated according to: vi,g+1 = xr1,g + f × (xr2,g −xr3,g) (5.7) with random indices r1,r2,r3 ∈ {1, 2, · · · ,np} mutually different and f > 0. the randomly chosen integers r1,r2 and r3 are also chosen to be different from the running index i, so that np must be greater or equal to four to allow for this condition (optimization performance may be greatly impacted by these choices as recommended by the author of de). f is a real and constant factor in [0, 2]. which is called the differential weight and controls the amplification of the differential variation (xr2,g−xr3,g) with the same recommendation as for np . 5.2.3. crossover. in order to increase the diversity of the perturbed parameter vectors, crossover is introduced. to this end, the trial vector: ui,g+1 = (u1i,g+1,u2i,g+1, ...udi,g+1) (5.8) is formed, where: uji,g+1 = { vji,g+1, if randb (j) ≤ cr or j=randr(i) xji,g, otherwise j = 1, 2, ...,d. (5.9) in equation (5.9), randb(j) is the jth evaluation of a uniform random number generator with outcome in [0; 1]; cr is the crossover constant in [0; 1] which has to be determined by the user; ranbr(i) is a randomly chosen index in {1, 2, ...,d} which ensures that ui,g+1 gets at least one parameter from vi,g+1 5.2.4. selection. this last operation is called selection. each population vector has to serve once as the target vector so that np competitions take place in one generation. therefore, to decide whether or not it should become a member of generation g + 1, the trial vector ui,g+1 is compared to the target vector xi,g using the greedy criterion. if vector ui,g+1 yields a smaller cost function value than xi,g, then xi,g+1 is set to ui,g+1; otherwise, the old value xi,g is retained in a case of minimization. xji,g+1 = { uji,g+1, if f(ui,g+1) ≤ f(xi,g) xji,g, if otherwise (5.10) mixed integer linear programming water distribution system 123 finally, this process continues to reach new generations to the number of np. then the same process is repeated to reach termination condition. moreover, only the objective function values are compared between two feasible individuals, which could improve the quality of the overall population, so only the feasible individuals with optimal objective function values would be selected. 6. fuzzy approach for best compromise solution extraction a group of pareto optimal solutions can be obtained through the algorithm. however, fuzzy decisionmaking technique is used in our work to get the best compromise solution. upon having the paretooptimal set of non-dominated solution, the proposed approach presents one solution to the decision maker as the best compromise solution. due to imprecise nature of the decision maker’s judgment, the ith objective function fi is represented by a membership function “i” defined as: ui =   1, if fi ≤ fmini f mini −fi f max i −f min i , if fmini ≤ fi ≤ f max i 0, if fi ≤ fmaxi (6.1) where fmini and f max i are the respective minimum and maximum values of the i th objective function among all non-dominated solutions, respectively. the membership function values ranges from 0 to 1. moreover, the satisfaction degree of each pareto optimal solution is µk = ∑no i=1 µ k i∑m k=1 ∑no i=1 µ k i (6.2) where m is the number of pareto front solution and no is the number of the objective function, hence the solution with the greatest satisfaction will be selected as the best compromise solution. the best compromise solution is that having the maximum value of k. 7. implementation procedure of multi-objective milp differential evolution the basic elements of mode algorithm used in this research can be briefly stated as follows: to start the optimization process, define the control parameters and other parameters used by the algorithm. the algorithm steps can be summarized as follows: to begin the optimization process, the control parameters and other parameters employed by the algorithm are defined. the algorithmic steps are summarized as: step 1: initialization: generate an initial population using equation (5.6) and process the objective functions obtained from milp model according to equation (4.1) to (16) as the external pareto optimal set. step 2: termination criteria: the following tasks are conducted to meet the termination criterion: for each solution xi = x1,x2, · · · ,xn in the population, the following actions are done in step 3 and 4. step 3: external set updating: the external pareto optimal set is updated as follows: (i) search the population for the non-dominated individuals and copy them to an external set that is called external pareto set. (ii) search the external pareto set for the non-dominated individuals and remove all dominated solutions from the set. step 4: perform de mutation: perform the de mutation operations according to equation (5.7) to generate the donor vector vi for each i th member xi. 124 h. b. awwalu, n. abdullahi, and m. hussaini step 5: perform de crossover: perform the de crossover according to equation (5.9) and find the trial vector ui. step 6: selection: selection between trial vector (child) and target vector (parent) is carried out according to the dominance criteria as follows: (i) if all objectives of solution ui are better (or at least one) or equal to that of corresponding objectives of xi, then ui dominates xi and replaces it in the new population and visa versa. (ii) if all objectives of solutions xi are equal or some are better and some are worse, then ui and xi are not dominated by each other and ui is retained in the new population. step 7: check stopping criteria: check for the stopping criteria. in this paper, it is chosen to be reaching the maximum number of generations (gen). problem continuous again and pareto-optimal set of solutions are updated until the maximum number of generations is reached. step 8: select and output the best compromise solution with fuzzy decision-making technique according to equations (5.10) and (6.1). 8. milp verification and validation process the verification can be conducted as specification verification and implementation verification. specification verification is used to ensure that the design processes of programming specification and implementation for a mathematical model is satisfactory. whereas, implementation verification is used to ensure that the designed simulation model is implemented in accordance with its simulation specification [16]. the milp mathematical model is converted into matlab codes using supported programming specification of the environment ready for the verification exercise. the verification of the proposed milp model for water distribution system was developed for different matlab codes that were run individually for the identification of code errors or problem. the breakpoints were set to paused the execution of the codes so that an error can be examine and discovered were the issues might occur and take the necessary corrections. the external validation of simulation models is an important topic in many engineering problems. some methods are known to support this process, but there is no general method which can deal with a broad class of systems [9]. validation refers to the extent to which the model or algorithm is satisfying the expectations of the problem by comparing the solution with the current solution in the system. in short, model validation involves comparing simulation results with empirical evidence [7]. this can be done by using problem instances and compare the results with the best-known solutions form the literature, compare your algorithm with solutions obtained from solving the mathematical model using a solver. if there are no standard problem instances, create random instance according to the literature or completely from scratch, and perform the comparison. in this study, two result are compared one from the output of milp model of water distribution system executed in matlab environment and the other is the empirical result obtained from excel environment after executing the proposed milp multi objective functions as a solutions obtained from solving the mathematical model. after removing of errors from the milp programming codes the matlab runs the codes successfully with the desired output such as values and plots as shown in figure 2. the set of optimal solutions after the run of milp codes in matlab are plotted in figure 2 (a) for the first objective function to minimize the total cost of pipe, pump, and fuel. and figure 2 (b) shows the second objective function of maximization the reliability of water production. the result presented shows the model is validated as the result indicated the minimal and maximum curve. also, shows the proposed objective functions are conflicting function as shown in figure 2 (d). where three objective mixed integer linear programming water distribution system 125 functions were plotted in one graph. in short, figure 2 proved the verification and validation processes of the proposed wds model. 9. conclusion conclusively, the proposed conceptual milp model for multi-objective wds is presented with three objective functions, five constraints and two decision variables, also the de is proposed for the optimization of the model. however, despite the advancement in the field of wds, many researcher?s device de as a promising metaheuristic technique for the optimization of water distribution. the optimal design of water distribution system is aimed to minimize the network design cost, usually through the determination of pipe, pump and reservoir sizes. moreover, the early applications of de for the optimization of wds depend on single-objective function for cost minimization, also the finding of optimal control strategies for the network operational problem are still rarely seen in the water research community. therefore, future research is needed to implement the proposed conceptual model with three objective functions using de. also, more emphasis and consideration is needed to the wds problem as a multi-objective problem with minimizing the cost function for maintenance and pipelining and maximizing the reliability or production benefit function of wds, with related functions available 126 h. b. awwalu, n. abdullahi, and m. hussaini that can strengthen the practical solution of wds. moreover, finding the optimal cost and production reliability for the implementation of wds networks is need to be considered due economically helpful in the water supply systems design. conflict of interest: we declare that there is no conflict of interests whatsoever. references 1. h. b. awwalu, n. abdullahi, and m. hussaini, single and multi-objective optimization of water distribution system: a review and bibliographic analysis, nda j. mil. sci. discip. stud. 1 (2022), 64–74. 2. b. m. brentan, e. jr. luvizotto, m. herrera, j. izquierdo, and r. pérez-garćıa, hybrid regression model for near real-time urban water demand forecasting, j. comput. appl. math. 309 (2017), 532–541. 3. b. m. brentan, e. jr. luvizotto, i. montalvo, j. izquierdo, and r. pérez-garćıa, near real time pump optimization and pressure management, procedia eng. 186 (2017), 666–675. 4. g. chiandussi, m. codegone, s. ferrero, and f. e. varesio, comparison of multi-objective optimization methodologies for engineering applications, comput. math. with appl. 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water resour. manag. 32(14) (2018), 4779–4791. 21. x. zhuan and x. xiao, optimal operation scheduling of a pumping station with multiple pumps, appl. energy, 104 (2013), 250–257. mixed integer linear programming water distribution system 127 department of mathematics and statistics, hussaini adamu federal polytechnic kazaure, jigawa state, nigeria email address: iyamibbg@gmail.com corresponding author, department of mathematical sciences, nigerian defence academy, kaduna nigeria email address: nabdullahi@nda.edu.ng department of mathematics and statistics, hussaini adamu federal polytechnic kazaure, jigawa state, nigeria email address: muktar.hussaini@hafedpoly.edu.ng 1. introduction 2. related works 3. conceptual model of proposed milp wds 4. proposed milp wds objective functions 4.1. cost objective function 4.2. reliability of production objective function 4.3. head loss function 5. constraints 5.1. decision variables 5.2. differential evolutionary (de) algorithm 6. fuzzy approach for best compromise solution extraction 7. implementation procedure of multi-objective milp differential evolution 8. milp verification and validation process 9. conclusion references mathematics in applied sciences and engineering https://doi.org/10.5206/mase/8196 volume 1, number 1, march 2020, pp.16-26 https://ojs.lib.uwo.ca/mase a prey-predator system with herd behaviour of prey in a rapidly fluctuating environment guruprasad samanta, ashok mondal, debgopal sahoo, and pralay dolai abstract. a statistical theory of non-equilibrium fluctuation in damped volterra-lotka prey-predator system where prey population lives in herd in a rapidly fluctuating random environment has been presented. the corresponding results have also been obtained in absence of herd behaviour. the method is based on the technique of perturbation approximation of non-linear coupled stochastic differential equations. the characteristic of group-living of prey population has been emphasized using square root of prey density in the functional response. numerical results have also been obtained by varying some of the vital system parameters. 1. introduction the classical volterra-lotka equations are generally used to describe the time-evolution of interacting prey-predator system. the introduction of intraspecific competition among the prey, resulting from the limited resources, makes the volterra-lotka model rough and the system is known as the damped volterra-lotka system. in natural ecosystems, it has been observed that most of the prey populations live forming groups, and all members of a group do not interact at a time (bera et al. 2015, 2016a, 2016b). major predators of zebra, buffalo, kongoni, toki and thomsons gazelle are hyena, wild dog, lion, leopard and cheetahs and so to defend against predators they form groups. the underlying reasons behind group formation more likely depend upon self-defence, group-defence, group alertness within a group and speed, to avoid being killed by a predator (khan et al. 2004). models of group formation are analysed to study environmental and social forces, and individual decision rules that lead to formation of swarms, flocks, schools, herds, and other groups (deangelis and mooij 2005). it is pointed out by fryxell et al. (2007) that group formation profoundly reduce food intake rates below the expected levels. as a consequence, suitable form of functional response was searched by researchers to describe the social behaviour of such populations. freedman and wolkowitz (1986) analysed the characteristics of group defence in this regard. now, when a population lives forming groups, then all members of a group do not interact at a time and some of the reasons for this herd behaviour are for searching food resources, defending the predators. to explore the consequence of spatial group formation of fixed shape by predators, cosner et al. (1999) introduced the idea that the square root of the predator variable is to be used in the function describing the encounter rate in two-dimensional systems. unfortunately, such an idea has not been incorporated by the researchers for about a decade. the most significant works of ajraldi et al. (2011) and braza (2012) gave such modelling a new dimension. the central ideas are as follows: let x be the density of a population that gathers in herds, and suppose that herd occupies an area a. the number of individuals staying at outermost positions in the herd is proportional to received by the editors 4 august 2019; revised 12 november 2019; accepted 20 november 2019, published online 6 december, 2019. 2000 mathematics subject classification. 92d25; 37h10. key words and phrases. herd, random environment, coloured noise, white noise, stability. 16 a prey-predator system with herd behaviour of prey in a rapidly fluctuating env. 17 the length of the perimeter of the patch where the herd is located. clearly, its length is proportional to √ a. since x is distributed over a two-dimensional space, √ x would therefore count the individuals at the edge of the patch. as a result, the encounter rate e(x,y) = βxy should change its form to e(x,y) = β √ xy in two dimension. rapidly fluctuating environmental variations usually cause random variations in system parameters, in particular, in the natural growth coefficient of the prey and in the natural mortality of the predator. since these are the main parameters subject to coupling of a prey-predator pair with its environment (dimentberg 1988). bera et al. (2016a) considered a prey-predator model, where the ‘functional response’ is of the form holling type-ii, but the prey density is replaced by its square root. they studied the stochastic version of the model, which takes into account the effect of fluctuating environment characterized by gussion white noises. a prey-predator model was proposed in the work of maiti et al.(2016), where both the prey and the predator show herd behaviour. the effect of fluctuating environment was analyzed by them incorporating gaussian white noises. they concluded that, to keep ecological balance in a fluctuating environment, the system has to maintain some restrictions. in the present article we have developed a general stochastic analysis of the behaviour of the damped volterra-lotka prey-predator system with herd behaviour in prey population in a rapidly fluctuating random environment. the method is based on the technique of perturbation approximation of nonlinear coupled stochastic differential equations. numerical results have also been obtained by varying some of the vital system parameters. we have derived the corresponding results in absence of herd behaviour. 2. damped volterra-lotka system with herd behaviour of prey: basic stochastic differential equations for damped volterra-lotka system with herd behaviour of prey, the prey population represented by its biomass x(t) and that of predator population represented by y(t) satisfy the following deterministic equation: d dt x(t) = f(x(t)), (1) where x(t) = [ x(t) y(t) ] , f =  x(t)(α−kx(t)) −β √ x(t)y(t) y(t)(−m + ηβ √ x(t))   , and α,m,k,η,β > 0. it is assumed that fluctuations in the environment will manifest themselves mainly as fluctuations in the natural growth coefficient of the prey (α) and in the natural mortality of the predator (−m), because these are the main parameters subject to coupling of a prey-predator pair with its environment (samanta and maiti 2003). in a random environment, the parameters α and −m are replaced by α + θ1(t) and −m + θ2(t) respectively, where θ1(t) and θ2(t) are random fluctuating terms. we assume that these fluctuations are rapid and we express this fact by writing θ(τ) ≡ (θ1(τ),θ2(τ)) where τ = t/� and 0 < � << 1 is a small, non-random parameter. in a rapidly fluctuating random environment, the stochastic modification of (1) is as follows: d dt x(�,t) = f(x(�,t),θ(t/�)), (2) 18 g. p. samanta, a. mondal, d. sahoo, and p. dolai where x(�,t) = [ x(�t) y(�,t) ] , f(x(�,t),θ(t/�)) =  x(�,t)(α + θ1(t/�) −kx(�,t)) −β √ x(�,t)y(�,t) y(�,t)(−m + θ2(t/�) + ηβ √ x(�,t))   , x(�,t),y(�,t) represent the prey and predator population respectively and α,m,k,η,β > 0; 0 < � << 1. the meaning of this is as follows: as the natural time t changes by a typical amount δt, θ(t/�) fluctuates considerably, since it experiences an elapsed time δτ = δt � which is large when � is small. we assume that the perturbed terms θ1(τ),θ2(τ); τ = t/�, are coloured noises or ornstein-uhlenbeck processes. the mathematical expectations and correlation functions of these processes are given by 〈θi(τ)〉 = 0, 〈θi(τ1)θi(τ2)〉 = σ2i 2γi exp(−γi|τ1 − τ2|), 〈θ1(τ1)θ2(τ2)〉 = σ exp(−|τ1 − τ2|)(1 + |τ1 − τ2|), (γi > 0, i = 1, 2), (3) where 〈·〉 represents the average over the ensemble of the stochastic process. this is motivated by the fact that lim |τ1−τ2|→∞ 〈θ1(τ1)θ2(τ2)〉 = 0 ⇒ θ1(τ1),θ2(τ2) tend to independent random processes. it is also noted that as σi,γi → ∞ keeping σ2i γ2i =constant= d2i (say), then θi(τ) → diηi(τ) where ηi(τ) are standard white noises, i.e., 〈ηi(τ)〉 = 0, 〈ηi(τ1)ηi(τ2)〉 = δ(τ1 − τ2). 3. perturbation approximation and non-equilibrium fluctuation we shall now use a two term perturbation approximation to x(�,t) (white 1977): x(�,t) ∼ x0(t) + √ �y 0(t). (4) the first approximation x0(t) = [ x0(t) y0(t) ] , satisfies d dt x0(t) = f(x0(t)), (5) where f(x0(t)) = lim t→∞ 1 t ∫ t 0 〈f(x0(t), θ(τ))〉dτ =  x0(t)(α−kx0(t)) −β √ x0(t)y0(t) y0(t)(−m + ηβ √ x0(t))   . these are just the equations of the damped volterra-lotka system with herd behaviour of prey in a fixed environment. this system has a unique non-trivial equilibrium (both components of which are non-zero) at the point: x = [ x? y? ] , where x? = m2 η2β2 , y? = αη2β2 −km2 η3β4 . (6) it is immediately apparent that in the absence of predators the limit value of prey population will be x′ = α k . the realization of the obvious condition: a prey-predator system with herd behaviour of prey in a rapidly fluctuating env. 19 x? < x′ ⇒ αη2β2 −km2 > 0 (7) which makes y? positive, and hence this equilibrium exists. we assume that the system is at the initial time t = 0 at x, therefore we have x0(t) = x. here y 0(t) =  y 01 (t) y 02 (t)   is a gaussian random process which satisfies the linear equation d dt y 0(t) = cy 0(t) + w(t), (8) where y 0(0) = 0, c = ∂f ∂x (x) =  −a − m η b 0   , (9) where a = 2m(2km2 −αη2β2) + (αη2β2 −km2) 2mη2β2 , b = αη2β2 −km2 2mηβ2 (10) and 〈w(t)〉 = 0, 〈w(t)wt (t′)〉 = aδ(t− t′), a = lim t→∞ 1 t ∫ t 0 ∫ t 0 〈[f(x,θ(τ1)) −〈f(x,θ(τ1))〉] [f(x,θ(τ2)) −〈f(x,θ(τ2))〉]t〉dτ1dτ2 = [ a11 a12 a21 a22 ] , a11 = m4σ21 η4β4γ21 , a12 = a21 = 4m2(αη2β2 −km2)σ η5β6 , a22 = (αη2β2 −km2)2σ22 η6β8γ22 . now the solution of (8) is given by y 0(t) = y (t) ∫ t 0 y −1(s)w(s)ds, (11) where y (t) satisfies the linear equation: d dt y (t) = cy (t), y (0) = i. (12) therefore, 〈y 0(t)〉 = 0, since 〈w(s)〉 = 0. the solution of (12) is given by y (t) =   1√ ∆ ( λ1e λ1t −λ2eλ2t ) − m η √ ∆ ( eλ1t −eλ2t ) b√ ∆ ( eλ1t −eλ2t ) − 1√ ∆ ( λ2e λ1t −λ1eλ2t )   , where (a and b are given by (10)): ∆ = a2 − 4m η b, λ1 = −a + √ ∆ 2 , λ2 = −a− √ ∆ 2 . 20 g. p. samanta, a. mondal, d. sahoo, and p. dolai let us assume that 2km2 −αη2β2 > 0 ⇒ km2 < αη2β2 < 2km2 ⇒ a > 0. (13) hence the system (1) is locally asymptotically stable at the unique non-trivial equilibrium point given by (6). now the expression of the strength of the fluctuation d(t), the covariance at one instant of time, is given by d(t) = 〈y 0(t)y 0 t (t)〉 = y (t) [∫ t 0 y −1(s)ay −1 t (s)ds ] y t (t) = [ d11(t) d12(t) d21(t) d22(t) ] , where d11(t) = e2λ1t 2∆b φ(λ1,λ2) + e2λ2t 2∆b φ(λ2,λ1) + 2m ∆η e−atψ + 1 2ab { a11b + a22 m η } , d12(t) = d21(t) = e2λ1t 2∆λ1 φ(λ1,λ2) + e2λ2t 2∆λ2 φ(λ2,λ1) − e−at 2∆b θ − a22 2b , d22(t) = e2λ1t 2∆λ1 z(λ2) + e2λ2t 2∆λ2 z(λ1) + 2b ∆ e−atψ + m, φ(x,y) = a11bx− 2a12xy + a22 m η y, ψ = 1 a { a11b + a22 m η } + a12, θ = 2a11b 2 + 2a12ab + a22 { a2 + 2 m η b } , z(x) = a11b 2 − 2a12bx + a22x2, m = a11bη 2am + a22η 2amb { a2 + m η b } + a12 m η. therefore, d(t) converges exponentially to the limiting variance d(∞) =  d11(∞) d12(∞) d21(∞) d22(∞)   , (14) where a prey-predator system with herd behaviour of prey in a rapidly fluctuating env. 21 d11(∞) = m3 {2m(2km2 −αη2β2) + (αη2β2 −km2)}η2β2 [ m2 σ21 γ21 + 2(αη2β2 −km2) η2β2 σ22 γ22 ] , d12(∞) = d21(∞) = − m(αη2β2 −km2) η5β6 σ22 γ22 , d22(∞) = m3(αη2β2 −km2) 2{2m(2km2 −αη2β2) + (αη2β2 −km2)}η2β4 σ21 γ21 + [ (αη2β2 −km2){2m(2km2 −αη2β2) + (αη2β2 −km2)} 2mη6β8 + m(αη2β2 −km2)2 {2m(2km2 −αη2β2) + (αη2β2 −km2)}η4β6 ] σ22 γ22 + 4m(αη2β2 −km2)σ η4β6 . (15) this convergence is rapid. in the case of high-amplitude fluctuations d11(∞) and d22(∞) are large. this indicates that for high-amplitude fluctuations the system demonstrates statistical parametric shatter as a result of rapidly fluctuating environmental conditions, and the equilibrium, which is stable in absence of these fluctuations, becomes unstable. it is also evident from (15) that the prey population is more sensitive in rapidly fluctuating environmental conditions and the natural growth coefficient (α) of the prey play a significant role in instability. this is due to the herd behaviour of the prey population. 4. numerical simulation for system (2) parameter k β m η σ1 γ1 σ2 γ2 σ value 0.001 0.4 0.09 0.5 0.4 0.16 0.8 0.64 0.1 table-1: parameter values used for numerical simulation of system (2) for figure-1 if we change the parameter value of α continuously from 0.5 to 2.0, limiting variance d11(∞) decreases whereas d22(∞) increases rapidly (see figure-1), i.e., in a rapidly fluctuating environment, the interior equilibrium of system (2) which is stable in absence of these fluctuations, becomes unstable. parameter α k m η σ1 γ1 σ2 γ2 σ value 1.1 0.001 0.09 0.5 0.4 0.16 0.8 0.64 0.1 table-2: parameter values used for numerical simulation of system (2) for figure-2 now taking α = 1.1 and varying the parameter value of β from 0.2 to 2, both the limiting variance d11(∞) and d22(∞) decreases and converges to a positive value which is very close to zero (see figure2), i.e, the prey and the predator population coexist in stable equilibrium because the perturbation terms around the interior (coexistence) equilibrium given by (6) tend to zero. so, it can be concluded that for increasing consumption rate of predator the system develops an internal mechanism for coexistence steady state free from fluctuation though the environment is fluctuating rapidly. it is a very interesting result to maintain ecological balance, that is, to maintain the balance of nature. 22 g. p. samanta, a. mondal, d. sahoo, and p. dolai 0.5 1 1.5 2 1.74 1.76 1.78 1.8 1.82 α d 1 1 (∞ ) 0.5 1 1.5 2 0 2000 4000 6000 α d 2 2 (∞ ) figure 1. limiting variance d11(∞) and d22(∞) with respect to the parameter value α. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 β d 1 1 (∞ ) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 x 10 4 β d 2 2 (∞ ) figure 2. limiting variance d11(∞) and d22(∞) with respect to the parameter value β. parameter α k β η σ1 γ1 σ2 γ2 σ value 1.1 0.001 0.4 0.5 0.4 0.16 0.8 0.64 0.1 table-3: parameter values used for numerical simulation of system (2) for figure-3 now taking α = 1.1, β = 0.4 and varying the predator death rate m from 0.05 to 0.5, both the limiting variance d11(∞) and d22(∞) increases rapidly near m = 0.5 (see figure-3), i.e, the system becomes unstable near m = 0.5. again around m = 0.2 which is not too far from the rate ηβ, the limiting variances are small and so the predator and prey coexist in stable equilibrium which is free from fluctuation though the environment is fluctuating rapidly. this agrees with our theoretical findings. 5. damped volterra-lotka system without herd behaviour: basic stochastic differential equations in absence of herd behaviour of prey, system (1) takes the following form: a prey-predator system with herd behaviour of prey in a rapidly fluctuating env. 23 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 2 4 6 8 x 10 4 m d 1 1 (∞ ) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 5000 10000 15000 m d 2 2 (∞ ) figure 3. limiting variance d11(∞) and d22(∞) with respect to the parameter value m. d dt x(t) = f(x(t)), (16) where x(t) = [ x(t) y(t) ] , f =  x(t)(α−kx(t) −βy(t)) y(t)(−m + ηβx(t))   , and α,m,k,η,β > 0. in a rapidly fluctuating random environment, system (16) is modified as follows: d dt x(�,t) = f(x(�,t),θ(t/�)), (17) where x(�,t) = [ x(�t) y(�,t) ] , f(x(�,t),θ(t/�)) =  x(�,t)(α + θ1(t/�) −kx(�,t) −βy(�,t)) y(�,t)(−m + θ2(t/�) + ηβx(�,t))   , x(�,t),y(�,t) represent the biomass of prey and predator population respectively and α,m,k,η,β > 0; 0 < � << 1. as in section 2, the perturbed terms θ1(τ),θ2(τ); τ = t/�, are coloured noises or ornstein-uhlenbeck processes characterized by (3). proceeding as in section 3, the expression of the strength of the fluctuation d(t), the covariance at one instant of time, is given by d(t) = 〈y 0(t)y 0 t (t)〉 = y (t) [∫ t 0 y −1(s)ay −1 t (s)ds ] y t (t) = [ d11(t) d12(t) d21(t) d22(t) ] , where (18) d11(t) = e2λ1t 2∆b φ(λ1,λ2) + e2λ2t 2∆b φ(λ2,λ1) + 2m ∆η e−atψ + 1 2ab { a11b + a22 m η } , (19) 24 g. p. samanta, a. mondal, d. sahoo, and p. dolai d12(t) = d21(t) = e2λ1t 2∆λ1 φ(λ1,λ2) + e2λ2t 2∆λ2 φ(λ2,λ1) − e−at 2∆b θ − a22 2b , (20) d22(t) = e2λ1t 2∆λ1 z(λ2) + e2λ2t 2∆λ2 z(λ1) + 2b ∆ e−atψ + m, (21) φ(x,y) = a11bx− 2a12xy + a22 m η y, ψ = 1 a { a11b + a22 m η } + a12, θ = 2a11b 2 + 2a12ab + a22 { a2 + 2 m η b } , z(x) = a11b 2 − 2a12bx + a22x2, m = a11bη 2am + a22η 2amb { a2 + m η b } + a12 m η, (22) where a11 = m2σ21 η2β2γ21 , a12 = a21 = 4m(ηαβ −km)σ η2β3 , a22 = (ηαβ −km)2σ22 η2β4γ22 , and ∆ = a2 − 4m η b, a = km ηβ , b = ηαβ −km β , λ1 = −a + √ ∆ 2 , λ2 = −a− √ ∆ 2 . therefore, d(t) converges exponentially to the limiting variance: d(∞) =   1 2ηβ2k { mβσ21 γ21 + (ηαβ−km)σ22 ηγ22 } (km−ηαβ)σ22 2η2β3γ22 (km−ηαβ)σ22 2η2β3γ22 (ηαβ−km) β2 { σ21 2kγ21 + (k2m+ηβ(ηαβ−km))σ22 2η2β2kmγ22 + 4σ η }   . (23) this convergence is rapid except when k is close to zero. in the case of high-amplitude fluctuations d11(∞) and d22(∞) are large. this indicates that for high-amplitude fluctuations the system demonstrates statistical parametric shatter as a result of rapidly fluctuating environmental conditions, and the equilibrium, which is stable in absence of these fluctuations, becomes unstable. this parametric shatter may occur not only for high-amplitude fluctuations, but also for high fertility to the prey and for small η,β. 5.1. special case: volterra-lotka system. now using (19) to (22), we have lim k→0 d11(t) = 1 4 √ m α 1 η2β2 [ 8σ √ mα { cos(2t √ mα) − 1 } + { mσ21 γ21 − ασ22 γ22 } sin(2t √ mα) ] + m(mσ21γ 2 2 + ασ 2 2γ 2 1 ) 2η2β2γ21γ 2 2 t (24) and a prey-predator system with herd behaviour of prey in a rapidly fluctuating env. 25 lim k→0 d22(t) = − 1 4 √ α m 1 β2 [ 8σ √ mα { cos(2t √ mα) − 1 } + { mσ21 γ21 − ασ22 γ22 } sin(2t √ mα) ] + α(mσ21γ 2 2 + ασ 2 2γ 2 1 ) 2β2γ21γ 2 2 t. (25) from the above results, we see that as k → 0+ the damped volterra-lotka system tends to a classical volterra-lotka system in a random environment which demonstrates statistical parametric shatter with increasing time with a periodic background noise. 6. discussion and conclusion in this paper, we have studied the stability behaviour of the damped volterra-lotka prey-predator system where prey population lives in herd in a rapidly fluctuating random environment. the method is based on the technique of perturbation approximation of non-linear coupled stochastic differential equations. the characteristic of group-living of prey population has been emphasized using square root of prey density in the functional response. the assumption of condition (13) implies that the system (in deterministic environment) is locally asymptotically stable at the unique non-trivial equilibrium point given by (6). from (13) it is evident that for predator death rates m less than yet not too far from the rate ηβ it consumes prey, the predator and prey coexist in stable equilibrium. this is reasonable since, because m is only moderate in size (in deterministic environment), the predator can sufficiently sustain itself yet not grow too much so as to wipe out the prey. ultimately though, the coexistence necessarily becomes unstable when the predator death rate gets smaller in deterministic environment. in real environment, the coexistence of populations has immense importance for ecological balance in nature. from this viewpoint, study of the stability of the interior equilibrium is emphasized. the analysis indicates that for high-amplitude fluctuations the system demonstrates statistical parametric shatter as a result of rapidly fluctuating environmental conditions, and the equilibrium, which is stable in the absence of these fluctuations, becomes unstable. it is also evident from (15) that the prey population is more sensitive in rapidly fluctuating environmental conditions and the natural growth coefficient (α) of the prey play a significant role in instability (causing ecological imbalance in nature). this is due to the herd behaviour of the prey population. in section 5, it is derived that in absence of herd behaviour, the proposed system tends to a classical volterra-lotka system for large carrying capacity (of the prey population) in a rapidly fluctuating random environment which demonstrates statistical parametric shatter with increasing time with a periodic background noise. these results are in good agreement with the results obtained by baishya and chakrabarti (1987), samanta (1996), samanta and maiti (2003). it is pointed out that this phenomenon does not occur in presence of herd behaviour among the individuals of prey population. so, we have come to the conclusion that when the individuals of prey population act collectively as part of a group, the system develops an internal mechanism to resist this statistical parametric shatter (causing ecological imbalance in nature). it is of course a new result. acknowledgements the authors are grateful to the anonymous referees and the chief editor prof. xingfu zou for their careful reading, valuable comments and helpful suggestions, which have helped them to improve the presentation of this work significantly. 26 g. p. samanta, a. mondal, d. sahoo, and p. dolai references [1] v. ajraldi, m. pittavino, e. venturino, modelling herd behavior in population systems, nonlinear analysis rwa 12(2011), 2319-2338. 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[14] g. p. samanta, influence of environmental noises on the gomatam model of interacting species, ecological modelling 91(1996), 283-291. [15] g. p. samanta, m. maiti, stochastic gomatam model of interacting species: non-equilibrium fluctuation and stability, systems analysis modelling simulation 43(2003), 683-692. [16] b. s. white, the effects of a rapidly fluctuating random environment on systems of interacting species, siam j. appl. math. 32(1977), 666-693. corresponding author. department of mathematics, indian institute of engineering science and technology, shibpur, howrah 711103, india e-mail address: g p samanta@yahoo.co.uk, gpsamanta@math.iiests.ac.in department of mathematics, indian institute of engineering science and technology, shibpur, howrah 711103, india e-mail address: ashoke.2012@yahoo.com department of mathematics, indian institute of engineering science and technology, shibpur, howrah 711103, india e-mail address: debgopalsahoo94@gmail.com department of mathematics, indian institute of engineering science and technology, shibpur, howrah 711103, india e-mail address: pralay dolai@yahoo.in mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase online first, pp.1-19 https://doi.org/10.5206/mase/xxxxx a study on the mild solution of special random impulsive fractional differential equations sayooj aby jose1,2, varun bose c. s.3, bijesh p. biju4, and abin thomas5 abstract. in this article, we deal with mild solution of special random impulsive fractional differential equations. initially, we present the existence of the mild solution via leray-schauder fixed point method. after that, we establish the exponential stability of the system. finally, we give examples to illustrate the effectiveness of the theoretical results. 1. introduction impulsive differential equations are very adaptive mathematical model that replicate the evolution of large classes of real process. recently, in the fields of science and technology, we use fractional differential equations and impulsive fractional differential equations as a great mathematical tool in modelling. the stabilities like continuous dependence mittag – leffler stability, hyers ulam stability and hyers-ulamrassins stability for fractional differential equations and impulsive fractional differential equations made curiosity in the mind of many researchers in the field of mathematics [10, 8, 14]. for impulsive differential systems, most researchers concentrate on the problems related to fixed time impulses [5, 21, 29]. but the actual jumps happen mostly at random points. the solutions of the random impulsive differential equations are a stochastic process. now a day, the characteristics of solutions to some integer order differential equations with random impulses have been analysed [25, 2, 15, 28]. anguraj et al. [2] established the existence and exponential stability of semilinear functional differential equations with random impulses under non-uniqueness. yong and wu [28] investigated the solutions of stochastic differential equations with random impulse using lipschitz condition. wu et al. [26, 27] discussed the exponential stability and boundedness of differential equations with random impulses. sayooj et al.[17] have studied some characteristics of random integro differential equations with non local conditions. in [16, 18, 19], the author found sufficient conditions for the existence as well as stability of special random impulsive differential system with non local conditions using contraction mapping principle and continuous dependence on initial conditions. now a days, fractional calculus has a lot of advanced research work has been done. and also it have proved to be valuable tools in the modeling on many phenomina in various field of science and engineering [4, 12, 13, 20, 22, 30]. the study about impulsive fractional differential equations have a great attention, akram ben alissa et al. [1] study the impulsive wave equation and analysis of this problem from different angles to prove some results about impulsive controlability and obervability without any geometrical condition on space ω. in many researchers studies about the existence, stability and uniqueness of fractional differential equations without random impulses [5, 9]. in this paper we make received by the editors 25 may 2022; 26 november 2022; published online 15 december 2022. 2010 mathematics subject classification. 35r12; 60h99; 34d20; 35b10. key words and phrases. existence; leray-schauder alternative fixed point; fractional differential equation; random impulses; exponential stability. 1 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/xxxxx 2 x. one, x. two, x. three, and x. four a first attempt to study the existence and exponential stability results for special random impulsive fractional differential systems by make use of the leray – schauder alternative fixed point theorem. the main contributions of this work are given below: ↪→ we substantiate sufficient conditions for the existence of solutions for special random impulsive fractional differential equations entangling the caputo fractional derivative. ↪→ we prove the results on existence of solutions of special random impulsive fractional differential equations by the use of the leray – schauder alternative fixed point theorem. this problem helps to solve many complicated random impulsive fractional systems. ↪→ we find exponential stability in the quadratic mean of special random impulsive fractional differential equations. ↪→ we provide examples of special random impulsive fractional differential systems as well as random impulsive fractional differential systems. it helps to interpret the effectiveness of the proposed results. and the remaining work is constructed as follows: this paper consist of 4 sections. in section 1 we present few preliminaries, hypotheses results about fractional derivatives. section 2 will be concerned with existence and followed by exponential stability in the quadratic mean of special random impulsive fractional differential equations in section 3. the last section is allocated to examples illustrating the applicability of the imposed conditions. 2. preliminaries consider a real separable hilbert space x and a non empty set ω. let %k be a random variable and %k maps ω to dk, where dk = (0,dk) for every k ∈ n ( collection of natural numbers ) and 0 < dk < +∞. also for i,j = 1, 2, . . . assume that if i 6= j then %i and %j are independent with each other. also assume %k follow erlang distribution. let % be a real constant, denote <% = [%, +∞), <+ = [0, +∞). consider the semilinear functional special random impulsive differential equations of the form cdat x(t) = ax(t) + f(t,x(t),ux(t),v x(t)) t 6= ξk, t ≥ t0, x(ξk) = bk(%k)x(ξ − k ),k = 1, 2, 3, . . . , (2.1) x(t0) = xt0 cdat is the caputo fractional derivative of order 0 < a < 1. a is the infinitesimal generator of a strongly continuous semi group of bounded linear operators t(t), t ∈ x. the function f : <% ×x × x × x → x,bk : dk → x for each k ∈ n; ξ0 = t0 and ξk = ξk−1 + %k for each k ∈ n. obviously 0 < t0 = ξ0 < ξ1 < ξ2 < ξ3 · · · < ξk < ... ; x(ξk−) = limt↑ξk x(t) according to their path with the norm ‖x‖ = supt0≤u≤t | x(u) | for every t satisfying t ∈ [t0,t]. ux(t) = ∫ t t0 k (t,r)x(r)dr, k ∈ c[d,<+], v x(t) = ∫ t t0 h (t,r)x(r)dr, h ∈ c[d0,<+], where d = {(t,r) ∈ <2 : t0 ≤ r ≤ t ≤ t}, d0 = {(t,r) ∈ <2 : t0 ≤ t,r ≤ t}. let {bt, t ≥ 0} be the simple counting process generated by {ξn}, this implies {bt ≥ t} = {ξn ≤ t}, also ft is the notation for the σ− algebra generated by {bt, t ≥ 0}. the probability space denoted as (ω,p,{ft}). and the hilbert space of all {ft}− measurable square integrable random variables with values in x is denoted as l2 = l2(ω,{ft},x). your running title 3 let b represent banach space b([t0,t], l2), the family of all {ft}measurable random variable ψ with the norm ‖ψ‖2 = sup t∈[t0,t] e‖ψ(t)‖2 definition 2.1. the fractional integral of the order a with the lower limit 0 for a function f is defined as iaf(t) = 1 γ(a) ∫ t 0 f(r) (t−r)1−a dr, t > 0,a > 0, provided the right-hand side is pointwise defined on [0,∞), where γ is a gamma function. definition 2.2. the riemann–liouville derivative of order a with the lower limit 0 for a function f : [0,∞) → r can be written as ldaf(t) = 1 γ(n−a) dn dtn ∫ t 0 f(r) (t−r)a+1−n dr, t > 0,n− 1 < a < n. definition 2.3. the caputo derivative of order a for a function f : [0,∞) → r can be written as cdaf(t) =l da [ f(t) − n−1∑ k=0 tk k! f(k)(0) ] , t > 0,n− 1 < a < n. definition 2.4. a semigroup {t(t), t ≥ t0} is said to be uniformly bounded if ‖t(t)‖≤w for all t ≥ t0, where w ≥ 1 is some constant. if w = 1, then the semigroup is said to be contraction semigroup. definition 2.5. for a given t ∈ (t0, +∞), a stochastic process {x(t) ∈ b, 0 < t0 ≤ t ≤ t} is called a solution to the equation (2.1) in (ω,p,{ft}), if (i) x(t) ∈ b is ftadapted; (ii) x(t) = +∞∑ k=0 [ k∏ i=1 bi(%i)t(t− t0)xt0 + 1 γ(a) k∑ i=1 k∏ j=i bj(%j) ∫ ξi ξi−1 (t−r)a−1t(t−r)f(r,x(r),ux(r),v x(r))dr + 1 γ(a) ∫ t ξk (t−r)a−1t(t−r)f(r,x(r),ux(r),v x(r))dr ] i[ξk,ξk+1)(t), t ∈ [t0,t], where t ∈ (t0, +∞), n∏ j=m (·) = 1 as m > n, k∏ j=i bj(%j) = bk(%k)bk−1(%k−1) . . .bi(%i), and ia(·) is the index function. remark: the proof of mild solution similar to [3, 23, 29], so we omit it. hypotheses. some hypotheses which are used for proving the main results are given below; (h1) there exist a continuous non-decreasing function h : <+ → (0,∞) and l1, l2, l3 ∈ l1([%,t],<+) so that e‖f(t,ζ1,ζ2,ζ3)‖2 ≤ l1(t)h ( e‖ζ1‖ )2 + l2(t)h ( e‖ζ2‖ )2 + l3(t)h ( e‖ζ3‖ )2 4 x. one, x. two, x. three, and x. four (h2) e { maxi,k {∏k j=i‖bj(%j)‖ }} is uniformly bounded if, e { max i,k {∏k j=i‖bj(%j)‖ }} ≤ θ, for each %j ∈ dj,j ∈ n, θ > 0 a constant (h3) define l ,k ∗ and h∗ such that, l = max{l1, l2, l3}, k∗ = sup t∈[t0,t] ∫ t t0 |k (t,r)|2dt < ∞, and h∗ = sup t∈[t0,t] ∫ t t0 |h (t,r)|2dt < ∞. our existence and exponential stability theorems are based on the succeeding theorem, which is a version of the topological transversal theorem. lemma 2.1. let e be a convex subset of a banach space x, and assume that 0 ∈ e. let f : e → e be a completely continuous operator, and let u(f) = {x ∈ e : x = λfx for some 0 < λ < 1}, then either u(f) is unbounded or f has a fixed point. 3. existence here, we presents the results on existence of solutions of special random impulsive fractional differential equations by make use of the leray – schauder alternative fixed point theorem. theorem 3.1. assume (h1), (h2), and (h3) hold, then the system (2.1) has mild solution x(t), defined on [t0,t], provided the following inequality is satisfied: γ ∫ t t0 l (r)dr < ∫ ∞ γ1 dr h(r) , (3.1) where γ = 2w2 max{1, θ2}(t−t0) 2a−1(1+k∗+h∗) (2a−1)γ(a) ,γ1 = 2w 2θ2e‖ϕ‖2 and wθ ≥ 1√ 2 . proof. let ψ be an operator from b to b and the arbitrary positive number t ∈ (t0,∞): ψx(t) = +∞∑ k=0 [ k∏ i=1 bi(%i)t(t− t0)xt0 + 1 γ(a) k∑ i=1 k∏ j=i bj(%j) ∫ ξi ξi−1 (t−r)a−1t(t−r)f(r,x(r),ux(r),v x(r))dr + 1 γ(a) ∫ t ξk (t−r)a−1t(t−r)f(r,x(r),ux(r),v x(r))dr ] i[ξk,ξk+1)(t), t ∈ [t0,t] your running title 5 first we deduce the solution of the integral equation and assume λ ∈ (0, 1): x(t) = λ +∞∑ k=0 [ k∏ i=1 bi(%i)t(t− t0)xt0 + 1 γ(a) k∑ i=1 k∏ j=i bj(%j) ∫ ξi ξi−1 (t−r)a−1t(t−r)f(r,x(r),ux(r),v x(r))dr + 1 γ(a) ∫ t ξk (t−r)a−1t(t−r)f(r,x(r),ux(r),v x(r))dr ] i[ξk,ξk+1)(t), t ∈ [t0,t] hence by (h1), (h2) and (h3) ‖x(t)‖2 ≤ λ2 [ +∞∑ k=0 [∥∥∥∥ k∏ i=1 bi(%i) ∥∥∥∥‖t(t− t0)‖‖xt0‖ + 1 γ(a) k∑ i=1 ∥∥∥∥ k∏ j=i bj(%j) ∥∥∥∥ ∫ ξi ξi−1 (t−r)a−1‖t(t−r)f(r,x(r),ux(r),v x(r))‖dr + 1 γ(a) ∫ t ξk (t−r)a−1‖t(t−r)f(r,x(r),ux(r),v x(r))‖dr ] i[ξk,ξk+1)(t) ]2 ≤ 2 [ +∞∑ k=0 [∥∥∥∥ k∏ i=1 bi(%i) ∥∥∥∥2‖t(t− t0)‖2‖xt0‖2 ] + 2 [ ∞∑ k=0 1 γ(a) k∑ i=1 ∥∥∥∥ k∏ j=i bj(%j) ∥∥∥∥ ∫ ξi ξi−1 (t−r)a−1‖t(t−r)f(r,x(r),ux(r),v x(r))‖dr + 1 γ(a) ∫ t ξk (t−r)a−1‖t(t−r)f(r,x(r),ux(r),v x(r))‖dr ] i[ξk,ξk+1)(t) ]2 ≤ 2w2θ2 ∥∥∥∥xt0 ∥∥∥∥2 + 2w2 max{1, θ2}(t − t0)2a−1γ(a)(2a− 1) ∫ t t0 ∥∥∥∥f(r,x(r),ux(r),v x(r)) ∥∥∥∥2dr, ‖x(t)‖2 ≤ 2w2θ2 ∥∥∥∥ϕ ∥∥∥∥2 + 2w2 max{1, θ2}(t − t0)2a−1γ(a)(2a− 1) ∫ t t0 ∥∥∥∥f(r,x(r),ux(r),v x(r)) ∥∥∥∥2dr, and e‖x(t)‖2 ≤ 2w2θ2e [ ‖ϕ‖2 ] + 2w2 max{1, θ2} (t − t0)2a−1 γ(a)(2a− 1) ∫ t t0 e [ ‖f(r,x(r),ux(r),v x(r))‖2 ] dr ≤ 2w2θ2e [ ‖ϕ‖2 ] + 2w2 max{1, θ2} (t − t0)2a−1 γ(a)(2a− 1) ∫ t t0 [ l1(r)h ( e‖x(r)‖2 ) + l2(r)h ( e‖ux(r)‖2 ) + l3(r)h ( e‖v x(r)‖2 )] dr, sup t0≤υ≤t e‖x(υ)‖2 ≤ 2w2θ2e‖ϕ‖2 + 2w2 max{1, θ2} (t − t0)2a−1 γ(a)(2a− 1) ∫ t t0 l (r)(1 + k∗ + h∗)h ( sup t0≤υ≤r e‖x(υ)‖2 ) dr, ≤ 2w2θ2e‖ϕ‖2 + 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t t0 l (r)h(ω(r))dr 6 x. one, x. two, x. three, and x. four where ω(t) = sup t0≤υ≤t e [ ‖x(υ)‖2 ] , t ∈ [t0,t]. moreover, for any t ∈ [t0,t], ω(t) ≤ 2w2θ2e [ ‖ϕ‖2 ] + 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t t0 l (r)h(ω(r))dr. represent by the right hand side of the above inequality as v (t), then ω(t) ≤ v (t) for t ∈ [t0,t], v (t0) = 2w2θ2e‖ϕ‖2 = γ1 and v ′(t) = 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) l (t)h(ω(t)) ≤ 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) l (t)h(ω(t)), t ∈ [t0,t]. then v ′(t) h(v (t)) ≤ 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) l (t), t ∈ [t0,t] apply the change of variable and integrate the previous inequality from t0 to t, we get ∫ v (t) v (t0) dr h(r) ≤ 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t t0 l (r)dr ≤ 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t t0 l (r)dr < ∫ ∞ γ1 dr h(r) = ∫ ∞ v (t0) dr h(r) . by the mean value theorem and the above inequality, there is a constant υ such that v (t) ≤ υ, and therefore ω(t) ≤ υ. where as supt0≤υ≤t e‖x(υ)‖ 2 = ω(t) hold for each t ∈ [t0,t], we have supt0≤υ≤t e‖x(υ)‖ 2 ≤ υ, where υ depends on the function l and h and on t, therefore e‖x(t)‖2 = sup t0≤υ≤t e‖x(υ)‖2 ≤ υ in the following steps, we will show that ψ is continuous and completely continuous. step 1: we show that ψ is continuous. for every t ∈ [t0,t] and consider {xn} be a convergent sequence of elements of x ∈ b, then ψxn(t) = +∞∑ k=0 [ k∏ i=1 bi(%i)t(t− t0)ϕ(0) + 1 γ(a) k∑ i=1 k∏ j=i bj(%j) ∫ ξi ξi−1 (t−r)a−1t(t−r)f(r,xn(r),uxn(r),v xn(r))dr + 1 γ(a) ∫ t ξk (t−r)a−1t(t−r)f(r,xn(r),uxn(r),v xn(r))dr ] i[ξk,ξk+1)(t). your running title 7 so ψxn(t) − ψx(t) = +∞∑ k=0 [ 1 γ(a) k∑ i=1 k∏ j=i bj(%j) ∫ ξi ξi−1 (t−r)a−1t(t−r) [ f(r,xn(r),uxn(r),v xn(r)) −f(r,x(r),ux(r),v x(r)) ] dr + 1 γ(a) ∫ t ξk (t−r)a−1t(t−r) [ f(r,xn(r),ux(r),v x(r)) −f(r,x(r),ux(r),v x(r)) ] dr ] i[ξk,ξk+1)(t), and e‖ψxn − ψx‖2 ≤w2 max{1, θ2} (t − t0)2a−1 γ(a)(2a− 1) ∫ t t0 e‖f(r,xn(r),uxn(r),v xn(r)) −f(r,x(r),ux(r),v x(r))‖2dr. so ψ is continuous. step 2: we show that ψ is completely continuous operator. represent θm = {x ∈ b | ‖x‖2 ≤ m} where m ≥ 0. step 2.1: we prove that ψ maps to θm into an equicontinuous family. let t1, t2 ∈ [t0,t] and x ∈ θm. whenever t0 < t1 < t2 < t, by (h1), (h2), (h3) and condition (3.1), we obtain ψx(t2) − ψx(t1) = +∞∑ k=0 [ k∏ i=1 bi(%i)t(t2 − t0)xt0 + 1 γ(a) k∑ i=1 k∏ j=i bj(%j) ∫ ξ ξi−1 (t2 −r)a−1t(t2 −r)f(r,x(r),ux(r),v x(r))dr + 1 γ(a) ∫ t2 ξk (t2 −r)a−1t(t2 −r)f(r,x(r),ux(r),v x(r))dr ] i[ξk,ξk+1)(t2) − +∞∑ k=0 [ k∏ i=1 bi(%i)t(t1 − t0)xt0 + 1 γ(a) k∑ i=1 k∏ j=i bj(%j) ∫ ξi ξi−1 (t1 −r)a−1t(t1 −r)f(r,x(r),ux(r),v x(r))dr + 1 γ(a) ∫ t1 ξk (t1 −r)a−1t(t1 −r)f(r,x(r),ux(r),v x(r))dr ] i[ξk,ξk+1)(t1) = +∞∑ k=0 [ k∏ i=1 bi(%i)t(t2 − t0)xt0 + 1 γ(a) k∑ i=1 k∏ j=i bj(%j) ∫ ξi ξi−1 (t2 −r)a−1t(t2 −r)f(r,x(r),ux(r),v x(r))dr 8 x. one, x. two, x. three, and x. four + 1 γ(a) ∫ t2 ξk (t2 −r)a−1t(t2 −r)f(r,x(r),ux(r),v x(r))dr ][ i[ξk,ξk+1)(t2) − i[ξk,ξk+1)(t1) ] + +∞∑ k=0 [ k∏ i=1 bi(%i) [ t(t2 − t0) −t(t1 − t0) ] xt0 + 1 γ(a) k∑ i=1 k∏ j=i bj(%j) ∫ ξi ξi−1 [ (t2 −r)a−1t(t2 −r) − (t1 −r)a−1t(t1 −r) ] f(r,x(r),ux(r),v x(r))dr + 1 γ(a) ∫ t1 ξk [ (t2 −r)a−1t(t2 −r) − (t1 −r)a−1t(t1 −r) ] f(r,xn(r),ux(r),v x(r))dr + 1 γ(a) ∫ t2 t1 (t2 −r)a−1t(t2 −r)f(r,x(r),ux(r),v x(r))dr ] i[ξk,ξk+1)(t1). moreover e‖ψx(t2) − ψx(t1)‖2 ≤ 2e‖i1‖2 + 2e‖i2‖2, where i1 = +∞∑ k=0 [ k∏ i=1 bi(%i)t(t2 − t0)xt0 + 1 γ(a) k∑ i=1 k∏ j=i bj(%j) ∫ ξi ξi−1 (t2 −r)a−1t(t2 −r)f(r,x(r),ux(r),v x(r))dr + 1 γ(a) ∫ t ξk (t2 −r)a−1t(t2 −r)f(r,x(r),ux(r),v x(r))dr ][ i[ξk,ξk+1)(t2) − i[ξk,ξk+1)(t1) ] and i2 = +∞∑ k=0 [ k∏ i=1 bi(%i) [ t(t2 − t0) −t(t1 − t0) ] xt0 1 γ(a) k∑ i=1 k∏ j=i bj(%j) ∫ ξi ξi−1 [ (t2 −r)a−1t(t2 −r) − (t1 −r)t(t1 −r) ] f(r,x(r),ux(r),v x(r))dr + 1 γ(a) ∫ t1 ξk (t2 −r)a−1t[(t2 −r) − (t1 −r)a−1t(t1 −r)]f(r,x(r),ux(r),v x(r))dr + 1 γ(a) ∫ t2 t1 (t2 −r)a−1t(t2 −r)f(r,x(r),ux(r),v x(r))dr ] i[ξk,ξk+1)(t1) besides, e‖i1‖2 ≤ e [ +∞∑ k=0 [ k∏ i=1 ‖bi(%i)‖‖t(t2 − t0)‖‖xt0‖ + 1 γ(a) k∑ i=1 k∏ j=i ‖bj(%j)‖ ∫ ξi ξi−1 (t2 −r)a−1‖t(t2 −r)‖‖f(r,x(r),ux(r),v x(r))‖dr + 1 γ(a) ∫ t2 ξk (t2 −r)a−1‖t(t2 −r)‖‖f(r,x(r),ux(r),v x(r))‖dr ][ i[ξk,ξk+1)(t2) − i[ξk,ξk+1)(t1) ]]2 ≤2w2θ2e‖xt0| 2e(i[ξk,ξk+1)(t2) − i[ξk,ξk+1)(t1)) your running title 9 + 2 max{1, θ2} (t − t0)2a−1 γ(a)(2a− 1) e ∫ t2 t0 ‖t(t2 −r)‖‖f(r,x(r),ux(r),v x(r))‖dr ×e(i[ξk,ξk+1)(t2) − i[ξk,ξk+1)(t1)) ≤2w2θ2e‖xt0‖ 2e ( i[ξk,ξk+1)(t2) − i[ξk,ξk+1)(t1) ) + 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t2 t0 l (r)h(e‖x(r)‖2)dr ×e ( i[ξk,ξk+1)(t2) − i[ξk,ξk+1)(t1) ) ≤2w2θ2e‖xt0‖ 2e ( i[ξk,ξk+1)(t2) − i[ξk,ξk+1)(t1) ) + 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t2 t0 l ∗h(e(m))dre ( i[ξk,ξk+1)(t2) − i[ξk,ξk+1)(t1) ) → 0, as t1 → t2. where l ∗ = sup { l (t) : t ∈ [t0,t] } . e‖i2‖2 ≤e [ +∞∑ k=0 [ k∏ i=1 ‖bi(%i)‖‖t(t2 − t0) −t(t1 − t0)‖‖xt0‖ + 1 γ(a) k∑ i=1 k∏ j=i ‖bj(%j)‖ ∫ ξi ξi−1 ‖(t2 −r)a−1t(t2 −r) − (t1 −r)a−1t(t1 −r)‖‖f(r,x(r),ux(r),v x(r))‖dr + 1 γ(a) ∫ t1 ξk ‖(t2 −r)a−1t(t2 −r) − (t1 −r)a−1t(t1 −r)‖‖f(r,x(r),ux(r),v x(r))‖dr + 1 γ(a) ∫ t2 t1 (t2 −r)a−1‖t(t2 −r)‖‖f(r,x(r),ux(r),v x(r))dr‖ ] i[ξk,ξk+1)(t1) ]2 ≤3θ2‖t(t2 − t0) −t(t1 − t0)‖2e‖xt0‖ 2 + 3 max{1, θ2}(t1 − t0) 1 γ(a) e ∫ t1 t0 ‖(t2 −r)a−1t(t2 −r) − (t1 −r)a−1t(t1 −r)‖2‖f(r,x(r),ux(r),v x(r))‖2dr + 3w2 (t − t0)2a−1 γ(a)(2a− 1) e ∫ t2 t1 ‖f(r,x(r),ux(r),v x(r))‖2dr ≤3θ2‖t(t2 − t0) −t(t1 − t0)‖2e‖xt0‖ 2 + 3 max{1, θ2}(t1 − t0) 1 γ(a) ∫ t1 t0 ‖(t2 −r)a−1t(t2 −r) − (t1 −r)a−1t(t1 −r)‖2l ∗h(m)dr + 3w2 (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t2 t1 l ∗h(m)dr → 0 as t1 → t2. so, ψ maps θm into an equicontinuous family of functions. step 2.2: we prove that ψθm is uniformly bounded. 10 x. one, x. two, x. three, and x. four by (3.1), ‖x‖2 ≤ m, (h1), (h2) and (h3), we get ‖ψx(t)‖2 ≤ [ +∞∑ k=0 [ k∏ i=1 ‖bi(%i)‖‖t(t− t0)‖‖xt0‖ + 1 γ(a) k∑ i=1 k∏ j=i ‖bj(%j)‖ ∫ ξi ξi−1 (t−r)a−1‖t(t−r)‖‖f(r,x(r),ux(r),v x(r))‖dr + 1 γ(a) ∫ t ξi−1 (t−r)a−1‖t(t−r)‖‖f(r,x(r),ux(r),v x(r))‖dr ] i[ξk,ξk+1)(t) ]2 . ≤ 2w2θ2‖ϕ(0)‖2 + 2w2 max{1, θ2} (t − t0)2a−1 γ(a)(2a− 1) ∫ t t0 ‖f(r,x(r),ux(r),v x(r))‖2dr. thus, e‖ψx(t)‖2 ≤ 2w2θ2e‖ϕ(0)‖2 + 2w2 max{1, θ2} (t − t0)2a−1 γ(a)(2a− 1) ∫ t t0 e‖f(r,x(r),ux(r),v x(r))‖2dr. ≤ 2w2θ2e‖ϕ(0)‖2 + 2w2 max{1, θ2} (t − t0)2a(1 + k∗ + h∗) γ(a)(2a− 1) ‖bm‖l. therefore {(ψx(t)),‖x‖2 ≤ m} is uniformly bounded, so does {ψθm}. then by the arzela – ascoli theorem, ψ maps θm into a precompact set in x. step 2.3: we prove that ψθm is compact. let t ∈ (t0,t] be fixed, and let � be a real number such that � ∈ (0, t− t0) for x ∈ θm, we establish (ψ�x)(t) = +∞∑ k=0 [ k∏ i=1 bj(%j)t(t− t0)xt0 + 1 γ(a) k∑ i=1 k∏ j=i bj(%j) ∫ ξi ξi−1 (t−r)a−1t(t−r)f(r,x(r),ux(r),v x(r))dr + 1 γ(a) ∫ t−� ξk (t−r)a−1t(t−r)f(r,x(r),ux(r),v x(r))dr ] i[ξk,ξk+1)(t), t ∈ (t0, t− �). being t(t) is a compact operator, the set h�(t) = {(ψ�x)(t) : x ∈ θm} is precompact in x for each � ∈ (0, t− t0). furthermore, for each x ∈ θm, we attain (ψx)(t) − (ψ�x)(t) = +∞∑ k=0 [ 1 γ(a) ∫ t ξk (t−r)a−1t(t−r)f(r,x(r),ux(r),v x(r))dr ] i[ξk,ξk+1)(t) − +∞∑ k=0 [ 1 γ(a) ∫ t−� ξk (t−r)a−1t(t−r)f(r,x(r),ux(r),v x(r))dr ] i[ξk,ξk+1)(t). by making use of (h1), (h2), (h3), condtion 4.1, and ‖x(b)‖2 ≤ m, we obtain e‖(ψx)(t) − (ψ�x)(t))‖2t ≤w 2 (t − t0) 2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t t−� l ∗h(m)dr. hence, there exist precompact sets arbitrarily close to the set {(ψx)(t) : x ∈ θm} is precompact in x. so, ψ is completely continuous operator. your running title 11 furthermore, the set u(ψ) = {x ∈ b : x = λψx for some 0 < λ < 1} is bounded. hence, by lemma 2.1, the operator ψ has a fixed point in b. so, system (2.1) has a mild solution. � 4. exponential stability in the quadratic mean this section, we establish the exponential stability of a second moment of mild solution of system. for an ft adapted process, ψ(t) : [0,∞) → r is almost continuous in t. in order to attain the stability, we suppose that f(t, 0) ≡ 0 for any t ≤ t0 thus the system (2.1) accept a trivial solution. furthermore, e‖ψ‖2t → 0 as t →∞. definition 4.1. system (2.1) is said to be exponentially stable in the quadratic mean if there exist positive constants k1 > 0 and ν > 0 such that e‖x(t)‖≤ k1e‖ϕ‖2e−ν(t−t0), t ≥ t0. now we introduce the following hypothesis used in our discussion: (h4) µh(ψ) ≤ h(µψ) for all ψ ∈ r+, where µ > 1. (h5) ‖t(t)‖≤we−ξ(t−t0), t ≥ 1. theorem 4.1. assume that the hypothesis of theorem 2.1 and (h4) − (h5) hold. if the following inequality is satisfied, then the system (2.1) is exponentially stable in the quadratic mean: γ∗ ∫ t t0 l (r)dr < ∫ ∞ γ2 dr h(r) , (4.1) where γ∗ = 2w2 max{1, θ2}(t−t0) 2a−1(1+k∗+h∗) γ(a)(2a−1) ,γ2 = 2w 2θ2e‖ϕ‖2, and wθ ≥ 1√ 2 . proof. let ψ be defined in theorem 2.1. making use of hypotheses (h1) − (h5), we get ‖x(t)‖2 ≤ λ2 ( +∞∑ k=0 [∥∥∥∥ k∏ i=1 bi(%i) ∥∥∥∥‖t(t− t0)‖‖xt0‖ + 1 γ(a) k∑ i=1 ∥∥∥∥ k∏ j=i bj(%j) ∥∥∥∥ ∫ ξi ξi−1 (t−r)a−1‖t(t−r)f(r,x(r),ux(r),v x(r))‖dr + 1 γ(a) ∫ t ξk (t−r)a−1‖t(t−r)f(r,x(r),ux(r),v x(r))‖dr ] i[ξk,ξk+1)(t) )2 ≤ 2 +∞∑ k=0 [∥∥∥∥ k∏ i=1 bi(%i) ∥∥∥∥2‖w2e−2k(t−t0)‖‖xt0‖2i[ξk,ξk+1)(t) ] + 2   ∞∑ k=0   1 γ(a) k∑ i=1 ∥∥∥∥ k∏ j=i bj(%j) ∥∥∥∥ ∫ ξi ξi−1 (t−r)a−1‖we−ξ(t−r)f(r,x(r),ux(r),v x(r))‖dr + 1 γ(a) ∫ t ξk (t−r)a−1w2e−ξ(t−r)‖f(r,x(r),ux(r),v x(r))‖dr ] i[ξk,ξk+1)(t) )2 12 x. one, x. two, x. three, and x. four ≤ 2  max k { k∏ j=i ‖bj(%j)‖2 }2 w2e−2k(t−t0)‖xt0‖2 + 2  max i,k { 1, k∏ j=i ‖bj(%j)‖ }2 · · 1 γ(a) +∞∑ k=0 ∫ t t0 (t−r)a−1we−ξ(t−r)‖f(r,x(r),ux(r),v x(r))‖dr · i2[ξk,ξk+1)(t) ≤ 2w2θ2e−2k(t−t0)‖xt0‖ 2 + 2w2 max{1, θ2} (t − t0)2a−1 γ(a)(2a− 1) ∫ t t0 e−2ξ(t−r)‖f(r,x(r),ux(r),v x(r))‖2dr, ‖x(t)‖2 ≤ 2w2θ2e−2ξ(t−t0)‖ϕ‖2 + 2w2 max{1, θ2} (t − t0)2a−1 γ(a)(2a− 1) ∫ t t0 e−2k(t−r)‖f(r,x(r),ux(r),v x(r))‖2dr, ze‖x(t)‖2 ≤ 2w2θ2e−2k(t−t0)e‖ϕ‖2 + 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t t0 e−2ξ(t−r)l (r)h ( e‖x(r)‖2 ) dr, = 2w2θ2e−2ξ(t−t0)e‖ϕ‖2 + 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) e−2k(t−t0) ∫ t t0 e2ξ(r−t0)l (r)h ( e‖x(r)‖2 )2 dr. thus, e2ξ(t−t0)e‖x(t)‖2 ≤ 2w2θ2e‖ϕ‖2 + 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t t0 e2ξ(t−r)l (r)h ( e‖x(r)‖2 )2 dr. furhtermore, sup t0≤υ≤t e2ξ(υ−t0)e‖x(t)‖2 ≤ 2w2θ2e‖ϕ‖2 + 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t t0 l (r)h ( sup t0≤υ≤t e2ξ(υ−t0)e‖x(r)‖2 ) dr. take ω1(t) = sup t0≤υ≤t e2ξ(υ−t0)e‖x‖2, t ∈ [t0,t]. also, for any t ∈ [t0,t], we have ω1(t) ≤ 2w2θ2e‖ϕ‖2 + max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t t0 l (r)h ( ω1(r) ) dr. your running title 13 denote the right hand side of the above inequality v1(t), we obtain ω1(t) ≤ v1(t), t ∈ [t0,t], v1(t0) = 2w2θ2e‖ϕ‖2 = γ2 and v ′1 (t) ≤ 2w 2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) l (t)h ( ω1(t) ) ≤ 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) l (t)h ( v1(t) ) , t ∈ [t0,t]. that is, v ′1 (t) h ( v1(t) ) ≤ 2w2 max{1, θ2}(t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) l (t), t ∈ [t0,t]. apply the change of variable and integrate the previous inequality from t0 to t, we get∫ v1(t) v1(t0) dr h ( r ) ≤ 2w2 max{1, θ2}(t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t t0 l (r)dr ≤ 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) ∫ t t0 l (r)dr < ∫ ∞ γ2 dr h ( r ) = ∫ ∞ v1(t0) dr h ( r ), t ∈ [t0,t]. by the mean value theorem and above inequality there exist a constant υ1 such that v1(t) ≤ υ1, and therefore ω1(t) ≤ υ1. whereas supt0≤υ≤t e 2ξ(υ−t0)e‖x‖2 = ω1(t) holds for each t ∈ [t0,t], we have supt0≤υ≤t e 2ξ(υ−t0)e‖x‖2 ≤ k1, where υ1 depends on the function l and h. therefore, e2ξ(t−t0)e‖x‖2 = sup t0≤υ≤t e2ξ(υ−t0)e‖x‖2 ≤ υ1. as in the previous theorem, we will prove that ψ is completely continuous operator through the following steps. step 1: we show that ψ is continuous. for every t ∈ [t0,t] and consider {xn} be a convegent sequence of element of x ∈ b, we obtain e‖ψxn(t) − ψx(t)‖2 ≤w2 max{1, θ2} (t − t0)2a−1 γ(a)(2a− 1) e−2k(t−t0) ∫ t t0 e2ξ(r−t0)e‖f(r,xn(r),uxn(r),v xn(r)) −f(r,x(r),ux(r),v x(r))‖2dr → 0, as n →∞. so ψ is continuous. step 2: we show that ψ is completely continuous operator. represent θm1 = {x ∈ b | ‖x‖ 2 ≤ m1} 14 x. one, x. two, x. three, and x. four where m1 ≥ 0. step 2.1: we prove that ψ maps θm1 into an equicontinuous family. let x ∈ θm1 and t1, t2 ∈ [t0,t]. if t0 < t1 < t2 < t , then by making use of (h1) − (h5) and condition (3) and pursuing the similar process of step 2.1 of theorem 3.1, we obtain e‖ψx(t2) − ψx(t1)‖2 → 0 as t2 → t1. so, ψ maps to θm1 into an equicontinuous family of functions. step 2.2: we prove that ψθm1 is uniformly bounded. by the condition (3.1) and (h1) − (h5), we get ‖ψx(t)‖2 ≤ 2 [ max k { k∏ j=i ‖bj(%j)‖2 }] w2e−2k(t−t0)‖xt0‖ 2 + 2 [ max i,k { 1, k∏ j=i ‖bj(%j)‖ }]2 × ( 1 γ(a) +∞∑ k=0 ∫ t t0 (t−r)a−1we−ξ(t−r)‖f(r,x(r),ux(r),v x(r))‖dr ) i[ξk,ξk+1)(t) 2. so e‖ψx(t)‖2 ≤ 2w2θ2e−2k(t−t0)e‖xt0‖ 2 + 2w2 max{1, θ2} (t − t0)2a−1(1 + k∗ + h∗) γ(a)(2a− 1) e−2k(t−t0) × ∫ t t0 e2ξ(r−t0)l ∗h(m)dr, where l ∗ = sup{l (t) : t ∈ [t0,t]}. being e−2k(t−t0) → 0, the right hand side of the previous inequality tends to 0 as t →∞. ie, ‖(ψx)‖2 → 0 t →∞. therefore {(ψx(t)),‖x‖2b ≤ m1} is uniformly bounded, thus {ψθm1} is uniformly bounded. step 2.3: we prove that ψθm1 is compact. let t ∈ (t0,t] be fixed and � be a real number such that � ∈ (0, t− t0), for x ∈ θm1 , we establish (ψ�x)(t) = +∞∑ k=0 [ k∏ i=0 bj(%j)t(t− t0)xt0 + 1 γ(a) k∑ i=1 k∏ i=i bj(%j) ∫ ξi ξi−1 (t−r)a−1t(t−r)f(r,x(r),ux(r),v x(r))dr your running title 15 + 1 γ(a) ∫ t−� ξk (t−r)a−1t(t−r)f(r,x(r),ux(r),v x(r))dr ] i[ξk,ξk+1)(t), t ∈ (t0, t− �). being t(t) is a compact operator, the set h�(t) = {(ψ�x)(t) : x ∈ θm1} is precompact in x for each � ∈ (0, t− t0). using (h1) − (h5), condition (3), and ‖x‖2 ≤ m1, we obtain e‖ψxn(t) − ψx(t)‖2 ≤w2 (t − t0)2a−1 γ(a)(2a− 1) e−2k(t−t0) ∫ t t−� e2ξ(r−t0)l ∗h ( e‖x(r)‖2 ) dr. hence, there exist precompact sets arbitrarily close to the set {(ψx)(t) : x ∈ θm1}. thus the set {(ψx)(t) : x ∈ θm1} is precompact in x. so, ψ is a completely continuous operator. furthermore, the set u(ψ) = {x ∈ b : x = λψx for some 0 < λ < 1} is bounded. hence, by lemma 2.1, the operator ψ has a fixed point in b. so the system 2.1 has a mild solution and e‖ψ(t)‖2 → 0 as t →∞. hence the proof. � 5. applications example 5.1. consider random impulsive fractional differential equations,  cdat z(t,x) = zxx(x,t) + f1(t,z(t,x)) t 6= ξk, t ≥ % z(x,ξk) = q(k)%kz(x,ξ − k ) as x ∈4̂ z(t, 0) = z(t,π) = 0 z(t0,x) = z0(x), x ∈ ∂4̂ (5.1) consider 4̂⊂<n be a bounded domain with smooth boundary ∂4̂, x = l2(4̂), %k be random variable defined on dk ≡ (0,dk) for k ∈ n, dk ∈ (0, +∞) . also assume that %k follow erlang distribution and if i 6= j then %i and %j are independent with each other for i,j = 1, 2, . . . . here q is a function of k, ξk = ξk−1 + %k for k ∈ n, t0 ∈<+. let a be an operator on x by az = ∂2z ∂x2 with the domain d(a) = { z ∈ x | z and ∂z ∂x are absolutely continuous, ∂2z ∂x2 ∈ x,z = 0 on ∂4̂ } thus a generates a strongly continuous semigroup s(t) which is self adjoint,compact and analytic. furthermore the operator a can be represented by az = ∞∑ n=1 n2 < z,zn > zn, z ∈ d(a) here zn(ζ) = √ 2 π sin(nζ),n = 1, 2, . . . , forms the orthonormal set of eigenvectors of a. also for every z ∈ x,s(t)z = ∑∞ n=1 e (−n2t) < z,zn > zn, which holds ‖s(t)‖ ≤ e(−π 2(t−t0)), t ≥ t0. therefore s(t) is 16 x. one, x. two, x. three, and x. four a semigroup. consider that the following assumptions: (i) f : <% ×x → x, is a continuous function defined by f(t,z)(x) = f1(t,z(x)) t0 ≤ t ≤ t, 0 ≤ x ≤ π and also ∃ a continuous non-decreasing function h : <+ → (0,∞)x and l ∈ l1([%,t],<+) therefore e‖f(t,z)‖2 ≤ l (t)h ( e‖z‖2 ) (ii) e { maxi,k {∏k j=i‖bj(%j)‖ }} is uniformly bounded if, e { max i,k {∏k j=i‖bj(%j)‖ }} ≤ θ, for each %j ∈ dj,j ∈ n,θ > 0 a constant (iii) γ ∫ t t0 l (r)dr < ∫ ∞ γ1 dr h(r) , (5.2) where γ = 2w2 max{1, θ2}(t−t0) 2a−1 (2a−1)γ(a) ,γ1 = 2w 2θ2e‖ϕ‖2 and wθ ≥ 1√ 2 . assume that assumptions (i),(ii) and (iii) are satisfied, then the problem (5.1) becomes a random impulsive fractional differential equation. from all the above facts, in view of theorem 3.1, we conclude that (5.1) has a mild solution. remark 5.2. let the conditions of example 5.1 along with (h4) − (h5) be hold. if the following inequality is satisfied, γ∗ ∫ t t0 l (r)dr < ∫ ∞ γ2 dr h(r) , (5.3) where γ∗ = 2w2 max{1, θ2}(t−t0) 2a−1 γ(a)(2a−1) ,γ2 = 2w 2θ2e‖ϕ‖2, and wθ ≥ 1√ 2 . then the mild solution z of the example 5.1 is exponentially stable in the quadratic mean. example 5.3. consider special random impulsive fractional differential equations,  cdat zt(t,x) = zxx(x,t) + f1(t,z(t,x)) + ∫t 0 f2(θ,z(tsinθ,x))dθ t 6= ξk, t ≥ % z(x,ξk) = q(k)%kz(x,ξ − k ) as x ∈4̂ z(t, 0) = z(t,π) = 0 z(t0,x) = z0(x), x ∈ ∂4̂ (5.4) consider 4̂⊂<n be a bounded domain with smooth boundary ∂4̂, x = l2(4̂), %k be random variable defined on dk ≡ (0,dk) for k ∈ n, dk ∈ (0, +∞) . also assume that %k follow erlang distribution and if i 6= j then %i and %j are independent with each other for i,j = 1, 2, . . . . here q is a function of k, ξk = ξk−1 + %k for k ∈ n, t0 ∈<+. let a be an operator on x by az = ∂2z ∂x2 with the domain d(a) = { z ∈ x | z and ∂z ∂x are absolutely continuous, ∂2z ∂x2 ∈ x,z = 0 on ∂4̂ } your running title 17 thus a generates a strongly continuous semigroup s(t) which is self adjoint, compact and analytic. furthermore the operator a can be represented as az = ∞∑ n=1 n2 < z,zn > zn, z ∈ d(a) here zn(ζ) = √ 2 π sin(nζ),n = 1, 2, . . . , forms the orthonormal set of eigenvectors of a. also for every z ∈ x,s(t)z = ∑∞ n=1 e (−n2t) < z,zn > zn, which holds ‖s(t)‖ ≤ e(−π 2(t−t0)), t ≥ t0. therefore s(t) is a semigroup. consider that the following assumptions: (i) f : <% ×x → x,f1 : <% ×x → x is a continuous function defined by f(t,z)(x) = f1(t,z(x)) t0 ≤ t ≤ t, 0 ≤ x ≤ π f1(θ,x(t + θ))dθ = ∫ t 0 f2(θ,z(tsinθ,x))dθ and also function f and f1 satisfies the lipschitz condition. (ii) e { maxi,k {∏k j=i‖bj(%j)‖ }} is uniformly bounded if, e { max i,k {∏k j=i‖bj(%j)‖ }} ≤ θ, for each %j ∈ dj,j ∈ n,θ > 0 a constant (iii) γ ∫ t t0 l (r)dr < ∫ ∞ γ1 dr h(r) , (5.5) where γ = 2w2 max{1, θ2}(t−t0) 2a−1(1+k∗+h∗) (2a−1)γ(a) ,γ1 = 2w 2θ2e‖ϕ‖2 and wθ ≥ 1√ 2 . assume that assumptions (i), (ii) and (iii) are satisfied, then the problem (5.1) becomes a random impulsive fractional differential equation. from all the above facts, in view of theorem 3.1, we conclude that 5.4 has a mild solution. remark 5.4. let the conditions of example 5.3 along with (h4) − (h5) be hold. if the following inequality is satisfied, γ∗ ∫ t t0 l (r)dr < ∫ ∞ γ2 dr h(r) , (5.6) where γ∗ = 2w2 max{1, θ2}(t−t0) 2a−1(1+k∗+h∗) γ(a)(2a−1) ,γ2 = 2w 2θ2e[‖ϕ‖2], and wθ ≥ 1√ 2 . then the mild solution z of the example 5.3 is exponentially stable in the quadratic mean. 6. conclusion in this article we mainly focused on the existence and stability of the random impulsive fractional differential equations via leray-schauder fixed point method. firstly, we established the existence of mild solution and continued to prove the exponential stability of the system. finally, we provided an application to assist of our theory. in future, we will study controllability of random impulsive fractional differential system via fixed point approach. 18 x. one, x. two, x. three, and x. four references [1] a. b. alissa and w. zouhair, qualitative properties for 1-d impulsive wave equation: controllability and observability, quaestiones mathematicae, (2021). 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[30] y. zhou, basic theory of fractional differential equations, world scientific, singapore, (2014). 1 ramanujan centre for higher mathematics, alagappa university, karaikudi-630 003, india., 2 department of mathematics, alagappa university, karaikudi-630 003, india. 3 department of mathematics, school of advanced sciences, vellore institute of technology, vellore 632 014, tamil nadu, india 4 department of mathematics, muslim association college of engineering, trivandrum, india. 5 corresponding author, department of mathematics, marian engineering college, trivandrum, india. email address: thomas.abin49@gmail.com 1. introduction 2. preliminaries hypotheses 3. existence 4. exponential stability in the quadratic mean 5. applications 6. conclusion references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 4, december 2020, pp.275-285 https://doi.org/10.5206/mase/9314 a solution to a fractional order semilinear equation using variational method ramesh karki and young hwan you abstract. we will discuss how we obtain a solution to a semilinear pseudo-differential equation involving fractional power of laplacian by using a method analogous to the direct method of calculus of variations. more precisely, we will discuss the existence of a weak form of solutions to this equation as a minimizer of a suitable energy-type functional whose euler-lagrange equation is the semilinear equation, and also discuss the possibility of regularity of such a weak solution so that it will be a solution to the semilinear equation. 1. introduction we consider n as a fixed positive integer greater than 1 throughout this paper. we call a function u : rd → r a nzd-periodic function if and only if u(x + nej) = u(x) for all j = 1, 2, . . . , d, where each ej = (0, 0, . . . , 1, . . . , 0) is the j th standard unit vector in rd. we assume that f : rd ×r → r is an nzd+1-periodic continuous function such that∫ y+n 0 f(x,z) dz = ∫ y 0 f(x,z) dz (1.1) for all x ∈ rd, y ∈ r. under these assumptions on f, our main goal is to study the existence of a weak form of solutions to a semilinear pseudo-differential equation (ψde) of the form −(−∆)αu(x) = f(x,u(x)), x ∈ rd, (1.2) where ∆ is a d-dimensional laplacian and α is a fixed real number with 0 < α < 1. because of nonlocal behaviors of the fractional power of laplacian, we will consider the problem under the periodic boundary setting. this will allow us to reduce the problem into a variational type problem, or more precisely, a problem of finding a minimizer of an energy type functional i associated with equation (1.2). here, the main idea is to use a method that is analogous to a widely used classical method, the direct method of calculus of variations, for studying solutions to nonlinear partial differential equations (see [6, 9, 12] etc.). problems similar to the one we mentioned above have been widely studied. under a stronger assumption that f mentioned above is also infinitely smooth, r. de la llave and e. valdinnoci have studied the existence of a special (birkhoff) class of solutions to a problem similar to ours by using the steepest descent method in [8]. in this method, they consider u as a real-valued function of a spacial variable x ∈ rd and a forward time parameter t > 0, then consider a steepest descent equation corresponding to received by the editors 19 january 2020; firstly revised 23 may 2020, secondly revised 1 september 2020; accepted 28 october 2020; published online 09 november 2020. 2000 mathematics subject classification. primary 35s15. key words and phrases. energy-type functional, minimizer, weak solution, lower semicontinuity, minimizing sequence, regularity. 275 276 r. karki and y. h. you the problem, reduce this equation to the abstract cauchy problem (a type of evolutionary problem in an infinite dimensional space), and find a solution (in the birkhoff class) to the problem as an equilibrium solution. in [3], t. blass, r. de la llave and e. valdinnoci have studied the existence of this type of solution to the semilinear elliptic equation au + f(x,u) = 0, x ∈ rd (1.3) involving the positive definite uniformly elliptic self-adjoint operator a = − ∑d i,j=1(a ij(x)uxi)xj of order 2 with smooth symmetric coefficients aij by using the sobolev gradient descent method which is analogous to the steepest descent method but is more specific than the latter one. in [14], r. karki has generalized this method to the semilinear pseudo-differential equation aαu + f(x,u) = 0, x ∈ rd (1.4) involving the fractional power of the operator a. the advantage of studying these semilinear equations by using the steepest or (sobolev) gradient descent method is that the solution to the corresponding steepest or gradient descent equation converges to an equilibrium solution at a much faster rate, which can be seen via numerical simulation (see [2]). other advantages could be exploring the the dynamics of the gradient descent equations corresponding to the problems, which can be applicable in different fields such as mathematical physics, math finance, engineering, mechanics, and geology etc. despite having these advantages, we use neither of these methods to solve our problem. using the variational approach, j. moser has thoroughly discussed the existence, the regularity and many other properties of solutions to a problem analogous to ours in [15, 16, 17], especially the case when (−∆)α is replaced by −∆. now we use this variational approach to find a weak solution to our semilinear fractional poisson type equation (1.2). by obtaining a weak solution to equation (1.2), we will devise a powerful tool that is needed to obtain a classical solution to equation (1.2). fractional poisson type equations (special cases of equation (1.2)) are studied when exploring anomalous diffusion in the context of geophysical electromagnetics (see [24]) and near-surface geotechical engineering as non-local electromagnetic effects occur due to fractures and stratigraphic layering (see [11, 23]). for the case λ = 1 2 , equation (1.2) and the problem similar to ours appear in many fields, and have been widely studied for last several years. for instance, they appear in the theory of water waves when approximating the dirichlet problem to the neumann problem (see [7, 18]). their applications to the ultrarelativistic limit of quantum mechanics (see [10]) and recently, to phase transition problems involving fractional powers of laplacian have appeared (see [1, 4, 13]). the operator (−∆) 1 2 also plays a vital role in the thin obstacle problem (see [5, 22]). we will prove the existence of a weak solution to equation (1.2) in two main steps. first, we will show that a regular enough minimizer (i.e., a member of a suitable sobolev space) of the energy-type functional i is a weak solution to equation (1.2), and then show that the functional i does indeed have a minimizer. schematically, we will first discuss some function spaces with respective norms, and some operators such as laplacian and fractional laplacian defined on these spaces in section 2.1. in section 2.2, we will introduce a few key terms such as an energy-type functional i whose euler-lagrange equation is equation (1.2), a minimizer of i and a weak form of solution to equation (1.2), and show that a minimizer with sufficient regularity is indeed such a solution. in section 2.3, we will prove the existence of weak solutions to equation (1.2) in the form of minimizer of i. finally, we will comment on possibilities of proving regularity results related to a weak solution to equation (1.2), and also briefly discuss other stronger forms of solutions to equation (1.2) in section 3. a solution via variational method 277 2. existence of weak solutions in the form of minimizers 2.1. some preliminaries. in this subsection, we will characterize some function spaces and their norms using the ideas adopted in [14]. such characterizations will provide us some tools which will be really handy while working with these spaces and their norms later. we use the notation ntd to denote quotient space rd/nzd, which is the same as the d-dimensional square [0,n]d after identifying its opposite sides. in other words, ntd is a d-dimensional torus. we use c∞(ntd) to denote the space of all smooth functions u : rd → r that are nzd-periodic and equip c∞(ntd) with the uniform norm. any u in c∞(ntd) has a fourier series representation u(x) = ∑ j∈zd e 2π n i〈x,j〉ûj, x ∈ rd (2.1) where each ûj is the j th fourier coefficient of u and is given by ûj = 1 nd ∫ ntd e− 2π n i〈x,j〉u(x) dx. (2.2) we use l2(ntd) to denote the space of all square integrable functions u : rd → r that are nzdperiodic. then l2(ntd) is the completion of c∞(ntd) with respect to the norm ‖u‖l2(ntd) = ∑ j∈zd |ûj|2. (2.3) using (2.1) together with the convergence in l2(ntd), each u in l2(ntd) can be expressed as u(x) = ∑ j∈zd e 2π n i〈x,j〉ûj, x ∈ rd (2.4) where the equality needs to be understood in almost everywhere sense. we consider the sobolev space hs(ntd) with s > 0 as the completion of c∞(ntd) in l2(ntd) under the norm ‖u‖hs(ntd) = ∥∥∥(i + (−∆)s) 12 u∥∥∥ l2(ntd) . (2.5) throughout the rest of this subsection, we will be referring to s > 0. as we can express the operator −∆ : d(−∆) ⊆ l2(ntd) → l2(ntd) as (−∆)u(x) = ( 2π n )2 ∑ j∈zd |j|2ûje 2π n i〈x,j〉, (2.6) we can use the spectral integral from the spectral theory of unbounded self-adjoint operators on hilbert spaces (see [19, 20, 21]) to express the operator (−∆)s on l2(ntd) as (−∆)su(x) = ( 2π n )2s ∑ j∈zd |j|2sûje 2π n i〈x,j〉. (2.7) therefore, by the virtue of equations (2.3) (2.7), we have ‖u‖2hs(ntd) = ∑ j∈zd [ 1 + ( 2π n )2s |j|2s ] |ûj|2. (2.8) also, u ∈ d((−∆)s) if and only if u ∈ l2(ntd) and( 2π n )2s ∑ j∈zd |j|2s|ûj|2 < ∞, 278 r. karki and y. h. you which is true if and only if ‖u‖2hs(ntd) < ∞, meaning u ∈ h s(ntd). moreover, for each u ∈ d((−∆)s) we can define ‖u‖ḣs(ntd) =  (2π n )2s ∑ j∈zd |j|2s|ûj|2   1 2 , (2.9) to obtain a seminorm (not a norm) ‖·‖ḣs(ntd) on d((−∆) s). actually, the mean û0 = 1 nd ∫ ntd u(x) dx for any u ∈ d((−∆)s) over ntd has no contribution in ‖u‖ḣs(ntd) even if u is nonzero. 2.2. minimizers and weak solutions. in order to study solutions to the semilinear ψde (1.2), we first consider the energy-type functional i(u) = ∫ ntd { 1 2 [(−∆) α 2 u(x)]2 + f(x,u(x)) } dx, (2.10) defined on a subspace of l2(ntd), where f(x,y) = ∫ y 0 f(x,z) dz, x ∈ rd, y ∈ r, (2.11) and then study the critical values of i. among those critical values, we are basically interested on minimizers of i as discussed in section 1. we observe that i is naturally defined for all u ∈ hα(ntd). taking this into account, we introduce a minimizer of i and a weak solution to equation (1.2). definition 2.1 (minimizer). a u ∈ hα(ntd) is a minimizer of i if i(u + φ) ≥ i(u) for all φ ∈ c∞(ntd). it follows from definition 2.1 that if u ∈ hα(ntd) is a minimizer of i, then d dt i(u + tφ)|t=0 = 0 (2.12) for all φ ∈ c∞(ntd). definition 2.2 (weak solution). a u ∈ hα(ntd) is a weak solution to equation (1.2) if u satisfies 〈(−∆) α 2 u, (−∆) α 2 φ〉 + 〈f(.,u),φ〉 = 0 (2.13) for all φ ∈ c∞(ntd). now we establish a fundamental result that relates a minimizer of i to a weak solution to equation (1.2). theorem 2.1. let f : rd × r → r be an nzd+1-periodic continuous function such that equation (1.1) holds. if u ∈ hα(ntd) is a minimizer of i given by equation (2.10), then u is a weak solution to equation (1.2). proof. let u ∈ hα(ntd) be a minimizer of i and let φ ∈ c∞(ntd). then i(u + tφ) − i(u) = ∫ ntd { 1 2 [ (−∆) α 2 (u + tφ) ]2 + f(.,u + tφ) } − ∫ ntd { 1 2 [ (−∆) α 2 u ]2 + f(.,u) } = ∫ ntd 1 2 { 2t(−∆) α 2 u(−∆) α 2 φ + t2 [ (−∆) α 2 φ ]2} + ∫ ntd {f(.,u + tφ) −f(.,u)} , a solution via variational method 279 so we have i(u + tφ) − i(u) t = ∫ ntd (−∆) α 2 u(−∆) α 2 φ + t 2 ∫ ntd [ (−∆) α 2 φ ]2 (2.14) + ∫ ntd f(.,u + tφ) −f(.,u) t since u ∈ hα(ntd) and φ ∈ c∞(ntd), we have (−∆) α 2 u, (−∆) α 2 φ ∈ l2(ntd), so (−∆) α 2 u(−∆) α 2 φ,[ (−∆) α 2 φ ]2 ∈ l1(ntd). thus the first two integrals on the right side of equation (2.14) are finite. moreover, since f is nzd+1-periodic on rd × r, it is bounded there and, therefore, there exists a real number m > 0 such that |fy(x,y)| = |f(x,y)| ≤ m for all (x,y) ∈ rd ×r. thus we have∣∣∣∣f(.,u + tφ) −f(.,u)t ∣∣∣∣ ≤1t |f(.,u + tφ) −f(.,u)| = 1 t ∣∣∣∣ ∫ t 0 d ds f(.,u + sφ) ds ∣∣∣∣ = 1 t ∣∣∣∣ ∫ t 0 fy(.,u + sφ)φds ∣∣∣∣ ≤ 1 t ∫ t 0 |fy(.,u + sφ)||φ|ds ≤ m|φ| for all t > 0. since m|φ| ∈ l1(ntd), the dominated convergence theorem implies that lim t→0 ∫ ntd f(.,u + tφ) −f(.,u) t exists and equals ∫ ntd fy(.,u)φ = ∫ ntd f(.,u)φ. therefore, letting t → 0 on the both sides of equation (2.14), we obtain d dt i(u + tφ)|t=0 = 〈(−∆) α 2 u, (−∆) α 2 φ〉 + 〈f(.,u),φ〉. (2.15) also, equation (2.12) holds true for a minimizer u ∈ hα(ntd) of i. from equation (2.12) and equation (2.15), we obtain equation (2.13). hence u is a weak solutions to equation (1.2). � theorem 2.1 guarantees that in order to find weak solutions to equation (1.2), it suffices to prove the existence of a minimizer of i in hα(ntd). theorem 2.2. let f : rd×r → r be an nzd+1-periodic continuous function such that equation (1.1) holds. then there exists u ∈ hα(ntd) such that u is a minimizer of i given by equation (2.10), and hence a weak solution to equation (1.2). 2.3. proof of theorem 2.2. we will complete the proof of theorem 2.2 by subsequently proving a few results. the first of them is related to a coercive condition satisfied by i in hα(ntd). proposition 2.3 (coercivity). suppose i is given by equation (2.10). then there exists a positive constant λ depending on f, n and d such that i(u) ≥ 1 2 ( 1 −n−d )[ 1 + ( 2π n )2α]−1 ‖u‖2hα(ntd) − λ (2.16) for all u ∈ hα(ntd). 280 r. karki and y. h. you proof. notice that f given by equation (2.11) is continuous and ntd+1-periodic on rd × r. let λ0 be a positive real number such that |f(x,y)| ≤ λ0 for all (x,y) ∈ rd ×r. then the integrand l((−∆) α 2 u(x),x,u(x)) := 1 2 [ (−∆) α 2 u(x) ]2 + f(x,u(x)) (2.17) of i satisfies the condition l(p,x,z) := 1 2 p2 + f(x,z) ≥ 1 2 p2 − λ0 (2.18) for all (p,x,z) ∈ r×rd ×r. let u ∈ hα(ntd). first, using (2.18) into (2.10), then using (2.3), (2.7), we get i(u) ≥ 1 2 ∥∥(−∆) α2 u∥∥2 l2(ntd) − λ0n d = 1 2 ∑ j∈zd ( 2π n )2α |j|2α|ûj|2 − λ0nd and next using equation (2.9), we get i(u) ≥ 1 2 ‖u‖ḣα(ntd) − λ0n d. (2.19) taking j = 0 ∈ zd in equation (2.2) and using the cauchy-schwartz inequality, we get |û0| ≤ 1 nd ∫ ntd |u(x)| dx ≤n−d · (nd) 1 2 ‖u‖l2(ntd) =n− d 2 ‖u‖l2(ntd) using the last inequality into equation (2.8), we have ‖u‖2hα(ntd) ≤ ∑ j∈zd−{0} [ 1 + ( 2π n )2α |j|2α ] |ûj|2 + n−d‖u‖ 2 l2(ntd) ≤ ∑ j∈zd−{0} |j|2α [ 1 + ( 2π n )2α] |ûj|2 + n−d‖u‖ 2 l2(ntd) ≤ [ 1 + ( 2π n )2α] ∑ j∈zd |j|2α|ûj|2 + n−d‖u‖ 2 hα(ntd) = [ 1 + ( 2π n )2α] ‖u‖2ḣα(ntd) + n −d‖u‖2hα(ntd) , from which, we obtain ‖u‖2ḣα(ntd) ≥ ( 1 −n−d )[ 1 + ( 2π n )2α]−1 ‖u‖2hα(ntd) . (2.20) combining inequality (2.19) and inequality (2.20), we obtain i(u) ≥ 1 2 ( 1 −n−d )[ 1 + ( 2π n )2α]−1 ‖u‖2hα(ntd) − λ0n d and hence i(u) ≥ 1 2 ( 1 −n−d )[ 1 + ( 2π n )2α]−1 ‖u‖2hα(ntd) − λ a solution via variational method 281 for some positive constant λ depending on f, n and d. � the next important result required to prove theorem 2.2 is related to weakly lower semicontinuity of i in hα(ntd). recall that i is weakly lower semicontinous at u ∈ hα(ntd) if and only if for every sequence {uk} in hα(ntd) converging weakly in hα(ntd) to a limit u, i(u) ≤ lim inf k→∞ i(uk) and is weakly lower semicontinuous in hα(ntd) if and only if it is weakly lower semicontinuous at each u ∈ hα(ntd). proposition 2.4 (weakly lower semicontinuity). i is weakly lower semicontinuous in hα(ntd). proof. consider any u ∈ hα(ntd) and any sequence {uk} in hα(ntd) that converges weakly in hα(ntd) to u and set m = lim inf k→∞ i(uk). to complete the proof, we need to show that i(u) ≤ m. since a weakly convergent sequence is bounded, we have sup k ‖uk‖hα(ntd) < ∞. from definition of limit infimum, we may assume, by passing to a subsequence if necessary, that m = lim k→∞ i(uk). (2.21) by the sobolev embedding theorem, the inclusion hα(ntd) ↪→ l2(ntd) is compact. so, we may assume, by passing another subsequence of the last subsequence of our original sequence {uk} if necessary, that {uk} converges strongly in l2(ntd) to u, which then implies that uk → u a. e. in ω, where ω = ntd. let � > 0 be given. then, by egoroff’s theorem, there exists a measurable subset e� of ω with |ω −e�| ≤ � (here |ω −e�| denotes the lebesgue measure of ω −e�) such that uk → u uniformly on e�. define a subset f� of ω by f� = { x ∈ ω : |u(x)| + |(−∆) α 2 u(x)| ≤ 1 � } . as � → 0, |ω −f�|→ 0. define another subset g� of ω by g� = e� ∩f�. as � → 0, |ω −g�| ≤ |ω −e�| + |ω −f�|→ 0. since l given by equation (2.17) is bounded from below, we may assume without loss of generality that l ≥ 0 (otherwise, we may consider l̃ = l + λ0 ≥ 0 where λ0 is as in the proof of proposition 2.3). therefore, from i(uk) = ∫ ω 1 2 [ (−∆) α 2 uk ]2 + f(.,uk) 282 r. karki and y. h. you for all k, we have i(uk) ≥ ∫ g� 1 2 [ (−∆) α 2 uk ]2 + f(.,uk) (2.22) for all k. recall that if a map φ : r → r is convex, then φ(y) ≥ φ(x) + φ′(x)(y −x) for all y,x ∈ r. since the map p 7→ 1 2 p2 mapping r into itself is convex, 1 2 p2k ≥ 1 2 p2 + p(pk −p) for all pk,p ∈ r. so, we have 1 2 ( (−∆) α 2 uk(x) )2 ≥ 1 2 ( (−∆) α 2 u(x) )2 + (−∆) α 2 u(x) ( (−∆) α 2 uk(x) − (−∆) α 2 u(x) ) for all x ∈ rd and for all k, and thus∫ g� 1 2 [ (−∆) α 2 uk ]2 ≥ ∫ g� 1 2 [ (−∆) α 2 u ]2 (2.23) + ∫ g� (−∆) α 2 u ( (−∆) α 2 uk − (−∆) α 2 u ) . since uk → u uniformly on g�, for each x ∈ rd we have (−∆) α 2 uk(x) − (−∆) α 2 u(x) =(−∆) α 2 (uk −u)(x) = ∑ j∈zd ( 2π n )α |j|α ̂(uk −u)je 2π n i〈x,j〉 = ∑ j∈zd ( 2π n )α |j|α · 1 nd ∫ ω e− 2π n i〈y,j〉(uk(y) −u(y)) dy e 2π n i〈x,j〉 = ∑ j∈zd 1 nd ( 2π n )α |j|αe 2π n i〈x,j〉[ ∫ g� e− 2π n i〈y,j〉(uk(y) −u(y)) dy + ∫ ω−g� e− 2π n i〈y,j〉(uk(y) −u(y)) dy], which approaches 0 as � → 0 and k →∞ because of the facts that g� ⊆ ω with |ω −g�|→ 0 as � → 0 and uk → u uniformly on g�. this limit and the monotone convergence theorem then imply that∫ g� [( (−∆) α 2 uk − (−∆) α 2 u )]2 = ∫ ω χg� [( (−∆) α 2 uk − (−∆) α 2 u )]2 (2.24) approaches 0 as � → 0 and k →∞. by the cauchy-schwartz’s inequality,∣∣∣∣ ∫ g� (−∆) α 2 u ( (−∆) α 2 uk − (−∆) α 2 u )∣∣∣∣2 ≤ ∫ g� [ (−∆) α 2 u ]2 (2.25) · ∫ g� [( (−∆) α 2 uk − (−∆) α 2 u )]2 . from limit (2.24) and inequality (2.25), we can obtain that∫ g� (−∆) α 2 u ( (−∆) α 2 uk − (−∆) α 2 u ) → 0 (2.26) a solution via variational method 283 as � → 0 and k → ∞. letting � → 0 and k → ∞ on both sides of inequality (2.23) and using limit (2.26), we get lim �→0, k→∞ ∫ g� 1 2 [ (−∆) α 2 uk ]2 ≥ lim �→0 ∫ g� 1 2 [ (−∆) α 2 u ]2 . (2.27) since |ω − g�| → 0 as � → 0, χg�(x) → 1 for almost every x ∈ ω as � → 0. by the monotone convergence theorem, we have lim �→0 ∫ g� [ (−∆) α 2 u ]2 = lim �→0 ∫ ω χg� [ (−∆) α 2 u ]2 = ∫ ω [ (−∆) α 2 u ]2 . (2.28) from inequality (2.27) and equation (2.28), we have lim �→0, k→∞ ∫ g� 1 2 [ (−∆) α 2 uk ]2 ≥ ∫ ω 1 2 [ (−∆) α 2 u ]2 . (2.29) moreover, from the facts that f is continuously differentiable at the functional component y and fy(x,y) = f(x,y), x ∈ rd, y ∈ r, and uk → u uniformly on e� (⊇ g�), it follows that lim �→0 k→∞ ∫ g� f(.,uk) = ∫ ω f(.,u) (2.30) letting � → 0 and k →∞ on the both sides of (2.22), then applying (2.29) and (2.30), we get lim k→∞ i(uk) ≥ ∫ ω 1 2 [ (−∆) α 2 u ]2 + ∫ ω f(.,u) = ∫ ω [ 1 2 [(−∆) α 2 u ]2 + f(.,u)] =i(u), and hence, by limit (2.21), i(u) ≤ m as desired. � finally, we are ready to prove the existence of minmizer of i in hα(ntd). proposition 2.5 (existence of minimizer). i has a minimizer in hα(ntd). proof. set m0 = inf u∈hα(ntd) i(u). without loss of generality, assume that m0 < ∞ (otherwise, each u ∈ hα(ntd) becomes a minimizer of i). let {uk} be a minimizing sequence in hα(ntd), meaning a sequence satisfying lim k→∞ i(uk) = m0. then we have sup k i(uk) < ∞. by proposition 2.3, we have i(uk) ≥ 1 2 ( 1 −n−d )[ 1 + ( 2π n )2α]−1 ‖uk‖hα(ntd) − λ for all u ∈ hα(ntd), where λ is a positive constant depending on f, n and d. this implies that sup k ‖uk‖hα(ntd) < ∞. due to the weak compactness theorem, every bounded sequence in a hilbert space is weakly precompact, that is, every bounded sequence in a hilbert has a weakly convergent subsequence. therefore, 284 r. karki and y. h. you there exists a subsequence {ukj} of {uk} and an element u ∈ hα(ntd) such that ukj ⇀ u in hα(ntd). by lower semicontinuity of i from proposition 2.4, we have i(u) ≤ lim inf j→∞ i(ukj ) = m0. on the other hand, it is clear that m0 ≤ i(u). therefore, m0 = i(u). this shows that u is a minimizer of i in hα(ntd). � with proposition 2.5, we have completed the proof of theorem 2.2. in this way, we have proved that under the given assumptions on f, equation (1.2) does have a weak solution in hα(ntd). 3. some remarks on regularity a weak solution u ∈ hα(ntd) to equation (1.2) does not necessarily satisfy equation (1.2) unless u is sufficiently regular or smooth. but in order to have sufficient regularity or smoothness of u we may need to add further assumptions on the nonlinear functional f. for the moment, suppose we have a suitable f so that u ∈ hs(ntd) for s ≥ 2α. since for the weak solution u, we have 〈(−∆) α 2 u, (−∆) α 2 φ〉 + 〈f(.,u),φ〉 = 0 for all φ ∈ c∞(ntd) and (−∆) α 2 is self-adjoint on l2(ntd) (see [8, 20, 21]), we have 〈(−∆)αu + f(.,u),φ〉 = 0 for all φ ∈ c∞(ntd) and hence (−∆)αu + f(.,u) = 0 a. e. (3.1) this leads us to ask the following. remark 3.1. under what assumptions on f, does a minimizer u ∈ hα(ntd) of i belong to hs(ntd), s ≥ 2α so that u satisfies equation (1.2) in a pointwise almost everywhere sense (meaning u is a strong solution to equation (1.2))? furthermore, suppose we have even a better f so that the weak solution u ∈ hs(ntd), s ≥ 2α is smooth enough to become both u and (−∆)αu continuous on rd. then equation (3.1) reduces to (−∆)αu(x) + f(x,u(x)) = 0 ∀ x ∈ rd. this means u solves equation (1.2) in a classical sense and thus equation (1.2) is the euler-lagrange equation of i(u). so, it is natural to expect the following. remark 3.2. additionally, how smooth does f need to be so that a weak solution u of equation (1.2) will satisfy it in a pointwise (everywhere) sense (meaning u is a classical solution to equation (1.2))? references [1] g. alberti and g. bellettini, a nonlocal anisotropic phase transitions i. the optimal profile problem, math. ann. 310(1998), 527560. 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[8] r. de la llave and e. valdinnoci, a generalization of aubrey-mather theory to partial differential equations an pseudo differential equations, annals de l’institut henry poincare (c) non linear analysis 26(2009), 1309-1344. [9] l. c. evans, partial differential equations, vol. 19, graduate studies in mathematics, american mathematical society, providence, ri, 1998. [10] c. fefferman and r. de la llave, relativistic stability of matter. i, rev. mat. iberoamericana 2 (1986), 119-213. [11] j. ge, m. e. everett and c. j. weiss, fractional diffusion analysis of the electromagnetic field in fractured mediapart 2: 3d approach, geophysics 80(2015), 175-185. [12] m. giaquinta (ed.), topics in calculus of variations, lecture notes in mathematics 1365, springer-verlag, 1987. [13] a. garroni and g. palatucci, a singular perturbation result with a fraction norm, in: variational problems in material sience, progr. nonlin. diff. eqns. appl. 68(2006), 111-126. [14] r. karki, sobolev gradients & application to nonlinear pseudo-differential equations, neural, parallel & scientific computations 22(2014), 359-374. [15] j. moser, minimal solutions of variational problems on a torus, annals de l’i. h. p. analyse non linéaire 3(1986), 229-272. [16] j. moser, a stability theorem for minimal foliations on a torus, ergodic theory dynamical systems (charles conley memorial issue), 8(1988), 251-281. [17] j. moser, minimal foliations on a torus, in giaquinta m. (eds): topics in calculus of variations. lecture notes in mathematics, vol. 1365. springer, berlin, heidelberg. https://doi.org/10.1007/bfb0089178. [18] p. i. naumkin and i.a. shishmarev, nonlinear nonlocal equations in the theory of waves, american mathematical society, providence, ri, 1994. [19] l. roncal and p. r. stinga, fractional laplacian on the torus, communications in contemporary mathematics 18(3)(2016): 1550033. [20] k. schmudgen, unbounded self-adjoint operators on hilbert space, graduate text in mathematics 265, springerverlag, ny, 2012. [21] m. a. shubin, pseudodifferential operators and spectral theory, springerverlag, 2nd edition, 2001. [22] l. silvestre regularity of the obstacle problem for a fractional power of the laplace operator, commun. pure appl. math. 60 (2007), 67-112. [23] f. vallianatos a non-existensive view of electrical resistivity spatial distribution estimated using transient electromagnetic responses in a karstified formation (keritis basin, crete, greece), phys. a 505 (2018), 171-178. [24] c.j. weiss, b.g. van bloemen waanders and h. antil, fractional operators applied to geophysical electromagnetics, geophysical journal international 220 (2020), 1242-259. corresponding author, natural science and mathematics, indiana university east, richomond, in 47374, u.s.a. e-mail address: rkarki@iue.edu natural science and mathematics, indiana university east, richomond, in 47374, u.s.a. e-mail address: youy@iue.edu mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 2, number 4, december 2021, pp.235-272 https://doi.org/10.5206/mase/14112 prevailing winds and spruce budworm outbreaks: a reaction-diffusion-advection model abby anderson and olga vasilyeva abstract. we extend the classical reaction-diffusion model for spatial population dynamics of spruce budworm on a finite domain with hostile boundary conditions by including an advection term representing biased unidirectional movement of individuals due to a prevailing wind. we use phase plane techniques to establish existence of a critical value of advection speed that prevents outbreak solutions on any finite domain while possibly allowing an endemic solution. we obtain lower and upper bounds for this critical advection value in terms of biological parameters involved in the reaction term. we also perform numerical simulations to illustrate the effect of advection on the dependence of the domain size on the maximal population density of a steady state solution and on critical domain sizes for endemic and outbreak solutions. the results are also applicable to other ecological settings (rivers, climate change) where a logistically growing population is subject to predation by a generalist, diffusion and biased movement. 1. introduction this paper is inspired by classical 1979 work [11] of d. ludwig, d.g. aronson and h.f. weinberger where the authors introduced and analyzed a reaction-diffusion model for the spatial dynamics of spruce budworm (sbw) population in which the insects were assumed to disperse by diffusive movement only. on page 251 in [11], the authors state the following: “the limitation that dispersal is modelled by pure diffusion is a serious one. there are prevailing wind patterns over new brunswick which produce systematic, convective motions of the budworm. the addition of a convective term to the present model presents no particular difficulties for the treatment of the differential equation. the boundary conditions introduce serious complications.” in this paper, we extend the reaction-diffusion model from [11] to include the convective/advective motion in the form of an advection term. the boundary conditions remain hostile (lethal), as they were in [11]. the main difficulty in generalizing the results from [11] is the presence of the advection term which requires new techniques since as the first integral method and comparison methods used in [11] are not applicable anymore. we use the geometric method based on the phase plane analysis of a boundary value problem given by a system of two first order odes obtained from the second order ode describing the profile of a steady state solution of the original reaction-diffusion-advection (rda) problem. this approach was used in [1, 22, 21, 2] to study steady states or traveling wave solutions arising in nonlinear rda models. received by the editors 18 july 2021; 24 october 2021; published online 3 november 2021. 2010 mathematics subject classification. primary 35k57; secondary 34b15. key words and phrases. reaction-diffusion-advection models, steady state, critical advection, stable and unstable manifolds, spruce budworm modeling. a. anderson was supported in part by an undergraduate student research award from the natural sciences and engineering research council of canada. o. vasilyeva was supported in part by a discovery grant from the natural sciences and engineering research council of canada (rgpin-2017-04376). 235 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14112 236 a. anderson and o. vasilyeva historically, rda models have been used to adrress the well-known ”drift paradox” in river ecology: the ability of population to persist in finite river segments despite being constantly washed downstream [20, 13, 15, 4]. the rda approach had also been used to model the dynamics of sinking phytoplankton and populations experiencing a shift in habitat boundaries due to climate change [7, 17, 1]. a recent survey on a variety of rda models is given in [9]. the development of rda models progressed from the linear reaction term case in [20], to monostable logistic term (e.g. in [22, 10]), and to the bistable allee effect term [3, 23]. in some rda models, additional complexity was introduced by incorporating spatial dependence of the reaction term as in [23] or by considering river networks instead of the single river [18, 19, 21, 6, 2]. the mathematical motivation of our paper is in extending the rda techniques to the case of a reaction term that combines the logistic growth with predation by a generalist. the resulting nonlinear reaction term has up to four zeros. in the non-spatial setting considered in [12], zero state is an unstable equilibrium, and there are stable endemic and outbreak equilibria, with the latter two separated by an unstable threshold. as the complexity of the reaction term increases, it is natural to expect multiple steady state solutions, and indeed, in [11], the main focus was on the analysis of endemic and outbreak steady state solutions. in section 2 (preliminaries), we give an overview the classical nonspatial and spatial models introduced in [12] and [11]. first, we outline how by scaling the original nonspatial model one obtains a model involving two biological parameters q and r [12]. then we focus on the “cusped” region in the qr-plane for which the nonspatial model has exactly three positive equilibria (we denote this region by ω) . then we describe the spatial (reaction-diffusion) model [11] focusing on the subregion ω∗ of ω for which the spatial model admits both endemic and outbreak solutions. we also recall the behavior of the spatial model when the parameters (q,r) are taken outside of ω∗, as described in [11]. in section 3, we set up an extension of the spatial model from [11] obtained by adding an advection term. using our geometric approach, we study steady state solutions of the resulting boundary value problem by reducing it to a nonlinear ode system and analyzing its orbits in the phase plane. we identify the equilibria of the ode system and their nature depending on the advection velocity q. we introduce notation for stable and unstable manifolds of the two saddle point equilibria of the system that will play the crucial role in our analysis and find the slopes of the stable and unstable manifolds at these saddle points. in section 4, we focus on the behavior of unstable manifold of the first saddle point as we increase advection and prove technical lemmas that allow us to establish the existence of a critical value of advection q∗cr beyond which the outbreak solution is not possible on any finite domain. we also obtain an upper bound on q∗cr. we perform further geometric analysis of the vector field associated with the ode system and construct a piecewise linear approximation of the orbit connecting two saddle points that allows us to obtain a lower bound on q∗cr (the details are given in appendix a). in section 5, we perform a detailed analysis of behavior of various orbits in the phase plane using trapping region method as one of the main tools. this leads to our main result that describes the global picture of orbits in the phase plane. depending on the advection speed, we classify all possible scenarios in terms of orbits representing positive steady state solutions. endemic and outbreak orbits are classified according to the maximal density or the upstream or downstream density gradients. we also discuss the dependence of habitat length on the maximal density of a steady state solution. numerical simulations are performed in section 6. we test the estimates for critical advection q∗cr for outbreak solutions, and explore the effect of advection on the relation between the habitat length and the maximal population density, and how the increase of advection speed changes the nature of steady states. section 7 contains a discussion of our results and provides their biological interpretation. prevailing winds and spruce budworm outbreaks: an rda model 237 2. preliminaries: an overview of the classical non-spatial and spatial spruce budworm models in this section, we recall the classical non-spatial [12] and spatial [11] model for spruce budworm (sbw) dynamics, summarizing and illustrating the relevant results from these two papers. for the sake of completeness and for the reader’s convenience, we expand some of the arguments appearing in [12, 11] by adding details and providing plots and diagrams. 2.1. the non-spatial ludwig-jones-holling model. the non-spatial sbw model introduced in [12], is given by db dt = rb ( 1 − b k′s ) −β b2 (α′s)2 + b2 , (2.1) where b is the budworm density (larvae per acre) at time t (measured in years). the first term on the right gives the logistic population growth in the absense of predators. here r is the per capita growth rate at low density, k′ is the carrying capacity per branch, and s is the number of (standard size) branches per acre. the second term on the right is the predation term. it uses a type 3 functional response which takes into account the predation pattern between the spruce budworm and generalist predators (birds). namely, when the budworm density is low, the predation rate is very small as the birds tend to ignore the budworms and feed on other sources. when the budworm density increases, the birds start noticing their presence and start actively feeding on them. finally, if the density increases beyond a certain point, the birds become saturated and as the result the predation level stabilizes at a constant value β, the limiting consumption rate (larvae per year), or saturation rate. here α′s is the density of budworm (larvae per acre) that results in the predators consuming at half saturation rate (β 2 ). using u = b α′s , t = rt, r = rα ′s β and q = k ′ α′ (2.1) is reduced to the non-dimensional version du dt = φ(u; q,r), (2.2) where φ(u; q,r) = u− u2 q − 1 r u2 1 + u2 = u ( 1 − u q ) − 1 r u2 1 + u2 . (2.3) the number of equilibria of (2.2) varies between 2 and 4. indeed, they are the roots of the equation u ( 1 q u3 −u2 + ( 1 q + 1 r ) u− 1 ) = 0. we will denote the zero solution of the above equation by u0. as in [11], in this paper we will focus on the case when the above equation has three distinct positive roots. it was observed in [11] that the set of pairs (q,r) for which (2.2) has three positive equilibria corresponds to a region in the qr-plane bounded by two curves r = r1(q) and r = r2(q) where q > 3 √ 3 and r2(q) < r1(q). here, the lines passing through (q, 0) and (0,ri(q)), i = 1, 2, are tangent to the curve v = u 1 + u2 in the uv-plane. fixing q > 3 √ 3 and varying r from r2(q) to r1(q), one can observe that the line v = r ( 1 − u q ) 238 a. anderson and o. vasilyeva figure 2.1. curve in the qr-plane bounding the region ω corresponding to (2.2) having three positive equilibria. transitions from being tangent to the curve v = u 1 + u2 at a point ( ū2, ū2 1+ū22 ) to having three intersection points with the curve, and to being tangent to the curve at a point ( ū1, ū1 1+ū21 ) , where 1 < ū1 < ū2 < q. let ω ⊂ r2 be the region in the qr-plane defined by ω = {(q,r) : φ(u; q,r) has three positive zeros}. note that ω is an open set.to obtain the boundary of ω, one can simply trace a parametric curve (q(s),r(s)) where s > 1 and the tangent line to the graph of v = u u2+1 at u = s has the u-intercept q(s) and the v-intercept r(s). in fact, it is given by the following parametric equation: (q(s),r(s)) = ( s + s(s2 + 1) s2 − 1 , 2s3 (s2 + 1)2 ) , s > 1. a plot of (q(s),r(s)) is given in figure 2.1. from now on, we will be working with the case when (2.2) has three positive equilibria. for simplicity, we will denote φ(u) = φ(u; q,r). thus, we assume that φ(u) has four distinct roots u0 = 0 < u1 < u2 < u3, the equilibria of system (2.2). in this case, φ ′(u0),φ ′(u2) > 0 and φ ′(u1),φ ′(u3) < 0 (see the graph in figure 2.2 for q = 6.5 and r = 0.6). it follows that the equilibria u0,u2 of (2.2) are unstable and u1,u3 are stable. as in [11], we will refer to the equilibria u0 = 0,u1,u2,u3 as the extinction, endemic, threshold and outbreak equilibria, respectively. indeed, u ≡ 0 corresponds to the complete absence of individuals, the lower of the two stable states corresponds to the endemic situation, while the higher stable state describes the population outbreak. the unstable steady state separating the endemic and the outbreak equilibria serves as a threshold. prevailing winds and spruce budworm outbreaks: an rda model 239 figure 2.2. graph of φ(u) = u− 1 q u2 − 1 r u2 1+u2 for q = 6.5,r = 0.6. the following lemma describing the behavior of the function φ(·; q,r) will be useful in our further analysis. lemma 2.1. for (q,r) ∈ ω, the function φ(·) = φ(·; q,r) has exactly four inflection points ±û1,±û2 such that 0 < û1 < 1 < û2. proof. differentiating (2.3) three times we get: φ′(u) = 1 − 2 q u− 1 r 2u (u2 + 1)2 , φ′′(u) = − 2 q − 2 r · 1 − 3u2 (u2 + 1)3 , and φ(3)(u) = 1 r · 24u(1 −u2) (u2 + 1)4 . note that since φ(·) has four zeros 0 = u0 < u1 < u2 < u3 in each of which it changes sign, it has at least two positive inflection points û1 < û2. since φ ′′(·) is an even function, −û1 and −û2 are also inflection points of φ(·). note also that φ(3)(u) has three zeros, u = −1, 0, 1. clearly, φ′′(·) has local maxima at −1 and 1 and local minimum at 0, and therefore it has at most four zeros. thus, φ(·) has exactly four inflection points given by ±û1 and ±û2 where 0 < û1 < 1 < û2. � 2.2. the spatial ludwig-aronson-weinberger model. the spatial version of the sbw model described below was introduced in [11]. in this model, the habitat is represented by an infinite strip in the xy -plane of the form [ −l 2 , l 2 ] × r. the population density is assumed to depend only on x, and thus the authors reduce the setting to that of a finite interval [ −l 2 , l 2 ] of the real line (where the spatial variable x is measured in kilometers). thus, the quantity of interest becomes the linear density of sbw (individuals per kilometer) given by the function u(x,t) at location x ∈ [ −l 2 , l 2 ] at time t ≥ 0 (years). in addition to the birth and death dynamics described by the reaction term discussed 240 a. anderson and o. vasilyeva earlier in the non-spatial setting ([12]), the individuals are assumed to move randomly. the dispersal of individuals from a fixed location over a short time interval δt is described by a normal distribution with mean 0 and variance σ2δt. thus, the corresponding reaction-diffusion equation takes the form: ∂b ∂t = σ2 2 ∂2b ∂x2 + rb ( 1 − b k′s ) −β b2 (α′s)2 + b2 , (2.4) the boundary conditions considered in [11] are of the hostile type: u ( − l 2 ,t ) = u ( l 2 ,t ) = 0, for t > 0. introducing the scaled variables t = rt, x = 2r σ x, u = b α′s , one obtains the non-dimensionalized reaction-diffusion equation as follows: ∂u ∂t = ∂2u ∂x2 + φ(u), (2.5) where the reaction term is given by (2.3). the boundary conditions for the non-dimensional case will be u ( − l 2 , t ) = u ( l 2 , t ) = 0, (2.6) for t > 0, where l = 2r σ l. a steady-state solution u(x) of (2.5, 2.6) satisfies the following boundary value problem on [ − l 2 , l 2 ] : u′′ + φ(u) = 0, u ( − l 2 ) = u ( l 2 ) = 0. (2.7) for our further analysis we will use the following fact about the spatial dynamics of sbw model given by (2.5, 2.6) as observed in [11]. fact 2.2. ([11]) let u : [ − l 2 , l 2 ] → r be a positive steady state solution of (2.5, 2.6) (solution of (2.7)). also let µ = max− l 2 ≤x≤ l 2 u(x). let f(u) = ∫ u 0 φ(w)dw = 1 2 u2 − 1 3q u3 − u r + 1 r arctan u. note that since u′′(x) + φ(u(x)) ≡ 0, the expression 1 2 (u′(x))2 + f(u(x)) in constant for − l 2 < x < l 2 . thus, 1 2 (u′(x))2 + f(u(x)) = f(µ) for any − l 2 < x < l 2 . in particular, for any 0 < w < µ we have f(w) < f(µ). it follows from fact 2.2 that for any positive steady-state solution u(−) of (2.5, 2.6), or equivalently, a positive solution u(−) of (2.7), the maximal value of u(−), µ = max− l 2 ≤x≤ l 2 u(x) satisfies 0 < µ < u1 or u∗ < µ < u3, where u2 < u ∗ < u3 is such that f(u ∗) = f(u1) (see figure 2.3). note that such u ∗ exists only when f(u3) > f(u1). as we will see later in this paper, the value u ∗ will play a crucial role in our analysis. when such u∗ does not exist, we have 0 < µ < u1. moreover, it was shown in [11], as a consequence of fact 2.2, that the habitat length l can be expressed explicitly in terms of the maximal density µ of the positive steady state solution, as follows: l(µ) = √ 2 ∫ µ 0 dw√ f(µ) −f(w) . furthermore, it was shown in [11] that here exists a function r̃(q) such that r2(q) < r̃(q) < r1(q) for any q > 3 √ 3, so that for any (q,r) ∈ ω, we have f(u3) > f(u1) exactly when r̃(q) < r < r1(q). prevailing winds and spruce budworm outbreaks: an rda model 241 figure 2.3. graphs of f(u) = 1 2 u2 − 1 3q u3 − u r + 1 r arctan u (solid) and φ(u) = u − 1 q u2 − 1 r u2 1+u2 (dashed) for q = 8.5,r = 0.5. here u2 < u ∗ < u3 is such that f(u∗) = f(u1). figure 2.4. six subregions of qr-plane. in order to summarize the behavior of l = l(µ) and the nature of steady states as described in [11], we will consider subregions of the first quadrant of the qr-plane with boundaries given by q = 3 √ 3, r = r1(q), r = r2(q) and r = r̃(q), as well as r = r̄(q) (to be introduced below). we will enumerate the regions as i-vi. see figure 2.4. (i) {(q,r) : 0 < q ≤ 3 √ 3,r > 0} (this region was not studied in [11], we include it for completeness): here φ(u) has a unique positive zero u1; l(µ) is defined for 0 < µ < u1 and is increasing from its left limit of π to ∞ at u1. thus, there exists a critical domain size lc1 = π such that for any l > lc1 there is a unique positive steady state solution u1(x) of (2.5, 2.6), it is bounded above by u1. for l ≤ lc1 there is no positive steady state solution. (ii) {(q,r) : q > 3 √ 3, 0 < r ≤ r2(q)}: when r < r2(q) φ(u) has only one positive zero u1, for r = r2(q) a new positive (double) zero u2 = u3 appears. the behavior of l(µ) and the nature of steady states are identical to that in region (i). (iii) {(q,r) : q > 3 √ 3,r2(q) < r ≤ r̃(q)}: now φ(u) has three positive zeros u1 < u2 < u3 and f(u1) ≥ f(u3) (with strict inequality for r < r̃(q)); l(µ) behaves as in (i) and (ii). we still 242 a. anderson and o. vasilyeva figure 2.5. dependence of habitat length l on maximal denisty µ for different values of r (for a fixed q). have only one positive steady state u1(x) bounded above by u1 for l > l c 1 = π. for l ≤ lc1 there is no positive steady state solution. (iv) {(q,r) : q > 3 √ 3, r̃(q) < r < r1(q)}: in this case φ(u) still has three positive zeros u1 < u2 < u3 and f(u1) < f(u3) (so, u ∗ exists); l(µ) has two branches, the left branch behaves as in (i-iii), the right branch is defined for u∗ < µ < u3, and it has a minimal value of l c 2 and approaches infinity at u∗ and u3. thus, there exists another critical habitat length l c 2 > l c 1 = π such that for l > lc2, there exist three distinct positive steady-state solutions u1(x),u2(x),u3(x) of (2.5, 2.6) , where u1(x) < u2(x) < u3(x) for all x ∈ ( − l 2 , l 2 ) . denote the maximal value of ui(x) (maximal density) on [ − l 2 , l 2 ] by µi, i = 1, 2, 3. the smallest of the solutions, u1(x), satisfies µ1 < u1 and is referred to as an endemic solution. the largest of them, u3(x), satisfies u ∗ < µ3 < u3 and is called an outbreak solution. both u1(x) and u3(x) are asymptotically stable. the intermediate solution u2(x) satisfies u∗ < µ2 < µ3 and is unstable. if the initial density of organisms is below u2(x) (throughout the habitat) then in the long term the density approaches the endemic state u1(x) (when l > l c 1). if the initial density is above u2(x) then the density approaches the outbreak state u3(x) (provided l > lc2). for l c 1 < l ≤ lc2, there is only one (endemic) positive steady state solution bounded above by the constant u1. for l ≤ lc1 there is no positive steady state solution. (v) {(q,r) : q > 3 √ 3,r1(q) ≤ r < r̄(q)}: when r reaches r1(q), the first two positive zeros of φ, u1 and u2, coincide, then they disappear for r > r1(q). let u1 denote the only positive zero that φ(u) has in this case. any positive steady state will have its maximal density bounded by u1. as we keep increasing r until a certain value that we call r̄(q), we observe that l(µ) is defined for 0 < µ < u1, has limit π at 0, followed by local maximum (with value l c 3), local minimum (with value lc2) and goes to infinity at u1. thus, we still observe three distinct positive steady state solutions (stable endemic and outbreak, and unstable threshold) for l in the finite prevailing winds and spruce budworm outbreaks: an rda model 243 interval lc2 < l < l c 3, while for π = l c 1 < l ≤ lc2 there is only one (endemic) positive solution. for l ≥ lc3, only a positive outbreak solution is left. (vi) {(q,r) : q > 3 √ 3,r ≥ r̄(q)}: as we increase r further, φ(u) still has only one positive zero u1; now l(µ) is increasing on the interval 0 < µ < u1, has limit π at 0, and goes to infinity at u1. thus, we can see a single positive steady state solution for all l > l c 1 = π. therefore, r = r̄(q) separates the three steady state case from the single steady state. the scenarios above are illustrated by l vs. µ plots in figure 2.5. note that for r = r1(q), the two vertical asymptotes µ = u1 and µ = u ∗ collapse into one. one of our main goals in sections 5 and 6 will be to explore the effect of advection on the above scenarios. most of our analysis will be carried out in the case when φ(u) has three positive zeros and the system (2.5, 2.6) has both endemic and outbreak solutions for sufficiently large domains, that is, we will consider (q,r) in region (iv), which we will denote ω∗: (q,r) ∈ ω∗ = {(q,r) : q > 3 √ 3, r̃(q) < r < r1(q)}. in this case, there is a clear distinction between endemic and outbreak solutions since their maximal densities are separated by the zeros of φ(u). in the region (v), the difference is more subtle and the phase plane technique that we use in this paper is not as effective. however, we do perform numerical simulations in all six regions of the qr-plane. from now on, unless specified otherwise, we will assume that (q,r) ∈ ω∗. 3. model set up in our setting, all individuals from the population are subject to a unidirectional movement bias given in the form of wind transporting the larval population in one direction. this phenomenon is represented by an advection term −q∂u ∂x (where q ≥ 0). therefore, we add the advection term to the non-dimensionalized version of a reaction-diffusion equation (2.5) to get the following pde boundary value problem:{ ∂u ∂t = ∂ 2u ∂x2 −q∂u ∂x + φ(u), u ( − l 2 , t ) = u ( l 2 , t ) = 0. (3.1) where, as before, φ(u) = u− 1 q u2 − 1 r u2 1+u2 . the steady state solutions of (3.1) can be viewed as solutions of the following second order ode boundary value problem: { u′′ −qu′ + φ(u) = 0 u ( − l 2 ) = u ( l 2 ) = 0. (3.2) by introducing v(x) = u′(x), we rewrite (3.2) as the following system of first order ode:{ du dx = v dv dx = qv −φ(u) (3.3) subject to the boundary conditions u ( − l 2 ) = u ( l 2 ) = 0. (3.4) for any l > 0, a solution of (3.3, 3.4) is represented by an orbit in the uv-plane that starts at a point on the positive v-semiaxis (when x = − l 2 ) and ends at a point on the negative v-semiaxis (when x = l 2 ). observe that for any solution (u(x),v(x)), we have du dx > 0 (< 0) for v > 0 (v < 0), while for 244 a. anderson and o. vasilyeva figure 3.1. phase plane portrait for system (3.3) when α = 0.154,β = 1.67 and q = 0. the four equilibria of (3.3) are easily identified as two centers and two saddle points. the portions of the closed orbits in the first and fourth quadrants form endemic and outbreak solutions of (3.3) corresponding to different habitat lengths. v = 0 we have dv dx = −φ(u) < 0 exactly when 0 < u < u1 or u2 < u < u3. thus, for a fixed l > 0, the positive solutions of (3.3, 3.4) are of two kinds: those that cross the u-axis between 0 and u1 (endemic solutions/orbits) and those that cross the u-axis between u2 and u3 (outbreak solutions/orbits). note that the u-coordinate of the intersection of an orbit (u(x),v(x)) with the u-axis is the maximum value of u(x) on the interval − l 2 ≤ x ≤ l 2 (maximum density in the habitat). thus, we can characterize the endemic and outbreak solutions by max − l 2 ≤x≤ l 2 u(x) < u1 and max − l 2 ≤x≤ l 2 u(x) > u2, respectively. we will use the word “orbits” instead of “solutions” when the actual value of l > 0 is not specified. as in the non-advective case, the equilibria of (3.3) are given by zeros of φ(u), i.e. (u(x),v(x)) = (ui, 0) where 0 ≤ i ≤ 3. we will now determine their nature. consider the jacobian matrix of system (3.3): j(u,v) = [ 0 1 −φ′(u) q ] . its trace and determinant are given by tr(j(u,v)) = q ≥ 0 and det(j(u,v)) = φ′(u). we observe the following regarding each equilibrium point of (3.3): • (u0, 0) = (0, 0): when q = 0, the the second order differential equation in (3.2) has a first integral (see fact 2.2) and the phase plane portrait is symmetric with respect to the u-axis. in fact, when q = 0, the orbits of the system (3.3) are level curves of 1 2 v2 +f(u), where f(u) is given in the fact 2.2. thus, orbits are given by v = ±1 2 √ −f(u) + c, where c are constants. note that prevailing winds and spruce budworm outbreaks: an rda model 245 figure 3.2. phase plane portraits for system (3.3) when α = 0.154,β = 1.67, for q = 0.1, 0.2, 0.27, 0.33. the nature of saddle points remains the same, while both centers become unstable spirals. −f(·) has a local maximum at u = 0 (see figure 2.3). it follows that in a neighborhood of (0, 0) the phase portrait of the system (3.3) consists of closed curves. therefore, this equilibrium point is a center (see p. 296 in [8] for a similar argument). note that φ′(0) = 1. when 0 < q < 2, we have 4 det(j(0, 0)) − tr2(j(0, 0)) = 4 − q2 < 0 while tr(j(0, 0)) = q > 0, so the equilibrium is an unstable spiral. when q ≥ 2, we have an unstable node. • (u1, 0): since det(j(u1, 0)) = φ′(u1) < 0, this equilibrium point is a saddle for any value of q. • (u2, 0): similarly to (0, 0), when q = 0, this equilibrium point is also a center (note that −f(·) has a local maximum at u = u2 as well). note that φ ′(u2) > 0. when 0 < q < 2 √ φ′(u2), we have 4 det(j(u2, 0)) − tr2(j(u2, 0)) = 4(φ′(u2))2 − q2 < 0 while tr(j(u2, 0)) = q > 0, so the equilibrium is an unstable spiral. when q ≥ 2 √ φ′(u2), we have an unstable node. • (u3, 0): as in the case of (u1, 0), we have det(j(u3, 0)) = φ′(u3) < 0, and therefore this equilibrium point is a saddle point for any value of q. the changes in the phase plane portrait of system (3.3) as we increase the advection are illustrated in figure 3.2. as noted above, for any value of q, the system (3.3) has exactly two saddle points: (u1, 0) and (u3, 0). we will denote the upper and lower branches of the stable and unstable manifolds of (ui, 0) (where i = 1 246 a. anderson and o. vasilyeva or 3) by sst+i ,s st− i ,s unst+ i and s unst− i , respectively. here “+” corresponds to “upper” (first quadrant) and “-” corresponds to “lower” (fourth quadrant). thus, sst+i and s st− i have (ui, 0) as their ω-limit, while sunst+i and s unst− i have the same point as their α-limit. since the right hand side of (3.3) is given by c∞ functions of u and v, by the result of guysinsky et al. [5], the homeomorphisms involved in the grobman-hartman theorem can be chosen to be differentiable at the corresponding equilibrium point and having jacobian matrix equal to the identity matrix at that point. it follows that the lines through each saddle point spanned by the eigenvectors of the jacobian matrix j(ui, 0) are tangent to their stable and unstable manifolds (see also lemma 6.2 in [21]). alternatively, the slopes of the stable and unstable manifolds of these equilibria can be found by viewing them as solution curves of an ordinary differential equation involving v as a function of u. namely, since every solution (u(x),v(x)) of system (3.3) satisfies u′(x) = v(x), the portions of the solution curves of (3.3) located either in the upper half of the uv-plane {(u,v) : v > 0} or in the lower half {(u,v) : v < 0} satisfy the vertical line test. such a curve located in either half of the uv-plane can be viewed as the graph of a function v = v(u) defined on a certain interval. such a function will be a solution of the first order differential equation dv du = q − φ(u) v . (3.5) lemma 3.1. the slopes of the tangent lines to the stable and unstable manifolds at the saddle point (ui, 0) (i = 1, 3) of (3.3) are given by m = q ± √ q2 − 4φ′(ui) 2 . furthermore, the slope of sst+i and s st− i is given by m = q− √ q2−4φ′(ui) 2 < 0 and the slope of sunst+i and sunst−i is given by m = q+ √ q2−4φ′(ui) 2 > 0. proof. let v = v(u) be the function whose graph is the upper portion of the stable or unstable manifold of the saddle point (ui, 0). thus, v(u) is defined on an interval of the form (ui − ε,ui) or (ui,ui + ε), v(u) > 0 for all such u, limu→ui v(u) = 0, and v(u) satisfies (3.5) on its domain. our goal is to find m = limu→ui v ′(u). noticing that φ(ui) = 0 and limu→ui v(u) = 0 and using l’hospital’s rule, we get: m = lim u→ui v′(u) = q − lim u→ui φ(u) v(u) = q − lim u→ui φ′(u) v′(u) = q − φ′(ui) m . the above equation can be written as m2 −qm + φ′(ui) = 0. hence, the slope of the tangent line to the stable or unstable manifold at (ui, 0) is given by m = q± √ q2−4φ′(ui) 2 . the second statement of the lemma follows by noticing that du dx = v > 0 in the first quadrant and du dx = v < 0 in the fourth quadrant, and φ′(ui) > 0. � 4. the critical advection for outbreak orbits in this section we will study conditions for the existence of outbreak orbits in terms of the advection speed q. we establish existence of a critical value of advection beyond which the outbreak solutions do not exist. we obtain upper and lower bounds for this threshold value. prevailing winds and spruce budworm outbreaks: an rda model 247 figure 4.1. behaviour of sunst+1 : three possible cases. 4.1. existence and upper bound for critical advection for outbreaks. we will begin by describing the effect of increasing q on the behavior of the unstable manifold of the saddle point (u1, 0). due to the fact that the vector field on the u-axis is given by ( du dx , dv dx ) = (0,−φ(u)), and φ(u) < 0 for u1 < u < u2 and u > u3 and φ(u) > 0 for u2 < u < u3, there are three possible types of behavior of the upper portion of the unstable manifold sunst+1 of (u1, 0): • case 1: sunst+1 crosses the u-axis between u2 and u3; • case 2: sunst+1 becomes a saddle-to-saddle conection with the ω-limit at (u3, 0); • case 3: sunst+1 stays in the first quadrant without approaching any equilibrium. we will denote the point where sunst+1 intersects u-axis (if any) as (u ∗, 0). to indicate the dependence of u∗ on q we will use the notation u∗ = u∗(q). we will write u∗(q) = u3 in case 2, and u ∗(q) = ∞ in case 3. we will also write u∗(q) < ∞ in cases 1 and 2. see figure 4.1 for an illustration of the three cases. recall that the portion of the curve located in the first quadrant is the graph of a smooth function of u, which we denote v = v(u; q). it is defined for u1 ≤ u ≤ u∗(q) (or u ≥ u1 in case (3)) and is positive and satisfies the differential equation (3.5) for u1 < u < u ∗(q) (or for u > u1 in case (3)), and we have v(u1; q) = v(u ∗; q) = 0 (or just v(u1; q) = 0 in case (3)). the following lemma shows monotonicity of v(u; q) and u∗(q) with respect to q. lemma 4.1. suppose 0 ≤ q1 < q2 and u∗(q2) < ∞. then also u∗(q1) < ∞, u∗(q1) < u∗(q2), and for any u1 < u < u ∗(q1) we have v(u; q1) < v(u; q2). proof. by lemma 3.1, the slope of v(u; q) at u = u1 equals m(q) = q+ √ q2−φ′(u1) 2 , which is a strictly increasing funciton of q. thus, m(q1) < m(q2). hence, lim u→u+1 v(u; q1) −v(u1; q1) u−u1 = lim u→u+1 v(u; q1) u−u1 = m(q1) < m(q2) = lim u→u+1 v(u; q2) −v(u1; q2) u−u1 = lim u→u+1 v(u; q2) u−u1 . then for some ε > 0, we have v(u; q1) < v(u; q2) for all u1 < u < u1 + ε. next, we will show that this inequality holds for all values u > u1 for which both curves are in the first quadrant of the uv-plane, i.e. for all u1 < u < min(u ∗(q1),u ∗(q2)). suppose that for some u1 < ũ < min(u ∗(q1),u ∗(q2)) we have v(ũ; q1) = v(ũ; q2) (> 0). we may assume that ũ is the smallest such value. then for all u1 < u < ũ we have 0 < v(u; q1) < v(u; q2), 248 a. anderson and o. vasilyeva which implies that d du v(ũ; q1) ≥ dduv(ũ; q2). but d du v(ũ; q1) = q1 − φ(ũ) v(ũ; q1) = q1 − φ(ũ) v(ũ; q2) < q2 − φ(ũ) v(ũ; q2) = d du v(ũ; q2), a contradiction. thus, for all u1 < u < min(u ∗(q1),u ∗(q2)) we have (0 <)v(u; q1) < v(u; q2). then we must have u∗(q1) ≤ u∗(q2). we claim that the inequality is strict. indeed, suppose u∗(q1) = u∗(q2). then for some u2 < u < u ∗(q1) close to u ∗(q1) we have d du v(u; q1) > d du v(u; q2). since u > u2, we have φ(u) > 0. then d du v(u; q1) = q1 − φ(u) v(u; q1) < q2 − φ(u) v(u; q1) < q2 − φ(u) v(u; q2) = d du v(u; q2), a contradiction. thus, u∗(q1) < u ∗(q2), and for any u1 < u < u ∗(q1) we have v(u; q1) < v(u; q2), as needed. � lemma 4.2. for any u1 < u < u2, v(u; q) is a continuous function of q ≥ 0. proof. first, recall that φ(u) < 0 for all u1 < u < u2 let 0 ≤ q1 < q2 and u1 < u < u2. by lemma 4.1, 0 < v(u; q1) < v(u; q2). thus, d du (v(u; q2) −v(u; q1)) = q2 −q1 + φ(u) ( 1 v(u; q1) − 1 v(u; q2) ) < q2 −q1. by the mean value theorem applied to the function f(u) = v(u; q2) − v(u; q1) on the interval [u1,u], we have: 0 < v(u; q2) −v(u; q1) < (q2 −q1)(u−u1). it follows that for any q0 ≥ 0 we have lim q→q0 v(u; q) = v(u; q0), as needed. � the next lemma shows that for sufficiently large values of q, case 3 takes place (i.e. u∗(q) = ∞), and moreover, v(u; q) is an increasing function of u ≥ u1. lemma 4.3. let φ∗ = maxu2≤u≤u3 φ(u) and q̄ = √ φ∗ u2−u1 . then (i) for any q ≥ 0, v(u2; q) > q(u2 −u1); (ii) for any q ≥ q̄, v(u; q) is a nondecreasing function of u ≥ u1; (iii) for any q ≥ q̄ we have u∗(q) = ∞, and the system (3.3) has no outbreak orbits. proof. (i) first, note that for any q ≥ 0 and u1 < u < u2 we have φ(u) < 0 and v(u; q) > 0, and therefore d du v(u; q) = q − φ(u) v > q. let us define v(u1; q) = limu→u+1 v(u; q) = 0. then by the mean value theorem, for some u1 < ξ < u2, we have v(u2; q) = v(u2; q) −v(u1; q) = d du v(ξ; q)(u2 −u1) > q(u2 −u1). (ii) suppose q ≥ q̄. let g(u,v) = q − φ(u) v . then g(u,v) ≥ 0 for any (u,v) in the quadrant u ≥ u2, v ≥ q(u2 −u1). prevailing winds and spruce budworm outbreaks: an rda model 249 indeed, for any u ≥ u2, we have φ(u) ≤ φ∗ (note that φ∗ > 0 and φ(u) < 0 for u > u3). thus, for u ≥ u2, v ≥ q(u2 −u1) we have g(u,v) = q − φ(u) v ≥ q − φ∗ v ≥ q − φ∗ q(u2 −u1) = 1 q ( q2 − q̄2 ) ≥ 0, as needed. by (i), v(u; q) > q(u2 − u1) for u = u2. next, we wil show that this inequality holds for all u ≥ u2. if not, let û > u2 be the smallest u > u2 such that v(û; q) = q(u2 − u1). then, by the mean value theorem, there exists u2 < ξ < û such that d du v(ξ; q) < 0. but (ξ,v(ξ; q)) belongs to the quadrant u ≥ u2, v ≥ q(u2−u1), and, therefore, dduv(ξ; q) = g(ξ,v(ξ; q)) ≥ 0, a contradiction. thus, for any u ≥ u2, we have v(u; q) > q(u2 −u1) and hence dduv(u; q) = g(u,v(u; q)) ≥ 0 (in fact, ≥ 1 q (q2 − q̄2)). combining the above with the fact that d du v(u; q) > q > 0 for all u1 < u < u2, as established int he proof of (i), we conclude that the function v(u; q) is nondecreasing for all u > u1. (iii) clearly, v(u; q) > 0 for any q ≥ q̄ and u > u1, i.e. u∗(q) = ∞. therefore, the boundary value problem (3.3, 3.4) has no solution whose orbit crosses the u-axis to the right from u1. hence, there is no outbreak orbits for q ≥ q̄. � let γ = {q ≥ 0 : u∗(q) < ∞}. thus, γ is the set of all values of q for which either case 1 or case 2 occurs. by lemma 4.1, γ is an interval of the real line containing zero as its left endpoint. lemma 4.3 shows that q̄ 6∈ γ and q̄ therefore serves as an upper bound for γ. let us define q∗cr = sup γ. clearly, q∗cr ≤ q̄ = √ φ∗ u2−u1 . the following proposition clarifies the role of q∗cr as a critical value of q for existence of outbreak orbits. proposition 4.4. (i) for any 0 ≤ q < q∗cr we have u∗(q) < u3. (ii) for any q > q∗cr we have u ∗(q) = ∞. (iii) u∗(q∗cr) = u3 (i.e. for q = q ∗ cr, s unst+ 1 becomes a heteroclinic connection between (u1, 0) and (u3, 0) and coincides with s st+3). (iv) limq→q∗−cr u ∗(q) = sup u∗(γ) = u3. proof. (i) suppose 0 ≤ q < q∗cr . let q̃ = q+q∗cr 2 . then q < q̃ < q∗cr and q, q̃ ∈ γ. by lemma 4.1, u∗(q) < u∗(q̃) ≤ u3, as needed. (ii) follows by noticing that q > q∗cr = sup γ implies q 6∈ γ. (iii) let ū be any value between u1 and u2, e.g. ū = u1+u2 2 . for any q ≥ 0, let (u(x; µ,ν,q),v (x; µ,ν,q)), x ≥ 0, be the solution (u(x),v(x)) of the system (3.3) such that (u(0),v(0)) = (µ,ν). then by global continuity with respect to the initial conditions and parameters, for any fixed x > 0, (u(x; µ,ν,q),v (x; µ,ν,q)) is continuous with respect to (µ,ν,q). let (uq(x),vq(x)) = (u(x; ū,v(ū; q),q),v (x; ū,v(ū; q),q)), which defines the portion of sunst+1 starting at (ū,v(ū; q)) for the given value of q. by lemma 4.2, v(ū; q) is a continuous function of q. then for any fixed x = l > 0, (uq(l),vq(l)) is continuous with respect to q. to show u∗(q∗cr) = u3 we need to eliminate the two other cases: u ∗(q∗cr) < u3 and u ∗(q∗cr) = ∞. case (1): suppose u∗(q∗cr) < u3. then for q = q ∗ cr, the orbit s unst+ 1 crosses into the fourth quadrant, and therefore, for some l > 0 we will have vq∗cr (l) < 0. let ε = −vq∗cr (l) > 0. for any q > q ∗ cr, we have q 6∈ γ, and thus, the orbit sunst+1 stays in the first quadrant. therefore, for any x > 0 we will 250 a. anderson and o. vasilyeva have vq(x) > 0. in particular, for any q > q ∗ cr, we have ‖(uq(l),vq(l)) − (uq∗cr (l),vq∗cr (l))‖≥ ε, contradicting the continuity of (uq(l),vq(l)) at q = q ∗ cr. case (2): suppose u∗(q∗cr) = ∞. then for q = q∗cr, the orbit s unst+ 1 stays in the first quadrant and eventually crosses the vertical line u = u3. thus, for some l > 0, we have uq∗cr (l) > u3. let ε = uq∗cr (l) −u3 > 0. for any q < q ∗ cr, we have q ∈ γ, and thus, the orbit s unst+ 1 stays to the left from the line u = u3. therefore, for any x > 0 we will have uq(x) < u3. in particular, for any q < q ∗ cr, we have ‖(uq(l),vq(l)) − (uq∗cr (l),vq∗cr (l))‖≥ ε, contradicting the continuity of (uq(l),vq(l)) at q = q ∗ cr. (iv) in this proof we will use the notation (uq(x),vq(x)) as defined in the proof of (iii). note that by monotonicity of u∗(q) (lemma 4.1) and the fact that u∗(q) ≤ u3 for all q ∈ γ, the set u∗(γ) is bounded above by u3 and limq→q∗−cr u ∗(q) = sup u∗(γ) ≤ u3. suppose that sup u∗(γ) < u3. let ε = u3−sup u∗(γ) 2 . note that ε > 0. by (iii), we have uq∗cr (x) → u3 as x →∞. thus, for some l > 0, we have uq∗cr (l) > u3 −ε. in particular, for any q < q ∗ cr, we have ‖(uq(l),vq(l)) − (uq∗cr (l),vq∗cr (l))‖≥ ε, and as in (iii), it contradicts the continuity of (uq(l),vq(l)) at q = q ∗ cr. thus, sup u ∗(γ) = u3, as needed. � 4.2. lower bound for critical advection for outbreaks. we will now work towards obtaining a lower bound for q∗cr. recall that v(u; q) satisfies the differential equation (3.5) for u1 < u < u ∗(q) and v(u1; q) = v(u ∗(q); q) = 0 (when q ≤ q∗cr). multiplying both sides of (3.5) by v(u; q) and integrating from u1 to u ∗(q) gives: ∫ u∗(q) u1 (qv(u; q) −φ(u))du = ∫ u∗(q) u1 v(u; q) dv du du = ∫ v(u∗(q);q) v(u1;q) vdv = (v(u∗(q); q))2 2 − (v(u1; q)) 2 2 = 0 − 0 = 0. therfore, we obtain the following: q ∫ u∗(q) u1 v(u; q)du = ∫ u∗(q) u1 φ(u)du. since, ∫u∗(q) u1 v(u; q)du > 0, we get q = ∫u∗(q) u1 φ(u)du∫u∗(q) u1 v(u; q)du . (4.1) note that φ(u) < 0 for u1 < u < u2 and φ(u) > 0 for u2 < u < u3. let a1 = − ∫u2 u1 φ(u)du, a∗ =∫u∗(q) u2 φ(u)du and a2 = ∫u3 u2 φ(u)du. then a1,a ∗,a2 > 0 and a ∗ ≤ a2. notice that ∫u∗(q) u1 φ(u)du = a∗ − a1 and ∫u3 u1 φ(u)du = a2 − a1. recall that for q = 0, outbreak orbits exist exactly when u∗(0) < u3. this is also equivalent to existence of outbreak orbits for some q ≥ 0, or for all sufficiently small q ≥ 0. since the existence of an outbreak orbit for q = 0 is equivalent to f(u1) < f(u3), and f(u3)−f(u1) = a2 −a1, an outbreak orbit exists for q = 0 (or for sufficiently small q ≥ 0) if and only if a1 < a2. below we give an alternative proof of this fact based on the phase plane argument. prevailing winds and spruce budworm outbreaks: an rda model 251 proposition 4.5. suppose φ(u) has three positive zeros. then outbreak orbits exist for sufficiently small q ≥ 0 if and only if a2 > a1. proof. suppose that for some q ≥ 0, an outbreak solution exists (case 1 holds). then u2 < u∗(q) < u3, and by (4.1) 0 ≤ q = a∗ −a1∫u∗(q) u1 v(u; q)du , and thus, a∗ ≥ a1. note that since u∗(q) < u3, we have a2 > a∗. therefore a2 > a1. suppose that for q = 0, an outbreak solution does not exist (case 2 or case 3 holds), but a2 > a1. then u∗(0) = u3 or u ∗(0) = ∞. we have: 0 > a1 −a2 = − ∫ u3 u1 φ(u)du = ∫ u3 u1 v dv du du = (v(u3)) 2 2 − (v(u1)) 2 2 = (v(u3)) 2 2 − 0 = (v(u3)) 2 2 , a contradiction. thus, an outbreak solution exists. � figure 4.2 shows the phase plane of system (3.3) for q = 6.5 and q = 0 and three different values of β: (a) r = 0.588,a1 < a2, outbreak solutions exist (here r̃(q) < r < r1(q), i.e. we are in ω ∗ or region iv of the qr-plane); (b) r = 0.574,a1 = a2, no outbreak solution is possible (here r = r̃(q), which corresponds to the lower boundary of ω∗); (c) r = 0.571,a1 > a2, no outbreak solution is possible (here r2(q) < r < r̃(q), which corresponds to region iii). recall that limq→q∗cr− u ∗(q) = u3. passing to the limit as q → q∗cr − in (4.1), we get q∗cr = ∫u3 u1 φ(u)du∫u3 u1 v(u; q∗cr)du . (4.2) note that the denominator of the fraction in (4.2) represents the area of the region in the uv-plane bounded above by the curve v = v(u; q∗cr), which is the orbit s unst+ 1 = s st+ 3 , where u1 ≤ u ≤ u3, and the u-axis. we will denote this orbit by s13. our next objective is to find an upper bound for this area. let umin ∈ [u1,u2] be such that φ′(umin) is the minimal value of φ′ on [u1,u2]. if φ is concave up on [u1,u2], then umin = u1, otherwise it is the inflection point 0 < û1 < 1 of φ(·) (see lemma 2.1). the proof of the following proposition amounts to constructing a triangular “trapping region” enclosing the orbit s13 and having two of its vertices at the saddle points (u1, 0) and (u3, 0). proposition 4.6. for q ≥ 0, let a(q) = ( √ q2 − 4φ′(umin) + q)( √ q2 − 4φ′(u3) −q)(u3 −u1)2 4( √ q2 − 4φ′(umin) + √ q2 − 4φ′(u3)) . then ∫u3 u1 v(u; q∗cr)du < a(q ∗ cr). proof. see appendix a. � using the above proposition and the fact that ∫u3 u1 φ(u)du > 0, we obtain: q∗cr = ∫u3 u1 φ(u)du∫u3 u1 v(u,q∗cr)du > ∫u3 u1 φ(u)du a(q∗cr) . 252 a. anderson and o. vasilyeva figure 4.2. the phase plane of system (3.3) for q = 6.5,q = 0 and r = 0.588 (r̃(q) < r < r1(q)),r = 0.574 (r = r̃(q)), r = 0.571 (r2(q) < r < r̃(q)). graphs of φ(u) with the areas a1 and a2 indicated are shown on the right. note that a(q) is well-defined, continuous and positive for all q ≥ 0. for q ≥ 0, let g(q) = q − ∫u3 u1 φ(u)du a(q) , prevailing winds and spruce budworm outbreaks: an rda model 253 which is a continuous function for q ≥ 0. then g(0) = − ∫u3 u1 φ(u)du a(0) < 0, and g(q∗cr) > 0. let q > 0 be the smallest positive zero of g(q). then 0 < q < q ∗ cr, that is, q serves as a lower bound for q∗cr. 5. geometric analysis of endemic and outbreak orbits and steady state solutions in the first part of this section, we will use phase plane analysis to study conditions for existence of endemic and outbreak orbits. then we will discuss the dependence of habitat length on the maximal density of a steady state solution. 5.1. conditions for existence of endemic and outbreak orbits. first, we will show that for q ≥ 2 there is no positive solution (i.e. satisfying u(x) > 0 for − l 2 < x < l 2 ) to the boundary value problem (3.3, 3.4) for any l > 0, i.e. (3.3) has no endemic or outbreak orbits. proposition 5.1. suppose q ≥ 2 and l > 0. then (3.3, 3.4) has no positive solution. in particular, (3.3) has no endemic or outbreak orbits. proof. note that a positive solution of (3.3,3.4) satisfies v ( − l 2 ) > 0 and v (x∗) = 0 for some − l 2 < x∗ < l 2 . consider the half-line v = u, u > 0 in the uv-plane. to establish that there is no positive solution of (3.3,3.4) it suffices to show that the slope of vector field of the system (3.3) when restricted to this half-line is greater than 1. that is, qu−φ(u) u > 1 for all u > 0. we have φ(u) u = 1 − u q − 1 r · u u2 + 1 < 1 ≤ q − 1, for all u > 0, hence qu−φ(u) u = q − φ(u) u > q −q + 1 = 1, as needed. � the next lemma demonstrates that there is a horizontal line such that once an orbit of (3.3) passes above this line it will stay above it. recall that φ(u) < 0 outside the interval [0,u3], and thus φ(u) reaches its maximal value on r at some 0 ≤ u ≤ u3. lemma 5.2. let q > 0 and m = 1 q max0≤u≤u3 φ(u). then for any solution (u(x),v(x)) of (3.3), if for some x∗ we have v(x∗) > m, then v(x) > m for all x > x∗. proof. suppose for some x̂ > x∗ we have v(x̂) ≤ m. by the intermediate value theorem, there is x∗ < x̃ ≤ x̂ such that v(x̃) = m. we may assume that for all x∗ < x < x̃, v(x) > m. by the mean value theorem, there exists x∗ < ξ < x̃ such that v′(ξ) < 0. but v(ξ) > m, and thus v′(ξ) = qv(ξ) −φ(u(ξ)) > qm − max 0≤u≤u3 φ(u) = 0, a contradiction. � we will now show the existence of an orbit located in the first quadrant of the uv-plane, connecting a point on the positive v-semiaxis and the saddle point (u1, 0) (in fact, a portion of s st+ 1 ). 254 a. anderson and o. vasilyeva figure 5.1. region w used in the proof of lemma 5.3. lemma 5.3. let 0 < q < 2. then there exists a solution (u(x),v(x)), x ≥ 0, of (3.3) such that lim x→∞ (u(x),v(x)) = (u1, 0), (u(0),v(0)) = (0,v1) where 0 < v1 ≤ m, and (u(x),v(x)) ∈ (0,u1) × (0,m) for all x > 0. proof. let (u(x),v(x)), x ∈ r, be the solution of (3.3) such that lim x→∞ (u(x),v(x)) = (u1, 0) and v(x) > 0 as x → ∞. thus, it represents the upper portion of the stable manifold of the saddle point (u1, 0). note that u(x) is a non-decreasing function of x as long as v(x) ≥ 0. consider the region w = [0,u1] × [0,m] of the uv-plane (see figure 5.1). note that as x → ∞, (u(x),v(x)) ∈ (0,u1) × (0,m) (the interior of w). since the only other equilibrium point in w is (0, 0), as we decrease x, the only two possibilities are: (1) (u(x),v(x)) remains in w and approaches (0, 0) as x →−∞; (2) (u(x),v(x)) leaves w through (2a) the upper boundary of w (v = m, 0 < u < u1), (2b) the lower boundary of w (v = 0, 0 < u < u1), (2c) or the left boundary of w (u = 0, 0 < v ≤ m). the case (1) is not possible since (0, 0) is an unstable spiral, so there is no solution that approaches (0, 0) as x →−∞ entirely in the first quadrant. if (2a) holds then there is x∗ ∈ r such that v(x∗) = m and for some ε > 0, v(x) > m for all x∗ − ε < x < x∗, which contradicts lemma 5.2. the case (2b) contradicts the fact that dv dx = −φ(u(x)) < 0 on the lower boundary. thus, (2c) holds, i.e. there exists x∗ ∈ r such that (u(x∗),v(x∗)) = (0,v1) where 0 < v1 ≤ m, and for all x > x∗ we have (u(x),v(x)) ∈ w . shifting by −x∗ gives us x∗ = 0, as needed. � lemma 5.4. let 0 < q < 2 and suppose u∗(q) < u3 (case 1). then there exists a solution (u(x),v(x)), x ≥ 0, of (3.3) such that lim x→∞ (u(x),v(x)) = (u3, 0), (u(0),v(0)) = (0,v3) where v1 < v3 ≤ m, and (u(x),v(x)) ∈ (0,u3) × (0,m) for all x > 0. proof. consider the portions of the stable and unstable manifolds of the saddle point (u1, 0) of (3.3) located in the first quadrant (i.e. the first quadrant portions of the curves sst+1 and s usnt+ 1 ). note that both curves can be viewed as graphs of continuous functions of u, namely, v = vl(u) defined for prevailing winds and spruce budworm outbreaks: an rda model 255 figure 5.2. region w ′ = w1 ∪w2 ∪w3 used in the proof of lemma 5.4. 0 ≤ u ≤ u1 (by lemma 5.3) and v = vr(u) defined for u1 ≤ u ≤ u∗(q). by lemma 5.3, vl(u) > 0 for all 0 ≤ u < u1, and vl(u1) = 0. we also have vr(u1) = vr(u∗(q)) = 0 and vr(u) > 0 for u1 < u < u∗(q). by lemma 5.2, vl(u),vr(u) ≤ m throughout their respective domains. consider the following region of the uv-plane: w ′ = w1 ∪w2 ∪w3, where w1 = {(u,v) : 0 ≤ u < u1,vl(u) ≤ v ≤ m}, w2 = {(u,v) : u1 ≤ u < u∗(q),vr(u) ≤ v ≤ m} and w3 = [u ∗(q),u3] × [0,m]. note that w ′ is a closed simply connected plane region (see figure 5.2). let (u(x),v(x)), x ∈ r, be the solution of (3.3) such that limx→∞(u(x),v(x)) = (u3, 0) and v(x) > 0 as x → ∞ (it represents part of the upper portion of the stable manifold sst+3 of the saddle point (u3, 0)). recall that u(x) is a non-decreasing function of x as long as v(x) ≥ 0. note that as x → ∞, (u(x),v(x)) ∈ (u∗(q),u3) × (0,m) (the interior of w3). since the only other equilibrium point in w ′ is (u1, 0), as we decrease x, the only two possibilities are: (1) (u(x),v(x)) remains in w ′ and approaches (u1, 0) as x →−∞; (2) (u(x),v(x)) leaves w through (2a) the upper boundary of w ′ (v = m, 0 < u < u3), (2b) the lower boundary of w ′ (v = vl(u), 0 ≤ u < u1, or v = vr(u), u1 ≤ u ≤ u∗(q), or v = 0, u∗(q) < u < u3), (2c) or the left boundary of w ′ (u = 0,v1 < v ≤ m). the case (1) is not possible since (u1, 0) is a saddle point having v = v r(u) as its unstable manifold in w ′ (thus, no other orbit in w ′ can approach it as x →−∞). the case (2a) impossible for the same reason as in the proof of lemma 5.3. the case (2b) is impossible due to the uniqueness of the solution of an initial value problem (for lower boundaries of w1 or w2), or due to the fact that dv dx = −φ(u(x)) < 0 on the lower boundary of w3 (u ∗(q) < u < u3,v = 0). thus, (2c) holds, i.e. there exists x ∗ ∈ r such that (u(x∗),v(x∗)) = (0,v3) where v1 < v3 ≤ m, and for all x > x∗ we have (u(x),v(x)) ∈ w ′. shifting by −x∗, we may assume that x∗ = 0, as needed. 256 a. anderson and o. vasilyeva figure 5.3. region tim used to “capture” the curve s unst− i . � in order to establish the existence of endemic and outbreak orbits, we will now turn our attention to the fourth quadrant of the uv-plane. let i = 1 or 3. recall that (ui, 0) is a saddle point, and by lemma 3.1 the slope of sunst−i (the lower branch of its unstable manifold) is given by q+ √ q2−4φ′(ui) 2 . our goal is to find a closed triangular region tim = {(u,v) : 0 ≤ u ≤ ui, m(u−ui) ≤ v ≤ 0} bounded by positive u-semiaxis, negative v-semiaxis and the line through (ui, 0) with a positive slope m, such that the portion of sunst−i in the fourth quadrant will lie entirely in that region (see figure 5.3). to guarantee that the small portion of this curve near the saddle point is located inside this region, we will need m > q+ √ q2−4φ′(ui) 2 . to guarantee that sunst−i does not cross the hypothenuse of t i m, we will require that the vector field of the system (3.3), when restricted to the line segment v = m(u−ui), 0 < u < ui, forms an acute angle with the normal vector (−m, 1) pointing towards the origin. thus, the condition is (−m, 1) ·(v,qv−φ(u)) > 0 where v = m(u−ui) and 0 < u < ui. equivalently, we need m(u−ui)(q −m) −φ(u) > 0, or m2 −mq > − φ(u) u−ui . this inequality has to hold for all 0 < u < ui. for i = 1 or 3, define a function hi : [0,ui] → r as follows hi(u) = { − φ(u) u−ui , u < ui −φ′(ui), u = ui. prevailing winds and spruce budworm outbreaks: an rda model 257 clearly, hi is continuous on [0,ui], and thus, it attains its maximal value, h i max(> 0) in [0,ui]. note that by the definition of himax, we have − φ(u) u−ui ≤ himax for all 0 < u < ui. hence, it suffices to require m2 −mq −himax > 0. let mi be the positive root of the quadratic equation m 2 −mq −himax = 0, i.e. mi = q + √ q2 + 4himax 2 . clearly, m2 −mq −himax > 0 holds for m > mi. we are now ready to analyze the behavior of the curves sunst−1 , s unst+ 1 and s unst− 3 in the fourth quadrant of the uv-plane. lemma 5.5. suppose 0 ≤ q < 2. the fourth quadrant portion of the curve sunst−1 is contained in the region t 1m1 . it connects its α-limit (u1, 0) with a point (0,v − 1 ) where −m1u1 < v − 1 < 0. proof. since t 1m1 = ⋂ m>m1 t 1m, it suffices to prove the first statement for all m > m1. we have m > m1 ≥ q+ √ q2−4φ′(u1) 2 . hence, near the point (u1, 0), the curve s unst− 1 stays above the line v = m(u−u1) (as its tangent at (u1, 0) has slope < m). since m > m1, by the vector field arguments preceding this lemma, the curve cannot cross this line at any point (u,m(u−u1)) where 0 < u < u1. therefore the curve is contained in t 1m for any m > m1, and thus, also in t 1 m1 . the second statement follows easily by noticing that sunst−1 cannot approach the origin (an unstable equilibrium) and cannot cross the u-axis (as the vector field is directed downward on the u-axis between u = 0 and u = u1). � for q ≥ 0, let sst+1 (q) and s st− 1 (q) denote the curves s st+ 1 and s st− 1 for this particular value of q. lemma 5.6. suppose 0 < q < 2. the fourth quadrant portion of the curve sunst+1 (q) is contained in the region t 3m3\d, where d is the region bounded by the curve s unst− 1 (q), the positive u-semiaxis and negative v-semiaxis. the curve sunst+1 connects (u ∗(q), 0) with the point (0,v∗) where −m3u3 < v∗ < v−1 . proof. as in the previous lemma, it suffices to prove the first statement for t 3m for any m > m3. let (u(x),v(x)),x ∈ r be a solution of (3.3) representing sunst+1 (q). thus, limx→−∞(u(x),v(x)) = (u1, 0) and for some x∗, we have (u(x∗),v(x∗)) = (u∗(q), 0). as we increase x > x∗, u(x) will decrease and, by poincare-bendixson theorem, the point (u(x),v(x)) either leaves the region t 3m or approaches an equilibrium point in t 3m (note that by the bendixson-dulac criterion, the system (3.3) has no closed orbits for q > 0, as the vector field has positive divergence q). recall that the vector field of the system (3.3) points downward on the u-axis for 0 < u < u1 and u2 < u < u3. note that s st− 1 (0) is the lower portion of the right-side homoclinic orbit of (u1, 0) (for q = 0), it connects the point (u∗(0), 0) to (u1, 0) in the fourth quadrant, and is a graph of a function of u. recall that for any q ≥ 0 and point (u,v) where v 6= 0, the slope of the solution curve of (3.3) passing through (u,v) is given by dv du = q − φ(u) v . then for any q > 0, the vector field of (3.3) on the curve sst−1 (0) will point outside the region above the curve. in addition, by lemma 3.1, the slope of the curve sst−1 (q) at (u1, 0) is given by m(q) = q − √ q2 −φ′(u1) 2 , and for q > 0, m(q) > m(0). 258 a. anderson and o. vasilyeva figure 5.4. regions d and t 3m3\d from the proof of lemma 5.6. thus, we conclude that (u(x),v(x)) cannot cross the u-axis again for any x > x∗, and cannot approach (u1, 0) as x → ∞. clearly, it cannot approach (u2, 0) either. note that sunst+1 (q) cannot cross the curve sunst−1 (0) or the segment v = m(u−u3), 0 < u < u3 (by the vector field argument). since there are no more equilibria in the region t 3m\d, the only possibility for sunst+1 (q) is to leave the region t 3 m\d (and thus, the fourth quadrant) through the v-axis between v = v−1 and −m3u3. this proves the lemma (see figure 5.4 for an illustration). � lemma 5.7. suppose 0 < q < 2. the fourth quadrant portion of the curve sunst−3 is contained in the region t 3m3\e, where e is the region bounded by the curve s unst+ 1 , the positive u-semiaxis and negative v-semiaxis. it connects its α-limit (u3, 0) with a point (0,v − 3 ) where −m3u3 < v − 3 < v ∗. proof. the proof is similar to the proof of the previous two lemmas. � see figure 5.5 for an illustration of lemma 5.7. remark 5.8. note that when q = 0, we have sunst+1 = s st− 1 forming a loop symmetric with respect to the u-axis. note also that lemma 5.5 includes the case q = 0, and lemma 5.7 can be modified to include the case q = 0. moreover, because of the symmetry of the phase plane about the u-axis, this gives us the existence of orbits described in lemmas 5.3 and 5.4 for the non-advective case, which agrees with the analysis of the steady state solutions in the non-advective case done in [11]. we are now ready to state our main results. theorem 5.9. suppose (q,r) ∈ ω∗. let u1 < u2 < u3 be the three positive zeros of φ(u). (i) suppose 0 < q < min(2,q∗cr). then there exist v − 3 < v ∗ < v−1 < 0 < v1 < v3 with the following properties: prevailing winds and spruce budworm outbreaks: an rda model 259 figure 5.5. regions e and t 3m3\e from the proof of lemma 5.7. (1) sst+1 connects (0,v1) with (u1, 0) (in the first quadrant); (2) sunst−1 connects (u1, 0) with (0,v − 1 ) (in the fourth quadrant); (3) sunst+1 connects (u1, 0) with (0,v ∗), passing through the first and the fourth quadrants and crossing the u-axis at the point (u∗(q), 0) where u2 < u ∗(q) < u3; (4) sst+3 connects (0,v3) with (u3, 0), passing above s st+ 1 and s unst+ 1 (in the first quadrant); (5) sunst−3 connects (u3, 0) with (0,v − 3 ), passing below s unst+ 1 (in the fourth quadrant); (6) for any 0 < µ < u1 there exists l = l(µ; q) > 0 and a solution (u(−),v(−)) : [ − l 2 , l 2 ] → r2 of (3.3, 3.4) such that u(x) > 0 for all x ∈ ( − l 2 , l 2 ) , 0 < v ( − l 2 ) < v1 and v − 1 < v ( l 2 ) < 0, and (u(x∗),v(x∗)) = (µ, 0) for some x∗ ∈ ( − l 2 , l 2 ) (endemic orbits); (7) for any u∗(q) < µ < u3 there exists l = l(µ; q) > 0 and a solution (u(−),v(−)) : [ − l 2 , l 2 ] → r2 of (3.3, 3.4) such that u(x) > 0 for all x ∈ ( − l 2 , l 2 ) , v1 < v ( − l 2 ) < v3 and v − 3 < v ( l 2 ) < v∗, and (u(x∗),v(x∗)) = (µ, 0) for some x∗ ∈ ( − l 2 , l 2 ) (outbreak orbits); 260 a. anderson and o. vasilyeva figure 5.6. phase plane in the case when both endemic and outbreak orbits exist. (8) there are no other positive solutions of (3.3, 3.4) for any l > 0; (9) endemic orbits provide monotone bijections between intervals (0,v1) and (v − 1 , 0) of the v-axis via the interval (0,u1) of the u-axis; (10) outbreak orbits provide monotone bijections between intervals (v1,v3) and (v − 3 ,v ∗) of the v-axis via the interval (u∗(q),u3) of the u-axis. (ii) suppose q∗cr < 2, and q ∗ cr < q < 2. then there exist v − 1 < 0 < v1 such that (i)(1,2,6,8,9) hold. in this case, sunst+1 lies entirely in the first quadrant and passes above the point (u3, 0), and the system does not admit outbreak orbits. (iii) suppose q ≥ 2. then (3.3, 3.4) has no positive solutions for any l > 0 (no endemic or outbreak orbits). proof. parts (i)(1-5) follow from lemmas 5.3, 5.5, 5.6 and 5.7. parts (i)(6-10) follow by noticing the direction of the vector field of (3.3) on both u and v-axes, and the fact that the stable and unstable manifolds form “trapping regions” in the uv-plane. see figure 5.6. the proof of part (ii) is similar to that of part (i). see figure 5.7. part (iii) is given by proposition 5.1. � remark 5.10. for (q,r) is region iii (that is, ω\ω∗, see figure 2.4), the function φ(u) still has three positive roots, and for q ≥ 0, the unstable manifold sunst+1 remains in the first quadrant of the prevailing winds and spruce budworm outbreaks: an rda model 261 figure 5.7. phase plane in the case (ii) of theorem 5.9 (when only endemic orbits exist). figure 5.8. phase plane when (q,r) 6∈ ω and q < 2. uv-plane. thus, there are no outbreak orbits in this case. the equilibrium (u1, 0) remains a saddle point, and the phase portrait is similar to the one in figure 5.7. therefore, the same argument as the one used in the proof of theorem 5.9(ii) shows that for q < 2, there are endemic orbits with maximal density given by values 0 < µ < u1. for q ≥ 2, the conclusion of theorem 5.9(iii) holds as well. when (q,r) is outside of ω (that is, in regions i, ii, v or vi ), the function φ(u) has only one positive zero, we will still denote it u1, and we have φ(u) > 0 for 0 < u < u1 and φ(u) < 0 for u > u1. in the uv-plane, the point (u1, 0) is still a saddle point (for any q ≥ 0). in this case, the situation is again completely analogous to theorem 5.9(ii, iii). see figure 5.8 for a typical orbit when q < 2. 5.2. habitat length vs. maximal density. now that we have established conditions for existence of orbits representing positive solutions of (3.3, 3.4) for some l > 0, we can focus on the function l = l(µ; q) 262 a. anderson and o. vasilyeva that provides the length of the habitat corresponding to such an orbit passing through the point (µ, 0) on the u-axis; we can refer to l(µ; q) as the parametric length of the orbit. recall, that the function l(µ; q) is defined for 0 ≤ q < 2 and 0 < µ < u1 or, in the case the conditions of theorem 5.9(i) are satisfied (i.e. (q,r) ∈ ω∗ and q < q∗cr), u∗(q) < µ < u3. recall from section 2, that in the non-advective case, there is an explicit integral formula for l(µ; 0) derived in [11], however, the first integral method does not apply in the advective case. we can compute l(µ; q) as follows. let (uµ(x),vµ(x)) be the solution of the initial value problem consisiting of the system (3.3) and the condition (uµ(0),vµ(0)) = (µ, 0), defined on r. let l+(µ; q) be the smallest positive solution of uµ(−x) = 0 (the parametric length of the upper part of the orbit) and l−(µ; q) be the smallest positive solution of uµ(x) = 0 (the parametric length of the lower part of the orbit). then l(µ; q) = l+(µ; q) +l−(µ; q). note that (uµ(x),vµ(x)) is a continuous function of x and µ, and therefore, both l+(µ; q) and l−(µ; q) are continuous functions of µ (as solutions of uµ(−x) = 0 and uµ(x) = 0). thus, l(µ; q) is also a continuous function of µ. in the non-advective case, the behavior of l(µ; 0) was described in [11], and is summarized in section 2 according to the choice of (q,r) is one of the five regions of the qr-plane (see figure 2.4). as we increase q, we expect that the general shape of l(µ; q) as a function of µ for q > 0 will remain of the same four types outlined in figure 2.5, but with some vertical and horizontal transformations and transitions from one type of shape to another. of the main interest to us is the behavior of the critical domain lengths for the endemic and outbreak solutions, lc1(q) and l c 2(q), respectively, as we vary q. recall that in all the cases, lc1 = limµ→0 l(µ; 0) = π. in the advective case, linearizing (3.1) at zero steady state, we can find lc1(q) = lim µ→0 l(µ; q) = 2π√ 4 −q2 . note that lc1(q) increases with q and lim q→2 lc1(q) = ∞. it has been shown in [11] that lc2(0) > l c 1(0). we conjecture that the same inequality holds for 0 ≤ q < min(q∗cr, 2), and moreover, q ∗ cr < 2. in the next section, we will use numerics to investigate the effect of advection on the behavior of l(µ; q) and the nature of steady state solutions. 6. numerical simulations in this section, we will use numerics to test our approximations of the critical advection q∗cr for existence of outbreak orbits. for different choices of (q,r) ∈ ω∗, we obtain the values of q∗cr and its lower and upper estimates q and q̄, respectively. we will also explore the dependence of the habitat length l(µ; q) on the maximal population density µ of a steady state solution for different value of advection speed q. we will perform these numerical simulations for (q,r) in regions ii, iv and v (see figure 2.4). 6.1. estimating critical advection. we test our lower and upper estimates for q∗cr as follows. to obtain the lower estimate q, recall that q is the smallest positive root of the equation g(q) = 0, where g(q) = q − ∫ u3 u1 φ(u)du a(q) and a(q) is given by (a.1). recall that he upper estimate q̄ is given by q̄ = √ φ∗ u2 −u1 prevailing winds and spruce budworm outbreaks: an rda model 263 where φ∗ = maxu2≤u≤u3 φ(u). finally, we can approximate the actual value of q ∗ cr by using euler’s method to obtain the solution curve v = v(u; q) of the differential equation dv du = q − φ(u) v with the property lim u→u+1 v(u) = 0, which is the unstable manifold sunst+1 . since we know that the slope of s unst+ 1 at (u1, 0) is given by m = q+ √ q2−4φ′(u1) 2 (see lemma 3.1), we can use this information to perform the first step of the euler’s method to avoid division by zero. we determine whether v(u; q) intersects the u-axis (i.e. whether u∗(q) < ∞). increasing the advection q by small increments, we iterate until we reach the situation when u∗(q) = ∞ (the stopping criterion is when v(u; q) takes a value above the threshold m = 1 q max 0 ≤ u ≤ u3φ(u)). the following table summarizes the numerical results for estimating critical advection. all values of (q,r) are chosen in region iv (ω∗). q r q q∗cr q̄ 6.5 0.6 0.22 0.325 0.5507 10 0.5556 0.48 0.705 1.6122 10 0.5128 0.3 0.455 0.7454 15 0.5128 0.53 0.855 1.6031 15 0.4 0.3 0.475 0.7687 figures b.1 b.5 illustrate the lower and upper estimates for q∗cr for different choices of parameters. we observe that q∗cr < 2 for each choice of (q,r), and q < q ∗ cr < q̄. 6.2. dependence of habitat length on maximal density and advection. to investigate the effect of advection on the habitat length l = l(µ; q) we choose (q,r) = (15, 0.1) (region ii), (q,r) = (15, 0.5128) (region iv or ω∗), and (q,r) = (15, 0.7) (region v). in each case we consider three values of q: q = 0, 0.35, 0.7 for region iv (recall that in this case q∗cr = 0.855) and q = 0, 0.7, 1.4 for regions ii and v. in each case, we observe that l = l(µ; q) is monotonously increasing with respect to q. moreover, we make the following observations in each of the three examples. • (q,r) = (15, 0.1) (region ii, see figure b.6): as we increase q, the curve l = l(µ; q) (in the (µ,l)-plane) retains its monotonously increasing concave up shape, but moves upward. as a result, for a fixed l > π, we see that the maximal density µ of the unique positive steady state decreases, and for q > 2 √ 1 − π2 l2 only the the extinction steady state left. • (q,r) = (15, 0.5128) (region iv, see figure b.7): as we increase q, the left component of the curve l = l(µ; q), defined for 0 > µ < u1, retains its monotonously increasing concave up shape, but moves upward. the right component of the curve l = l(µ; q), defined for u∗(q) < µ < u3, retains its concave up shape with two vertical asymptotes. the vertical asymptote µ = u∗(q) is moving to the right as we increase q (thus reducing the domain of l = l(−; q)). the right component moves upward and is compressed horizontally. the minimal value lc2(q) of l = l(µ; q) on the interval (u∗(q),u3) is increasing accordingly. note that for q = q ∗ cr = 0.855, 264 a. anderson and o. vasilyeva the two vertical asymptotes collapse into one, and the right component of the curve l = l(µ; q) disappears. as a result, for a fixed l > π we see that the density of the endemic equilibrium is decreasing, and if l > lc2(q), the density of the threshold steady state is increasing and the density of the outbreak state is decreasing. at some value of q < q∗cr, the threshold and outbreak states are lost, and only endemic and extinction states are left. for q > 2 √ 1 − π2 l2 only the extinction steady state is left. in this example, lc2(q) > l c 1(q), as expected. • (q,r) = (15, 0.7) (region v, see figure b.8): we observe that for all values of q, the curve l = l(µ; q) is defined for 0 < µ < u1 and has a vertical asymptote at µ = u1. as in the previous cases, it moves upward as we increase q. for smaller values of q it still has two internal extrema: the local maximum value lc3(q) and the local minimum l c 2(q). however, for larger values of q, l(µ; q) becomes an increasing function of µ, similar to the situation in region vi. as a result, for a fixed π < l < lc2(0) we observe only an endemic state with its maximal density decreasing with q. for relatively small values of l > lc2(0), we can have the following scenarios as we increase q: (1) outbreak (large density) only with decreasing maximal density; (2) three states (endemic, threshold, outbreak) with endemic and outbreak maximal densities decreasing and threshold maximal density increasing; (3) endemic (small density) only, with decreasing maximal density; moreover, depending on l, the following transitions are possible as we increase q: (2 → 3), (1 → 2 → 3) or (1 → 3). as before, for q > 2 √ 1 − π2 l2 only the extinction steady state is left. 7. discussion in this paper we have considered an advective version of the classical non-dimensional reactiondiffusion model for population dynamics of spruce budworm introduced in [11]. biologically, the goal was to understand the possible effect of biased movement caused by prevailing winds on the spatial distribution and occurrence of sbw outbreaks. as in [11], we have used hostile boundary conditions; an appropriate setting could be that of an island or a peninsula (e.g. newfoundland, cape breton, anticosti) that experiences relatively strong prevailing winds and is known to have occasional sbw outbreaks, such as the 1970’s outbreak on the island newfoundland, as well as the current (2021) increase in sbw numbers observed on the island [14, 16]. mathematically, we explored the new rda setting where the reaction term φ(u) = u− 1 q u2 − 1 r u2 1 + u2 combines the logistic growth and the loss due to predation, and results in richer dynamics than in the logistic rda setting and allows multiple steady states. here q and r are non-dimensional biological parameters: q represents the carrying capacity of the tree branches and r characterizes the generalist predation impact (increasing r leads to decrease in predation). the main focus of this study was on the effect that advection can have on outbreaks as opposed to the endemic states. the original nonspatial spruce budworm model introduced in [12] focused on the case of three positive equilibria 0 < u1 < u2 < u3 given by the zeros of the reaction term φ(u), where u1 was the stable endemic equilibrium, u2 was the unstable threshold, and u3 was the stable outbreak equilibrium. we identify the endemic and outbreak solutions of the corresponding rda model given by ∂u ∂t = ∂2u ∂x2 −q ∂u ∂x + φ(u) prevailing winds and spruce budworm outbreaks: an rda model 265 (where q is the advection speed) with hostile boundary conditions as positive steady state solutions, where the maximal density of the endemic solution is bounded above by u1, and the maximal density of the outbreak solution is bounded below by u2. as in [22] and [21], here we employed the geometric approach to the study of steady state solutions of the rda model, by representing them as orbits in the phase plane a system of first order odes{ du dx = v dv dx = qv −φ(u) obtained by setting ∂u ∂t = 0 and letting v = ∂u ∂x . the orbits that originate on the positive v-semiaxis and terminate at the negative v-semiaxis are viewed as steady state solutions of the original rda equation subject to the hostile (dirichlet) conditions on some finite domain [0, l]. if the orbit intersects the u-axis at the point (µ, 0), we can identify µ as the maximal density of the corresponding steady state. one can then distinguish endemic and outbreak steady state solutions by noticing that the former are represented by orbits with 0 < µ < u1 while the latter correspond to u2 < µ < u3. the resulting system has four equilibria: (0, 0), (u1, 0), (u2, 0), (u3, 0). in the non-advective case considered in [11], the phase portrait is symmetric about the u-axis, equilibria (0, 0) and (u2, 0) are centers, while (u1, 0) and (u3, 0) are saddle points. in this case, there is an explicit formula for habitat length l(µ) as a function of maximal density µ. once we introduce advection, the phase plane portrait changes: it is no longer symmetric, and the two centres become unstable spirals for relatively small values of q, and become unstable nodes with the further increase of q. the other two equilibria remain saddle points. the first integral method that lead to the explicit formula for l(µ) in the non-advective case is no longer applicable for q > 0. the key element of our analysis was the fact that when the non-advective model admits an outbreak solution, the increase of advection forces the u-intercept (u∗, 0) of the upper branch of the unstable manifold of (u1, 0) to move towards (u3, 0), and at a certain critical value of advection speed q ∗ cr, we observe the appearance of a heteroclinic orbit connecting (u1, 0) and (u3, 0). as a consequence, outbreak orbits can only exist for q < q∗cr, as they have to cross the u-axis between u ∗ and u3. after establishing the existence of this critical advection value for outbreaks, we obtained upper and lower bounds for q∗cr, expressed in terms of the non-dimensional biological parameters q and r. we also observed that for q ≥ 2, no positive steady states exist on any finite domain. based on numerical evidence, we conjecture that q∗cr < 2 for any choice of the biological parameters q and r for which φ(u) has three positive zeros and non-advective model admits an outbreak solution. for 0 ≤ q < min(2,q∗cr), we showed, using phase plane techniques such as trapping regions, that the rda model has both endemic and outbreak steady state solutions for some finite domains, while for q satisfying min(2,q∗cr) < q < 2 (if any) only endemic solutions remain. our main objective in this paper was to establish and estimate the critical value of advection for existence of outbreak solutions on some finite domains. further exploration of the advective model for spruce budworm will require an analysis of the behaviour of the habitat length l(µ; q) as the function of both the maximal density µ and the advection speed q. in this paper, we made the first step in this direction by exploring l(µ; q) numerically and making some observations regarding its behaviour. in particular, we observe that l(µ; q) is increasing with respect to q and the critical domain size for outbreak solutions is always greater than that of the endemic solution (the latter given by lc1(q) = 2π√ 4−q2 ). most of our analysis focused on the case when in the non-advective setting we have three positive steady states (endemic, threshold and outbreak). in that case, the numerics suggest that when we fix the habitat size and let the advection increase, the outbreak and threshold solutions always disappear first, followed by disappearance of the endemic state. this agrees well with our analytical results indicating 266 a. anderson and o. vasilyeva that increasing advection beyond q∗cr prevents existence of outbreak solutions on any finite domain while still admitting endemic (low denisity) states, provided q∗cr < 2. we have also observed that in the case when the non-advective setting only allows an outbreak (high-density) solution, the increase of advection may lead to appearance of three states, replaced by a single endemic (low-density) state. exploring this case analytically is an interesting direction for future research. while our focus in this paper was on the model for sbw in a “windy island” habitat, there are many other advective settings where the reaction term that we considered would be appropriate. the same model can be applied to other defoliating insect species (e.g. gypsy moth), but more importantly, it is also applicable to logistically growing aquatic organisms subject to generalist predation. in this case, different boundary conditions may have to be imposed. a typical upstream boundary condition will be the zero flux condition which, in our non-dimensionized setting, can be visualized by the half line v = qu in the first quadrant of the uv-plane. the typical downstream conditions considered in such models are either hostile (u = 0) or outflow (v = 0). in either case, we can still talk about outbreak orbits versus endemic orbits, but the situation with existence of outbreak orbits is more complicated: even if the unstable manifold of (u1, 0) stays in the first quadrant, there still might be orbits connecting the half line v = qu and the half line u = 0 or v = 0 representing the downstream boundary condition. this setting will require further exploration and additional phase-plane analysis. other settings that our analysis can be applicable to are the effect of shifting habitat boundaries due to climate change on population dynamics as in [17, 1] and sinking phytoplankton as in [7], if one assumes the presence of a generalist predator. acknowledgements. the second author would like to thank andre arsenault, jean-noel candau, frithjof lutscher and sebastien portalier for valuable discussions. the authors also thank the anonymous referee for valuable comments and suggestions. appendix a. proof of proposition 4.6 consider the line passing through (u1, 0) with the slope m > 0. thus, its equation is given by v = m(u − u1). in order to guarantee that the line is above the curve s13 given by v = v(u; q∗cr) for u1 ≤ u ≤ u3, it suffices to require that the slope of the line is greater than the slope of the direction field of the differential equation (3.5) at every point on the line for u1 ≤ u ≤ u3, i.e. m ≥ dv du |v=m(u−u1) = ( q∗cr − φ(u) v ) |v=m(u−u1) = q ∗ cr − φ(u) m(u−u1) , for every u1 ≤ u ≤ u3. note that by lemma 3.1, the slope of the tangent line to v = v(u; q∗cr) at u = u1 is given by m1 = q∗cr+ √ (q∗cr) 2−4φ′(u1) 2 . clearly, we must have m ≥ m1. recall that φ(u) ≥ 0 for u2 ≤ u ≤ u3. thus, since m1 > q ∗ cr, for any u2 ≤ u < u3 and any m ≥ m1, we have m ≥ q∗cr ≥ q∗cr − φ(u) m(u−u1) . it remains to consider the case when u1 < u < u2. by the mean value theorem, for any u1 < u < u2 there exists u1 < ũ < u such that φ(u) = φ(u) −φ(u1) = φ′(ũ)(u−u1). thus, we need q − 1 m φ′(ũ) ≤ m, for all u1 < u < u2. recall that umin ∈ [u1,u2] is such that φ′(umin) is the minimal value of φ′ on [u1,u2]. then for any u1 < ũ < u2 and m > 0 we have q − 1 m φ′(ũ) ≤ q∗cr − 1 m φ′(umin). prevailing winds and spruce budworm outbreaks: an rda model 267 thus, it suffices to choose the smallest m > 0 such that q∗cr − 1 m φ′(umin) ≤ m. such m is given by m = q∗cr + √ (q∗cr) 2 − 4φ′(umin) 2 . note that m ≥ m1 (and m = m1 if umin = u1). therefore, the line v = q∗cr + √ (q∗cr) 2 − 4φ′(umin) 2 (u−u1) lies above the curve s13. we will now follow similar argument to construct another side of a triangular “trapping region” for s13 with the base given by the segment [u1,u3] of the u-axis. namely, we are looking for a line v = k(u−u3) that stays above s13. similar to the previous case, it suffices to ensure that the slope of the line is less than the slope of the direction field of the differential equation (3.5) at every point on the line where u1 ≤ u ≤ u3, i.e. k ≤ dv du |v=k(u−u3) = ( q∗cr − φ(u) v ) |v=k(u−u3) = q ∗ cr − φ(u) k(u−u3) , for all u1 ≤ u ≤ u3. note that by lemma 3.1, the slope of the tangent line to v = v(u) at u = u3 is given by m3 = q∗cr− √ (q∗cr) 2−4φ′(u3) 2 . note that we need k ≤ m3. recall that φ(u) ≤ 0 for u1 ≤ u ≤ u2. also note that m3 < 0 and u−u3 < 0. thus, for any u1 ≤ u ≤ u2 and any k ≤ m3, we have k ≤ 0 ≤ q − φ(u) k(u−u3) . it remains to consider the case when u2 < u < u3. by the mean value theorem, for any u2 < u < u3 there exists u < ũ < u3 such that φ(u) = φ(u) −φ(u3) = φ′(ũ)(u−u3). thus, we need q − 1 k φ′(ũ) ≥ k, for all u2 < u < u3. by lemma 2.1, φ(·) has only two positive inflection points: 0 < û1 < û2. clearly, û1 < u2 and φ′(û2) > 0. therefore, the mimimum of φ ′(u) on [u2,u3] occurs at u = u3. then for any u2 < ũ < u3 and k < 0 we have q − 1 k φ′(ũ) ≥ q∗cr − 1 k φ′(u3). thus, it suffices to choose the largest k < 0 such that q∗cr − 1 k φ′(u3) ≥ k. such k is given by k = q∗cr − √ (q∗cr) 2 − 4φ′(u3) 2 = m3. therefore, the line v = q∗cr+ √ (q∗cr) 2−4φ′(u3) 2 (u−u3) lies above the curve s13 (given by v = v(u), u1 ≤ u ≤ u3). hence, we have constructed a triangular “trapping region” for the curve s13 with the base given by the segment connecting (u1, 0) and (u3, 0) and the sides given by v = m(u−u1) and v = m3(u−u3) (the latter is the tangent line to s13 at (u3, 0)). note that if s13 is concave down (which seems to be the case according to the plots), or if û1 < u1 (e.g., if u1 ≥ 1), m is the slope of the tangent line to s13 at (u1, 0) as well. let θ1 = arctan m1 and θ2 = arctan m3. then we have tan θ1 = m1 = q∗cr + √ q∗cr 2 − 4φ′(umin) 2 268 a. anderson and o. vasilyeva figure a.1. estimating ∫u3 u1 v(u)du using a triangular trapping region. and tan θ2 = m3 = −q∗cr + √ q∗cr 2 − 4φ′(u3) 2 . thus, the area of the triangular region, is given by a = 1 2 (b1 + b2)h = 1 2 (u3 −u1)h. to find h, we note that h = b1 tan θ1 = b2 tan θ2 = (u3 −u1 − b1) tan θ2. solving for b1, we get b1 = (u3 −u1) tan θ2 tan θ1 + tan θ2 , and, therefore, h = (u3 −u1) tan θ1 tan θ2 tan θ1 + tan θ2 . hence, a = 1 2 · (u3 −u1)2 tan θ1 tan θ2 tan θ1 + tan θ2 = a(q∗cr), where a(q) = ( √ q2 − 4φ′(umin) + q)( √ q2 − 4φ′(u3) −q)(u3 −u1)2 4( √ q2 − 4φ′(umin) + √ q2 − 4φ′(u3)) (a.1) is viewed as a function of q ≥ 0. by the construction of the triangular region, we conclude that∫ u3 u1 v(u; q∗cr)du < a(q ∗ cr). appendix b. figures for section 6 prevailing winds and spruce budworm outbreaks: an rda model 269 figure b.1. biological parameter values: q = 6.5,r = 0.6. left: the graph of g(q) with the values q = 0.22 (circle), q∗cr = 0.325 (star) and q̄ = 0.5507 (square). right: orbits sunst+1 for q ≥ q with increments of ∆q = 0.01. roots u0 = 0 < u1 < u2 < u3 of φ(u) are marked with stars. figure b.2. biological parameter values: q = 10,r = 0.5556. left: the graph of g(q) with the values q = 0.48 (circle), q∗cr = 0.705 (star) and q̄ = 1.6122 (square). right: orbits sunst+1 for q ≥ q with increments of ∆q = 0.01. roots u0 = 0 < u1 < u2 < u3 of φ(u) are marked with stars. figure b.3. biological parameter values: q = 10,r = 0.5128. left: the graph of g(q) with the values q = 0.3 (circle), q∗cr = 0.455 (star) and q̄ = 0.7454 (square). right: orbits sunst+1 for q ≥ q with increments of ∆q = 0.01. roots u0 = 0 < u1 < u2 < u3 of φ(u) are marked with stars. 270 a. anderson and o. vasilyeva figure b.4. biological parameter values: q = 15,r = 0.5128. left: the graph of g(q) with the values q = 0.53 (circle), q∗cr = 0.855 (star) and q̄ = 1.6031 (square). right: orbits sunst+1 for q ≥ q with increments of ∆q = 0.05. roots u0 = 0 < u1 < u2 < u3 of φ(u) are marked with stars. figure b.5. biological parameter values: q = 15,r = 0.4. left: the graph of g(q) with the values q = 0.3 (circle), q∗cr = 0.475 (star) and q̄ = 0.7687 (square). right: orbits sunst+1 for q ≥ q with increments of ∆q = 0.05. roots u0 = 0 < u1 < u2 < u3 of φ(u) are marked with stars. figure b.6. habitat length l vs. maximal density µ for q = 15,r = 0.1 and q = 0, 0.7, 1.4. prevailing winds and spruce budworm outbreaks: an rda model 271 figure b.7. habitat length l vs. maximal density µ for q = 15,r = 0.5128 and q = 0, 0.35, 0.7. the positive zeros u1 < u2 < u3 of φ(u) are marked by stars. figure b.8. habitat length l vs. maximal density µ for q = 15,r = 0.7 and q = 0, 0.7, 1.4. 272 a. anderson and o. vasilyeva references [1] h. berestycki, o. diekmann, c.j. nagelkerke, p.a. zegeling, can a species keep pace with a shifting climate? bul. math. biol. 71 (2009), 399–429. 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(2019), 2093–2140. department of mathematics and statistics, university of guelph, 50 stone road east, room 437 macnaughton building, guelph, on n1g 2w1, canada e-mail address: aander20@uoguelph.ca corresponding author, memorial university of newfoundland, grenfell campus, corner brook, nl a2h 5g4, canada e-mail address: ovasilyeva@grenfell.mun.ca https://www.pc.gc.ca/en/pn-np/nl/grosmorne/decouvrir-discover/sb 1. introduction 2. preliminaries: an overview of the classical non-spatial and spatial spruce budworm models 2.1. the non-spatial ludwig-jones-holling model. 2.2. the spatial ludwig-aronson-weinberger model. 3. model set up 4. the critical advection for outbreak orbits 4.1. existence and upper bound for critical advection for outbreaks. 4.2. lower bound for critical advection for outbreaks. 5. geometric analysis of endemic and outbreak orbits and steady state solutions 5.1. conditions for existence of endemic and outbreak orbits 5.2. habitat length vs. maximal density 6. numerical simulations 6.1. estimating critical advection 6.2. dependence of habitat length on maximal density and advection 7. discussion appendix a. proof of proposition 4.6 appendix b. figures for section 6 references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 4, number 1, march 2023, pp.40-60 https://doi.org/10.5206/mase/15355 chaotic dynamics of the fractional order schnakenberg model and its control md. jasim uddin and s. m. sohel rana abstract. the schnakenberg model is thought to be the caputo fractional derivative. a discretization process is first used to create caputo fractional differential equations for the schnakenberg model. the fixed points in the model are categorized topologically. then, we show analytically that a fractional order schnakenberg model supports a neimark-sacker (ns) bifurcation and a flip-bifurcation under certain parametric conditions. using central manifold and bifurcation theory, we demonstrate the presence and direction of ns and flip bifurcations. the parameter values and the initial conditions have been found to profoundly impact the dynamical behavior of the fractional order schnakenberg model. numerical simulations demonstrate chaotic behaviors like bifurcations, phase portraits, period 2, 4, 7, 8, 10, 16, 20 and 40 orbits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. to support the system’s chaotic characteristics, we also quantitatively compute the maximal lyapunov exponents and fractal dimensions. finally, the chaotic trajectory of the system is stopped using the ogy approach, hybrid control method, and state feedback method. 1. introduction differentiation and integration to arbitrary order, commonly known as fractional calculus, has received a lot of interest from researchers. a mathematical notion from the 17th century is fractional calculus. but it may be regarded as a new research subject. due to their close resemblance to memory-based systems, which are present in most biological systems, fractional-order differential equations (fd) are the most often employed [13]. it is possible to successfully explain fractional-order differential equations in several fields, including science, engineering, finance, economics, and epidemiology[16, 17, 18, 19, 30]. switching from an integer-order model to a fractional-order one necessitates accuracy in the order of differentiation; even a slight change in α might greatly influence the final result[7]. fractional differential equations can describe phenomena that ides can’t wholly model [20]. the complicated dynamics of chaos and bifurcation may be seen in a nonlinear fractional differential system, much like in a nonlinear differential system. it’s fascinating and engaging to study disorder in fractional-order dynamical systems[1, 4, 8, 9, 13].there are several strategies to apply the concept of differentiation to arbitrary order. the most popular definitions are those by riemann-liouville, caputo, and grünwald-letnikov[35]. along with these definitions, researchers constantly look for the most effective strategy when developing or altering their models, including specific numerical approaches[6, 21, 31]. numerous discrete systems have aroused the interest of academics investigating the neimark-sacker and flip bifurcations, stable orbits, and chaotic attractors (see [24, 25, 36, 37]). the center manifold theory and standard form can mathematically quantify these phenomena. received by the editors 3 october 2022; accepted 3 march 2023; published online 22 march 2023. 2020 mathematics subject classification. 37c25, 37d45, 39a28, 39a33. key words and phrases. fractional order schnakenberg model, flip and neimark-sacker bifurcations, maximum lyapunov exponent, fractal dimension, chaos control. 40 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15355 chaotic dynamics of the fractional order schnakenberg model and its control 41 there are numerous cyclical and oscillatory processes in nature. circadian rhythms, present in almost every aspect of life, are possibly the most well-known of these recurring occurrences. a device made in 1979 by j. schnakenberg demonstrated sustained oscillations for a straightforward glycolysis model (a metabolic process that transforms glucose into cellular energy) [38], remarkably similar to a system of four reactions called the ”brusselator”[11]. a semi-analytical approach (see [5]) was used to investigate the schnakenberg model of a reaction-diffusion cell. a chemical schnakenberg model was investigated [23], which depicts autochemical processes with rhythmic behavior that may have various biological and biochemical applications. a variable-order space-time fractional reaction-diffusion schnakenberg model’s numerical solutions were studied in [15]. in addition,the schnakenberg model was described in [23, 29, 28], where a variety of numerical methods were employed to get approximations of solutions. the schnakenberg model is an oscillatory chemical reaction model that schnakenberg developed from a number of hypothetical tri-molecular autocatalytic processes. in order to find the fewest possible reactions and reactants that display limit-cycle behavior, this model has been developed. schnakenberg established that this kind of model needs at least three reactions, including one auto-catalytic reaction. thus, the following reaction scheme is developed for generic chemicals a, x, y , and b: x m1 m−1 a,b m2→ y, 2x + y m3→ 3x, where x,y represent chemicals with different concentrations and a,b have constant concentrations. the following differential equations are implied by mass action theory dnx dτ = m1na −m−1nx + m3n2xny , dny dτ = m2nb −m3n2xny , (1.1) where na,nb,nx and ny represent numbers of molecules for fixed and varying concentrations a,b,x and y respectively. the non-dimensional form of hypothetical schnackenberg model comes from the above system. ẋ = a−x + x2y, ẏ = b−x2y, (1.2) where x = √ m3 m−1 nx, y = √ m3 m−1 ny , τ −→ t = τm−1, a = m1 m−1 √ m3 m−1 na, b = √ m2 m−1 nb. due to their effective computational outcomes and complex dynamical behavior, discrete-time models governed by difference equations are preferable to continuous ones[32, 2]. additionally, this reasoning holds true for nonlinear oscillatory behavior associated with chemical reactions[22, 34, 14]. as a result, we research the stability, chaos management, and bifurcation analysis of discrete counterparts of(1.2) . recently, a few authors[1, 4, 8, 9, 13, 3, 12] use the well-known caputo fractional derivative instead of ordinary derivatives in continuous model. instead of using regular derivatives, the schnakenberg model can employ fractional derivatives because it works the same way. in one sense, this may interpret that the rate of change for the concentrations of the chemical products will be slower and this may lead a 42 md. jasim uddin and s. m. sohel rana better mathematical approximation. the fractional order schnackenberg model is given as follows dαx(t) = a−x(t) + x2(t)y(t), dαy(t) = b−x2(t)y(t), (1.3) where α is the fractional order that satisfies α ∈ (0, 1] and t > 0. there are a lot of approaches for discretize such kind of system. the piecewise constant approximation[3, 12] is one of them. the model is discretized by using this method. the steps are listed below: let system (1.2) ’s starting conditions be x(0) = x0,y(0) = y0. the discretized version of system (1.3) is given as: dαx(t) = a−x([ t ρ ]ρ) + x2([ t ρ ]ρ)y([ t ρ ]ρ), dαy(t) = b−x2([ t ρ ]ρ)y([ t ρ ]ρ). (1.4) first, let t ∈ [0,ρ), so t ρ ∈ [0, 1). thus, we obtain dαx(t) = a−x0 + x20y0; dαy(t) = b−x20y0. (1.5) the solution of (1.5) is simplified to x1(t) = x0 + j α ( a−x0 + x20y0 ) = x0 + tα αγ(α) ( a−x0 + x20y0 ) , y1(t) = y0 + j α ( b−x20y0 ) = y0 + tα αγ(α) ( b−x20y0 ) . (1.6) second, let t ∈ [ρ, 2ρ), so t ρ ∈ [1, 2). then dαx(t) = a−x1 + x21y1 dαy(t) = b−x21y1 (1.7) which have the following solution x2(t) = x1(ρ) + j α ρ ( a−x1 + x21y1 ) = x1(ρ) + (t−ρ)α αγ(α) ( a−x1 + x21y1 ) , y2(t) = y1(ρ) + j α ρ ( b−x21y1 ) = y1(ρ) + (t−ρ)α αγ(α) ( b−x21y1 ) , (1.8) where jαρ ≡ 1 γ(α) ∫ t ρ (t− τ)α−1dτ, α > 0. the result of doing the discretization process n times over xn+1(t) = xn(nρ) + (t−nρ)α αγ(α) ( a−xn(nρ) + x2n(nρ)yn(nρ) ) , yn+1(t) = yn(nρ) + (t−nρ)α αγ(α) ( b−x2n(nρ)yn(nρ) ) , (1.9) where t ∈ [nρ, (n + 1)ρ). for t −→ (n + 1)ρ, system (1.9) is reduced to chaotic dynamics of the fractional order schnakenberg model and its control 43 xn+1 = xn + ρα γ(α + 1) ( a−xn + x2nyn ) , yn+1 = yn + ρα γ(α + 1) ( b−x2nyn ) . (1.10) an ecological system in the real world is not always stable. a small change of control parameter may destabilize the system from locally stable coexistence to producing chaotic orbits. in a discrete dynamical system, flip bifurcation and neimark-sacker bifurcation are the important mechanisms for the generation of complex dynamics. because in the discrete predator-prey system, both bifurcations cause the system to jump from stable window to chaotic states through periodic and quasi-periodic states, and trigger a route to chaos. when a system undergoes a flip bifurcation, a sequence of period-doubling cascades leads the system from steady state to chaos. on the other hand, when system undergoes neimark-sacker bifurcation, it instigates a route to chaos, through a dynamic transition from a stable state, to invariant closed cycle, with periodic and quasi-periodic states occurring in between, to chaotic sets. this paper’s remaining sections are organized as follows.: sect. 2 investigates the fixed point topological classifications. in sect. 3, we explore analytically the possibility that the system (1.10) would experience a flip or ns bifurcation under a certain parametric condition. in sect.4, we numerically show system dynamics that includes bifurcation diagrams, phase portraits, and mles to support our analytical conclusions.to stabilize the chaos of the unmanaged system, we employ the ogy approach, hybrid control method, and state feedback technique in sect. 5. sect. 6 presents a brief discussion. 2. stability of fixed point the system (1.10) has a unique fixed point e(x∗,y∗), where x∗ = a + b and y∗ = b/(a + b)2, which always exists for all permissible parameter values. system (1.10)’s jacobian matrix, evaluated at e(x∗,y∗), is as follows: j (x∗,y∗) =   ( 1 + (−1 + 2x∗y∗) ρ α γ(α+1) ) −x∗ 2 ρα γ(α+1) −2x∗y∗ ρ α γ(α+1) ( 1 −x∗ 2 )   . (2.1) now at e(x∗,y∗), the jacobian matrix is given by je = ( 1 + (−a+b) a+b ρα γ(α+1) (a + b)2 ρ α γ(α+1) − 2b a+b ρα γ(α+1) ( 1 − (a + b)2 ρ α γ(α+1) ) ) . (2.2) the characteristic polynomial of the jacobian matrix can be written as f̃(λ) := λ2 − tr(je) λ + det(je) = 0, (2.3) where tr(je) is the trace and det(je) is the determinant of the jacobian matrix je, and is given by tr(je) =2 + (−a + b) a + b − (a + b)2 ρα γ(α + 1) , det(je) = a3(−1 + ρ̂)ρ̂ + 3a2b(−1 + ρ̂)ρ̂ + a(−1 + ρ̂)(−1 + 3b2ρ̂) + b(1 + ρ̂− b2ρ̂ + b2ρ̂2) a + b . (2.4) where ρ̂ = ρα/γ(α + 1). 44 md. jasim uddin and s. m. sohel rana the eigenvalues of the system(2.3) can be written as λ1,2 = tr(je) ± √ (tr(je))2 − 4det(je) 2 . by the jury criterion, the stability condition for the equlibrium point e(x∗,y∗) is given as follows: f̃(1) > 0, f̃(−1) > 0, f̃(0) − 1 < 0. let pde =  (a,b,ρ,α) : ρ = ( γ(1 + α) −a2e ± √ l a1e ) 1 α = ρ±,l ≥ 0   where a1e = (a + b) 2, a2e = − a + a3 − b + 3a2b + 3ab2 + b3 a + b = − a− b + (a + b)3 a + b = − a4e a + b , a3e = 4 a4e = a− b + (a + b)3, l = a22e −a1ea3e. the system (1.10), however, undergoes a flip bifurcation at e when the parameters (a,b,ρ,α) fluctuate within a narrow region of pde. also let nse = { (a,b,ρ,α) : ρ = ( γ(1 + α) −a2e a1e ) 1 α = ρns,l < 0 } . if the parameters (a,b,ρ,α) vary around the set nse, system (1.10) will suffer an ns bifurcation at that point. lemma 2.1. for every parameter value selection, the fixed point e is a − sink if (i)l ≥ 0, ρ < ρ−(stable node), (ii)l < 0, ρ < ρns(stable focus), − source (i)l ≥ 0, ρ > ρ+(unstable node), (ii)l < 0, ρ > ρns(unstable focus), − non-hyperbolic (i)l ≥ 0, ρ = ρ−or ρ = ρ+( saddle with flip), (ii)(i)l < 0, ρ = ρns( focus), − saddle: otherwise 3. bifurcation analysis in this section, we will study the presence, direction, and stability analysis of flip and ns bifurcations near to the fixed point e using center-manifold and bifurcation theory[26, 39, 40]. in other words, we take ρ to be the bifurcation parameter. 3.1. flip bifurcation. we arbitrarily select the parameters (a,b,ρ,α) to locate in pde. consider the system (1.10) at equilibrium point e(x∗,y∗). assume the parameters lie in pde. let l ≥ 0 and ρ = ρ− = ( γ(1 + α) −a2e − √ l a1e ) 1 α . then, the eigenvalues of j(e) are given as λ1 = −1 and λ2 = 3 + a2ρ− . chaotic dynamics of the fractional order schnakenberg model and its control 45 also note that the condition |λ2 6= 1| is equivalent to a2ρ− 6= −2,−4. (3.1) next, we use the transformation x̂ = x−x+, ŷ = y−y+ and set a(ρ) = j(x∗,y∗). we shift the fixed point of system(1.10) to the origin. so the system(1.10) can be written as ( x̂ ŷ ) → a(ρ−) ( x̂ ŷ ) + ( f1(x̂, ŷ,ρ−) f2(x̂, ŷ,ρ−) ) (3.2) where x = (x̂, ŷ)t and f1(x̂, ŷ,ρ−) = 1 (a + b)5 [ a− b + (a + b)3 − √ l ] x̂ ( 2b3ŷ + a2(2a + x̂)ŷ + b2(6a + x̂)ŷ ) + 1 (a + b)5 [ a− b + (a + b)3 − √ l ] x̂ ( b(x̂ + 6a2ŷ + 2ax̂ŷ) ) , f2(x̂, ŷ,ρ−) = − 1 (a + b)5 [ a− b + (a + b)3 − √ l ] x̂ ( 2b3ŷ + a2(2a + x̂)ŷ + b2(6a + x̂)ŷ ) − 1 (a + b)5 [ a− b + (a + b)3 − √ l ] x̂ ( b(x̂ + 6a2ŷ + 2ax̂ŷ) ) . (3.3) the system(3.2) can be expressed as xn+1 = axn + 1 2 b (xn,xn) + 1 6 c (xn,xn,xn) + o ( ‖xn‖ 4 ) where b(x,y) = ( b1(x,y) b2(x,y) ) and c(x,y,u) = ( c1(x,y,u) c2(x,y,u) ) are symmetric multi-linear vector functions of x,y,u ∈ r2 and defined as follows: b1(x,y) = 2∑ j,k=1 δ2f1(ξ,ρ) δξjδξk ∣∣∣∣∣∣ ξ=0 xjyk = 1 (a + b)5 [ a− b + (a + b)3 − √ l ]( a3(x2y1 + x1y2) ) + 1 (a + b)5 ( 3a2b(x2y1 + x1y2) ) + 1 (a + b)5 ( b3(x2y1 + x1y2) + 3a 2b(x2y1 + x1y2) + bx1y1 ) , b2(x,y) = 2∑ j,k=1 δ2f2(ξ,ρ) δξjδξk ∣∣∣∣∣∣ ξ=0 xjyk = 1 (a + b)5 [ a− b + (a + b)3 − √ l ]( (a + b)3(x2y1 + x1y2) + bx1y1 ) , 46 md. jasim uddin and s. m. sohel rana and c1(x,y,u) = 2∑ j,k,l=1 δ2f1(ξ,ρ) δξjδξkδξl ∣∣∣∣∣∣ ξ=0 xjykul = 1 (a + b)3 ( a− b + (a + b)3 − √ l ) (u1x1y2) + 1 (a + b)3 ( a− b + (a + b)3 − √ l ) (u1x1y2) , c2(x,y,u) = 2∑ j,k,l=1 δ2f1(ξ,ρ) δξjδξkδξl ∣∣∣∣∣∣ ξ=0 xjykul = − 1 (a + b)3 ( a− b + (a + b)3 − √ l ) (u2x1y1 + u1x2y1) − 1 (a + b)3 ( a− b + (a + b)3 − √ l ) (u1x1y2) . let q1,q2 ∈ r2 be two eigenvectors of a and at for eigenvalue λ1 (ρ−) = −1 such that a (ρ−) q1 = −q1 and at (ρ−) q2 = −q2. then by direct calculation we get q1 = ( −(a+b) 3(−a−3b−+(a+b)3− √ l) 2b(a−b+(a+b)3− √ l) 1 ) = ( q11 1 ) , q2 = ( −a+3b−(a+b) 3+ √ l a−b+(a+b)3− √ l 1 ) = ( q21 1 ) . in order to get 〈q1,q2〉 = 1, where 〈q1,q2〉 = q11q21 + q12q22, we have to use the normalized vector q2 = γfq2, with γf = 1/(1 + q11p11). to obtain the direction of the flip bifurcation, we have to check the sign of l1(ρ−), the coefficient of critical normal form ([26]) and is given by l1 (ρ−) = 1 6 〈q2,c(q1,q1,q1)〉− 1 2 〈 q2,b ( q2, (a− i)−1b(q1,q1) )〉 . (3.4) in light of the reasoning above, the following theorem may be used to demonstrate the direction and stability of flip bifurcation. theorem 3.1. suppose (3.1) holds and l1(ρ−) 6= 0, then flip bifurcation at fixed point e(x∗,y∗) for system (2.1) if the ρ changes its value in small neighbourhood of pde. moreover, if l1(ρ−) < 0 (resp. l1(ρ−) > 0), then there is a smooth closed invariant curve that can bifurcate from e(x ∗,y∗), and the bifurcation is sub-critical (resp. super-critical). 3.2. neimark-sacker bifurcation. when the parameters (a,b,α,ρ) ∈ nse and l < 0, then the eigenvalues of system (2.3) are given by λ,λ̄ = tr(je) ± i √ 4det(je) −tr(je)2 2 (3.5) where tr(je) = 2 − (a− b)(a− b + (a + b)3) (a + b)4 − a− b + (a + b)3 a + b , det(je) = 1. let ρns = −a2e/a1e. then, the transversality and non-resonance conditions respectively read d|λi(ρ)| dρ |ρ=ρns = − a2e 2 6= 0; −(tr(je))|ρ=ρns 6= 0 ⇒ a22 a1e 6= 2, 3. (3.6) chaotic dynamics of the fractional order schnakenberg model and its control 47 using the transformation x̂ = x−x+, ŷ = y −y+ and setting a(ρ) = j(x∗,y∗), we move the system( 1.10)’s fixed point to the origin. so the system(1.10) can be written as x → a(ρ)x + f (3.7) where x = (x̂, ŷ)t and f = (f1,f2) t are given by f1(x̂, ŷ,ρns) = (a− b + (a + b)3)x̂ ( 2b3ŷ + a2(2a + x̂)ŷ + b2(6a + x̂)ŷ + b(x̂ + 6a2ŷ + 2ax̂ŷ) ) (a + b)5 , f2(x̂, ŷ,ρns) = − (a− b + (a + b)3)x̂ ( 2b3ŷ + a2(2a + x̂)ŷ + b2(6a + x̂)ŷ + b(x̂ + 6a2ŷ + 2ax̂ŷ) ) (a + b)5 . (3.8) the system(3.2) can be expressed as xn+1 = axn + 1 2 b (xn,xn) + 1 6 c (xn,xn,xn) + o ( ‖xn‖ 4 ) where b1(x,y) = 2(a− b + (a + b)3) (a + b)5 [ (a + b)3(x2y1 + x1y2) + bx1y1 ] , b2(x,y) = − 2(a− b + (a + b)3) (a + b)5 [ (a + b)3(x2y1 + x1y2) + bx1y1 ] , and c1(x,y,u) = 2(a− b + (a + b)3) (a + b)5 (u2x1y1 + u1x2y1 + u1x1y2) , c2(x,y,u) = − 2(a− b + (a + b)3) (a + b)5 (u2x1y1 + u1x2y1 + u1x1y2) . let q1,q2 ∈ c2 be two eigenvectors of a(ρns) and at (ρns)for eigenvalue λ(ρns) and λ̄(ρns) such that a (ρns) q1 = λ (ρns) q1, a (ρns) q̄1 = λ̄ (ρns) q̄1, at (ρns) q2 = λ̄ (ρns) q2, a t (ρns) q̄2 = λ (ρns) q̄2. (3.9) therefore, using a straight calculation, we obtain q1 = ( −−a+b+(a+b) 3+ √ l 4b 1 ) = ( ζ1 + iζ2 1 ) , q2 = ( −a+b+(a+b)3− √ l 2(a+b)3 1 ) = ( ξ1 + iξ2 1 ) . where ζ1 = − −a + b + (a + b)3 4b , ζ2 = − √ −l 4b , ξ1 = −a + b + (a + b)3 2(a + b)3 , ξ2 = − √ −l 2(a + b)3 . for 〈q1,q2〉 = 1 to hold, we can take normalized vector as q2 = γnsq2 where γns = 1 1 + (ξ1 + iξ2)(ζ1 − iζ2) . the following is how the eigenvectors are calculated: 48 md. jasim uddin and s. m. sohel rana q1 = ( ζ1 + iζ2 1 ) , q2 = ( ξ1+iξ2 1+(ξ1+iξ2)(ζ1−iζ2) 1 1+(ξ1+iξ2)(ζ1−iζ2) ) . we break down x ∈ r2 as x = zq1 + z̄q̄1 by considering ρ vary near to ρns and for z ∈ c. the precise formulation of z is z = 〈q2,x〉. as a result, the system (2.3) changed to the following system for |ρ| near ρns: z 7−→ µ(ρ)z + ĝ(z, z̄,ρ) (3.10) where λ(ρ) = (1 + ϕ̂(ρ))eiθ(ρ) with ϕ̂ (ρns) = 0 and ĝ(z, z̄,ρ) is a smooth complex-valued function. applying taylor expansion to the function ĝ, we obtain ĝ(z, z̄,ρ) = ∑ k+l≥2 1 k! l! ĝkl(ρ)z k−l with ĝkl ∈ c,k, l = 0, 1, . . . . it is possible to define the taylor coefficients by means of symmetric multi-linear vector functions. ĝ20 (ρns) = 〈q2,b(q1,q1)〉, ĝ11 (ρns) = 〈q2,b(q1, q̄1)〉, ĝ02 (ρns) = 〈q2,b(q̄1, q̄1)〉, ĝ21 (ρns) = 〈q2,c(q1,q1, q̄1)〉. (3.11) the sign of the first lyapunov coefficient l2(ρns) determines the direction of ns bifurcation and is defined by l2 (ρns) = re ( λ2ĝ21 2 ) − re ( (1−2λ1)λ22 2(1−λ1) ĝ20ĝ11 ) − 1 2 |ĝ11| 2 − 1 4 |ĝ02| 2 . (3.12) according to the above discussion, the direction and stability of ns bifurcation can be presented in the following theorem. theorem 3.2. suppose (3.6) holds and l2(ρns) 6= 0, then ns bifurcation at fixed point e(x∗,y∗) for system (1.10) if the ρ changes its value in small neighbourhood of nse. moreover, if l2(ρns) < 0 (resp. l2(ρns) > 0), then there is a smooth closed invariant curve that can bifurcate from e(x ∗,y∗), and the bifurcation is sub-critical (resp. super-critical). 4. numerical simulations in this part, we will carry out numerical simulations to corroborate our theoretical findings for system (1.10) that include bifurcation diagrams, phase portraits, mles and fds. example 1: these are the chosen parameter values: a = 1.5,b = 0.5,α = 0.58 and, ρ fluctuates in the range 0.3 ≤ ρ ≤ 0.5. also the initial condition is (x0,y0) = (1.998, 0.121). we identify a fixed point e(x∗,y∗) = (2, 0.125) and bifurcation point for the system (1.10) is evaluated at ρ− = 0.349411. and the eigenvalues of a(ρ−) are λ1,2 = −1, 0.256747. let q1,q2 ∈ r2 be two eigenvectors of a(ρ−) and at (ρ−) corresponding to λ1,2, respectively. therefore, q1 ∼ (−1.43845, 1)t and q2 ∼ (0.179806, 1)t . for 〈q1,q2〉 = 1 to told, we take normalized vector as q2 = γfq2 where, γf = 1.34887. then we get chaotic dynamics of the fractional order schnakenberg model and its control 49 (a) (b) (c) (d) figure 1. flip bifurcation diagram in (a) (ρ,x) plane, (b) (ρ,y) plane, (c) mles, (d) fds . q1 ∼ (−1.43845, 1)t and q2 ∼ (0.24253, 1.34887)t . from (3.4), we obtain the lyapunov coefficient l1(ρ−) = 3.93558 > 0. therefore, the flip bifurcation is sub-critical and the requirements of theorem 3.1 are established. the bifurcation diagrams in fig. 1 (a-b) demonstrate that fixed point e stability takes place for ρ < ρ−, loses stability at ρ = ρ−, and a period doubling phenomenon results in chaos for ρ > ρ−. fig.1 (c-d) displays the calculated mles and fds associated to fig.1(a-b).we see that different choices of ρ result in the chaotic set with period −2,−4,−8, and −16 orbits. dynamical states that are stable, periodic, or chaotic are compatible with the sign in fig.1(c-d), in accordance with the highest lyapunov exponent. for various values of ρ, the phase portraits of the bifurcation diagrams in fig.1(a-b) are shown in fig.2. example 2: the following parameter values are chosen: a = 0.5,b = 0.5,α = 0.58 and, ρ fluctuates in the range 0.45 ≤ ρ ≤ 0.6. also the initial condition is (x0,y0) = (0.998, 0.498). we obtain a fixed point e(x∗,y∗) = (1, 0.5) and bifurcation point of the system (1.10) is evaluated at ρns = 0.820228. and the eigenvalues are λ1,2 = 0.5 ± 0.866025i. also d |λi(ρ)| dρ |ρ=ρns = − a2e 2 = 0.5 6= 0, 50 md. jasim uddin and s. m. sohel rana (a) (b) (c) (d) (e) (f) (g) (h) (i) figure 2. phase picture for various ρ values matching to figure 1 a,b. red ∗ is the fixed point e0. −(tr(je))|ρ=ρns 6= 0 ⇒ a22e a1e = 1 6= 2, 3. let q1,q2 ∈ c2 be two complex eigenvectors corresponding to λ1,2, respectively. therefore, q1 ∼ (−0.5 − 0.866025i, 1)t and q2 ∼ (0.5 − 0.866025i, 1)t . for 〈q1,q2〉 = 1 to hold, q2 is normalized vector q2 = γnsq2 where, γns = 0.5 − 0.288675i. also, by (3.11), the taylor coefficients are ĝ20 = 2.0 + 0.57735i, ĝ11 = 0.5 − 0.288675i, ĝ02 = 0.5 − 2.02073i, ĝ21 = −2 + 0.i. from (3.12),the lyapunov coefficient is obtained as l2(ρns) = −0.75 < 0. as a result, the conditions of theorem 3.2 are met and the ns bifurcation is super-critical. the ns bifurcation diagrams are shown in fig.3(a,b), which reveals that the fixed point e is stable while ρ < ρns, but loses stability when ρ = ρns, and exhibits an attractive closed invariant curve when ρ > ρns. because of the presence of mles (3(c)), system dynamics are not stable. the behavior of the smooth invariant curve as it separates from the stable fixed point and expands in radius is nicely illustrated by the phase portraits (fig.4) of the bifurcation diagrams in fig.4 for various values of rho.the closed curve abruptly vanishes as ρ increases, and for different values of ρ, orbits with periods of −7,−10,−20, and −40 arise. example 3: as other parameter values change (e.g. parameter a), the schnakenberg model may exhibit more dynamic behavior in the neimark-sacker bifurcation diagram. a new neimark-sacker bifurcation diagram is created when the parameter values are set as in example 2 with ρ = 0.53145 and a range between 0.1 ≤ a ≤ 0.3, as shown in figure (5) (a-b). the system experiences a neimark-sacker bifurcation at a = ans = 0.11. the maximal lyapunov exponent, which corresponds to figure5(a-b), chaotic dynamics of the fractional order schnakenberg model and its control 51 (a) (b) (c) (d) figure 3. ns bifurcation diagram in (a) (ρ,x) plane, (b) (ρ,y) plane, (c) mles, (d) fds. is calculated and shown in figure5(c), confirming the existence of chaos and the period window as a act as a variable parameter. a thorough explanation of the behavior of the smooth invariant curve may be found in the phase portrait of the bifurcation diagrams for different values of a illustrated in figure6. this diagram illustrates how the stable fixed point breaks off the smooth invariant curve as its radius rises. figure(7) displays the plot of the maximum lyapunov exponents for two control parameters as a 2d projection onto the (ρ,a) and (ρ,b) plane. we note that the dynamics of system (1.10) shift from chaotic to non-chaotic phase when the values of the control parameters a and b grow. 4.1. fractal dimension. the measurement of the fractal dimensions (fd) serves to identify a system’s chaotic attractors and is defined by [10] d̂l = k + ∑k j=1 tj |tk+1| (4.1) where k is the largest integer such that k∑ j=1 tj ≥ 0 and k+1∑ j=1 tj < 0, and tj’s are lyapunov exponents. now, the fractal dimensions for the system (1.10) take the following form: d̂l = 2 + t1 |t2| . (4.2) 52 md. jasim uddin and s. m. sohel rana (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) figure 4. phase picture for various ρ values matching to figure 3 a,b. red ∗ is the fixed point e0. because the chaotic dynamics of the system (1.10) ( see fig. 2) are quantified with the sign of fd (see fig. 3 (d)), it is certain that as the value of the parameter ρ increases, the dynamics of the fractional order schnakenberg system become unstable. 5. chaos control it is believed that dynamical systems will be optimized in relation to some performance criterion and will prevent chaos. in physics, biology, ecology, chemical engineering, telecommunications, and other fields, chaotic behavior is studied. additionally, the practical chaos management techniques can be used in a variety of fields, including physics labs, biochemistry, turbulence, and cardiology, as well as in communication systems. recently, a lot of scholars have become quite interested in the problem of managing chaos in discrete-time systems. controlling chaos is a challenging issue. for controlling chaos in fractional order schnakenberg model we introduce ogy [33], hybrid control strategy [41] and state feedback[27] . in ogy method we can not use ρ as a control parameter. we use a as a control parameter to implement ogy method. chaotic dynamics of the fractional order schnakenberg model and its control 53 (a) (b) (c) (d) figure 5. ns bifurcation diagram in (a) (a,x) plane, (b) (a,y) plane, (c) mles , (d) fds we can modify the system (1.10) as follows to apply ogy method: xn+1 = xn + ρα γ(α + 1) ( a−xn + x2nyn ) = f̃1(x,y,a), yn+1 = yn + ρα γ(α + 1) ( b−x2nyn ) = f̃2(x,y,a). (5.1) where a represents the chaos control parameter. in addition, it is presumable that |a−a0| < ν̃, where ν̃ > 0 and a0 represents the nominal parameter, lies in the chaotic areas. we employ a stabilizing feedback control strategy to steer the trajectory toward the desired orbit. if the system (1.10) has an unstable fixed point at (x+,y+) in a chaotic zone created by the formation of a neimark-sacker bifurcation, the system (5.1) can be approximated in the area surrounding the unstable fixed point at (x+,y+) by the following linear map: [ xn+1 −x+ yn+1 −y+ ] ≈ ãee [ xn −x+ yn −y+ ] + b̃ee [a−a0] (5.2) where ãee = [ ∂f̃1(x,y,a) ∂x ∂f̃1(x,y,a) ∂y ∂f̃2(x,y,a) ∂x ∂f̃2(x,y,a) ∂y ] = [ ( 1 + (−a+b) (a+b) ρα γ(α+1) ) (a + b)2 ρ α γ(α+1) −2b a+b ρα γ(α+1) 1 − (a + b)2 ρ α γ(α+1) ] . 54 md. jasim uddin and s. m. sohel rana (a) (b) (c) (d) (e) (f) (g) figure 6. phase picture for various a values matching to figure 5 a,b. red ∗ is the fixed point e0. (a) (b) figure 7. maximum lyapunov exponents projected in two dimensions onto the (a) (ρ,a) plane (b) (ρ,b) plane and b̃ee = [ ∂f̃1(x,y,a) ∂a ∂f̃2(x,y,a) ∂a ] = [ ρα γ(α+1) 0 ] . following that, we define the system’s 5.1 controllability matrix as follows: c̃ee = [ b̃ee : ãeeb̃ee ] =   ραγ(α+1) ( 1 + (−a+b) (a+b) ρα γ(α+1) ) ρα γ(α+1) 0 −2b a+b ( ρα γ(α+1) )2   . chaotic dynamics of the fractional order schnakenberg model and its control 55 the rank of c̃ee is then obvious to perceive to be 2. considering that [a−a0] = −k̃ee [ xn −x+ yn −y+ ] where k̃ee = [σ̃e1 σ̃e2], system (5.1) becomes [ xn+1 −x+ yn+1 −y+ ] ≈ [ãee − b̃eek̃ee] [ xn −x+ yn −y+ ] . additionally, the suitable controlled system of (1.10) is offered by xn+1 = xn + ρα γ(α + 1) ( (a0 − σ̃e1(xn −x+) − σ̃e2(yn −y+)) −xn + x2nyn ) , yn+1 = yn + ρα γ(α + 1) ( b−x2nyn ) . (5.3) the fixed point (x+,y+) is additionally locally asymptotically stable if and only if both of the matrix’s eigenvalues (ãee − b̃eek̃ee) are located within an open unit disk. also, ãee − b̃eek̃ee = [ ( 1 + (−a+b) (a+b) ρα γ(α+1) − ρ α γ(α+1) σ̃e1 ) (a + b)2 ρ α γ(α+1) − ρ α γ(α+1) σ̃e2 −2b a+b ρα γ(α+1) 1 − (a + b)2 ρ α γ(α+1) ] . the jacobian matrix (ãee − b̃eek̃ee) has the following characteristic equation: λe 2 − ( 2 − ( (a + b)2 + (−a + b) (a + b) − σ̃e1 ) ρα γ(α + 1) ) λe + 1 a + b ( a + b− ( a + a3 − b− 3a2b + 3ab2 ) ρα γ(α + 1) − ( b3 − (a + b)3 )( ρα γ(α + 1) )2) + 1 a + b ( (a + b) ρα γ(α + 1) ( −1 + (a + b)2 ρα γ(α + 1) ) σ̃e1 − 2bσ̃e2 ( ρα γ(α + 1) )2) = 0. (5.4) if we consider eigenvalues λe1 and λe2 of the characteristic equation (5.4), we obtain λe1 + λe2 = ( 2 − ( (a + b)2 + (−a + b) (a + b) − σ̃e1 ) ρα γ(α + 1) ) , λe1λe2 = 1 a + b ( a + b− ( a + a3 − b− 3a2b + 3ab2 ) ρα γ(α + 1) ) + 1 a + b ( − ( b3 − (a + b)3 )( ρα γ(α + 1) )2 + (a + b) ρα γ(α + 1) ( −1 + (a + b)2 ρα γ(α + 1) ) σ̃e1 ) + 1 a + b ( −2bσ̃e2 ( ρα γ(α + 1) )2) . (5.5) then, we must work out the solutions to the equations λe1 = ±1 and λe1λe2 = 1 to get the lines of marginal stability. furthermore, these restrictions ensure that λe1 and λe2 are located inside the open unit disk. assume λe1λe2 = 1 and from (5.5), we get 56 md. jasim uddin and s. m. sohel rana ke1 = 1 a + b ρα γ(α + 1) ( −(a + b) − (a + b)3 + (a + b)3 ρα γ(α + 1) ) + 1 a + b ρα γ(α + 1) ( (a + b) ( −1 + (a + b)2 ρα γ(α + 1) ) σ̃e1 ) − 1 a + b ρα γ(α + 1) ( 2bσ̃e2 ( ρα γ(α + 1) )2) . next, consider λe1 = 1, we obtain ke2 = 1 a + b ( 4(a + b) − 2(a− b + (a + b)3) ρα γ(α + 1) + (a + b)3 ( ρα γ(α + 1) )2) + 1 a + b ( (a + b) ρα γ(α + 1) ( −2 + (a + b)2 ρα γ(α + 1) ) σ̃e1 − 2bσ̃e2 ( ρα γ(α + 1) )2) . lastly, if λe1 = −1 , then ke3 = − ρα γ(α + 1) ( (a + b)3 + (a + b)3σ̃e1 − 2bσ̃e2 ) a + b . stable eigenvalues are then found in the triangle in the σ̃e1, σ̃e2 plane enclosed by the straight lines ke1,ke2,ke3 for a given parametric value. hybrid control strategy is applied to system (1.10) to control chaos. we rewrite our uncontrolled system (1.10) as xn+1 = g(xn,δ) (5.6) where δ ∈ r is bifurcation parameter and xn ∈ r2, g(.) is non-linear vector function. applying hybrid control strategy, the controlled system of (5.2) becomes xn+1 = ωeg(xn,δ) + (1 −ωe)xn (5.7) where the control parameter is 0 < ωe < 1. now, if we implement the above mentioned control strategy to system (1.10), then we get the following controlled system xn+1 = ωe ( xn + pα γ(α + 1) ( a−xn + x2nyn )) + (1 −ωe)xn, yn+1 = ωe ( yn + sα γ(α + 1) ( b−x2nyn )) + (1 −ωe)yn. (5.8) an approach called state feedback control is used to stabilize chaos at the stage of the system’s (1.10) unstable trajectories. system (1.10) may be made to take on a controlled form by introducing a feedback control law as the control force uee and is given below. xn+1 = xn + ρα γ(α + 1) ( a−xn + x2nyn ) + uee, yn+1 = yn + ρα γ(α + 1) ( b−x2nyn ) , uee = −k1(xn −x+) −k2(yn −y+), (5.9) where (x+,y+) represents the positive fixed point of the system (1.10) and k1 and k2 stand for the feedback gains. example 4: we use (a0,b,α,ρ) = (0.15, 1.8, 0.58, 0.53145) to implement the ogy feedback control technique for system (1.10).the unstable system (1.10) in this case has a single positive fixed point chaotic dynamics of the fractional order schnakenberg model and its control 57 (x+,y+) = (1.95, 0.473373). then, in accordance with these parametric values, we provide the following controlled system xn+1 = xn + 0.777474 ( (0.15 − σ̃e1(xn − 1.95) − σ̃e2(yn − 0.473373)) −xn + x2nyn ) , yn+1 = yn + 0.777474 ( 1.8 −x2nyn ) . (5.10) where k̃ = [σ̃e1 σ̃e2] serve as the gain matrix. we also get, ãee = [ 1.65786 2.95634 −1.43534 −1.95634 ] , b̃ee = [ 0.777474 0 ] , c̃ee = [ 0.777474 1.28895 0 −1.11594 ] . the rank of the matrix c̃ee is therefore easily verified to be 2. consequently, it is possible to control the system (5.10) and the jacobian matrix of the controlled system is given by ãee − b̃eek̃ee = [ 1.65786 − 0.777474σ̃e1 2.95634 − 0.777474σ̃e2 −1.43534 −1.95634 ] . for marginal stability, the lines ke1,ke2 and ke3 are provided by ke1 = −0.000000986495 + 1.52101σ̃e1 − 1.11594σ̃e2 = 0, ke2 = 1.70152 + 0.743533σ̃e1 − 1.11594σ̃e2 = 0, ke3 = −2.29848 − 2.29848σ̃e1 + 1.11594σ̃e2 = 0. the marginal lines ke1,ke2 and ke3 define the stable triangular area for the controlled system (5.10) is displayed in figure(8). determining the hybrid control technique’s efficiency in reducing chaotic (unstable) system dynamics, we are taken the parameter values discussed for ogy method except ρ = 0.5812 > ρns. as a result, it demonstrates that the fixed point e(1.95, 0.473373) of system (1.10) is unstable, however for the controlled system(5.8), this fixed point is stable iff 0 < ωe < 0.9494221636969571. by taking ωe = 0.85, which demonstrates that the fixed point e is a sink for the controlled system(5.8) and removes the unstable system dynamics around e. the stable region and stable trajectories are shown in figure(8). we have carried out numerical simulations (in figure(8)) to examine the operation of the feedback control method’s control of chaos in an unstable condition. the parameters values will be same as we choose for ogy method. the feedback gains are choosen as k1 = −0.42 and k2 = −0.35. 6. conclusions in this study, a novel fractional order schnakenberg model is discussed. the caputo fractional derivative idea is the basis for such a fractional order model. we show that the system (1.10) can experience a bifurcation (flip or ns) at a specific positive fixed point e if fluctuations about the sets pde or nse. the center manifold theorem and bifurcation theory are used to achieve this. the model displays a range of complex dynamical behaviors as ρ and α are changed, including the appearance of flip and ns bifurcations, period 2, 4, 6, 7, 8, 10, 16, 20 and 40 orbits, and quasi-periodic orbits, as well as attracting invariant circles and chaotic sets. by computing the maximal lyapunov exponents and fractal dimension, we can verify the existence of chaos. typically, one of the traits that suggests the existence of chaos is a positive lyapunov exponent.we demonstrate that the values of the lyapunov exponent 58 md. jasim uddin and s. m. sohel rana (i) (ii) (iii) (iv) (v) (vi) figure 8. (i-iii) stable region in ogy method, hybrid method and state feedback methods, (iv-vi) stable system trajectories can alternately be negative and positive. by creating pathways from periodic and quasi-periodic states, the two bifurcations lead the system to quickly transition from steady state to chaotic dynamical behavior. alternatively, chaotic dynamics might occur or vanish concurrently with the appearance of bifurcations. finally, we reduce unstable system trajectories by employing an ogy, hybrid, and state feedback control technique. though it remains a challenging topic, investigating multiple parameter bifurcation in the system. references [1] m. a. m. abdelaziz, a. i. ismail, f. a. abdullah and m. h. mohd, bifurcations and chaos in a discrete si epidemic model with fractional order, advances in difference equations, 2018(2018), 1-19. 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[41] l. g. yuan and q. g. yang, bifurcation, invariant curve and hybrid control in a discrete-time predator–prey system, applied mathematical modelling, 39(2015), 2345–2362. 60 md. jasim uddin and s. m. sohel rana md. jasim uddin, corresponding author, department of mathematics, university of dhaka, dhaka-1000, bangladesh. email address: jasim.uddin@du.ac.bd s. m. sohel rana, department of mathematics, university of dhaka, dhaka-1000, bangladesh. email address: srana.mthdu@gmail.com 1. introduction 2. stability of fixed point 3. bifurcation analysis 3.1. flip bifurcation 3.2. neimark-sacker bifurcation 4. numerical simulations 4.1. fractal dimension. 5. chaos control 6. conclusions references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 4, number 2, june 2023, pp.79-99 https://doi.org/10.5206/mase/15231 diffusion-driven instability and pattern formation in a prey-predator model with fear and allee effect debjit pal, dipak kesh, and debasis mukherjee abstract. this paper analyses a predator-prey model with holling type ii response function incorporating allee and fear effect in the prey. the model includes intra species competition among predators. we find out the local dynamics as well as hopf bifurcation by considering level of fear as bifurcation parameter. the condition for diffusion-driven instability and patterns are then demonstrated in relation to the system’s ecological parameters and diffusion coefficients. intra-specific competition affects the dynamics of the system and turing pattern formation. moreover, output of results are verified through numerical simulation. thus, from a dynamical standpoint, the considered model seems to be relevant in the field of ecology. 1. introduction the investigation of diverse systems relating to the interaction between predators and preys, which are the building blocks of an ecosystem, is among the most intriguing research areas for ecologists, biologists, and even mathematicians. the most important aspect of a predator-prey interaction is that predator development is determined by its functional response to the prey species, which quantifies how much prey each predator consumes per unit of time. there are two techniques to determining a predator’s influence on prey in predator-prey interactions [3, 4]. one involves a predator killing its prey directly, while the other involves an indirect tactic such as fear of predation, allee effect caused by predation, and so on. in comparison to direct predation, indirect predation has a more efficient and long-lasting evolutionary effect. in reality, all animals exhibit a variety of anti-predator responses, such as habitat use, alterations in hunting behaviour, physiology etc. in a predator-prey system, the well-established idea is that predators can only impact prey by destroying prey populations. direct prey killing by predators is far more common in ecology compared to indirect influence. however, certain experimental findings demonstrate that, in addition to direct killing, fear of predator has a significant influence on prey behaviour and physiology [7]. because of their fear of predators, preys are often forced to relocate their grazing region to a safer location by surrendering their most preferred regions. as a result, their chances of survival rise immediately, but their long-term fitness drops. the prey’s fecundity rate is reduced as a result of this. as an illustration of indirect dread zanette et al. [36] investigated a field study on song sparrows over a full mating season, utilising electric fences and seines to deliberately prevent direct predation and managing predation risk through predator call playbacks. they studied the effect of fear on demography and discovered a 40% decrease in offspring generated each year in free-living song sparrows population. sasmal and takeuchi [17] investigated a predatorprey system including fear effect and explored the multi-stability of the system and the occurrence of received by the editors 16 september 2022; accepted 13 march 2023; published online 2 april 2023. 2020 mathematics subject classification. primary 35b32, 35b36; secondary 92d25. key words and phrases. allee effect, fear effect, diffusion, hopf bifurcation, turing instability, pattern. d.pal was supported by csir (file no. 09/0096(12492)/2021-emr-i), government of india. 79 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15231 80 d. pal, d. kesh, and d. mukherjee hopf bifurcation. xiao and li [31] studied a predator-prey mutual interaction model with fear impact. several work in predator-prey model was also done incorporating fear effect [16, 37, 40, 28, 20, 11, 33]. one of the most intriguing phenomena in ecology is the allee effect, which was put forth by allee [2]. the allee effect is supported by the correlation between species density and relative growth rate at low population densities which have been observed in many natural populations, including plants, insects, marine invertebrates, birds, and mammals. some of the factors underlying the allee effect include difficulties in finding a mate, reproductive enablement, predation, inbreeding anxiety, environment conditioning, low density social disorder, and so on. allee effects are broadly separated into two types: strong allee effects and weak allee effects. the strong allee effect has a population density threshold that causes the species to become extinct if it falls below this population density [9]. the weak allee effect, on the other hand, happens when the growth rate slows but remains positive at low population density. ecologists have given close attention to this issue as it connects to species extinction. sen and banerjee [19] investigated several sorts of bifurcation behaviour in the holling type ii model with a strong allee effect in growth function of prey and a density-dependent mortality rate for predators. with the use of the holling type-iii functional response, vishwakarma and sen [25] looked into the influence of the allee effect on prey and hunting cooperation in predators. several scientists have also examined the allee effect in predator-prey models [1, 18, 35]. many scholars have shown interest in predator-prey model incorporating both fear effect and allee effect. liu [13] considered a predator-prey model that included fear and allee effects and studied qualitative nature of the system. lie et al. [12] explored the global stability, saddle-node bifurcation, transcritical bifurcation, and hopf bifurcation in a predator-prey model, taking into account the fear effect and additive allee effect. xie [32] discussed the influence of the allee and fear effects on a holling type-ii predator-prey model by considering the system as  du dt = u ( ru (a + u) (1 + fv) −d1 −ku ) − euv 1 + bu , dv dt = −d2v + ceuv 1 + bu . (1.1) where u (t) and v (t) are the respective densities of prey and predator populations at time t. as allee effect also acts on prey’s birth rate so birth rate of prey is expressed as q(a,u) = ru a+u , where u denotes density of prey at time t (u > 0), r(> 0) denotes prey’s maximum birth rate and a(> 0) presents the level of allee and q(a,u) satisfied limu→0 q(a,u) = 0, limu→+∞q(a,u) = r, lima→0 q(a,u) = r, lima→+∞q(a,u) = 0 and ∂q(a,u) ∂a < 0. since certain experimental tests revealed that the fear effect can significantly slow down the reproduction process so prey’s birth rate, q (a,u) is scaling by a term n(f,v) = 1 (1+fv) , this speaks of the price of an anti-predator response brought on by fear, and the parameter f indicates fear level. considering the experimental results and biological viewpoint, the factor n(f,v) must satisfy the following conditions: (1) n (0,v) = 1 : there will be no reduction in prey production if prey exhibits no anti-predator activities(fear of predation). (2) n (f, 0) = 1 : there is no decrease in prey production due to anti-predator behaviours if there is no predator. (3) limf→+∞n(f,v) = 0 : prey production decreases to zero when anti-predator behaviour is extremely high. (4) limv→+∞n(f,v) = 0 : due to intense anti-predator responses, if the predator population is sufficiently large, prey reproduction is decreased to zero. (5) ∂n(f,v) ∂f < 0 : the production of prey declines as anti-predator behaviours rise. diffusion-driven instability and pattern formation in a prey-predator system 81 (6) ∂n(f,v) ∂v < 0 : prey productivity declines as the size of the predator population rises. inspired by the preceding model (1.1), in a prey-predator system with a holling type-ii functional response, we wish to examine the combined impact of include both fear and the allee effect in addition to predator’s intra-specific competition which is added with model (1.1). the new model can be written as ;   du dt = u ( ru (a + u) (1 + fv) −d1 −ku ) − euv 1 + bu , dv dt = −d2v −hv2 + ceuv 1 + bu . (1.2) where it is assumed that all of the parameters are positive and are shown in table 1. parameters meaning value u prey density at time t − v predator density at time t − a level of allee 0.1 b handling time of predators 0.9 c conversion efficiency 0.6 d1 mortality of prey 0.9 d2 mortality of predator 0.2 e consumption rate of predator 0.85 k intraspecific competition among the prey species 0.1 h intraspecific competition among the predator species − r maximum birth rate of prey − f level of fear induced by predator − table 1. description of parameters and values of some parameters in model (1.1) and (1.2) . the idea of spatial pattern development in predator-prey systems by turing instability plays a crucial role in ecology as well as other real-world issues in science, engineering, and technology. turing instability refers to the fact that diffusion converts a stable system to an unstable one. patterns are formed by discontinuities in population distribution over time and space, and they reveal environmental irregularities. diffusion significantly affects the development of patterns. it is self-evident that spatial and spatiotemporal pattern formation is critical in ecology. the patterns we see in biological and ecological communities are those that are required for their survival. however, the processes that produce patterns are still unknown. the reaction-diffusion hypothesis of pattern creation was initially put forward by alan turing in 1952 in his seminal study on the chemical basis of morphogenesis [24]. later, wolpert [29] suggested a phenomenological idea of pattern creation and gave a very concise, non-technical explanation of how animal pattern generation occurs [30], which has resulted in an immense number of enlightening experimental research, many of which are also related to marine ecology. chen and wang [6] investigate qualitative features of a predator-prey model with diffusion system and demonstrate the presence and absence of positive equilibrium point. in recent times, many researchers give attention in the field of pattern formation [15, 5, 21, 34, 10, 23, 27] such as sun et al. have studied several dynamics of pattern in a spatial prey predator model incorporating allee [22], zhang et.al. examined diffusive ratio dependent predator-prey model taking delay in fear effect and discussed the different bifurcations 82 d. pal, d. kesh, and d. mukherjee and patterns induced by delay [38], duan et.al. [8] looked into hopf-hopf bifurcation and chaotic attractors considering fear effect in delayed diffusive predator-prey model. zhang, zhao and yuan [39] explored the effect of discontinuous harvesting in diffussive pre-predator model in presence of allee and fear effect. jun ma et al. [14, 26] investigate an interesting study of patterns in neuronal network. in general, reaction-diffusion equations are frequently used to represent spatio-temporal dynamics, and the study of such systems adds to our understanding of the performance of spatio-temporal instabilities in ecosystems with a large degrees of freedom. when we extend model (1.2) by integrating random dispersal of both species, the diffusive model becomes:   ∂u ∂t = u ( ru (a + u) (1 + fv) −d1 −ku ) − euv 1 + bu + d1∇2u, ∂v ∂t = −d2v −hv2 + ceuv 1 + bu + d2∇2v. (1.3) considering the twodimensional spatial domain ω = [0,l] × [0,l] ∈ r2, ∇2 = ∂ 2 ∂x2 + ∂ 2 ∂y2 and subject to non-negative initial conditions and zero flux boundary conditions. here for ecological feasibility, prey and predator’s respective diffusion coefficients d1 and d2 are choosen to be positive, and u (x,y,t) and v (x,y,t) denote the respective densities of prey and predator, at spatial location (x,y) ∈ ω and time instant t. all the other parameters of the model are described in previous section in table 1. in this paper, for simplicity of calculation, birth rate of the prey, r is considered to be constant, not density dependent. the prime objective of our analysis is to examine the continuous model without diffusion as well as diffusion-driven instability and pattern creation in 2d coordinates. this article is arranged as follows: in section 2, we address the existence and local stability of the equilibrium points. the occurrence of hopf bifurcation around the interior equilibrium point is established in section 4. section 5 reveals numerical simulations of the model in absence of diffusion. in section 6, we explore the model with diffusion with initial and boundary conditions and in two subsections we establish the condition for diffusion-driven instability and the numerical simulation of pattern formation. finally, the article is ended in section 7 with a discussion where a short comparison between the presence and absence of intra-specific competition among predator is also shown. 2. equilibrium points here we shall find all the equilibrium points of system (1.2). firstly, e0 (0, 0) is always an equilibrium point of system (1.2) . searching for predator free equilibrium points, let us take (u′, 0) be such an equilibrium point. then, u′ is the positive root of ku2 + (d1 + ak −r) u + ad1 = 0. the solution is, u = −(d1 + ak −r) ∓ √ ∆1 2k . (2.1) where, ∆1 = (d1 + ak −r) 2 − 4akd1. when (d1 + ak −r) < 0 and ∆1 > 0 i.e. 0 < a < a1 = (√ r − √ d1 )2 k diffusion-driven instability and pattern formation in a prey-predator system 83 holds, then model (1.2) has two predator free or axial equilibrium points i.e. two values of u′, given by e1 (u1, 0) = ( −(d1 + ak −r) − √ ∆1 2k , 0 ) , e2 (u2, 0) = ( −(d1 + ak −r) + √ ∆1 2k , 0 ) . an interior equilibrium point e∗ (u∗,v∗) will be the positive solution of the two equations given by:  ru (a + u) (1 + fv) −d1 −ku− ev 1 + bu = 0, −d2 −hv + ceu 1 + bu = 0. (2.2) from second equation of (2.2), v = mu−d2 h(1 + bu) . (2.3) where m = ce− bd2. to find u, putting the expression of v from (2.3) into the first equation of (2.2) we get a polynomial of degree 5 in u, in the form (eg) u5 + ( ef + dg−rb3h2 ) u4 + ( cg + df − 3rb2h2 ) u3 + ( bg + cf − 3rbh2 ) u2. + ( ag + bf −rh2 ) u + af = 0. (2.4) where, a = a (d1h−ed2) b = 2abhd1 + ahk + hd1 + mea−ed2, c = 2abhk + ab2hd1 + 2bhd1 + hk + me, d = b2hd1 + ab 2hk + 2bhk, e = b2hk, f = h−fd2, g = bh + fm. we know that a polynomial of degree 5 must have a real root, positive or negative. if constant term af < 0 i.e. either a > 0 and f < 0 or a < 0 and f > 0 ⇒ either ed2 d1 < h < fd2 or ed2 d1 > h > fd2 , then the equation (2.4) must have at least one positive root, say u∗. putting that u∗ in equation (2.3) we can get positive v = v∗(say) if mu∗ − d2 > 0 ⇒ u∗ > d2m and m > 0 ⇒ ce > bd2 and an interior equilibrium point of given model (1.2) exists only if both u∗ and v∗ are positive. again one can also give conclusive evidence about the existence of the interior equilibrium point from graphs of isoclines as shown in fig 1. 3. local stability analysis of equilibrium points: we first calculate the jacobian matrix of system (1.2) at any arbitrary point (u,v) and it is given by; j(u,v) =   ru(2a+u) (a+u)2(1+fv) −d1 − 2ku− ev(1+bu)2 −rfu2 (a+u)(1+fv)2 − eu (1+bu) cev (1+bu)2 −d2 − 2hv + ceu(1+bu)   . (3.1) now we shall discuss the local stability of the system (1.2) near each of the equilibrium point. for e0 (0, 0), from (3.1) we get 84 d. pal, d. kesh, and d. mukherjee figure 1. position of x-isoclines(red) and y-isoclines(blue) and interior equilibrium point is indicated by a circle, for a particular set of values of parameters. j (0, 0) =   −d1 0 0 −d2   . the eigenvalues are −d1 and −d2, both are real negative hence extinction steady state e0 (0, 0) become stable i.e., when concentration of prey and predator is belonged in the attraction region of e0, both the populations will die out. now from (3.1) for e1 (u1, 0) we get, j(u1, 0) =   ru1(2a+u1) (a+u1) 2 −d1 − 2ku1 −rfu21 (a+u1) − eu1 (1+bu1) 0 −d2 + ceu1(1+bu1)   . (3.2) now from equation (2.1) we can write√ ∆1 = r −d1 −ak − 2ku1, =⇒ u1 √ ∆1 = ad 2 1 −ku 2 1. using ru ′ a+u′ −d1 −ku′ = 0, and using the last expression, the jacobian takes the form j(u1, 0) =   u1 √ ∆1 (a+u1) −rfu21 (a+u1) − eu1 (1+bu1) 0 −d2 + ceu1(1+bu1)   . (3.3) the two eigenvalues are given by λ1 = u1 √ ∆1 (a + u1) > 0, λ2 = −d2 + ceu1 (1 + bu1) . diffusion-driven instability and pattern formation in a prey-predator system 85 so λ2 > 0 if −d2 + ceu1(1+bu1) > 0 ⇒ b < b1 = ( ce d2 − 1 u1 ) ; and λ2 < 0 if b > b1. thus, e1 (u1, 0) is a unstable node for 0 < b < b1 and e1 (u1, 0) is a saddle point for b > b1. similarly for e2 (u2, 0), we get, j(u2, 0) =   −u2 √ ∆1 (a+u2) −rfu22 (a+u1) − eu2 (1+bu2) 0 −d2 + ceu2(1+bu2)   . (3.4) the two eigenvalues are given by λ1 = −u2 √ ∆1 (a + u2) < 0, λ2 = −d2 + ceu2 (1 + bu2) . so λ2 > 0 if −d2 + ceu2(1+bu2) > 0 =⇒ b < b2 = ( ce d2 − 1 u2 ) , and λ2 < 0 if b > b2. therefore, e2 (u2, 0) is a saddle point for 0 < b < b2 and e2 (u2, 0) is a stable node for b > b2. summarizing the above, for the predator free equilibrium points e1 and e2 , the results are represented in table2 existence criterion equilibria nature 0 < a < a1 e1 (u1, 0) unstable node if 0 < b < b1 and saddle if b > b1 0 < a < a1 e2 (u2, 0) saddle if 0 < b < b2 and locally asymptotically stable if b > b2 table 2. existence and stability condition of predator extinction steady state of model (1.2). . from the above discussion we can conclude that if allee effect is very weak i.e. value of the parameter a is smaller than threshold value a1, then predator free equilibrium points exit. in addition to it also if the handling time of predator b is rather greater than a threshold value, then e1 is a saddle point (b > b1) and e2 is locally asymptotically stable (b > b2). the quadrant i is separated into two regions, attraction regions of e0 and e2, respectively. on the other hand if the handling time of predator b is rather smaller than a threshold value, then e1 is unstable (0 < b < b1) and e2 is saddle (0 < b < b2). if system (2) has no coexisting equilibrium point then based on on the level of allee, handling time of the predator and initial population density, either predator species will become extinct (when trajectory tends to e2 ) or both prey and predator populations will go to extinct (when trajectory tends to e0). now to determine the stability of e∗ (u∗,v∗), we have to consider the sign of determinant and trace of j at (u∗,v∗) . it is known that the system will be stable if tr < 0 and det > 0. from (3.1) for e∗ (u∗,v∗), we get, j (u∗,v∗) =   ru∗(2a+u∗) (a+u∗)2(1+fv∗) −d1 − 2ku∗ − ev ∗ (1+bu∗)2 −rfu∗2 (a+u∗)(1+fv∗)2 − eu ∗ (1+bu∗) cev∗ (1+bu∗)2 −d2 − 2hv∗ + ceu ∗ (1+bu∗)   =:   j11 j12 j21 j22   (3.5) 86 d. pal, d. kesh, and d. mukherjee where j12 = −rfu∗2 (a + u∗) (1 + fv∗) 2 − eu∗ (1 + bu∗) < 0, j21 = cev∗ (1 + bu∗) 2 > 0, j22 = −d2 − 2hv∗ + ceu∗ (1 + bu∗) = −hv∗ < 0 (by second equation of (2.2) ), j11 = ru∗ (2a + u∗) (a + u∗) 2 (1 + fv∗) −d1 − 2ku∗ − ev∗ (1 + bu∗) 2 = u∗ ( ra (a + u∗) 2 (1 + fv∗) −k + bev∗ (1 + bu∗) 2 ) (by first equation of (2.2) ). the characteristic equation is, λ2 − (j11 + j22) λ + (j11j22 −j12j21) = 0. where det j= (j11j22 −j12j21) and trj = (j11 + j22) = v ∗ ( ae (1 + bu∗) (a + u∗) + beu∗ (1 + bu∗) 2 ) − k(u∗)2 −ad1 a + u∗ −hv∗ by equation (2.2). we first consider the stability condition on trj. for stability of the system we must have trj < 0, that is, v∗ ( ae (1 + bu∗) (a + u∗) + beu∗ (1 + bu∗) 2 ) < ( k(u∗)2 −ad1 a + u∗ + hv∗ ) =⇒ v∗ ( ae (1 + bu∗) (a + u∗) + beu∗ (1 + bu∗) 2 ) + ad1 a + u∗ < ( k(u∗)2 a + u∗ + hv∗ ) . (3.6) then from the equation (3.6), we have e < e∗ where e∗ = (1 + bu∗) 2 [ (a + u∗) hv∗ + ( ku∗2 −ad1 )] v∗ (a + 2abu∗ + bu∗2) . (3.7) thus, trj < 0 if e < e∗. we next consider the stability condition on detj. it is known that for e∗ to be stable, we must have detj > 0, that is, ⇒ j11j22 −j12j21 > 0. (3.8) since (–j12j21) > 0 , so equation (3.8) holds good in two cases below. case-i: j11j22 > 0. since j22 < 0, the above holds if j11 < 0, that is, rau∗ (a + u∗) 2 (1 + fv∗) + beu∗v∗ (1 + bu∗) 2 < ku∗. (3.9) case-ii: j11j22 < 0 and in addition |j11j22| < |j12j21|. since j22 < 0, the above holds if j11 > 0 and |j11j22| < |j12j21|. =⇒ rau ∗ (a+u∗)2(1+fv∗) + beu ∗v∗ (1+bu∗)2 > ku∗ and |j11j22| < |j12j21|. (3.10) diffusion-driven instability and pattern formation in a prey-predator system 87 summarizing the above conditions on trj and det j, we conclude that e∗ (u∗,v∗) is stable when e < e∗ and equation (3.9) or (3.10) hold. 4. hopf bifurcation in the study of continuous prey-predator models, bifurcation is possibly the most crucial topic. here, we use the predator-induced fear level, f, as a bifurcation parameter in the investigation of the hopf bifurcation of the system (1.2). for hopf bifurcation, we need to find out a critical value of f = f1 such that (1) trj = j11 + j22 = 0 at f = f1. (2) detj = j11j22–j12j21 > 0 at f = f1. the characteristic equation of j (u∗,v∗) takes the form λ2 + detj = 0 at f = f1 whenever e∗ (u∗,v∗) is feasible. as a result, two eigenvalues are conjugate to one another and are purely imaginary. (3) if the eigenvalues of (3.1) around e∗ (u∗,v∗) have the general form λ = λ1 +iλ2 , then dλ1 df |f=f1 6= 0. the characteristic equation at e∗ is given by λ2 − (trj) λ + detj = 0. equating the real and imaginary parts of the both sides, we get( λ21 −λ 2 2 ) − (trj) λ1 + detj = 0, 2λ1λ2 − (trj) λ2 = 0. (4.1) differentiating the aforementioned equation with respect to the bifurcation parameter f, we obtain (2λ1 − trj) dλ1 df − 2λ2 dλ2 df −λ1 d df (trj) + d df (detj) = 0, 2λ2 dλ1 df + (2λ1 − trj) dλ2 df −λ2 d df (trj) = 0. (4.2) now eliminating dλ2 df from the above two equation of (4.2) we obtain dλ1 df as dλ1 df = 1 p 2 + 4λ22 [ p ( λ1 d df (trj) − d df (detj) ) + 2λ22 d df (trj) ] , where p = (2λ1 − trj). using the facts trj|f=f1 = 0 and λ1|f=f1 = 1 2 trj|f=f1 = 0, we have, dλ1 df |f=f1 = 1 2 d df (trj) |f=f1 = 1 2 d df [( u∗ ( ra (1 + fv∗) (a + u∗) 2 + bev∗ (1 + bu∗) 2 ) −k ) −hv∗ ] f=f1 6= 0. as a result, the system admits a hopf bifurcation near e∗ (u∗,v∗) at f = f1, where f1 is the positive solution of the equation[( u∗ ( ra (1 + fv∗) (a + u∗) 2 + bev∗ (1 + bu∗) 2 ) −k ) −hv∗ ] f=f1 = 0. in this section jacobian j is calculated at e∗ (u∗,v∗). 88 d. pal, d. kesh, and d. mukherjee 5. numerical simulation now we examine the analytical results through numerical simulation by taking different values of parameters. here we fix seven parameters out of ten as b = 0.9, c = 0.6, e = 0.85, a = 0.1, k = 0.1, d1 = 0.9, d2 = 0.2. example 1 : set h = 0.18, f = 1.2, r = 1. for these choice of parameters e0 (0, 0) becomes stable (fig. 2). (a) phase diagram of prey and predator (b) phase diagram of prey and predator figure 2. for h = 0.18, f = 1.2, r = 1 with initial value (1, 0.3) example 2 : choose h = 0.18, f = 1.2, r = 1.5. then 0 < a < a1 = 2.7606 which implies that two predator free equilibrium points e1 (0.1567, 0) and e2 (5.7433, 0) exist along with the extinct equilibrium point e0 (0, 0). since b1 = −3.8316 and b2 = 2.3759 so from table 2 we conclude that e1 and e2 both are saddle points hence the trajectory goes to (0, 0) (fig. 3). moreover if we increase the value of k from 0.1 to 0.7 then 0 < a < a1 = 0.3944. hence two predator free equilibrium points e1 (0.2571, 0) and e2 (0.5, 0) exist along with the extinct equilibrium point e0 (0, 0). here b1 = −1.3395 and b2 = 0.55 so table 2 concludes that e0 and e2 are locally asymptotically stable and e1 is saddle. the attraction zones of e0 and e2 divide the first quadrant into two sections. both populations will become extinct as the initial population of prey lies in the attraction region of e0. whereas the predator population will only begin to decline when initial population of prey lies in the attraction region of e2 instead of e0. diffusion-driven instability and pattern formation in a prey-predator system 89 (a) phase diagram of prey and predator (b) phase diagram of prey and predator figure 3. for h = 0.18, f = 1.2, r = 1.5, k = 0.1 with initial value (1, 0.01) (a) phase diagram of prey and predator (b) phase diagram of prey and predator figure 4. for h = 0.18, f = 1.2, r = 1.5, k = 0.7 with initial value (1, 0.01) (blue) and (1, 0.1) (red). now if we also fixed h = 0.2, r = 2.2 then for increasing values of level of fear f, the following dynamics will obtain. example 3 : for f = 1.05 the system (1.2) has an equilibrium point e∗(1.4339, 0.5963). here e∗ = 1.1893, hence e < e∗ so e∗ becomes stable (fig. 5). 90 d. pal, d. kesh, and d. mukherjee (a) phase diagram of prey and predator (b) phase diagram of prey and predator figure 5. for h = 0.2, f = 1.05, r = 2.2 with initial value (1, 0.4) example 4: for f = 1.84 the system (1.2) possesses unique equilibrium point e∗(1.0658, 0.3872). in this case, the criterion of hopf bifurcation are fulfilled i.e. trj = 0 at f = f1, detj > 0 at f = f1 and dλ1 df |f=f1 6= 0 (f1 = 1.84) and the model admits stable limit cycle with increasing f (figs. 6 and 7). (a) phase diagram of prey and predator (b) phase diagram of prey and predator figure 6. for h = 0.2, f = 1.84, r = 2.2 with initial value (1, 0.4) diffusion-driven instability and pattern formation in a prey-predator system 91 (a) prey(u) vs fear effect(f) (b) predator(v) vs fear effect(f) (c) 3-d picture figure 7. hopf bifurcation example 5 : for f = 2.2 the system (1.2) has unique equilibrium point e∗(0.9906, 0.3354). here e∗ = 0.7466 hence e > e∗ and e∗ is an unstable source and the trajectory goes to a limit cycle (fig. 8). (a) phase diagram of prey and predator (b) phase diagram of prey and predator figure 8. for h = 0.2, f = 2.2, r = 2.2 with initial value (1, 0.365) 92 d. pal, d. kesh, and d. mukherjee 6. model with diffusion in this part, we consider the model (1.3) and examine the impact of diffusion about e∗ (u∗,v∗), subject to non-negative initial conditions: u (x,y, 0) = u0 (x,y) and v (x,y, 0) = v0 (x,y) for (x,y) ∈ ω, and zero flux boundary conditions:( ∂ ∂x u (x,y,t) ) |x=0 = 0, ( ∂ ∂x u (x,y,t) ) |x=l = 0,( ∂ ∂y u (x,y,t) ) |y=0 = 0, ( ∂ ∂y u (x,y,t) ) |y=l = 0,( ∂ ∂x v (x,y,t) ) |x=0 = 0, ( ∂ ∂x v (x,y,t) ) |x=l = 0,( ∂ ∂y v (x,y,t) ) |y=0 = 0, ( ∂ ∂y v (x,y,t) ) |y=l = 0. 6.1. diffusion-driven instability. the formation of patterns can occur for a variety of reasons, but the focus of this article is on the pattern formation caused by diffusion-driven instability. the criterion for stability of uniform steady state e∗ (u∗,v∗) without diffusion are given below: tr (j (u∗,v∗)) = (j11 + j22) < 0 and det (j (u ∗,v∗)) = j11j22–j12j21 > 0. now to determine the turing instability we linearize the system (1.3) ( we expand u and v in spatial domain as u (x,y,t) = u∗ + u1 (x,y,t) and v (x,y,t) = v ∗ + v1 (x,y,t) where |u1| << u∗ and |v1| << v∗ and ignoring all the non linear terms in u and v). patterns are time independent and spatially heterogeneous solution of system (1.3). let us take the solution is separable, so take u1 (x,y,t) = a (t) e i(kxx+kyy), and v1 (x,y,t) = b (t) e i(kxx+kyy), where k = √ k2x + k 2 y denotes the wave number of the solution. the additional two diffusive terms which are incorporated in model (1.2) to get model (1.3) are d1 ( ∂2 ∂x2 ( a (t) ei(kxx+kyy) ) + ∂2 ∂y2 ( a (t) ei(kxx+kyy) )) = −d1k2 ( a (t) ei(kxx+kyy) ) and d2 ( ∂2 ∂x2 ( b (t) ei(kxx+kyy) ) + ∂2 ∂y2 ( b (t) ei(kxx+kyy) )) = −d2k2 ( b (t) ei(kxx+kyy) ) . then the corresponding jacobian matrix for the system (1.3) is given by j (k) =   j11 −d1k2 j12 j21 j22 −d2k2   . the characteristic equation is, µ2 − tr (j (k)) µ + det (j (k)) = 0. where tr (j (k)) = j11 −d1k2 + j22 −d2k2 = tr (j (u∗,v∗)) − (d1 + d2) k2 diffusion-driven instability and pattern formation in a prey-predator system 93 and det (j (k)) = ( j11 −d1k2 )( j22 −d2k2 ) −j12j21 = det (j (u∗,v∗)) − (d1j22 + d2j11) k2 + d1d2k4. applying routh-hurwitz criteria from the characteristic equation we can conclude that e∗ (u∗,v∗) is locally asymptotically stable for all k > 0 if tr (j (k)) = tr (j (u∗,v∗)) − (d1 + d2) k2 < 0 and det (j (k)) = det (j (u∗,v∗)) − (d1j22 + d2j11) k2 + d1d2k4 > 0. from the stability of diffusion free system we get that tr (j (u∗,v∗)) < 0 and since d1,d2 and k are all positive so tr (j (k)) < 0 always. since in case of forming turing pattern the system have to be unstable with respect to spatially heterogeneous perturbations. so the condition to induce turing instability is that, for some k > 0, det (j (k)) = det (j (u∗,v∗)) − (d1j22 + d2j11) k2 + d1d2k4 < 0. (6.1) now we take derivative to det (j (k)) with respect to k2 and equate to zero so that we can obtain following critical value k2c = d1j22 + d2j11 2d1d2 . to get such feasible critical value k2c , the necessary condition is d1j22 + d2j11 > 0. for this critical value k2c , det (j (k)) will be minimum and that minimum value is obtained by putting the value of k2c in the expression of det (j (k)) in (6.1) and that minimum value is thus obtained is, det (j (kc)) = det (j (u ∗,v∗)) − (d1j22 + d2j11) 2 4d1d2 . for turing instability the minimum value of det (j (k)) i.e det (j (kc)) must also be negative when d1j22 + d2j11 > 2 √ d1d2 (det (j (u∗,v∗))) =⇒ d1j22 + d2j11 > 2 √ d1d2 (j11j22–j12j21). now we are in a position to list the conditions for diffusion-driven instability and the list is as follows: (1) j11 + j22 < 0, (2) j11j22–j12j21 > 0, (3) d1j22 + d2j11 > 0, (4) d1j22 + d2j11 > 2 √ d1d2 (j11j22–j12j21). 6.2. numerical simulation. in this section, in support of our analytic and theoretical finding, we describe numerical observations of system (1.3) in ω ∈ r2. to simulate system (1.3), we must employ the forward time centered space (ftcs) technique to convert an infinite dimensional continuous model to a finite dimensional discrete model. simulation is performed to the model (1.3) for all (x,y) ∈ ω = [0,l] × [0,l] where δt = 0.05 (time step size) and δx = 0.5, δy = 0.5 (space step sizes) and initial data with a random perturbations around e∗ (u∗,v∗). numerical trials are conducted till the pattern achieves a stationary state, at which time the system’s behaviour ceases to change. by studying several sorts of spatial dynamics using numerical evaluations, we discovered that the distributions of both species are consistently identical. other parameters value are taken as d1 = 0.9, d2 = 0.2, k = 0.1, b = 0.9, e = 0.85, c = 0.6, r = 2.2, a = 0.1, h = 0.2, d1 = 0.01, d2 = 1.2 and different input parameter f are chosen as 1.05, 1.20, 1.35, 1.50, 1.65 and 1.80 and in all the cases the conditions of diffusion driven instability are satisfied. figs. 9 and 10 display respective spatial patterns of prey and predator species around e∗ (u∗,v∗) in 2d space. an interesting finding from figs. 9 and 10 is that the bounds for prey 94 d. pal, d. kesh, and d. mukherjee population u and predator population v over 2d space is varying with the change of system parameter f. spots pattern (h -pattern) is generated for prey species, while for predator species, larger values of f (f = 1.65,f = 1.8) form a spot dominated combination of spots and very short stripes. we now provide some biological justifications for the trend shown in the aforementioned figures. for prey, the spots pattern consists of blue (low population density) quadrilaterals on a red (high population density) backdrop, indicating that the prey species are concentrated in isolated areas having low population density, whereas the rest of the region is densely populated. (a) f = 1.05 (b) f = 1.2 (c) f = 1.35 (d) f = 1.5 (e) f = 1.65 (f) f = 1.8 figure 9. evolution of prey species for d1 = 0.9, d2 = 0.2, k = 0.1, b = 0.9, e = 0.85, c = 0.6, r = 2.2, a = 0.1, h = 0.2, d1 = 0.01, d2 = 1.2 and different values of f. 7. discussion the present investigation deals with a predator-prey system with allee and fear effects on prey and holling type-ii functional response. intra-specific competition among predators is also taken under consideration. we extend the concept of generalized predator–prey model to spatially extended systems, in which the motion can be described by a diffusion term to the system (1.2). firstly, we have studied the model without diffusion and have found the equilibrium points and deduce that interior equlibrium point is stable when consumption rate of predator, e less than a threshold value, e∗ and equation (3.9) or (3.10) hold. then we have examined the existence of hopf bifurcation and limit cycle taking level of fear, f as bifurcation parameter (figs. 58). xie [32] discussed system (1.1) but the difference between the system (1.1) and (1.2) is that intraspecific competition among predators, h is considered in system (1.2). consideration of h does not affect the trivial and axial equilibrium points, it only affects the stability of interior equilibrium point e∗ and it is illustrated by an example. if we take h = 0 and the other parameters value same as example 3 in section 5, then simulation result, fig. 11 shows that if h is not taken into consideration (for system diffusion-driven instability and pattern formation in a prey-predator system 95 (a) f = 1.05 (b) f = 1.2 (c) f = 1.35 (d) f = 1.5 (e) f = 1.65 (f) f = 1.8 figure 10. evolution of predator species for d1 = 0.9, d2 = 0.2, k = 0.1, b = 0.9, e = 0.85, c = 0.6, r = 2.2, a = 0.1, h = 0.2, d1 = 0.01, d2 = 1.2 and different values of f. (a) for h = 0.2 (b) for h = 0 figure 11. the effect of intra-specific competition among predators on the stability of interior equilibrium point. for f = 1.05, r = 2.2, b = 0.9, c = 0.6, e = 0.85, a = 0.1, k = 0.1, d1 = 0.9, d2 = 0.2. (1.1) ), e∗ becomes unstable and trajectory goes to extinct. but when h is considered (for system (1.2)), for the same value of parameters, e∗ becomes stable and both species exist simultaneously. so the stability of e∗ is affected when intra-specific competition among predators is considered. another numerical simulation in fig. 12 shows that for various initial populations, in absence of h the trajectory 96 d. pal, d. kesh, and d. mukherjee always goes to (0, 0) i.e. both the population goes to extinction but in presence of h, for some initial populations (red coloured region in fig. 13) trajectory goes to (0, 0) and for some initial populations ( blue coloured region in fig. 13) trajectory goes to e∗(1.4339, 0.5963) and both the populations survive. so the region is divided into two parts, attraction region of e0 and attraction region of e ∗. hence interior equilibrium point becomes stable when intra-specific competition among predators is considered. (a) for h = 0.2 (b) for h = 0 figure 12. role of h on stability of interior equilibrium point for the parameters set in example 3 of section 4 with various initial points (1, 0.4), (0.8, 0.8), (0.3, 0.8), (1, 1) and (1.9, 0.9). we then introduced diffusion in the model and spatial pattern generation mentioned in sect. 5, the conditions necessary for diffusion-driven instability and stimulating turing pattern formation corresponding to the reaction-diffusion system have been achieved successfully. we have explored the presence of turing patterns by varying the control parameter f. the contour image of the spatial patterns for preys and predators are depicted in figs. 9 and 10 which show the temporal evolution of interacting populations at t = 10000 for various values of f. the color-bars represent that the density of the prey population drops with rising fear level which supports the fact that as level of fear increases, prey moving away from the predator, but all prey responds to predation danger in different ways, including habitat changes, feeding, reduced mating, vigilance, and various physiological changes. such anti-predator efforts may lower reproduction in the long run, but they are immediately helpful since they raise the likelihood of adult survival. in addition, fearful prey tend to forage less, which reduces their growth and reproduction rate of prey. as a result due to less-availability of food, predator population density also decreases. we observe that intra-specific competition among predators has also crucial role in the formation of turing patterns. conditions (1) and (3) of section 6.1 indicate that j11 and j22 must be of opposite signs for turing instability. but in absence of h that is for system (1.1), j22 = 0 hence diffusion driven instability does not occur in system (1.1) in presence of self-diffusion only. but in presence of h turing patterns appear for suitable choice of parameters. hence h operates as an underlying mechanism capable of generating non-uniform spatial distributions of predators and prey via diffusion-driven instability. but the considered model (1.2) also have some limitations which also give us scopes for future works such as birth rate of the prey, r is considered to be constant, not density dependent. these kind of diffusion-driven instability and pattern formation in a prey-predator system 97 figure 13. basin of attraction of equilibrium points e0 (red coloured region) and e∗ (blue coloured region) for parameter values b = 0.9, k = 0.1, r = 2.2, a = 0.1, e = 0.85, h = 0.2, f = 1.05, d1 = 0.9, d2 = 0.2, c = 0.6. assumptions are proper for some unicellular and bacterial species. the species which suffer from the fear effect and the population suffers from decline in their growth, growth rate is density dependent. so one can look into the dynamics of the system considering density dependent growth rate of prey. in the non-spatial model, we use 1 1+fv to represent the fear effect, however, in a 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email address: dkeshju@gmail.com d. mukherjee, department of mathematics, vivekananda college, thakurpukur, kolkata-700063, india email address: mukherjee1961@gmail.com 1. introduction 2. equilibrium points 3. local stability analysis of equilibrium points: 4. hopf bifurcation 5. numerical simulation 6. model with diffusion 6.1. diffusion-driven instability 6.2. numerical simulation 7. discussion references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 4, number 2, june 2023, pp.128-143 https://doi.org/10.5206/mase/15658 on the weak solution of the von-karman model with thermoelastic plates jaouad oudaani and mustapha raïssouli abstract. in this article we aim to study the dynamic von-karman model coupled with thermoelastic equations without rotational terms, subject to a thermal dissipation. we establish the existence as well as the uniqueness of a weak solution related to the dynamic model. at the end, we apply the finite difference method for approximating the solution of our problem. 1. introduction nonlinear oscillation of an elastic model, for dynamic von-karman model without rotational terms, subject to a thermal dissipation [4] describes the phenomenon of small nonlinear vibration with a vertical displacement to the elastic plates. the case of nonlinear thermoelastic plate interaction coupled with thermal dissipation plays an interesting place in this subject and will be our fundamental target in the present paper. the model with clamped boundary conditions, in the note account of rotational terms, can be formulated as follows ([4]): find (u,φ,θ) ∈ l2 ( [0,t] ,h20 (ω) ) ×h20 (ω) ×h10 (ω) such that (p0)   utt + ∆ 2u + µ∆θ − [φ + f0,u] = p(x) in ω × [0,t] , kθt −η∆θ −µ∆ut = 0 in ω × [0,t] , u|t=0 = u0, (ut)|t=0 = u, θ|t=0 = θ0 in ω, u = ∂νu = 0 on γ × [0,t] , θ = 0 on γ, and (q) { ∆2φ + [u,u] = 0 in ω × [0,t] , φ = 0, ∂νφ = 0 on γ × [0,t] . here, u is the displacement, φ denotes the airy stress function and θ is the thermal function, ω is the surface plate, u0, u1, θ0 refer to the initial data and [., .] stands for the monge-ampère symbol defined through ([1]) [φ,u] = ∂11φ∂22u + ∂11u∂22φ− 2∂12φ∂12u. (1.1) the parameters µ,η > 0 are fixed real numbers and k > 0 measures the capacity of the heat/thermal. the plate is subject to the internal force f0 and the external force p. in [4], the authors studied the problem of the von-karman model for the case 0 ≤ k ≤ 1. our fundamental target in this paper is to explore a condition that should be satisfied by the external/internal loads and the initial data for ensuring the existence and the uniqueness of a weak solution for to the von-karman evolution, without rotational terms nor clamped boundary conditions, subject to the thermal dissipation when k > 0 and 0 < µ ≤ 2η. the present approach turns out of to received by the editors 31 december 2022; accepted 25 april 2023; published online 9 june 2023. 2020 mathematics subject classification. primary 74b20,74f10; secondary 74f05,74k05. key words and phrases. von-karman model, thermoelastic plate, rotational inertia, finite difference method. 128 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15658 on the weak solution of the von-karman model with thermoelastic plates 129 construct an iterative process converging, in an appropriate sense, to the unique solution of the initial problem. this paper will be organized as follows. in section 2 we present the mathematical structure of the model that will be studied in the sequel together with some basic tools and results. in section 3 we use an iterative method that will be a good tool for establishing the existence and the uniqueness of a weak solution of the dynamical plate problem without rotational terms, subject to a thermal dissipation. section 4 deals with a numerical simulation for approaching the solution of the initial problem. 2. preliminaries and main results throughout this paper, we denote by ω a nonempty bounded domain in r2 with regular boundary γ = ∂ω. we suppose that the parameters k,µ,η in the problem (p0) are such that k > 0 and 0 < µ ≤ 2η. let p ≥ 1 be a real number and m ≥ 1 be an integer. the notation |.|p,ω refers to the standard norm of lp(ω) while ‖.‖m,ω stands for the classical norm of hm(ω). for u ∈ h20 (ω), we put ‖u‖ =: |∆u|2,ω = (∫ ω (∆u)2 )1 2 , which is obviously a norm in h20 (ω) ([4, 5]). otherwise, we set ‖u‖20 =: ‖u‖ 2 + |ut| 2 2,ω . (2.1) we state the following result which will be needed in the sequel, see [7]. theorem 2.1. let f ∈ l2(ω). then the following problem (r)   ∆2v = f in ω, v = 0 on γ, ∂νv = 0 on γ, has one and only one solution v ∈ h20 (ω) ∩h4(ω) satisfying ‖v‖≤ c0 |f|2,ω , where c0 > 0 is a constant depending only on mes(ω). let us mention the following remark. remark 2.1. (i) under the condition that f ∈ l2 ( [0,t] ,l2(ω) ) , the solution of (r) belongs to the set l2 ( [0,t] ,h20 (ω) ∩h4(ω) ) . (ii) we mention again that the constant c0 > 0 in theorem 2.1 depends only on mes(ω) and does not depend on f. the following result will be needed later, see [7, 3]. theorem 2.2. let g ∈ l2 ( [0,t] ,l2(ω) ) , u0 ∈ l2(ω) and k,η,µ > 0. then the following problem : (d)   kut −η∆u = µg in ω × [0,t] , u|t=0 = u0 in ω, u = 0 on γ × [0,t] , has one and only one solution u ∈ c ( [0,t] ; h2(ω) ∩h10 (ω) ) ∩c1 ( [0,t] ; l2(ω) ) . we have the following result as well. 130 j. oudaani and m. raïssouli proposition 2.3. let f ∈ h2(ω), k > 0 and 0 < µ ≤ 2η. then the solution u of (d), when g = −∆f, satisfies the following inequality ∀t ∈ [0,t], 0 ≤ k |u|22,ω + (2η −µ) ∫ t 0 |∇u|22,ω ≤ k |u0| 2 2,ω + µ ∫ t 0 |∇f|22,ω . (2.2) proof. since u is the solution of the problem (d), with g = −∆f, then kut − η∆u = −µ∆f and so k〈ut,u〉−η〈∆u,u〉 = −µ〈∆f,u〉, where 〈., .〉 refers to the standard inner product of l2(ω). this latter equation is equivalent to k 2 d dt |u|22,ω + η |∇u| 2 2,ω = µ〈∇f,∇u〉. (2.3) by hölder inequality in l2(ω) and the standard inequality ab ≤ 1 2 a2 + 1 2 b2, valid for any a,b > 0, we can write 〈∇f,∇u〉≤ ∣∣∣〈∇f,∇u〉∣∣∣ ≤ |∇f|2,ω |∇u|2,ω ≤ 12 |∇f|22,ω + 12 |∇u|22,ω . substituting this in (2.3) we get k 2 d dt |u|22,ω + η |∇u| 2 2,ω ≤ µ 2 |∇f|22,ω + µ 2 |∇u|22,ω . integrating side by side this latter inequality with respect to t > 0, and using the fact that (u)|t=0 = u0 in ω, we obtain k 2 |u|22,ω + η ∫ t 0 |∇u|22,ω ≤ k 2 |u0| 2 2,ω + µ 2 ∫ t 0 |∇f|22,ω + µ 2 ∫ t 0 |∇u|22,ω . this implies (2.2) and the proof is finished. � the following result will be needed as well, see [4]. theorem 2.4. let f ∈ l2([0,t] ,l2(ω)) and (u0,u1) ∈ h20 (ω) ×l2(ω). then the problem (s1)   utt + ∆ 2u = f in ω × [0,t] , u = ∂νu = 0 on γ × [0,t] , u|t=0 = u0, (ut)|t=0 = u in ω, has a unique solution such that (u,ut) ∈ c0([ 0,t ] ,h20 (ω) ×l2(ω)). for the sake of simplicity, we set f1(u,φ) = [ φ + f0,u ] . (2.4) with this, the following result may be stated. proposition 2.5. let u,v ∈ h20 (ω) be with small norms and f0 ∈ h4(ω) be such that ‖f0‖4,ω < 1 4 . let φ,ϕ ∈ h20 (ω) be the solutions of ∆2φ = − [u,u] and ∆2ϕ = − [v,v], respectively. then there exists 0 < c1 < 1 such that ∣∣∣ [u,φ] − [v,ϕ] ∣∣∣ 2,ω ≤ c1‖u−v‖ and ∣∣∣f1(u,φ) −f1(v,ϕ)∣∣∣ 2,ω ≤ c1 ‖u−v‖ . proof. according to [4], we have∣∣∣ [u,φ] − [v,ϕ] ∣∣∣ 2,ω ≤ c0 ( ‖u‖2 + ‖v‖2 ) ‖u−v‖ , on the weak solution of the von-karman model with thermoelastic plates 131 for some c0 > 0 depending only on mes(ω). if we assume that ‖u‖ ≤ c and ‖v‖ ≤ c, for some c > 0 enough small, then we get ∣∣∣ [u,φ] − [v,ϕ] ∣∣∣ 2,ω ≤ 2c0c2 ‖u−v‖ . otherwise, by using (1.1) it is not hard to check that∣∣∣ [f0,u−v ] ∣∣∣ 2,ω ≤ 4‖f0‖4,ω . it follows that we have∣∣∣f1(u,φ) −f1(v,ϕ)∣∣∣ 2,ω ≤ ∣∣∣ [φ + f0,u] − [ϕ + f0,v ] ∣∣∣ 2,ω , ≤ ∣∣∣ [φ,u] − [ϕ,v ] ∣∣∣ 2,ω + ∣∣∣ [f0,u−v ] ∣∣∣ 2,ω , ≤ ( 2c0c 2 + 4‖f0‖4,ω ) ‖u−v‖ . if ‖f0‖4,ω < 1 4 and 0 < c < √ 1 − 4‖f0‖4,ω 2c0 , then 0 < 2c0c 2 < c1 =: 2c0c 2 + 4‖f0‖4,ω < 1. in summary, the proposition is completely proved. � remark 2.2. according to remark 2.1,(ii), the constant c1 in proposition 2.5 depends only on mes(ω) and ‖f0‖4,ω. now, we are in the position to state and establish the following main result. theorem 2.6. let f ∈ l2 ( [0,t],l2(ω) ) , θ0 ∈ h10 (ω) and (u0,u) ∈ h20 (ω) × l2(ω). the following problem: (s)   utt + ∆ 2u + µ∆θ = f in ω × [0,t] , kθt −η∆θ = µ∆ut in ω × [0,t] , u = ∂νu = θ = 0 on γ × [0,t] , (u)|t=0 = u0, (ut)|t=0 = u, (θ)|t=0 = θ0 in ω, has one and only one solution (u,θ) ∈ l2([ 0,t ] ,h20 (ω)×h10 (ω)) satisfying that ut ∈ l2([ 0,t ] ,l2(ω)) and, for any t ∈ [0,t], ‖u‖20 + k |θ| 2 2,ω + 2η ∫ t 0 |∇θ|22,ω ≤ e t ( ‖u0‖ 2 + |u|22,ω + k |θ0| 2 2,ω + ∫ t 0 |f|22,ω ) . (2.5) further, the so-called energy equality holds true: ‖u‖20 + 2η ∫ t 0 |∇θ|22,ω + k |θ| 2 2,ω = ‖u0‖ 2 + |u|22,ω + k |θ0| 2 2,ω + 2 ∫ t 0 ∫ ω fut. (2.6) proof. to prove our result, we will study the problem (s) by considering the nth-order approximate solution and its associate variational problem. we divide the proof into fourth steps. step 1: let { ek,e1k } be a basis in the space h 2 0 (ω) × h10 (ω). the n-order galerkin approximate solution to the problem (s1), with clamped boundary conditions on the interval [ 0,t ], is a function (un(t),θn(t)) of the form, [1, 6], un(t) = n∑ k=1 hk(t)ek and θ n(t) = n∑ k=1 lk(t)e 1 k, n = 1, 2, 3, ..., 132 j. oudaani and m. raïssouli where (hk, lk) ∈ w2,∞(0,t; r) ×w1,∞(0,t; r). let (un,φn,θn) be a solution of (p0) and (q) corresponding to the initial data (un0,θn0) and un1 such that the two following requirements are satisfied: (un0,θn0) converges to (u0,θ0) in l 2([ 0,t ] ,h20 (ω) ×h 1 0 (ω)) (2.7) (un1) converges to u in l 2([ 0,t ] ,l2(ω)). (2.8) now, let us consider the iterative problem (sn) associated to the problem (s) given by: (sn)   untt + ∆ 2un + µ∆θn = f in ω × [0,t] , kθnt −η∆θn = µ∆unt in ω × [0,t] , un = ∂νu n = θn = 0 on γ × [0,t] , (un)|t=0 = un0, (u n t )|t=0 = un1, (θ n)|t=0 = θn0 in ω, we now multiply the first equation of (sn) by unt and the second equation by θn and we then integrate both them over ω, with the help of some standard integral rules, we get  ∫ ω unttu n t + ∫ ω ∆un∆unt + µ ∫ ω ∆θnunt = ∫ ω funt k ∫ ω θnt θ n + η ∫ ω (∇θn)2 = µ ∫ ω ∆unt θ n. since (unt ,θ n) ∈ h10 (ω) ×h10 (ω) and ∫ ω ∆θnunt = ∫ ω θn∆unt , the two last equations imply that   1 2 d dt ( |unt | 2 2,ω + ‖u n‖2 ) + µ ∫ ω θn∆unt = ∫ ω funt , k 2 d dt |θn|22,ω + η |∇θ n|22,ω = µ ∫ ω θn∆unt . from these two latter equalities we deduce that we have 1 2 d dt ( |unt | 2 2,ω + ‖u n‖2 ) + k 2 d dt |θn|22,ω + η |∇θ n|22,ω = ∫ ω funt . integrating this latter equality over [0, t], and using (2.1) with the fact that un|t=0 = un0, (u n t )|t=0 = un1, θ n |t=0 = θn0, we get 1 2 ( ‖un‖20 + k |θ n|22,ω ) + η ∫ t 0 |∇θn|22,ω = 1 2 ( |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω ) + ∫ t 0 ∫ ω funt . (2.9) let s ∈ [0,t]. by the hölder inequality in l2(ω), with (2.1), we have∫ t 0 ∫ ω funt ≤ ∫ t 0 |f|2,ω |u n t |2,ω ≤ 1 2 ∫ t 0 |f|22,ω + 1 2 ∫ t 0 |unt | 2 2,ω ≤ 1 2 ∫ t 0 |f|22,ω + 1 2 ∫ t 0 ( ‖un‖20 + k |θ n|22,ω + 2η ∫ s 0 |∇θn|22,ω ) . (2.10) on the weak solution of the von-karman model with thermoelastic plates 133 combining (2.9) and (2.10), we have shown that ‖un‖20 + k |θ n|22,ω + 2η ∫ t 0 |∇θn|22,ω ≤ |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω + ∫ t 0 |f|22,ω + ∫ t 0 ( ‖un‖20 + k |θ n|22,ω + 2η ∫ s 0 |∇θn|22,ω ) . (2.11) step 2: for 0 ≤ s ≤ t, we put i(s) = ‖un‖20 + k |θ n|22,ω + 2η ∫ s 0 |∇θn|22,ω . the inequality (2.11), yields i(s) − ∫ s 0 i(z)dz = ‖un‖20 + k |θ n|22,ω + 2η ∫ s 0 |∇θn|22,ω − ∫ s 0 ( ‖un‖20 + k |θ n|22,ω + 2η ∫ z 0 |∇θn|22,ω ) dz ≤ |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω + ∫ t 0 |f|22,ω . it follows that d ds ( e−s ∫ s 0 i(z)dz ) = e−s ( i(s) − ∫ s 0 i(z)dz ) ≤ e−s ( |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω + ∫ t 0 |f|22,ω ) . (2.12) now, if we remark that |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω + ∫ t 0 |f|22,ω = i(0) + ∫ t 0 |f|22,ω does not depend on s, and we integrate (2.12) over [0, t], then we get∫ t 0 d ds ( e−s ∫ s 0 i(z)dz ) ds ≤ (∫ t 0 e−sds )( |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω + ∫ t 0 |f|22,ω ) , from which we deduce e−t ∫ t 0 i(z)dz ≤ (1 −e−t) ( |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω + ∫ t 0 |f|22,ω ) . it follows that∫ t 0 i(z)dz ≤ (1−e −t) e−t ( |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω + ∫ t 0 |f|22,ω ) = (et − 1) ( |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω + ∫ t 0 |f|22,ω ) ≤ (et − 1) ( |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω + ∫ t 0 |f|22,ω ) . this, with (2.11), yields ‖un‖20 + k |θ n|22,ω + 2η ∫ t 0 |∇θn|22,ω ≤ ( |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω + ∫ t 0 |f|22,ω ) + (et − 1) ( |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω + ∫ t 0 |f|22,ω ) , and therefore ‖un‖20 + k |θ n|22,ω + 2η ∫ t 0 |∇θn|22,ω ≤ e t ( |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω + ∫ t 0 |f|22,ω ) . (2.13) 134 j. oudaani and m. raïssouli according to (2.7) and (2.8), the sequences (un0,θn0) and (un1) are, respectively, bounded in the spaces l2 ( [0,t] ,h20 (ω) × h10 (ω) × l2(ω) ) and l2 ( [0,t] ,l2(ω) × l2(ω) ) . this, with (2.13), imply that the sequences (un,θn) and (unt ) are also bounded, respectively, in l 2 ( [0,t] ,h20 (ω) × h10 (ω) × l2(ω) ) and l2 ( [0,t] ,l2(ω) × l2(ω) ) . these latter banach spaces are reflexive and therefore there exists a subsequence (unl,θnl ) such that (unl,θnl ) ⇀ (u,θ) weakly in l2 ( [0,t] ,h20 (ω) × l2(ω) ) and( (unl )t,∇θnl ) ⇀ ( (u)t,∇θ ) weakly in l2 ( [0,t] ,l2(ω) ×l2(ω) ) . step 3: in this step, we will establish that (u,θ), previously defined, is a weak solution of the problem (s), by following the same way as in [8]. let ϕj ∈ c1(0,t) be such that ϕj(t) = 0 for any 1 ≤ j ≤ j0, and we set ψ = j0∑ j=1 ψj ⊗ej, ϕ = j0∑ j=1 ϕj ⊗e1j. according to (sn), with further elementary manipulations and operations, we may infer that − ∫ t 0 ∫ ω u nl t ψt + µ ∫ t 0 ∫ ω ∇θnl∇ψ + ∫ t 0 ∫ ω ∆unl ∆ψ = ∫ t 0 ∫ ω fψ − ∫ ω unl1ψ(0) (2.14) and ∫ t 0 ( −k ∫ ω θnlϕt + η ∫ ω ∇θnl∇ϕ−µ ∫ ω ∇unl∇ϕt ) = −k ∫ ω θnl0ϕ(0) −µ ∫ ω ∇unl1∇ϕ(0). (2.15) letting nl → +∞ in (2.14) and (2.15) we deduce that the two following equalities − ∫ t 0 ∫ ω utψt + µ ∫ t 0 ∫ ω ∇θ∇ψ + ∫ t 0 ∫ ω ∆u∆ψ = ∫ t 0 ∫ ω fψ − ∫ ω uψ(0) and ∫ t 0 ( −k ∫ ω θϕt + η ∫ ω ∇θ∇ϕ−µ ∫ ω ∇u∇ϕt ) = −k ∫ ω θ0ϕ(0) −µ ∫ ω ∇u∇ϕ(0), hold true for all ψ ∈ l2([0,t],h20 (ω)), with ψt ∈ l2([0,t],h1(ω)) and ϕ ∈ l2([0,t],h10 (ω)), with ϕt ∈ l2([0,t],l2(ω)), such that ψ(t) = ϕ(t) = 0. this means that (u,θ) is a weak solution of the problem (s). further, by analogous way as for proving (2.13), we may show that for all t ∈ [0,t], we have the following inequality ‖u‖20 + k |θ| 2 2,ω + 2η ∫ t 0 |∇θ|22,ω ≤ e t ( |u|22,ω + ‖u0‖ 2 + k |θ0| 2 2,ω + ∫ t 0 |f|22,ω ) . (2.16) step 4: we now show the uniqueness. let (u1,θ1) and (u2,θ2) are two solutions of (s). then (u1 −u2,θ1 −θ2) is a solution of the following problem  (u1 −u2)tt + ∆2(u1 −u2) + µ∆(θ1 −θ2) = 0 in ω × [0,t] , k(θ1 −θ2)t −η∆(θ1 −θ2) = µ∆(u1 −u2)t in ω × [0,t] , θ1 −θ2 = u1 −u2 = ∂ν(u1 −u2) = 0 on γ × [0,t] , (u1 −u2)|t=0 = 0, ((u1 −u2)t)|t=0 = 0, (θ1 −θ2)|t=0 = 0 in ω. according to (2.16) we have ‖u1 −u2‖0 + k |θ1 −θ2| 2 2,ω + 2η ∫ t 0 |∇(θ1 −θ2)| 2 2,ω ≤ et ( |u1 −u2| 2 2,ω + ‖(u1)0 − (u2)0‖ 2 + k |(θ1)0 − (θ2)0| 2 2,ω ) . then we deduce u1 = u2 and θ1 = θ2. finally, by (2.9) we have for all n ≥ 0 1 2 ( ‖un‖0 + k |θ n|22,ω ) + η ∫ t 0 |∇θn|22,ω = 1 2 ( |un1| 2 2,ω + ‖un0‖ 2 + k |θn0| 2 2,ω ) + ∫ t 0 ∫ ω funt , on the weak solution of the von-karman model with thermoelastic plates 135 from which by letting n → +∞ we get (2.6), so completing the proof. � 3. iterative approach and convergence result to establish the existence and the uniqueness of a weak solution for (p0), without rotational terms i.e. α = 0, we will use the following iterative approach. let n ≥ 2 and let 0 6= u1 ∈ h20 (ω) be given. we define φn−1 ∈ h20 (ω) as the unique solution of ∆2φn−1 = − [un−1,un−1] and (un,θn) as the solution of the following problem : (pn)   (un)tt + ∆ 2un = f(un−1,φn−1,θn) in ω × [0,t] , k(θn)t −η∆θn = µ∆(un)t in ω × [0,t] , un = ∂νun = θn = 0 on γ × [0,t] , (un)|t=0 = u0, ((un)t)|t=0 = u, (θn)|t=0 = θ0 in ω, where we set f(u,φ,θ) = f1(u,φ) −µ∆θ + p and, f1 is defined by (2.4). the main result of this section is recited in the following. theorem 3.1. let p ∈ l2(ω), (u0,u) ∈ h20 (ω) × l2(ω) and θ0 ∈ h10 (ω). we suppose that all the following norms ‖f0‖4,ω , |p|2,ω , ‖u0‖ 2 , |u|22,ω and ‖θ0‖ 2 1,ω are small enough, and 0 < µ ≤ 2η. then the problem (p0), without rotational forces, has one and only one weak solution (u,φ,θ) in l2 ( [0,t] ,h20 (ω) ×h20 (ω) ×h10 (ω) ) such that ut ∈ l2 ( [0,t] ,l2(ω) ) . proof. we divide it into four steps. step 1: let us consider the problem (pn) with 0 6= u1 does not depend on t. for the sake of simplicity we use the notation ‖(u,θ)‖∗ = ‖u‖ 2 0 + k |θ| 2 2,ω + 2η ∫ t 0 |∇θ|22,ω , where, ‖.‖0 is defined by (2.1). let c0 > 0 be the constant defined by proposition 2.5. for ‖f0‖4,ω < 1 4 we choose c =: c(‖f0‖4,ω ,c0,t) > 0 such that 0 < 4c0c < 1, and 0 < c < √ 1 − 4‖f0‖4,ω 2c0 . we also choose u1 (independent on t) such that 0 < ‖u1‖2,ω < c < 1. by using an induction method, we will prove that the two following inequalities ‖u‖20 =: ‖un‖ 2 + |(un)t| 2 2,ω ≤‖u1‖ 2 2,ω and ‖φn‖2,ω ≤‖u1‖2,ω (3.1) are satisfied for all n ≥ 1 and t ∈ [0,t]. since u1 does not depend on t, then we have ‖u1‖ 2 0 =: ‖u1‖ 2 + |(u1)t| 2 2,ω = ‖u1‖ 2 2,ω . now, let φ1 be the solution of ∆ 2φ1 = − [ u1,u1 ]. theorem 2.1 tells us that there exists c0 > 0 such that ‖φ1‖2,ω ≤ c0 |[ u1,u1 ]|1,ω , and by using the same way as in the proof of proposition 2.5, with ‖u1‖2,ω < c and 0 < 4c0c < 1, we may deduce that ‖φ1‖2,ω ≤ 4c0 ‖u1‖ 2 2,ω ≤ 4c0c‖u1‖2,ω ≤‖u1‖2,ω . hence, the inequalities (3.1) are satisfied for n = 1. assume that for k = 2, ...,n and t ∈ [0,t], we have ‖uk‖ 2 0 ≤‖u1‖ 2 2,ω and ‖φk‖2,ω ≤‖u1‖2,ω . 136 j. oudaani and m. raïssouli according to theorem 2.1 and proposition 2.5, with remark 2.1 and remark 2.2, we have ‖φn‖2,ω ≤ c0 |[ un,un ]|1,ω ≤ 4c0 ‖un‖ 2 ≤ 4c0c‖un‖≤ c1 ‖un‖ . since un+1 is a solution of (pn+1), proposition 2.6, proposition 2.5 and theorem 2.1 imply that, there exist c1 > 0, with 0 < c1 =: 2c0c 2 + 4‖f0‖4,ω < 1, (3.2) such that ‖(un+1,θn+1)‖∗ ≤ e t ( ‖u0‖ 2 + k |θ0| 2 2,ω + |u| 2 2,ω + ∫ t 0 |f1(un,φn) + p|22,ω ) ≤ et ( ‖u0‖ 2 + k |θ0| 2 2,ω + |u| 2 2,ω + 2 ∫ t 0 ( |f1(un,φn)|22,ω + |p| 2 2,ω )) ≤ et ( ‖u0‖ 2 + k |θ0| 2 2,ω + |u| 2 2,ω + 2c 2 1 ∫ t 0 ‖un‖ 2 + 2t |p|22,ω ) ≤ et ( ‖u0‖ 2 + |u|22,ω + k |θ0| 2 2,ω + 2tc 2 1 ‖u1‖ 4 2,ω + 2t |p| 2 2,ω ) . this, with the fact that 0 < c1 < 1, ‖u1‖ < 1 and c21 ‖u1‖ 4 2,ω ≤ c1 ‖u1‖ 2 2,ω, implies that ‖(un+1,θn+1)‖∗ ≤ e t ( ‖u0‖ 2 + |u|22,ω + k |θ0| 2 2,ω + 2tc1 ‖u1‖ 2 2,ω + 2t |p| 2 2,ω ) . if we choose c > 0 and ‖f0‖4,ω small enough then c1 defined by (3.2) is also small enough and so 0 < c2 =: 2te tc1 < 1. we can then write ‖(un+1,θn+1)‖∗ ≤ e t ( ‖u0‖ 2 + |u|22,ω + k |θ0| 2 2,ω + 2t |p| 2 2,ω ) + c2 ‖u1‖ 2 2,ω . (3.3) in another part we can write ‖u0‖ 2 + |u|22,ω + k |θ0| 2 2,ω + 2t |p| 2 2,ω ≤ (1 − c2) et ‖u1‖ 2 2,ω , (3.4) since the left quantity of this inequality was assumed to be small enough. otherwise, it is not hard to check that ‖un+1‖ 2 0 =: ‖un+1‖ 2 + |(un+1)t| 2 2,ω ≤‖(un+1,θn+1)‖∗ (3.5) and ‖φn‖2,ω ≤ c1 ‖un‖2,ω ≤‖u1‖2,ω . according to (3.3), (3.5) and (3.5) we deduce that we have ‖un+1‖ 2 0 ≤ e t ( ‖u0‖ 2 + |u|22,ω + k |θ0| 2 2,ω + 2t |p| 2 2,ω ) + c2 ‖u1‖ 2 2,ω ≤ et (1 − c2) et ‖u1‖ 2 2,ω + c2 ‖u1‖ 2 2,ω = ‖u1‖ 2 2,ω . furthermore, we have ‖φn+1‖2,ω ≤ c0 |[ un+1,un+1 ]|1,ω , which with, ‖u1‖2,ω < c and 0 < 4c0c < 1, immediately yields ‖φn+1‖2,ω ≤ 4c0 ‖un+1‖ 2 ≤ 4c0 ‖u1‖ 2 2,ω ≤ 4c0c‖u1‖2,ω ≤‖u1‖2,ω . in summary, we have shown that for all n ≥ 1 and any t ∈ [0,t] one has ‖un‖ 2 0 ≤‖u1‖ 2 2,ω and ‖φn‖2,ω ≤‖u1‖2,ω . moreover we have k |θn| 2 2,ω + 2η ∫ t 0 |∇θn| 2 2,ω ≤‖(un,θn)‖∗ ≤‖u1‖ 2 2,ω . on the weak solution of the von-karman model with thermoelastic plates 137 step 2: for n ≥ 2, let (un,θn) be a solution of (pn). let 2 ≤ m ≤ n. it is not hard to see that θnm =: θn −θm and unm =: un −um satisfy the following:  unmtt + ∆ 2unm + µ∆θnm = f1(un−1,φn−1) −f1(um−1,φm−1) in ω × [0,t] , kθnmt −η∆θnm = µ∆(unm)t in ω × [0,t] , unm = θnm = ∂νu nm = 0 on γ × [0,t] , (unm)|t=0 = ((u nm)t)|t=0 = ((θ nm)t)|t=0 = 0 in ω. according to proposition 2.5 and theorem 2.1 we deduce that ‖φn−1 −φm−1‖2,ω ≤ 4c0c‖un−1 −um−1‖ . using proposition 2.6 and proposition 2.5 again we have ‖(un −um,θn −θm)‖∗ ≤ e t ∫ t 0 |f1(un−1,φn−1) −f1(um−1,φm−1)| 2 2,ω ≤ c1et ∫ t 0 ‖un−1 −um−1‖ 2 . this, with 0 < c3 = e tc1 < 1 t , yields ‖(un −um,θn −θm)‖∗ ≤ c3 ∫ t 0 ‖(un−1 −um−1,θn−1 −θm−1)‖∗ ≤ (c3)m−2 ∫ t 0 ... ∫ t 0 ‖(un−m+2 −u1,θn−m−2 −θ1‖∗ ≤ (c3)m−2 ∫ t 0 ... ∫ t 0 n−m+1∑ k=0 (c3) k ∫ t 0 ... ∫ t 0 ‖(u2 −u1θ2 −θ1)‖∗ ≤ (c3)m−2 ∫ t 0 ... ∫ t 0 n−m+1∑ k=0 (c3) k ∫ t 0 ... ∫ t 0 ( ‖(u2,θ2)‖∗ + ‖(u1,θ1)‖∗ ) ≤ (c3t)m−2 n−m+1∑ k=0 (c3t) k ( 4‖u1‖ 2 2,ω ) . it follows that we have∫ t 0 ‖(un −um,θn −θm)‖∗ ≤ t(c3t) m−2 n−m+1∑ k=0 (c3t) k(4‖u1‖ 2 2,ω), and so we infer that ‖φn −φm‖2,ω ≤ 4c0c‖un −um‖ . the sequence (un,φn−1)n≥2 is a cauchy sequence in the banach space l 2 ( [0,t] ,h20 (ω) ×h20 (ω) ) . it follows that (un,φn−1) converges to (u,φ) in l 2 ( [0,t] ,h20 (ω)×h20 (ω) ) and (un)t converges to (u)t in l2 ( [0,t] ,l2(ω) ) . step 3: now, let us rewrite that θnm =: θn −θm is a solution of the following problem  kθnmt −η∆θnm = µ∆(unm)t in ω × [0,t] , θnm = 0, on γ × [0,t] , ((θnm)t)|t=0 = 0 in ω. using theorem 2.2, proposition 2.3 and inequality (2.2), we have k |θn−1 −θm−1|22,ω + (2η −µ) ∫ t 0 |∇(θn−1 −θm−1)|22,ω ≤ µ ∫ t 0 (|∇(un−1 −um−1)t|2,ω) 2. 138 j. oudaani and m. raïssouli we deduce that (θn) is a cauchy sequence in the banach space l 2([0,t] ,h10 (ω)) and so (θn) converges to θ in l2([0,t] ,h10 (ω)). otherwise, by proposition 2.5 we may deduce that f1(un−1,φn−1) converges to f1(u,φ) in (l 2(ω))2. thanks to theorem 2.4, we have (un, (un)t) ∈ c0([0,t] ,h20 (ω) × l2(ω)) with (un)|t=0 = u0 and ((un)t)|t=0 = u1, and so (u)|t=0 = u0, ((u)t)|t=0 = u. for showing that (u,θ) is a weak solution of the problem (p0), we follow the same way as in [8]. let { ej,e 1 j } be a basis in the space h20 (ω) × h10 (ω) and let ϕj ∈ c1(0,t) , 1 ≤ j ≤ j0, be such that ϕj(t) = 0. we set ψ = j0∑ j=1 ϕj ⊗ej, ϕ = j0∑ j=1 ϕj ⊗e1j. as in the proof of theorem 2.6 (step 3), we have − ∫ t 0 ∫ ω (un)tψt +µ ∫ t 0 ∫ ω ∇θn∇ψ+ ∫ t 0 ∫ ω ∆un∆ψ = ∫ t 0 ∫ ω (f1(un−1,φn−1)+p)ψ− ∫ ω u1ψ(0) (3.6) and ∫ t 0 ( −k ∫ ω θnϕt + η ∫ ω ∇θn∇ϕ−µ ∫ ω ∇un∇ϕt ) = −k ∫ ω θ0ϕ(0) −µ ∫ ω ∇u1∇ϕ(0). (3.7) letting n →∞ in (3.6) and (3.7) we deduce that, for all ψ ∈ l2([0,t],h20 (ω)), ψt ∈ l2([0,t],l2(ω)), ϕ ∈ l2([0,t],h10 (ω)) and ϕt ∈ l2([0,t],l2(ω)) with ψ(t) = ϕ(t) = 0, we have − ∫ t 0 ∫ ω utψt + µ ∫ t 0 ∫ ω ∇θ∇ψ + ∫ t 0 ∫ ω ∆u∆ψ = ∫ t 0 ∫ ω (f1(u,φ) + p)ψ − ∫ ω u1ψ(0), and ∫ t 0 ( −k ∫ ω θϕt + η ∫ ω ∇θ∇ϕ−µ ∫ ω ∇u∇ϕt ) = −k ∫ ω θ0ϕ(0) −µ ∫ ω ∇u1∇ϕ(0). hence, (u,θ) is a weak solution of the problem (s1) with f = f1(u,φ) + p. summarizing, we have proved that (u,φ,θ) is a weak solution of the thermoelastic von-karman evolution. step 4: we now prove the uniqueness. assume that there exist two weak solutions (u1,φ1,θ1) and (u2,φ2,θ2) in l2([0,t] ,h10 (ω) × h20 (ω) × h10 (ω)) such that, for some c > 0 small enough, we have∥∥u1∥∥ ≤ c and ∥∥u2∥∥ ≤ c. then u12 =: u1 −u2 and θ12 =: θ1 −θ2 satisfy the following problem (p3)   u12tt + ∆ 2u12 = f(u1,φ1,θ1) −f(u2,φ2,θ2) in ω × [0,t] , kθ12t −η∆θ12 = µ∆u12t in ω × [0,t] , u12 = ∂νu 12 = θ12 = 0 on γ × [0,t] , u12(x1,x2, 0) = 0, (u 12)t(x1,x2, 0) = 0 in ω. (θ12)t(x1,x2, 0) = 0 in ω. it follows that (u1 −u2,θ1 −θ2) is a solution of the problem (p3). proposition 2.5, proposition 2.6 and theorem 2.1 ensure that there exists c1 > 0 such that ∥∥(u1 −u2,θ1 −θ2)∥∥∗ ≤ et ∫ t 0 ∣∣f1(u1,φ1) −f1(u2,φ2)∣∣22,ω ≤ c1et ∫ t 0 ∥∥u1 −u2∥∥2 ≤ c1et ∫ t 0 ∥∥(u1 −u2,θ1 −θ2)∥∥∗ . on the weak solution of the von-karman model with thermoelastic plates 139 since c1 is small enough so 0 < c3 = e tc1 < 1 t and therefore∫ t 0 ∥∥(u1 −u2,θ1 −θ2)∥∥∗ ≤ tc3 ∫ t 0 ∥∥(u1 −u2,θ1 −θ2)∥∥∗ , which, with 0 < tc3 < 1, immediately yields u 1 = u2, θ1 = θ2 and then φ1 = φ2. in conclusion, the dynamic von-karman equations coupled with thermal dissipation, without rotational inertia, has one and only one weak solution (u,φ,θ) in l2 ( [0,t] ,h20 (ω)×h20 (ω)×h10 (ω) ) . the proof of the theorem is finished. � we end this section by stating the following result. proposition 3.2. let (u,φ,θ) ∈ l2 ( [0,t] ,h20 (ω) ×h20 (ω) ×h10 (ω) ) be the unique solution of (p0). let φ0 ∈ h20 (ω) be the unique solution of ∆2φ0 = − [u0,u0]. then the following equality ẽ ( u(t),ut(t),φ ) + k |θ|22,ω + 2η ∫ t 0 |∇θt| 2 2,ω = ẽ1(u0,u,φ0) + k |θ0| 2 2,ω , holds true for any t ∈ [0,t], where we set ẽ ( u(t),ut(t),φ ) =: |ut| 2 2,ω + ‖u‖ 2 + 1 2 ∫ ω ( |∆φ|2 − 2 [u,f0] u− 4pu ) and ẽ1(u0,u,φ0) =: |u| 2 2,ω + ‖u0‖ 2 + 1 2 ∫ ω ( |∆φ0| 2 − 2 [u0,f0] u0 − 4pu0 ) . proof. by virtue of theorem 2.6, for any t ∈ [0,t] we have the following equality ‖u‖20 + 2η ∫ t 0 |∇θ|22,ω + k |θ| 2 2,ω = ‖u0‖ 2 + |u|22,ω + k |θ0| 2 2,ω + 2 ∫ t 0 ∫ ω f1(u,φ)ut + 2 ∫ t 0 ∫ ω p(x1,x2)ut. (3.8) first, let us observe that we have∫ t 0 ∫ ω p(x1,x2)ut = ∫ ω p(x1,x2)u(t) − ∫ ω p(x1,x2)u0. otherwise, with ∆2φ = [u,u], we have∫ t 0 ∫ ω f1(u,φ)ut = ∫ t 0 ∫ ω [u,φ + f0] ut = ∫ t 0 ∫ ω [u,φ] ut + ∫ t 0 ∫ ω [u,f0] ut, = 1 2 ∫ t 0 ∫ ω d dt ([u,u] φ) + 1 2 ∫ t 0 ∫ ω d dt ([u,f0] u), = − 1 4 ∫ ω |∆φ|2 + 1 4 ∫ ω |∆φ0| 2 + 1 2 ∫ ω [u,u] f0 − 1 2 ∫ ω [u0,u0] f0. substituting these into (3.8), we get ‖u‖20 + 2η ∫ t 0 |∇θ|22,ω + k |θ| 2 2,ω + 1 2 ∫ ω |∆φ|2 − ∫ ω [u,u] f0 − 2 ∫ ω p(x1,x2)u = ‖u0‖ 2 + |u|22,ω + k |θ0| 2 2,ω + 1 2 ∫ ω |∆φ0| 2 − ∫ ω [u0,u0] f0 − 2 ∫ t 0 ∫ ω p(x1,x2)u0. the proof of the proposition is finished. � 140 j. oudaani and m. raïssouli 4. numerical application in this section we will investigate a numerical resolution of our initial problem in the aim to illustrate the previous study. 4.1. preliminaries. we take ω =]0, 1[×]0, 1[⊂ r2 and let t > 0. for solving numerically the problem (p0), we use the finite difference method by considering a uniform mesh of width h. for this, let us denote by ωh the set of all mesh points inside the domain ω with internal points: (xi,yj) = (ih,jh), i,j = 1, ...n−1, h = 1/(n + 1), ∆t = 1/t. otherwise, we denote by ωh the set of boundary mesh points and by uh the finite-difference that approximates u. in [2], the author discussed a numerical study about the convergence and stability for the conservative finite difference schemes related to the dynamic von karman plate equations. in the aim to approximate numerically the unique weak solution of our problem, we use the discrete model of von-karman evolution presented in [2, 9]: (∗)   δ2t u n ij + µ(δ 2 x + δ 2 y)θ n ij + ∆ 2 hu n ij = [ u n ij v n ij + fij ] + pij in ωh, kδtθ n ij −η(δ 2 x + δ 2 y)θ n ij −µδt(δ 2 x + δ 2 y)u n ij = 0 in ωh, ∆2hv n ij = − [ u n ij u n ij ] in ωh, u0ij = (ϕ0)ij, δtu 0 ij = (ϕ1)ij, θ 0 ij = (θ0)ij in ωh, unij = v n ij = θ n ij = 0 on ωh, ∂νu n ij = ∂νv n ij = 0 on ωh, with the following discrete differential operators: δ2t u n ij = un+1ij − 2u n ij + u n−1 ij (∆t)2 , δtu n ij = un+1ij −u n ij ∆t , ∆2hu n ij = h −4 [ uij−2 + uij+2 + ui−2j + ui+2j − 8(uij−1 + uij+1 + ui−1j + ui+1j) + 2(ui−1j−1 + ui−1j+1 + ui+1j−1 + ui+1j+1) − 20uij ] , δ2xu n ij = uni+1j − 2u n ij + u n i−1j h2 , δ2yu n ij = unij+1 − 2u n ij + u n ij−1 h2 , δ2xyu n ij = uni+1j+1 −u n i+1j−1 −u n i−1j+1 + u n i−1j−1 (2h)2 , [ unij, v n ij ] = δ 2 xu n ijδ 2 yv n ij − 2δ 2 xyu n ijδ 2 xyv n ij + δ 2 yu n ijδ 2 xv n ij. summarizing the above, we have in fact transformed the above problem to the numerical resolution into 2 steps, as itemized below: step 1: we first utilize the numerical procedure of 13-point formula of finite difference discussed in [6]. this method is used for illustrating the weak solution of the next problem:  ∆2v = f1 in ω, v = g1 on γ, ∂νv = g2 on γ. step 2: afterwards, we adopt the discrete model of the von-karman evolution (∗) for approaching the solution of the thermoelastic model coupled with the dynamic von-karman evolution. on the weak solution of the von-karman model with thermoelastic plates 141 4.2. non-coupled approach. in [6], the author discussed a numerical analysis of finite-difference method about the numerical resolution of the biharmonic equation. such method, which is known as the non-coupled method of 13-point, may be summarized by the following result: proposition 4.1. the 13-point approximation of the biharmonic equation for approaching the unique solution v of the problem (p) is defined by: lhvij = h −4 { vij−2 + vij+2 + vi−2j + vi+2j − 8 ( vij−1 + vij+1 + vi−1j + vi+1j ) + 2 ( vi−1j−1 + vi−1j+1 + vi+1j−1 + vi+1j+1 ) − 20vij } = f1 ( xi,yj ) , for i,j = 1, 2, ...,n − 1, where we set vij = v(xi,yj). when the mesh point (xi,yj) is adjacent to the boundary ωh, then the undefined values of vh are conventionally calculated by the following approximation of ∂νv: vi−2,j = 1 2 vi+1,j −vij + 3 2 vi−1,j −h(∂xv)i−1,j, vi,j−2 = 1 2 vi,j+1 −vij + 3 2 vi,j−1 −h(∂yv)i,j−1, vi+2,j = 1 2 vi+1,j −vij + 3 2 vi−1,j −h(∂xv)i+1,j, vi,j+2 = 1 2 vi,j+1 −vij + 3 2 vi,j−1 −h(∂yv)i,j+1. the following example illustrates the previous theoretical study. example 1. let us consider the following body forces: p(x,y) = 10−2(x− 1)2(y − 1)2(e−x 2−y2 ), u1 = 10 −3(y3(x− 4)2)(e−x 2−y2 ), θ0 = 10x 2(x−y − 1)(e−(x−1) 2−(y−1)2 ), u0 = 10 −3x2(y − 3)3(e−x 2−y2 ), f0 = 10 −3x(e−x 2−y2 ) sin2(πx). displacement of plate, t = 0.1s displacement of plate, t = 50s 142 j. oudaani and m. raïssouli thermal value, t = 0.1s thermal value, t = 50s contour displacement valuet = 50s contour thermal valuet = 50s acknowledgements: the authors would like to thank the anonymous referee for his/her valuable comments which improved the final version of this manuscript. references [1] s. s. antman and t. von-karman, a panorama of hungarian mathematics in the twentieth centuray i, bolyai soc. math. studies 14(2006), 373-382. [2] s. bilbao, a family of conservative finite difference schemes for the dynamical von karman plate equations, numer. meth. partial. diff. eqns. 24(1) (2007), 193-218. https://www.doi.org/10.1002/num.20260. [3] h. brezis, analyse fonctionnelle, théorie et application., masson, paris (1983). [4] i. chueshov and i. lasiecka, von karman evolution, well-posedness and long time dynamics, new york, springer, 2010. [5] p. g. ciarlet and r. rabier, les equations de von karman., lecture notes in mathematics., vol. 826, new york, springer, 1980. [6] m. m. gubta and r. p. manohar, direct solution of biharmonic equation using non coupled approach, j. computational physics 33/2 (1979) 236-248. http://dx.doi.org/10.1016/0021-9991(79)90018-4. [7] j. l. lions., e. magenes, problèmes aux limites non homogènes et applications., vol.1, gauthier-villars, dunod, paris (1968). [8] j. l. lions, quelques methodes de résolution des problèmes aux limites non linéaires., dunod, paris (2002). [9] d. c. pereira, c. a. raposo, a. j. avila, numerical solution and exponential decay to von karman system with frictional damping, int. j. math. information sci. 8(4) (2014), 1575-1582. http://dx.doi.org/10.12785/amis/080411. https://www.doi.org/10.1002/num.20260 http://dx.doi.org/10.1016/0021-9991(79)90018-4 http://dx.doi.org/10.12785/amis/080411 on the weak solution of the von-karman model with thermoelastic plates 143 j. oudaani, ibn zohr university, poly-disciplinary faculty, department of mathematics and managment, code postal 638, ouarzazate, morocco. email address: oudaani1970@gmail.com m. räıssouli, corresponding author, department of mathematics, science faculty, moulay ismail university, morocco. email address: raissouli.mustapha@gmail.com 1. introduction 2. preliminaries and main results 3. iterative approach and convergence result 4. numerical application 4.1. preliminaries 4.2. non-coupled approach references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 4, number 1, march 2023, pp.61-78 https://doi.org/10.5206/mase/15536 energy criteria of global existence for the hartree equation with coulomb potential na tang, chenglin wang, and jian zhang abstract. this paper studies a class of hartree equations with coulomb potential. combined with the conservation of mass and energy, we analyze the variational characteristics of the corresponding nonlinear elliptic equation. according to the range of parameters, we construct the evolution invariant flows of the equation in different cases. then the sharp energy thresholds for global existence and blowup of solutions are discussed in detail. 1. introduction in this paper, we study a class of hartree equations with coulomb potential: iϕt + ∆ϕ + β|x|−1ϕ + (|x|−γ ∗ |ϕ|2)ϕ + |ϕ|pϕ = 0, t > 0,x ∈ rn, (1.1) where n ≥ 3, 2 < γ < min{4, n}, 0 ≤ β < (n− 2)2(γ − 2) 2(γ − 1) , 0 < p < 4 n− 4 , and ϕ = ϕ(t,x) is a complex value wave function of (t,x) ∈ r+ ×rn. equation (1.1) is considered as the first-principle model for beam-matter interaction in x-ray free electron lasers (xfel)[1, 4, 9]. the parameter β denotes the strength of an electron beam interaction with external coulomb force. recent developments using xfel include the motion of atoms, measuring the dynamics of atomic vibrations and biomolecular imaging [3, 8, 23]. besides, in the context of bec, such a model equation is also known as the gross-pitaevskii for dipole bose-einstein condensation with coulomb potential[24]. for (1.1), the local well-posedness was established in [6, 10]. feng and zhao [10] obtained the global well-posedness for (1.1) under some assumptions. in [15], authors proved the existence of ground states and normalized solutions for (1.1) with harmonic potential. if we remove the term β|x|−1 in (1.1), this equation may occur blow up in finite time for the whole range of p, see [25, 26]. to our knowledge, the existence of blowup and the sharp criteria of global existence for (1.1) has not been studied in the literature. we recall the hartree equation: iϕt + ∆ϕ + (|x|−(n−2) ∗ |ϕ|α)|ϕ|α−2ϕ = 0, t > 0,x ∈ rn. (1.2) when α = 2, the equation (1.2) becomes choquard-pekar equation, which occurs in the modelling of quantum semiconductor devices, the electron transport and the electron-electron interaction(see [17]). there are numerous results for equation (1.2). when n ≥ 3, 2 ≤ α ≤ 1 + 4 n−2 , genev and venkov [13] proved the local and global well-posedness and the existence of blow-up solutions. the dynamics received by the editors 23 november 2022; accepted 24 march 2023; published online 30 march 2023. 2010 mathematics subject classification. primary 54c40, 14e20; secondary 46e25, 20c20. key words and phrases. hartree equation, coulomb potential, blow-up, global existence, sharp energy threshold. 61 62 n. tang, c. wang, and j. zhang of blow-up solutions was investigated in [5, 20, 22, 28, 29]. in [2, 12, 21], they showed the sharp criteria for blow-up and scattering in h1(rn). huang, zhang, chen [16] and tian, yang, zhou [25] showed the sharp criteria of global existence for the hartree equation with subcritical perturbations. and leng, li, zheng [18] showed the sharp criteria of global existence for the hartree equation with supercritical perturbations. in [26], they detected the dynamical properties of blow-up solutions. lieb [17] showed the uniqueness of the radial symmetric standing wave in r3. the nonlinear schrödinger equation with coulomb potential is as follows: iϕt + ∆ϕ + β|x|−1ϕ = λf(|ϕ|2)ϕ, t > 0,x ∈ rn. (1.3) when β > 0, it provides a quantum mechanical description of coulomb force between two charged particles and corresponds to having an external attractive long-range potential due to the presence of a positively charged atomic nucleus(see [19]). when β ≤ 0 and f(|ϕ|2) = |x|−1 ∗ |ϕ|2 , chadam, glassey [5] obtained the existence of the unique global solution in h1(r3). hayashi, ozawa [14] showed the global existence and a decay rate of solutions when the initial data belongs to a weighted-l2 space. for (1.1), we construct different invariant flows under different parameter ranges. then we obtain the sharp energy thresholds for global existence and blow-up of solutions for (1.1). we mainly consider the following cases: (1) 0 < p < 2 n , 2 < γ < min{n, 4}; (2) p = 2 n , 2 < γ < min{n, 4}; (3) 2 n < p < 4 n , 2 < γ < min{n, 4}; (4) p = 4 n , 2 < γ < min{n, 4}; (5) 4 n < p < 4 n− 2 , 2 < γ < np 2 ; (6) 4 n < p < 4 n− 2 , np 2 ≤ γ < min{4,n}. this paper is organized as follows: in section 2, we establish some basic facts including local wellposedness, the conservation laws of mass and energy, and sharp inequalities. in section 3, we give the sharp energy thresholds of blow-up and global existence for (1.1). 2. preliminaries we impose the initial data of (1.1) as follows ϕ(0,x) = ϕ0, x ∈ rn. (2.1) for the cauchy problem (1.1) and (2.1), we define the energy space as h1(rn) := {v : v ∈ l2(rn),∇v ∈ l2(rn)}, (2.2) and introduce the inner product (u,v) := ∫ ∇u ·∇v + uvdx, (2.3) whose associated norm denoted by ‖ · ‖h1 . here and hereafter, for simplicity, we use ∫ ·dx to denote∫ rn ·dx. lemma 2.1. [6, 10] assume ϕ0 ∈ h1(rn), there exists a unique solution ϕ(t) of the cauchy problem (1.1) and (2.1) in c([0,t); h1(rn)) for some t ∈ (0,∞] (maximal existence time). we have the global existence and blowup of the hartree equation with coulomb potential 63 alternatives t = ∞ (global existence) or else t < ∞ and lim t→t ‖ϕ(t)‖h1 = ∞ (blow up). moreover for all t ∈ [0,t), the solution ϕ(t) satisfies the following: (i) conservation of mass: ∫ |ϕ(t)|2dx = ∫ |ϕ0|2dx. (2.4) (ii) conservation of energy: e(ϕ(t)) = ∫ 1 2 |∇ϕ(t)|2 − β 2 |x|−1|ϕ(t)|2 − 1 4 (|x|−γ ∗ |ϕ(t)|2)|ϕ(t)|2 − 1 p + 2 |ϕ(t)|p+2dx = e(ϕ0). (2.5) by a direct calculation, we have the following result. lemma 2.2. let ϕ0 ∈ h1(rn), ∫ |x|2|ϕ0|2dx < ∞ and ϕ(t,x) be a solution of the cauchy problem (1.1) and (2.1). put j(t) := ∫ |x|2|ϕ(t,x)|2dx, then one has j ′′ (t) = ∫ 8|∇ϕ|2 − 4β|x|−1|ϕ|2 − 2γ(|x|−γ ∗ |ϕ|2)|ϕ|2 − 4np p + 2 |ϕ|p+2dx = 8γe(ϕ0) + ∫ 8γ − 4np p + 2 |ϕ|p+2 − 4(γ − 2)|∇ϕ|2 + (4γ − 4)β|x|−1|ϕ|2dx. (2.6) lemma 2.3. [27] let ϕ0 ∈ h1(rn) and ∫ |x|2|ϕ0|2dx < ∞. then the following estimate holds:∫ |ϕ|2dx ≤ 2 n ( ∫ |∇ϕ|2dx) 1 2 ( ∫ |x|2|ϕ|2dx) 1 2 . (2.7) lemma 2.4. [27] for 0 < p < 4 n− 2 and v ∈ h1(rn), ‖v‖p+2p+2 ≤ 2(p + 2) np‖∇r‖p2 ‖v‖ 4 − (n− 2)p 2 2 ‖∇v‖ np 2 2 , (2.8) where r is the unique positive ground state solution of equation: −∆r + 4 − (n− 2)p np r−|r|pr = 0,r ∈ h1(rn). (2.9) lemma 2.5. [7, 29] for 0 < γ < min{4,n} and v ∈ h1(rn), one has ‖(|x|−γ ∗ |v|2)|v|2‖1 ≤ 4 γ‖∇w‖22 ‖v‖4−γ2 ‖∇v‖ γ 2, (2.10) where w is a positive ground state solution of equation: −∆w + 4 −γ γ w − (|x|−γ ∗ |w|2)w = 0,w ∈ h1(rn). (2.11) lemma 2.6. [11] assume 1 < α < n,v ∈ w1,α(rn), then∫ |v|α |x|α dx ≤ ( α n−α )α ∫ |∇v|αdx. (2.12) in the end, for simplicity, we denote c0 = 1 2 + β 4 − β (n− 2)2 , 1 a0 = 1 2 − β (n− 2)2 . 64 n. tang, c. wang, and j. zhang 3. sharp energy thresholds in this section, we state the sharp criteria for global existence and blow up of (1.1). according to the range of parameters p and γ, we show the results in the following six cases. case i: 0 < p < 2 n , 2 < γ < min{n, 4}. in this case, we have three theorems. let a1 = 2 2 np 2 np‖∇r‖p2 ‖ϕ0‖ 4−2np+2p 2 2 ,a2 = 1 2γγ‖∇w‖22 ‖ϕ0‖ 4−2γ 2 , d1 = ( 2 −np 2γ − 2 ) np−2 2γ−np + ( 2 −np 2γ − 2 ) 2γ−2 2γ−np , d2 = np 2 [ np(2 −np) 4γ(γ − 1) ] np−2 2γ−np + γ[ np(2 −np) 4γ(γ − 1) ] 2γ−2 2γ−np , b1 = [ 22−np(np)2γ−2γ2−np‖∇r‖2pγ−2p2 ‖∇w‖ 4−2np 2 (a0d1)2γ−np ] 1 4−2np+2pγ−2p , b2 = [ 22−np(np)2γ−2γ2−np‖∇r‖2pγ−2p2 ‖∇w‖ 4−2np 2 (a0d2)2γ−np ] 1 4−2np+2pγ−2p , k1 = np− 2 4γ [ (n− 2)2(2γa1 −npa1) 2 np np (2γ − 4)(n− 2)2 − 4β(γ − 1) ] np 2−np . under the constraint : ‖ϕ0‖2 < b1, we define two invariant sets: g1 = {ϕ ∈ h1 : e(ϕ) + c0‖ϕ‖22 < k1,‖ϕ‖2h1 < y1}, b1 = {ϕ ∈ h1 : e(ϕ) + c0‖ϕ‖22 < k1,‖ϕ‖2h1 > y1}, where y1 is the unique positive maximizer of : f1(y) := 1 a0 y −a1y np 2 −a2yγ. (3.1) let ỹ1 > 0 be the first positive root of equation f ′ 1(y) = d dy f1(y) = 0. theorem 3.1. for 0 < p < 2 n and 2 < γ < min{n, 4}. assume ‖ϕ0‖2 < b1, then the following facts are true: (i) when ϕ0 ∈ g1 ∪{0} and f1(ỹ1) < k1, the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞). (ii) when ϕ0 ∈ b1 and |x|ϕ0 ∈ l2(rn), the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. proof. firstly, according to (2.8) , (2.10) and (2.12), we estimate the energy functional e(ϕ), for all t ∈ (0,t], e(ϕ(t)) + c0‖ϕ(t)‖22 ≥ 1 2 ‖∇ϕ‖22 − β (n− 2)2 ‖∇ϕ‖22 − β 4 ‖ϕ‖22 + c0‖ϕ‖ 2 2 − 1 γ‖∇w‖22 ‖ϕ‖4−γ2 ‖∇ϕ‖ γ 2 − 2 np‖∇r‖p2 ‖ϕ‖ 4−(n−2)p 2 2 ‖∇ϕ‖ np 2 2 ≥ [ 1 2 − β (n− 2)2 ]‖ϕ‖2h1 − 1 2γγ‖∇w‖22 ‖ϕ‖4−2γ2 ‖ϕ‖ 2γ h1 (3.2) − 2 2 np 2 np‖∇r‖p2 ‖ϕ‖ 4−2np+2p 2 2 ‖ϕ‖ np h1 . global existence and blowup of the hartree equation with coulomb potential 65 let y = ‖ϕ(t)‖2 h1 ≥ 0, for all t ∈ (0,t], e(ϕ(t)) + c0‖ϕ(t)‖22 ≥ f1(‖ϕ(t)‖ 2 h1 ) = f1(y), (3.3) where f1 is defined in (3.1). secondly, we claim that the maximum of f1(y) on [0, +∞) is greater than 0. let g(y) = 1 a0 −a1y np 2 −1 −a2yγ−1. it follows that f1(y) = yg(y), lim y→0+ g(y) = lim y→+∞ g(y) = −∞ and g′(y) has only one zero point y0 = [ (2 −np)a1 2(γ − 1)a2 ] 2 2γ−np . thus the maximum of g(y) on [0, +∞) is g(y0). from ‖ϕ0‖2 < b1, we can obtain a 2γ−2 1 a 2−np 2 < (a0d1) np−2γ, which implies g(y0) = 1 a0 −a1y np 2 −1 0 −a2y γ−1 0 > 0. note that f1(y) → 0− as y → 0+ and f1(y) → −∞ as y → +∞. therefore, f1(y) has the unique positive maximizer y1 on [0,∞) and f1(y1) ≥ y0g(y0) > 0. thirdly, we prove the invariance of g1 and b1. when f1(ỹ1) < k1, combined with the structure of f1(y), we can easily know that g1 is a nonempty set. if ϕ0 ∈ g1, f1(ỹ1) < k1 and ϕ(t,x) is the corresponding solution of the cauchy problem (1.1) and (2.1), then by lemma 2.1, we have for all t ∈ [0,t), f1(‖ϕ‖2h1 ) ≤ e(ϕ) + c0‖ϕ‖ 2 2 < k1. (3.4) we only need to prove ‖ϕ‖2 h1 < y1. otherwise, by the continuity of ϕ(t) there exists t ∈ [0,t) such that ‖ϕ(t)‖2 h1 = y1, and then f1(‖ϕ(t)‖2h1 ) = f1(y1) > k1, which contradicts (3.4). thus ‖ϕ‖2 h1 < y1, which implies the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞). we can obtain b1 is a nonempty invariant set by the same token. finally, we prove the statement (ii) of theorem 3.1. from (2.6), we have j′′(t) = 8γe(ϕ0) + ∫ 8γ − 4np p + 2 |ϕ|p+2 − 4(γ − 2)|∇ϕ|2 + (4γ − 4)β|x|−1|ϕ|2dx ≤ 8γe(ϕ) + 16γ − 8np np ‖ϕ‖ 4−(n−2)p 2 2 ‖∇ϕ‖ np 2 2 − 4(γ − 2)‖∇ϕ‖ 2 2 + (4γ − 4)β[ 2 (n− 2)2 ‖∇ϕ‖22 + 1 2 ‖ϕ‖22] ≤ 8γ[e(ϕ) + c0‖ϕ‖22] + h1(y), (3.5) where h1(y) = (8γ − 4np)a1y np 2 + [ 8β(γ − 1) (n− 2)2 − 4γ + 8]y. h′1(y) has only one zero point y ∗ on [0, +∞), y∗ = [ (n− 2)2(2γ −np)npa1 (n− 2)2(2γ − 4) − 4β(γ − 1) ] 2 2−np . 66 n. tang, c. wang, and j. zhang h1(y) is increasing on(0,y ∗) and decreasing on (y∗, +∞), so the maximum of h1(y) is h1(y ∗) = (4 − 2np)[ (n− 2)2(2γa1 −npa1) 2 np np (n− 2)2(2γ − 4) − 4β(γ − 1) ] np 2−np = −8γk1. (3.6) by the invariance of b1, if ϕ0 ∈ b1 then for all t ∈ [0,t), f1(‖ϕ‖2h1 ) ≤ e(ϕ) + c0‖ϕ‖ 2 2 < k1. inserting the results into (3.5), we obtain j′′(t) ≤ 8γ[e(ϕ) + c0‖ϕ‖22] + h1(y ∗) < 0. therefore from lemma 2.1 and 2.3, it must be the case t < ∞, which implies that the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. this completes the proof of theorem 3.1. � under the constraint : b1 ≤‖ϕ0‖2 < b2, we define two invariant sets: g2 = {ϕ ∈ h1 : e(ϕ) + c0‖ϕ‖22 < k1,‖ϕ‖2h1 < y2}, b2 = {ϕ ∈ h1 : e(ϕ) + c0‖ϕ‖22 < k1,‖ϕ‖2h1 > y2}, where y2 is the unique positive maximizer of equation (3.1).let ỹ2 > 0 be the first positive root of the equation f′1(y) = d dy f1(y) = 0 under the constraint b1 ≤‖ϕ0‖2 < b2. theorem 3.2. for 0 < p < 2 n and 2 < γ < min{n, 4}. assume b1 ≤ ‖ϕ0‖2 < b2, then the following facts are true: (i) when ϕ0 ∈ g2 ∪{0} and f1(ỹ2) < k1, the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞). (ii) when ϕ0 ∈ b2 and |x|ϕ0 ∈ l2(rn), the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. proof. firstly, we claim that f1(y) ≤ 0 and f1(y) has two extrema on [0, +∞). when b1 ≤‖ϕ0‖2 < b2, we have f′1(y) = 1 a0 − npa1 2 y np 2 −1 −γa2yγ−1, and f′′1 (y) = np(2 −np)a1 4 y np 2 −2 −γ(γ − 1)a2yγ−2. then f′1(y) →−∞ as y → 0+ or y → +∞, and f′′1 (y) has only one zero point ym, ym = [ (2 −np)npa1 4(γ − 1)γa2 ] 2 2γ−np , so the maximum of f′1(y) on [0,∞) is f′1(ym) = 1 a0 − npa1 2 [ (2 −np)npa1 4(γ − 1)γa2 ] np−2 2γ−np −γa2[ (2 −np)npa1 4(γ − 1)γa2 ] 2γ−2 2γ−np . (3.7) by b1 ≤‖ϕ0‖2 < b2, we can get (a0d1) np−2γ ≤ a2γ−21 a 2−np 2 < (a0d2) np−2γ, which implies that f1(y) ≤ 0 and f′1(ym) > 0. note that f′1(y) is increasing on (0,ym) and decreasing on (ym, +∞). therefore f′1(y) has two zero points on [0, +∞), it follows that f1(y) has two extrema on [0, +∞). let y3 represent the minimal point and y2 represent the maximal point. it is not hard to find global existence and blowup of the hartree equation with coulomb potential 67 y3 < y2 , f1(y2) > k1. and then, the same as the proof of theorem 3.1, we can verify that both g2 and b2 are nonempty invariant sets. thus we obtain that the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞). besides, we can also verify j′′(t) ≤ 8γ[e(ϕ) + c0‖ϕ‖22] + h1(y ∗) < 0. therefore from lemma 2.1 and 2.3, it must be the case t < ∞, which implies the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. this completes the proof of theorem 3.2. � under the constraint : ‖ϕ0‖2 ≥ b2, we define the following invariant set: b3 = {ϕ ∈ h1 : e(ϕ) + c0‖ϕ‖22 < k1,‖ϕ‖2h1 > yk}, where yk is the unique positive solution of f1(y) = k1. then we get a sufficient condition for blow-up of solutions. theorem 3.3. let 0 < p < 2 n , 2 < γ < min{n, 4} and |x|ϕ0 ∈ l2(rn). when ‖ϕ0‖2 ≥ b2 and ϕ0 ∈ b3, the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. proof. firstly, we claim that f1(y) ≤ 0 and f1(y) has no extrema on [0, +∞). when ‖ϕ0‖2 ≥ b2, we have f′1(y) = 1 a0 − npa1 2 y np 2 −1 −γa2yγ−1, and f′′1 (y) = np(2 −np)a1 4 y np 2 −2 −γ(γ − 1)a2yγ−2. then f′1(y) →−∞ as y → 0+ or y → +∞, and f′′1 (y) has only one zero point ym, ym = [ (2 −np)npa1 4(γ − 1)γa2 ] 2 2γ−np , so the maximum of f′1(y) on [0,∞) is f′1(ym) = 1 a0 − npa1 2 [ (2 −np)npa1 4(γ − 1)γa2 ] np−2 2γ−np −γa2[ (2 −np)npa1 4(γ − 1)γa2 ] 2γ−2 2γ−np . (3.8) by ‖ϕ0‖2 ≥ b2, we can get a 2γ−2 1 a 2−np 2 ≥ (a0d2) np−2γ, it follows that f1(y) ≤ 0 and f′1(ym) < 0. therefore f1(y) is decreasing on [0, +∞), which implies f1(y) has no extrema on [0, +∞). by the monotonicity of f1(y), there exists unique yk ∈ (0, +∞) such that f1(y) = k1 . and then, the same as the proof of theorem 3.1 and 3.2, we can verify that b3 is a nonempty invariant set. besides, we can also verify j′′(t) ≤ 8γ[e(ϕ) + c0‖ϕ‖22] + h1(y ∗) < 0. therefore from lemma 2.1 and 2.3, it must be the case t < ∞, which implies the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. this completes the proof of theorem 3.3. � case ii: p = 2 n , 2 < γ < min{n, 4}. denote y4 = γ‖∇w‖22[((n− 2)2 − 2β)‖∇r‖ 2 n 2 − (n− 2) 2‖ϕ‖ 2 n 2 ] 2γ−2(n− 2)2‖∇r‖ 2 n 2 , 68 n. tang, c. wang, and j. zhang k2 = ‖∇w‖22[((n− 2)2(γ − 2) − 2β(γ − 1))‖∇r‖ 2 n 2 − (n− 2) 2(γ − 1)‖ϕ‖ 2 n 2 ] 2γ−1(n− 2)4‖∇r‖ 4 n 2 × [((n− 2)2 − 2β)‖∇r‖ 2 n 2 − (n− 2) 2‖ϕ‖ 2 n 2 ]. we define two invariant sets: g4 = {ϕ ∈ h1 : e(ϕ) + c0‖ϕ‖22 < k2,‖ϕ‖2h1 < y4,‖ϕ‖ 2 n 2 < ( γ − 2 γ − 1 − 2β (n− 2)2 )‖∇r‖ 2 n 2 }, b4 = {ϕ ∈ h1 : e(ϕ) + c0‖ϕ‖22 < k2,‖ϕ‖2h1 > y4,‖ϕ‖ 2 n 2 < ( γ − 2 γ − 1 − 2β (n− 2)2 )‖∇r‖ 2 n 2 }. theorem 3.4. for p = 2 n and 2 < γ < min{n, 4}, the following facts are ture: (i) when ϕ0 ∈ g4 ∪{0}, the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞). (ii) when ϕ0 ∈ b4 and |x|ϕ0 ∈ l2(rn), the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. proof. firstly, according to (2.8) , (2.10) and (2.12), we estimate the energy functional e(ϕ), for all t ∈ (0,t], e(ϕ(t)) + c0‖ϕ(t)‖22 ≥ 1 2 ‖∇ϕ‖22 − β (n− 2)2 ‖∇ϕ‖22 − β 4 ‖ϕ‖22 + c0‖ϕ‖ 2 2 − 1 24−γγ‖∇w‖22 ‖ϕ‖4h1 − 1 2‖∇r‖ 2 n 2 ‖ϕ‖ 2 n 2 ‖ϕ‖ 2 h1 = [ 1 2 − β (n− 2)2 ]‖ϕ‖2h1 − 1 24−γγ‖∇w‖22 ‖ϕ‖4h1 − 1 2‖∇r‖ 2 n 2 ‖ϕ‖ 2 n 2 ‖ϕ‖ 2 h1. (3.9) let y = ‖ϕ(t)‖2 h1 ≥ 0, for all t ∈ (0,t], e(ϕ(t)) + c0‖ϕ(t)‖22 ≥ f2(‖ϕ(t)‖ 2 h1 ) = f2(y), (3.10) where f2(y) = [ 1 2 − β (n− 2)2 − ‖ϕ0‖ 2 n 2 2‖∇r‖ 2 n 2 ]y − 1 24−γγ‖∇w‖22 y2, f′2(y) = [ 1 2 − β (n− 2)2 − ‖ϕ0‖ 2 n 2 2‖∇r‖ 2 n 2 ] − 1 23−γγ‖∇w‖22 y. by ‖ϕ‖ 2 n 2 < ( γ − 2 γ − 1 − 2β (n− 2)2 )‖∇r‖ 2 n 2 , we know ‖ϕ‖ 2 n 2 < (1 − 2β (n− 2)2 )‖∇r‖ 2 n 2 , so f′2(y) has only one zero point y4 on [0, +∞), y4 = γ‖∇w‖22[((n− 2)2 − 2β)‖∇r‖ 2 n 2 − (n− 2) 2‖ϕ‖ 2 n 2 ] 2γ−2(n− 2)2‖∇r‖ 2 n 2 . then the maximum of f2(y) is global existence and blowup of the hartree equation with coulomb potential 69 f2(y4) = γ‖∇w‖22[((n− 2)2 − 2β)‖∇r‖ 2 n 2 − (n− 2) 2‖ϕ‖ 2 n 2 ] 2 2γ(n− 2)4‖∇r‖ 4 n 2 . secondly, we prove the invariance of g4 and b4. combined with the structure of f2(y), we can easily know both g4 and b4 are nonempty sets. if ϕ0 ∈ g4, by lemma 2.1, the corresponding solution ϕ(t,x) of cauchy problem (1.1) and (2.1) satisfies: for all t ∈ [0,t), f2(‖ϕ(t)‖2h1 ) ≤ e(ϕ(t)) + c0‖ϕ(t)‖ 2 2 < k2, (3.11) and ‖ϕ‖ 2 n 2 < ( γ − 2 γ − 1 − 2β (n− 2)2 )‖∇r‖ 2 n 2 . we only need to prove ‖ϕ‖2 h1 < y4. otherwise, by the continuity of ϕ(t) there exists t ∈ [0,t) such that ‖ϕ(t)‖2 h1 = y4, then by computation we can get f2(‖ϕ(t)‖2h1 ) = f2(y4) > k2, which contradicts (3.11). thus ‖ϕ‖2 h1 < y4, which implies the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞). we can obtain the invariance of b4 by the same token. finally, we prove the statement (ii) of theorem 3.4. from (2.6), we have j′′(t) = 8γe(ϕ0) + ∫ 8γ − 4np p + 2 |ϕ|p+2 − 4(γ − 2)|∇ϕ|2 + (4γ − 4)β|x|−1|ϕ|2dx ≤ 8γ[e(ϕ) + c0‖ϕ‖22] + h2(y), (3.12) where h2(y) = [ 4(γ − 1)‖ϕ0‖ 2 n 2 ‖∇r‖ 2 n 2 − 4(γ − 2) + 8β(γ − 1) (n− 2)2 ]y. when ‖ϕ‖ 2 n 2 < ( γ − 2 γ − 1 − 2β (n− 2)2 )‖∇r‖ 2 n 2 , the maximum of h2(y) on [y4, +∞) is : h2(y4) = 24−γγ‖∇w‖22[((n− 2)2 − 2β)‖∇r‖ 2 n 2 − (n− 2) 2‖ϕ‖ 2 n 2 ] (n− 2)4‖∇r‖ 4 n 2 × [(n− 2)2(γ − 1)‖ϕ‖ 2 n 2 − ((n− 2) 2(γ − 2) − 2β(γ − 1))‖∇r‖ 2 n 2 ] = −8γk2. by the invariance of b4, if ϕ0 ∈ b4, then for all t ∈ [0,t), f2(‖ϕ‖2h1 ) ≤ e(ϕ) + c0‖ϕ‖ 2 2 < k2, ‖ϕ‖ 2 h1 > y4. inserting the results into (3.12), we obtain j′′(t) ≤ 8γ[e(ϕ) + c0‖ϕ‖22] + h2(y4) < 0. therefore from lemma 2.1 and 2.3, it must be the case t < ∞, which implies the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. this completes the proof of theorem 3.4. � 70 n. tang, c. wang, and j. zhang case iii: 2 n < p < 4 n , 2 < γ < min{n, 4}. denote k3 = np− 4 4npγ [ (n− 2)2(2γ −np)‖ϕ‖ 4−(n−2)p 2 2 [(2γ − 4)(n− 2)2 − 4β(γ − 1)] np 4 ‖∇r‖p2 ] 4 4−np , d3 = ( 4 −np 4γ − 4 ) np−4 4γ−np + ( 4 −np 4γ − 4 ) 4γ−4 4γ−np , b3 = [ ‖∇r‖4pγ−4p2 ‖∇w‖ 8−2np 2 2npγ−8γ+4(a0d3)4γ−np ] 1 8+4pγ−4p−2np , a3 = 1 2‖∇r‖p2 ‖ϕ0‖ 4−(n−2)p 2 2 ,a4 = 1 2γ‖∇w‖22 ‖ϕ0‖ 4−2γ 2 , f3(y) := 1 a0 y − 2 np‖∇r‖p2 ‖ϕ0‖ 4−(n−2)p 2 2 y np 4 − 1 2γγ‖∇w‖22 ‖ϕ0‖ 4−2γ 2 2 y γ. (3.13) let ỹ3 > 0 and y5 be the first and second positive roots of the equation f ′ 3(y) = d dy f3(y) = 0 respectively. then we define two invariant sets: g5 = {ϕ ∈ h1 : e(ϕ) + c0‖ϕ‖22 < k3,‖ϕ‖2h1 < y5,‖ϕ‖2 < b3}, b5 = {ϕ ∈ h1 : e(ϕ) + c0‖ϕ‖22 < k3,‖ϕ‖2h1 > y5,‖ϕ‖2 < b3}. theorem 3.5. for 2 n < p < 4 n and 2 < γ < min{n, 4}, the following facts are ture: (i) when ϕ0 ∈ g5 ∪{0} and f3(ỹ3) < k3, the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞). (ii) when ϕ0 ∈ b5 and |x|ϕ0 ∈ l2(rn), the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. proof. firstly, according to (2.8) , (2.10) and (2.12), we estimate the energy functional e(ϕ), for all t ∈ (0,t], e(ϕ(t)) + c0‖ϕ(t)‖22 ≥ 1 2 ‖∇ϕ‖22 − β (n− 2)2 ‖∇ϕ‖22 − β 4 ‖ϕ‖22 + c0‖ϕ‖ 2 2 − 1 γ‖∇w‖22 ‖ϕ‖4−γ2 ‖∇ϕ‖ γ 2 − 2 np‖∇r‖p2 ‖ϕ‖ 4−(n−2)p 2 2 ‖∇ϕ‖ np 2 2 ≥ [ 1 2 − β (n− 2)2 ]‖ϕ‖2h1 − 1 2γγ‖∇w‖22 ‖ϕ‖4−2γ2 ‖ϕ‖ 2γ h1 (3.14) − 2 np‖∇r‖p2 ‖ϕ‖ 4−(n−2)p 2 2 ‖ϕ‖ np 2 h1 . let y = ‖ϕ(t)‖2 h1 ≥ 0, for all t ∈ (0,t], e(ϕ(t)) + c0‖ϕ(t)‖22 ≥ f3(‖ϕ(t)‖ 2 h1 ) = f3(y), (3.15) where f3 is defined in (3.13). and then f′3(y) = 1 a0 − 1 2‖∇r‖p2 ‖ϕ0‖ 4−(n−2)p 2 2 y np 4 −1 − 1 2γ‖∇w‖22 ‖ϕ0‖ 4−2γ 2 y γ−1 = 1 a0 −a3y np 4 −1 −a4yγ−1, f′′3 (y) = − (np− 4)‖ϕ0‖ 4−(n−2)p 2 2 8‖∇r‖p2 y np 4 −2 − (γ − 1)‖ϕ0‖ 4−2γ 2 2γ‖∇w‖22 yγ−2. global existence and blowup of the hartree equation with coulomb potential 71 we can verify that f′′3 (y) has only one zero point y0 = [ 2γ−3(4 −np)‖∇w‖22 (γ − 1)‖ϕ0‖ 2−2γ+n−2 2 p 2 ‖∇r‖ p 2 ] 4 4γ−np , f′′3 (y) → +∞ as y → 0+ and f′′3 (y) → −∞ as y → +∞. thus the maximum of f′3(y) on [0,∞) is f′3(y0). by ‖ϕ0‖2 < b3, we can get a 4γ−4 3 a 4−np 4 < (a0d3) np−4γ, which implies f′3(y0) > 0. note that lim y→+∞ f′3 = −∞, so there exists a unique y5 ∈ (y0, +∞) such that f′3(y) = 0. thus f3(y) is increasing on (y0,y5) and decreasing on (y5, +∞). so the maximum of f3(y) on [0, +∞) is f3(y5). secondly, we prove the invariance of g5 and b5. when f3(ỹ3) < k3, combined with the structure of f3(y), we can easily know both g5 and b5 are nonempty sets. if ϕ0 ∈ g5, by lemma 2.1, the corresponding solution ϕ(t,x) of cauchy problem (1.1) and (2.1) satisfies: for all t ∈ [0,t), f3(‖ϕ(t)‖2h1 ) ≤ e(ϕ(t)) + c0‖ϕ(t)‖ 2 2 < k3, ‖ϕ‖2 < b3. (3.16) we only need to prove ‖ϕ‖2 h1 < y5. otherwise, by the continuity of ϕ(t) there exists t ∈ [0,t) such that ‖ϕ(t)‖2 h1 = y5, then by computation we get f3(‖ϕ(t)‖2h1 ) = f3(y5) > k3, which contradicts (3.16). thus ‖ϕ‖2 h1 < y5, which implies the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞). we can obtain b5 is a nonempty invariant set by the same token. finally, we prove the statement (ii) of theorem 3.5. from (2.6), we have j′′(t) = 8γe(ϕ0) + ∫ 8γ − 4np p + 2 |ϕ|p+2 − 4(γ − 2)|∇ϕ|2 + (4γ − 4)β|x|−1|ϕ|2dx ≤ 8γe(ϕ) + 16γ − 8np np ‖ϕ‖ 4−(n−2)p 2 2 ‖∇ϕ‖ np 2 2 − (4γ − 2)‖∇ϕ‖ 2 2 + (4γ − 4)β[ 2 (n− 2)2 ‖∇ϕ‖22 + 1 2 ‖ϕ‖22] ≤ 8γ[e(ϕ) + c0‖ϕ‖22] + h3(y), (3.17) where h3(y) = 16γ − 8np np‖∇r‖p2 ‖ϕ0‖ 4−(n−2)p 2 2 y np 4 + [−4(γ − 2) + 8β(γ − 1) (n− 2)2 ]y. then h′3 has only one zero point y ∗ on [0,∞), y∗ = [ (n− 2)2(2γ −np)‖ϕ0‖ 4−(n−2)p 2 2 [(n− 2)2(2γ − 4) − 4β(γ − 1)]‖∇r‖p2 ] 4 4−np . h3(y) is increasing on (0,y∗) and decreasing on (y∗, +∞). so the maximum of h3(y) on [0, +∞) is : h3(y∗) = 8 − 2np np [ (n− 2) np 2 (2γ −np)‖ϕ0‖ 4−(n−2)p 2 2 [(n− 2)2(2γ − 4) − 4β(γ − 1)] np 4 ‖∇r‖p2 ] 4 4−np = −8γk3. by the invariance of b5, if ϕ0 ∈ b5, then for all t ∈ [0,t), f3(‖ϕ‖2h1 ) ≤ e(ϕ) + c0‖ϕ‖ 2 2 < k3, ‖ϕ‖ 2 h1 > y5. inserting the results into (3.17), we obtain j′′(t) ≤ 8γ[e(ϕ) + c0‖ϕ‖22] + h3(y∗) < 0. 72 n. tang, c. wang, and j. zhang therefore from lemma 2.1 and 2.3, it must be the case t < ∞, which implies the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. this completes the proof of theorem 3.5. � case iv: p = 4 n , 2 < γ < min{n, 4}. denote y6 = γ‖∇w‖22[((n− 2)2 − 2β)‖∇r‖ 4 n 2 − (n− 2) 2‖ϕ‖ 4 n 2 ] 2γ−2(n− 2)2‖∇r‖ 4 n 2 , k4 = ‖∇w‖22[((n− 2)2(γ − 2) − 2β(γ − 1))‖∇r‖ 4 n 2 − (n− 2) 2(γ − 1)‖ϕ‖ 4 n 2 ] 2γ−1(n− 2)4‖∇r‖ 8 n 2 × [((n− 2)2 − 2β)‖∇r‖ 4 n 2 − (n− 2) 2‖ϕ‖ 4 n 2 ]. we define two invariant sets: g6 = {ϕ ∈ h1 : e(ϕ) + c0‖ϕ‖22 < k4,‖ϕ‖2h1 < y6,‖ϕ‖ 4 n 2 < ( γ − 2 γ − 1 − 2β (n− 2)2 )‖∇r‖ 4 n 2 }, b6 = {ϕ ∈ h1 : e(ϕ) + c0‖ϕ‖22 < k4,‖ϕ‖2h1 > y6,‖ϕ‖ 4 n 2 < ( γ − 2 γ − 1 − 2β (n− 2)2 )‖∇r‖ 4 n 2 }. theorem 3.6. for p = 4 n and 2 < γ < min{n, 4}, the following facts are ture: (i) when ϕ0 ∈ g6 ∪{0}, the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞). (ii) when ϕ0 ∈ b6 and |x|ϕ0 ∈ l2(rn), the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. proof. firstly, according to (2.8) , (2.10) and (2.12), we estimate the energy functional e(ϕ), for all t ∈ (0,t], e(ϕ(t)) + c0‖ϕ(t)‖22 ≥ 1 2 ‖∇ϕ‖22 − β (n− 2)2 ‖∇ϕ‖22 − β 4 ‖ϕ‖22 + c0‖ϕ‖ 2 2 − 1 24−γγ‖∇w‖22 ‖ϕ‖4h1 − 1 2‖∇r‖ 4 n 2 ‖ϕ‖ 4 n 2 ‖ϕ‖ 2 h1 = [ 1 2 − β (n− 2)2 ]‖ϕ‖2h1 − 1 24−γγ‖∇w‖22 ‖ϕ‖4h1 − 1 2‖∇r‖ 4 n 2 ‖ϕ‖ 4 n 2 ‖ϕ‖ 2 h1. (3.18) let y = ‖ϕ(t)‖2 h1 ≥ 0, for all t ∈ (0,t], e(ϕ(t)) + c0‖ϕ(t)‖22 ≥ f4(‖ϕ(t)‖ 2 h1 ) = f4(y), (3.19) where f4(y) = [ 1 2 − β (n− 2)2 − ‖ϕ0‖ 4 n 2 2‖∇r‖ 4 n 2 ]y − 1 24−γγ‖∇w‖22 y2, f′4(y) = [ 1 2 − β (n− 2)2 − ‖ϕ0‖ 4 n 2 2‖∇r‖ 4 n 2 ] − 1 23−γγ‖∇w‖22 y. global existence and blowup of the hartree equation with coulomb potential 73 by ‖ϕ‖ 4 n 2 < ( γ − 2 γ − 1 − 2β (n− 2)2 )‖∇r‖ 4 n 2 , we know ‖ϕ‖ 4 n 2 < (1 − 2β (n− 2)2 )‖∇r‖ 4 n 2 , so f′4(y) has only one zero point y6 on [0, +∞), y6 = γ‖∇w‖22[((n− 2)2 − 2β)‖∇r‖ 4 n 2 − (n− 2) 2‖ϕ‖ 4 n 2 ] 2γ−2(n− 2)2‖∇r‖ 4 n 2 . then the maximum of f4(y) is f4(y6) = γ‖∇w‖22[((n− 2)2 − 2β)‖∇r‖ 4 n 2 − (n− 2) 2‖ϕ‖ 4 n 2 ] 2 2γ(n− 2)4‖∇r‖ 8 n 2 . secondly, we prove the invariance of g6 and b6. combined with the structure of f4(y), we can easily know both g6 and b6 are nonempty sets. if ϕ0 ∈ g6, by lemma 2.1, the corresponding solution ϕ(t,x) of cauchy problem (1.1) and (2.1) satisfies: for all t ∈ [0,t), f4(‖ϕ(t)‖2h1 ) ≤ e(ϕ(t)) + c0‖ϕ(t)‖ 2 2 < k4, (3.20) ‖ϕ‖ 4 n 2 < ( γ − 2 γ − 1 − 2β (n− 2)2 )‖∇r‖ 4 n 2 . we only need to prove ‖ϕ‖2 h1 < y6. otherwise, by the continuity of ϕ(t) there exists t ∈ [0,t) such that ‖ϕ(t)‖2 h1 = y6. then by computation we get f4(‖ϕ(t)‖2h1 ) = f4(y6) > k4, which contradicts (3.20). thus ‖ϕ‖2 h1 < y6, which implies the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞). we can obtain the invariance of b6 by the same token. finally, we prove the statement (ii) of theorem 3.6. from (2.6), we have j′′(t) = 8γe(ϕ0) + ∫ 8γ − 4np p + 2 |ϕ|p+2 − 4(γ − 2)|∇ϕ|2 + (4γ − 4)β|x|−1|ϕ|2dx ≤ 8γ[e(ϕ) + c0‖ϕ‖22] + h4(y), (3.21) where h4(y) = [ 4(γ − 1)‖ϕ0‖ 4 n 2 ‖∇r‖ 4 n 2 − 4(γ − 2) + 8β(γ − 1) (n− 2)2 ]y. when ‖ϕ‖ 4 n 2 < ( γ − 2 γ − 1 − 2β (n− 2)2 )‖∇r‖ 4 n 2 , the maximum of h4(y) on [y6, +∞) is : h4(y6) = 24−γγ‖∇w‖22[((n− 2)2 − 2β)‖∇r‖ 4 n 2 − (n− 2) 2‖ϕ‖ 4 n 2 ] (n− 2)4‖∇r‖ 8 n 2 × [(n− 2)2(γ − 1)‖ϕ‖ 4 n 2 − ((n− 2) 2(γ − 2) − 2β(γ − 1))‖∇r‖ 4 n 2 ] = −8γk4. by the invariance of b6, if ϕ0 ∈ b6, then for all t ∈ [0,t), f4(‖ϕ‖2h1 ) ≤ e(ϕ) + c0‖ϕ‖ 2 2 < k4, ‖ϕ‖ 2 h1 > y6. inserting the results into (3.21), we obtain j′′(t) ≤ 8γ[e(ϕ) + c0‖ϕ‖22] + h4(y6) < 0. 74 n. tang, c. wang, and j. zhang therefore from lemma 2.1 and 2.3, it must be the case t < ∞, which implies the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. this completes the proof of theorem 3.6. � case v: 4 n < p < 4 n− 2 , 2 < γ < np 2 . denote k5 = (n− 2)2(γ − 2) − 2β(γ − 1) 2(n− 2)2γ y 2. then we have two invariant sets: g7 = {ϕ ∈ h1 : e(ϕ) + (β + 1)γ − 1 4γ ‖ϕ‖22 < k5,‖ϕ‖2 < 2 n− 2 y,‖∇ϕ‖2 < y}, b7 = {ϕ ∈ h1 : e(ϕ) + (β + 1)γ − 1 4γ ‖ϕ‖22 < k5,‖ϕ‖2 < 2 n− 2 y,‖∇ϕ‖2 > y}, where y is shown in the proof of the following theorem: theorem 3.7. for 4 n < p < 4 n− 2 and 2 < γ < np 2 , the following facts are ture: (i) when ϕ0 ∈ g7 ∪{0}, the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞). (ii) when ϕ0 ∈ b7 and |x|ϕ0 ∈ l2(rn), the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. proof. firstly, according to (2.8) , (2.10) and (2.12), we estimate the energy functional e(ϕ), for all t ∈ (0,t], e(ϕ(t)) + β(γ − 1) 4γ ‖ϕ(t)‖22 ≥ 1 2 ‖∇ϕ‖22 − β (n− 2)2 ‖∇ϕ‖22 − β 4 ‖ϕ‖22 + β(γ − 1) 4γ ‖ϕ‖22 − 1 γ‖∇w‖22 ‖ϕ‖4−γ2 ‖∇ϕ‖ γ 2 − 2 np‖∇r‖p2 ‖ϕ‖ 4−(n−2)p 2 2 ‖∇ϕ‖ np 2 2 = [ 1 2 − β (n− 2)2 ]‖∇ϕ‖22 − β 4γ ‖ϕ‖22 − 1 γ‖∇w‖22 ‖ϕ‖4−γ2 ‖∇ϕ‖ γ 2 − 2 np‖∇r‖p2 ‖ϕ‖ 4−(n−2)p 2 2 ‖∇ϕ‖ np 2 2 . (3.22) let y = ‖∇ϕ(t)‖2 ≥ 0, for all t ∈ (0,t], e(ϕ(t)) + β(γ − 1) 4γ ‖ϕ(t)‖22 ≥ ~1(‖∇ϕ(t)‖2) = ~1(y), (3.23) where ~1(y) = [ 1 2 − β (n− 2)2 ]y2 − β 4γ ‖ϕ‖22 − 1 γ‖∇w‖22 ‖ϕ‖4−γ2 y γ − 2 np‖∇r‖p2 ‖ϕ‖ 4−(n−2)p 2 2 y np 2 , ~′1(y) = [1 − 2β (n− 2)2 − 1 ‖∇w‖22 ‖ϕ‖4−γ2 y γ−2 − 1 ‖∇r‖p2 ‖ϕ‖ 4−(n−2)p 2 2 y np 2 −2]y = ~2(y)y. thus ~2(y) = 0 has only one positive solution, ~′2(y) = −(γ − 2) 1 ‖∇w‖22 ‖ϕ‖4−γ2 y γ−3 − np− 4 2‖∇r‖p2 ‖ϕ‖ 4−(n−2)p 2 2 y np 2 −3 < 0, which implies ~2 is decreasing on [0, +∞). note that ~2(0) = 1 − β (n− 2)2 > 0 and global existence and blowup of the hartree equation with coulomb potential 75 ~2[( ((n− 2)2 −β)‖∇w‖22 (n− 2)2‖ϕ‖4−γ2 ) 1 γ−2 ] = − ‖ϕ‖ 4−(n−2)p 2 2 ‖∇r‖p2 ( ((n− 2)2 −β)‖∇w‖22 (n− 2)2‖ϕ‖4−γ2 ) 1 γ−2 ( np 2 −2) < 0. since ~2 is continuous on [0, +∞), there exists a unique positive y, y ∈ [0, ( ((n− 2)2 −β)‖∇w‖22 (n− 2)2‖ϕ‖4−γ2 ) 1 γ−2 ], such that ~2(y ) = 0, thus the maximum of ~1(y) is ~1(y ). secondly, we prove the invariance of g7 and b7. combined with the structure of ~1(y), we can easily know both g7 and b7 are nonempty sets. if ϕ0 ∈ g7, by lemma 2.1 and ‖ϕ‖2 < 2 n− 2 y , the corresponding solution ϕ(t,x) of cauchy problem (1.1) and (2.1) satisfies: for all t ∈ [0,t), ~1(‖∇ϕ(t)‖22) ≤ e(ϕ) + β(γ − 1) 4γ ‖ϕ‖22 < (n− 2)2(γ − 2) − 2β(γ − 1) 2(n− 2)2γ y 2 < ~1(y ). (3.24) we only need to prove ‖∇ϕ‖2 < y . otherwise, by the continuity of ϕ(t) there exists t ∈ [0,t) such that ‖∇ϕ(t)‖2 = y , then by computation we get ~1(‖∇ϕ(t)‖2) = ~1(y ) ≤ e(ϕ) + β(γ − 1) 4γ ‖ϕ‖22, which contradicts (3.24). thus ‖∇ϕ‖2 < y , which implies the solution ϕ(t,x) of the cauchy problem (1.1) and(2.1) exists globally in t ∈ (0,∞). we can obtain the invariance of b7 by the same token. finally, we prove the statement (ii) of theorem 3.7. from (2.6), we have j′′(t) ≤ 8γe(ϕ0) + ∫ −4(γ − 2)|∇ϕ|2 + (4γ − 4)β[ 2 (n− 2)2 |∇ϕ|2 + 1 2 |ϕ|2]dx = 8γ[e(ϕ) + β(γ − 1) 4γ ‖ϕ‖22] − 4(n− 2)2(γ − 2) − 8β(γ − 1) (n− 2)2 ‖∇ϕ‖22 ≤ 8γ (n− 2)2(γ − 2) − 2β(γ − 1) 2γ(n− 2)2 y 2 − 4(n− 2)2(γ − 2) − 8β(γ − 1) (n− 2)2 y 2 = 0. therefore from lemma 2.1 and 2.3, it must be the case t < ∞, which implies the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. this completes the proof of theorem 3.7. � case vi: 4 n < p < 4 n− 2 , np 2 ≤ γ < min{4,n}. denote k6 = (n− 2)2(np− 4) −β(2np− 4) 2(n− 2)2np y ′2. then we have two invariant sets: g8 = {ϕ ∈ h1 : e(ϕ) + (β + 1)γ − 1 4γ ‖ϕ‖22 < k6,‖ϕ‖2 < 2 (n− 2) y ′,‖∇ϕ‖2 < y ′}, b8 = {ϕ ∈ h1 : e(ϕ) + (β + 1)γ − 1 4γ ‖ϕ‖22 < k6,‖ϕ‖2 < 2 (n− 2) y ′,‖∇ϕ‖2 > y ′}, where y ′ is shown in the proof of the following theorem: theorem 3.8. for 4 n < p < 4 n− 2 and np 2 ≤ γ < min{4,n}, the following facts are ture: (i) when ϕ0 ∈ g8 ∪{0} , the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞). 76 n. tang, c. wang, and j. zhang (ii) when ϕ0 ∈ b8 and |x|ϕ0 ∈ l2(rn), the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) blows up in a finite time. proof. firstly, according to (2.8) , (2.10) and (2.12), we estimate the energy functional e(ϕ), for all t ∈ (0,t], e(ϕ(t)) + β(np− 2) 4np ‖ϕ(t)‖22 ≥ 1 2 ‖∇ϕ‖22 − β (n− 2)2 ‖∇ϕ‖22 − β 4 ‖ϕ‖22 + β(np− 2) 4np ‖ϕ‖22 − 1 γ‖∇w‖22 ‖ϕ‖4−γ2 ‖∇ϕ‖ γ 2 − 2 np‖∇r‖p2 ‖ϕ‖ 4−(n−2)p 2 2 ‖∇ϕ‖ np 2 2 = [ 1 2 − β (n− 2)2 ]‖∇ϕ‖22 − β 2np ‖ϕ‖22 − 1 γ‖∇w‖22 ‖ϕ‖4−γ2 ‖∇ϕ‖ γ 2 − 2 np‖∇r‖p2 ‖ϕ‖ 4−(n−2)p 2 2 ‖∇ϕ‖ np 2 2 . (3.25) let y = ‖∇ϕ(t)‖2 ≥ 0, for all t ∈ (0,t], e(ϕ(t)) + β(np− 2) 4np ‖ϕ(t)‖22 ≥ ~3(‖∇ϕ(t)‖2) = ~3(y), (3.26) where ~3(y) = [ 1 2 − β (n− 2)2 ]y2 − β 2np ‖ϕ‖22 − 1 γ‖∇w‖22 ‖ϕ‖4−γ2 y γ − 2 np‖∇r‖p2 ‖ϕ‖ 4−(n−2)p 2 2 y np 2 , ~′3(y) = [1 − 2β (n− 2)2 − 1 ‖∇w‖22 ‖ϕ‖4−γ2 y γ−2 − 1 ‖∇r‖p2 ‖ϕ‖ 4−(n−2)p 2 2 y np 2 −2]y = ~2(y)y. the same as the proof of theorem 3.7, there exists a unique positive y ′ such that ~2(y ′) = 0, thus the maximum of ~3(y) is ~3(y ′). secondly, we prove the invariance of g8 and b8. combined with the structure of ~3(y), we can easily know both g8 and b8 are nonempty sets. if ϕ0 ∈ g8, by lemma 2.1 and ‖ϕ‖22 < 8 (n− 2)2 y ′2, the corresponding solution ϕ(t,x) of cauchy problem (1.1) and (2.1) satisfies: for all t ∈ [0,t), ~3(‖∇ϕ(t)‖22) ≤ e(ϕ) + β(np− 2) 4np ‖ϕ‖22 < (n− 2)2(np− 4) −β(2np− 4) 2(n− 2)2np y ′2 < ~3(y ′). (3.27) we only need to prove ‖∇ϕ‖2 < y ′. otherwise, by the continuity of ϕ(t) there exists t ∈ [0,t) such that ‖∇ϕ(t)‖2 = y ′, then by computation we get ~3(‖∇ϕ(t)‖2) = ~3(y ′) ≤ e(ϕ) + β(np− 2) 4np ‖ϕ‖22, which contradicts (3.27). thus ‖∇ϕ‖2 < y ′, which implies the solution ϕ(t,x) of the cauchy problem (1.1) and (2.1) exists globally in t ∈ (0,∞) . we can obtain the invariance of b8 by the same token. finally, we prove the statement (ii) of theorem 3.8. from (2.6), we have j′′(t) = 4npe(ϕ0) − ∫ (2np− 8)|∇ϕ|2 − (np− 2γ)(|x|−γ ∗ |ϕ|2)|ϕ|2 − (2np− 4)β|x|−1|ϕ|2dx ≤ 4npe(ϕ0) − ∫ (2np− 8)|∇ϕ|2 − (2np− 4)β[ 2 (n− 2)2 |∇ϕ|2 + 1 2 |ϕ|2]dx = 4np[e(ϕ) + β(np− 2) 4np ‖ϕ‖22] − (n− 2)2(2np− 8) −β(4np− 8) (n− 2)2 ‖∇ϕ‖22 ≤ 4np (n− 2)2(np− 4) −β(2np− 4) 2(n− 2)2np y ′2 − (n− 2)2(2np− 8) −β(4np− 8) (n− 2)2 y ′2 = 0. 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[29] j. zhang, s. h. zhu, stability of standing waves for the nonlinear fractional schrödinger equation, j. dyn differ equ. 29 (2017), 1017-1030. n. tang, school of mathematical sciences, university of electronic science and technology of china, chengdu, sichuan, 711731, p. r. china email address: tangna514@163.com c. wang, school of mathematical sciences, university of electronic science and technology of china, chengdu, sichuan, 711731, p. r. china email address: wangchenglinedu@163.com j. zhang, corresponding author, school of mathematical sciences, university of electronic science and technology of china, chengdu, sichuan, 711731, p. r. china email address: zhangjian@uestc.edu.cn mathematics in applied sciences and engineering https://doi.org/10.5206/mase/9451 volume 1, number 1, march 2020 , pp.27-38 https://ojs.lib.uwo.ca/mase global asymptotic stability of a delayed plant disease model yuming chen and chongwu zheng abstract. in this paper, we consider the following system of delayed differential equations,{ s′(t) = σϕ − βs(t)i(t − τ) − ηs(t), i′(t) = σ(1 − ϕ) + βs(t)i(t − τ) − (η + ω)i(t), which can be used to model plant diseases. here ϕ ∈ (0, 1], τ ≥ 0, and all other parameters are positive. the case where ϕ = 1 is well studied and there is a threshold dynamics. the system always has a disease-free equilibrium, which is globally asymptotically stable if the basic reproduction number r0 ≜ βσ/η(η + ω) ≤ 1 and is unstable if r0 > 1; when r0 > 1, the system also has a unique endemic equilibrium, which is globally asymptotically stable. in this paper, we study the case where ϕ ∈ (0, 1). it turns out that the system only has an endemic equilibrium, which is globally asymptotically stable. the local stability is established by the linearization method while the global attractivity is obtained by the lyapunov functional approach. the theoretical results are illustrated with numerical simulations. 1. introduction plant diseases can decrease the economic, aesthetic, and biological value of many types of plants. examples of plant diseases caused by plant viruses can be found in the journal, plant disease. as a result, integrated management concepts have been developed to combat various plagues suffered by crops. integrated management strategies combine available host resistance with cultural, chemical and biological control measures. for example, a cultural control strategy including replanting, and/or removing (roguing) diseased plants is a widely accepted treatment for plant epidemics. mathematical models of plant-virus disease epidemics are often built to provide a detailed exposition on how to describe, analyze, and predict epidemics of plant diseases. to name a few, see [7, 14, 16, 18, 19] and the references therein. based on the biological background in [2, 19], a simple model for plant diseases with a continuous cultural strategy can be built as follows. the plant population is divided into three compartments, susceptible plants (s), infected plants (i), and removed plants (r). removal occurs by death or by sanitation. the removed plants are assumed not to be infected again, so we need to focus only on s and i. we make the following assumptions. a) new plants enter the system at the rate σ with a proportion ϕ being susceptible and (1 − ϕ) being infected. b) the incidence rate is proportional to si and the transmission rate is β. c) plants are removed from the system at the rate η, in other words, 1/η is either the harvest time or the end of their reproductive lifetime. received by the editors 23 january 2020; revised 22 february 2020; accepted 22 february, 2020; published online 24 february 2020. 2000 mathematics subject classification. primary 34k20; secondary 92d30. key words and phrases. delay, equilibrium, stability, plant disease. the first author was supported in part by nserc of canada. 27 28 y. chen and c. zheng d) the rouging (removing infected plants from the system) rate is ω. figure 1 schematically sketches the transmission of a plant disease. the above assumptions lead to the is βsi ηs ωi ηi σ σϕ σ(1 − ϕ) figure 1. schematic diagram for the transmission of a plant disease following si model for plant diseases,  ds(t) dt = σϕ − βs(t)i(t) − ηs(t), di(t) dt = σ(1 − ϕ) + βs(t)i(t) − (η + ω)i(t). (1.1) here the removal is continual, which is common for plant diseases. for human epidemics, the removal can be impulsive at some certain time instants. for example, see de la sen et al. [4]. however, in disease transmission models, time delay is an important quantity for many epidemiological mechanisms, but this is not reflected in (1.1). we refer to van den driessche [20] for a brief review of delay differential equations arising from disease modeling. as in cooke [3], we incorporate a constant delay into (1.1) to obtain the following system of delayed differential equations,  ds(t) dt = σϕ − βs(t)i(t − τ) − ηs(t), di(t) dt = σ(1 − ϕ) + βs(t)i(t − τ) − (η + ω)i(t). (1.2) here τ is the average latent period of the plant disease, that is, the average time for a susceptible plant from getting infected to becoming infectious. an alternative approach, perhaps more realistic, is to incorporate a distributed delay as follows,  ds(t) dt = σϕ − βs(t) ∫ τ 0 i(t − s)dk(s) − ηs(t), di(t) dt = σ(1 − ϕ) + βs(t) ∫ τ 0 i(t − s)dk(s) − (η + ω)i(t), (1.3) where k : [0, τ] → r is nondecreasing and has bounded variation such that ∫ τ 0 dk(s) = k(τ) − k(0) = 1. the discussion for (1.3) is quite similar to that for (1.2). we deal with (1.2) in the sequel and in section 4 we will mention the necessary modification for a general system including (1.3). let c = c([−τ, 0], r2) be the banach space of continuous functions from [−τ, 0] to r2 equipped with the usual supremum norm ∥ · ∥. the initial condition of (1.2) is given as s(t) = φ1(θ) and i(t) = φ2(θ) for θ ∈ [−τ, 0], (1.4) a delayed plant disease model 29 where φ = (φ1, φ2) ∈ c such that φi(θ) ≥ 0 for θ ∈ [−τ, 0] and i = 1, 2. if (s(t), i(t)) is a solution of (1.2), then we have s(t) = s(0)e− ∫ t 0 [βi(s−τ)+η]ds + ϕσ ∫ t 0 e− ∫ t u [βi(s−τ)+η]dsdu (1.5) and i(t) = e−(η+ω)ti(0) + ∫ t 0 e−(η+ω)(t−u)[(1 − ϕ)σ + βs(u)i(u − τ)]du (1.6) for t ≥ 0. it follows that given an initial condition (1.4), we can use the step-by-step method to find s(t) first by (1.5) and then i(t) by (1.6). one can also easily see that any solution of (1.2) with initial condition (1.4) satisfies s(t) > 0 and i(t) > 0 for t > 0. without loss of generality, we need to consider only positive solutions of (1.2). sometimes, the solution of (1.2) with the initial condition (1.4) is also denoted by (s(t; φ), i(t; φ)). as usual, for a solution (s(t), i(t)) of (1.2) and t > 0, we define (st, it) ∈ c by st(θ) = s(t + θ) and it(θ) = i(t + θ) for θ ∈ [−τ, 0]. for more information on delay differential equations, we refer to hale and verduyn lunel [8] or diekmann et al. [5]. the case where ϕ = 1 of (1.2) is the standard delayed si or sir model. these standard models with discrete delays and distributed delays have been well studied. to name a few, see [1, 11, 12, 17]. the dynamics of (1.2) with ϕ = 1 is quite simple, which is a threshold dynamics. precisely, the system always has a disease free equilibrium e0 = ( σ η , 0), which is globally asymptotically stable if r0 ≜ βσ/η(η + ω) ≤ 1 and is unstable if r0 > 1; when r0 > 1, it also has an endemic equilibrium e∗ = ( η + ω β , βσ − η(η + ω) β(η + ω) ), which is globally asymptotically stable. r0 is called the basic reproduction number of (1.2). though (1.2) with ϕ = 1 and its modifications have been extensively studied, system (1.2) with ϕ ∈ (0, 1) has not been studied yet. one reason is that the epidemics considered in the literature are mainly human diseases. in such situations, in order to lower the economic cost, infected persons usually will be quarantined or hospitalized. therefore, it is reasonable to assume that ϕ = 1. but, for plant diseases, it is necessary and important to study the dynamics of (1.2) with ϕ ∈ (0, 1), which is the goal of this paper. for example, african small holders replant cassava and sweet potato with cuttings only from previous crop while commercial sweet potato growers in china use disease-free material from in vitro propagation programmes [6]. the first corresponds to ϕ ∈ (0, 1) and the latter corresponds to ϕ = 1 if we use (1.2) to model them, respectively. the remainder of the paper is organized as follows. we first study the existence of equilibria and their local stability in section 2. we then show that the system is globally asymptotically stable in section 3 by the lyapunov functional approach. the paper concludes with general remarks. it is mentioned that the approach here can be applied to deal with generalized versions of (1.2) with nonlinear incidence functions and distributed delays. 2. existence of equilibria and their local stability in this section, we study the local dynamics of (1.2) with ϕ ∈ (0, 1). we first consider the existence of equilibria. 30 y. chen and c. zheng proposition 2.1. suppose ϕ ∈ (0, 1). then (1.2) only has an endemic equilibrium e∗ϕ = (s ∗ ϕ, i ∗ ϕ), where s∗ϕ = βσ + η(η + ω) − √ ∆ϕ 2βη , i∗ϕ = σ − ηs∗ϕ η + ω , (2.1) ∆ϕ = [η(η + ω) + βσ] 2 − 4ϕβση(η + ω). proof. let (s, i) be an equilibrium of (1.2). then we have σϕ − βsi − ηs = 0 (2.2) and σ(1 − ϕ) + βsi − (η + ω)i = 0. (2.3) adding (2.2) and (2.3) yields σ − ηs − (η + ω)i = 0, or i = σ−ηs η+ω . then substituting it into (2.2), we obtain g(s) = βη η + ω s2 − ( βσ η + ω + η ) s + ϕσ = 0. note that g(0) = ϕσ > 0 and g(σ η ) = −(1 − ϕ)σ < 0. it follows that g(s) = 0 has two positive solutions, one less than σ η and the other larger than σ η . from i = (σ − ηs)/(η + ω) ≥ 0, we know that s ≤ σ/η. hence (1.2) has only one equilibrium, which is positive and is given by (2.1). this completes the proof. □ proposition 2.1 tells us that (1.2) has no disease-free equilibrium, which is due to the recruitment of infected plants. therefore, the basic reproduction number cannot be defined. in the following, we investigate the local stability of e∗ϕ. e ∗ ϕ is locally stable if for any ε > 0 there exists δ > 0 such that ∥(st(·; φ), it(·; φ)) − e∗ϕ∥ ≤ ε for t ≥ 0 and ∥φ − e∗ϕ∥ ≤ δ; otherwise, it is unstable. the local stability of e∗ϕ is obtained by the technique of linearization. precisely, if all roots of the characteristic equation associated with the linearized system about e∗ϕ have negative real parts then e∗ϕ is locally exponentially stable and hence stable; if at least one of the roots has positive real part then e∗ϕ is unstable. e ∗ ϕ is locally exponentially stable if there exist positive constants m, α, and δ such that ∥(st(·; φ), it(·; φ)) − e∗ϕ∥ ≤ me −αt∥φ − e∗ϕ∥ for all t ≥ 0 and ∥φ − e∗ϕ∥ ≤ δ. to prove the local stability of e∗ϕ, it is worthy to note that ∆ϕ > [η(η + ω) + βσ] 2 − 4βση(η + ω) = [βσ − η(η + ω)]2 as ϕ ∈ (0, 1). theorem 2.1. suppose ϕ ∈ (0, 1). then the equilibrium e∗ϕ of (1.2) is locally exponentially stable. proof. linearize (1.2) around e∗ϕ to obtain  ds(t) dt = −(βi∗ϕ + η)s(t) − βs ∗ ϕi(t − τ), di(t) dt = βi∗ϕs(t) + βs ∗ ϕi(t − τ) − (η + ω)i(t). the corresponding characteristic equation is det ( λ + βi∗ϕ + η βs ∗ ϕe −λτ −βi∗ϕ λ + η + ω − βs ∗ ϕe −λτ ) = 0, a delayed plant disease model 31 or λ2 + λ(2η + ω + βi∗ϕ) − βs ∗ ϕe −λτ (λ + η) + (βi∗ϕ + η)(η + ω) = 0. (2.4) to finish the proof, we only need to show that all roots of (2.4) have negative real parts. first, suppose that τ = 0. then (2.4) reduces to λ2 + λ(2η + ω + βi∗ϕ − βs ∗ ϕ) + [(βi ∗ ϕ + η)(η + ω) − βs ∗ ϕη] = 0. with (2.1), we get λ2 + pϕλ + √ ∆ϕ = 0, where pϕ = 2η3 + 3η2ω + ηω2 − βσω + (2η + ω) √ ∆ϕ 2η(η + ω) . if βσ ≥ η(η + ω), then √ ∆ϕ > βσ − η(η + ω) and hence pϕ ≥ 2η3 + 3η3ω + ηω2 − βσω + (2η + ω)[βσ − η(η + ω)] 2η(η + ω) = βσ η + ω > 0; if βσ < η(η + ω), then pϕ > 2η3 + 3η2ω + ηω2 − ωη(η + ω) + (2η + ω) √ ∆ϕ 2η(η + ω) = 2η3 + 2η2ω + (2η + ω) √ ∆ϕ 2η(η + ω) > 0. therefore, if τ = 0 then all roots of (2.4) have negative real parts. next, we claim that (2.4) has no roots on the imaginary axis. by way of contradiction, suppose that (2.4) has a root iξ with ξ ∈ r for some τ > 0. then substitute it into (2.4) and separate the real and imaginary parts to obtain (βi∗ϕ + η)(η + ω) − ξ 2 = βs∗ϕ(ξ sin ξτ + η cos ξτ) and (2η + ω + βi∗ϕ)ξ = βs ∗ ϕ(ξ cos ξτ − η sin ξτ). it follows that ξ4 + [(2η + ω + βi∗ϕ) 2 − 2(βi∗ϕ + η)(η + ω) − (βs ∗ ϕ) 2]ξ2 + [(βi∗ϕ + η) 2(η + ω)2 − (βs∗ϕη) 2] = 0. (2.5) note that (βi∗ϕ + η) 2(η + ω)2 − (βs∗ϕη) 2 = [(βi∗ϕ + η)(η + ω) + βs ∗ ϕη][(βi ∗ ϕ + η)(η + ω) − βs ∗ ϕη] = [(βi∗ϕ + η)(η + ω) + βs ∗ ϕη] √ ∆ϕ > 0 32 y. chen and c. zheng and (2η + ω + βi∗ϕ) 2 − 2(βi∗ϕ + η)(η + ω) − (βs ∗ ϕ) 2 = [(η + ω) + (η + βi∗ϕ)] 2 − 2(βi∗ϕ + η)(η + ω) − (βs ∗ ϕ) 2 = (η + ω)2 + (η + βi∗ϕ) 2 − (βs∗ϕ) 2 = (η + βi∗ϕ) 2 + (η + ω + βs∗ϕ)(η + ω − βs ∗ ϕ) = (η + βi∗ϕ) 2 + (η + ω + βs∗ϕ) η(η + ω) − βσ + √ ∆ϕ 2η > 0 (as √ ∆ϕ > |η(η + ω) − βσ|). it follows that (2.5) cannot hold and this proves the claim. note that the roots of (2.4) depend continuously on τ and that all roots of (2.4) with non-negative real parts are uniformly bounded (which is easy to see). this, combined with the claim and the result that all roots of (2.4) have negative real parts for the case where τ = 0, tells us that all roots of (2.4) have negative real parts for any τ ≥ 0. therefore, we have completed the proof. □ 3. global asymptotic stability of the endemic equilibrium in the previous section, we have shown that (1.2) has a unique endemic equilibrium e∗ϕ which is locally stable. indeed, e∗ϕ is also globally asymptotically stable, that is, e ∗ ϕ is locally stable and globally attractive. e∗ϕ is globally attractive if lim t→∞ (st, it) = e ∗ ϕ for every solution (s(t), i(t)) of (1.2). to prove the global attractivity, we first establish the permanence of (1.2). definition 3.1 ([9]). system (1.2) is said to be permanent if there are positive constants νi and mi (i = 1, 2) such that ν1 ≤ lim inf t→∞ s(t) ≤ lim sup t→∞ s(t) ≤ m1 and ν2 ≤ lim inf t→∞ i(t) ≤ lim sup t→∞ i(t) ≤ m2 hold for any solution of (1.2) with the initial condition (1.4). here νi and mi (i = 1, 2) are independent of (1.4). proposition 3.1. suppose ϕ ∈ (0, 1). then (1.2) is permanent. in fact, for every solution (s(t), i(t)), we have σϕη βσ + η2 ≤ lim inft→∞ s(t) ≤ lim sup t→∞ s(t) ≤ σ η , (3.1) σ(1 − ϕ) η + ω ≤ lim inft→∞ i(t) ≤ lim sup t→∞ i(t) ≤ σ η . (3.2) proof. let (s(t), i(t)) be any solution of (1.2). then d(s(t) + i(t)) dt = σ − ηs(t) − (η + ω)i(t) ≤ σ − η(s(t) + i(t)) and hence by the comparison principle (see, for example, [10]) we get s(t) + i(t) ≤ [ σ(eηt − 1) η + s(0) + i(0) ] e−ηt = σ η + η(s(0) + i(0)) − σ η e−ηt. a delayed plant disease model 33 it follows that lim sup t→∞ (s(t) + i(t)) ≤ σ η , which implies that lim sup t→∞ s(t) ≤ σ η and lim sup t→∞ i(t) ≤ σ η . (3.3) in particular, for any ε > 0, there exists a t > 0 such that i(t) ≤ σ η + ε for all t ≥ t. then, for t ≥ t + τ, ds(t) dt ≥ ϕσ − [ β ( σ η + ε ) + η ] s(t), which immediately yields from the comparison principle that lim inf t→∞ s(t) ≥ ϕσ β(σ η + ε) + η . as ε is arbitrary, we have lim inf t→∞ s(t) ≥ ϕση βσ + η2 . (3.4) finally, note that di(t) dt ≥ (1 − ϕ)σ − (η + ω)i(t) for t ≥ 0. again, by the comparison principle we get lim inf t→∞ i(t) ≥ (1 − ϕ)σ η + ω . (3.5) it follows from (3.3)–(3.5) immediately that (1.2) is permanent and both (3.1) and (3.2) hold. this completes the proof. □ now, we are ready to prove the main result of this section. theorem 3.2. suppose ϕ ∈ (0, 1). then the equilibrium e∗ϕ of (1.2) is globally asymptotically stable. proof. we have shown in theorem 2.1 that e∗ϕ is locally stable and hence it suffices to show that e ∗ ϕ is globally attractive. the proof is completed by using the lyapunov functional approach and is parallel to that of theorem 4.1 of mccluskey [13] (but we use the notation in hale and verduyn lunel [8]). to construct the lyapunov functional, we need the function g(x) = x − 1 − ln x for x > 0. it is well-known that g(x) ≥ 0 and g(x) = 0 if and only if x = 1. let d = { φ = (φ1, φ2) ∈ c ∣∣∣∣∣ φ1(θ) ∈ [ σϕη βσ+η2 , σ η ] and φ2(θ) ∈ [ (1−ϕ)σ η+ω , σ η ] for θ ∈ [−τ, 0] } . it is not difficult to see that d is an invariant set of (1.2). moreover, by proposition 3.1, d is attractive. therefore, to study the global attractivity of e∗ϕ, we only need to consider solutions starting in d. we define v : d → r by v (φ) = 1 βi∗ ϕ vs(φ1) + 1 βs∗ ϕ vi(φ2) + v+(φ2) for φ = (φ1, φ2) ∈ d, where vs(φ1) = g( φ1(0) s∗ϕ ), vi(φ2) = g( φ2(0) i∗ϕ ) and v+(φ2) = ∫ τ 0 g( φ2(−s) i∗ϕ )ds. 34 y. chen and c. zheng to find the derivative of v along solutions of (1.2), the following relationships between s∗ϕ and i ∗ ϕ will be helpful, ϕσ = βs∗ϕi ∗ ϕ + ηs ∗ ϕ, (3.6) (η + ω)i∗ϕ = (1 − ϕ)σ + βs ∗ ϕi ∗ ϕ. (3.7) for clarity, we first calculate the derivatives of vs, vi, and v+ along solutions of (1.2) one by one. firstly, v̇s(φ1) = lim sup h→0+ 1 h [vs(sh) − vs(φ1)] = dvs(sh) dh |h=0 = 1 s∗ϕ ( 1 − s∗ϕ s(0) ) ds(0) dt = 1 s∗ϕ ( 1 − s∗ϕ s(0) ) (ϕσ − βs(0)i(−τ) − ηs(0)) = 1 s∗ϕ ( 1 − s∗ϕ φ1(0) ) (ϕσ − βφ1(0)φ2(−τ) − ηφ1(0)). using (3.6) to replace ϕσ gives us v̇s(φ1) = 1 s∗ϕ ( 1 − s∗ϕ φ1(0) ) {η[s∗ϕ − φ1(0)] + β[s ∗ ϕi ∗ ϕ − φ1(0)φ2(−τ)]} = −η (φ1(0) − s∗ϕ) 2 φ1(0)s ∗ ϕ + βi∗ϕ ( 1 − s∗ϕ φ1(0) )( 1 − φ1(0) s∗ϕ φ2(−τ) i∗ϕ ) . for simplicity of notation, let x = φ1(0) s∗ϕ , y = φ2(0) i∗ϕ , and z = φ2(−τ) i∗ϕ . then we can get v̇s(φ1) = −η (φ1(0) − s∗ϕ) 2 φ1(0)s ∗ ϕ + βi∗ϕ ( 1 − xz − 1 x + z ) . (3.8) secondly, v̇i(φ2) = 1 i∗ϕ ( 1 − i∗ϕ φ2(0) ) [(1 − ϕ)σ + βφ1(0)φ2(−τ) − (η + ω)φ2(0)] = 1 i∗ϕ ( 1 − i∗ϕ φ2(0) )[ (1 − ϕ)σ + βφ1(0)φ2(−τ) − (η + ω)i∗ϕ φ2(0) i∗ϕ ] . with the help of (3.7), we get v̇i(φ2) = 1 i∗ϕ ( 1 − i∗ϕ φ2(0) )[ (1 − ϕ)σ ( 1 − φ2(0) i∗ϕ ) (3.9) +βs∗ϕi ∗ ϕ ( φ1(0) s∗ϕ φ2(−τ) i∗ϕ − φ2(0) i∗ϕ )] = −(1 − ϕ)σ (φ2(0) − i∗ϕ) 2 φ2(0)(i ∗ ϕ) 2 + βs∗ϕ ( 1 − y − xz y + xz ) . (3.10) a delayed plant disease model 35 finally, v̇+(φ2) = d dt ∫ τ 0 g ( i(t − s) i∗ϕ ) ds|t=0 = ∫ τ 0 d dt g ( i(t − s) i∗ϕ ) |t=0ds = ∫ τ 0 − d ds g ( i(−s) i∗ϕ ) ds = g ( i(0) i∗ϕ ) − g ( i(−τ) i∗ϕ ) = g(y) − g(z) = y − z + ln z − ln y. (3.11) now, we obtain from (3.8), (3.10), and (3.11) that v̇ (φ) = − η βi∗ϕs ∗ ϕ (φ1(0) − s∗ϕ) 2 φ1(0) − (1 − ϕ)σ βs∗ϕ(i ∗ ϕ) 2 (φ2(0) − i∗ϕ) 2 φ2(0) +2 − 1 x − xz y + ln z − ln y = − η βi∗ϕs ∗ ϕ (φ1(0) − s∗ϕ) 2 φ1(0) − (1 − ϕ)σ βs∗ϕ(i ∗ ϕ) 2 (φ2(0) − i∗ϕ) 2 φ2(0) + ( 1 − 1 x − ln x ) + ( 1 − xz y + ln xz y ) = − η βi∗ϕs ∗ ϕ (φ1(0) − s∗ϕ) 2 φ1(0) − (1 − ϕ)σ βs∗ϕ(i ∗ ϕ) 2 (φ2(0) − i∗ϕ) 2 φ2(0) −g ( 1 x ) − g ( xz y ) . then v̇ (φ) ≤ 0 and k = {φ ∈ d|v̇ (φ) = 0} = {φ = (φ1, φ2) ∈ c|φ1(0) = s∗ϕ, φ2(0) = i ∗ ϕ}. obviously, the largest set in k that is invariant with respect to (1.2) is the singleton {e∗ϕ}. by theorem 3.1 in chapter 5 of hale and verduyn lunel [8], every solution converges to e∗ϕ. this completes the proof. □ 4. concluding remarks in this paper, we considered a plant disease model (1.2) with ϕ ∈ (0, 1). unlike the case where ϕ = 1, there is no threshold dynamics. the model has a unique endemic equilibrium e∗ϕ, which is always globally asymptotically stable. figure 2 and figure 3 illustrate this result for the cases βσ < η(η + ω) and βσ > η(η + ω), respectively. in both cases, τ = 4, β = 0.0064, η = 0.002, ϕ = 0.75. but for figure 2, σ = 0.0015 and ω = 0.003; while for figure 3, σ = 0.015 and ω = 0.03. these parameter values are reasonable for cassava (figure 2) and sweet potatoes (figure 3), respectively. we refer to van den bosch et al [19] for more detail. the global asymptotic stability implies that the disease cannot be eradicated. this is realistic since, in practice, it is almost impossible to guarantee that all replanted plants are healthy. this result looks quite different from that for the case where ϕ = 1. however, when ϕ is very close to 1, the result is 36 y. chen and c. zheng 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 s i figure 2. when σ = 0.0015, ϕ = 0.75, β = 0.0064, η = 0.002, ω = 0.003 and τ = 4, the equilibrium e∗ϕ ≈ (0.3826, 0.1470) of (1.2) is globally asymptotically stable. 1.5 2 2.5 3 3.5 4 4.5 5 0.2 0.25 0.3 0.35 0.4 0.45 0.5 s i figure 3. when σ = 0.015, ϕ = 0.75, β = 0.0064, η = 0.002, ω = 0.03 and τ = 4, the equilibrium e∗ϕ ≈ (2.9428, 0.2848) of (1.2) is globally asymptotically stable. consistent with that for the case where ϕ = 1. in fact, i∗ϕ = βσ − η(η + ω) + √ ∆ϕ 2β(η + ω) . if βσ = η(η + ω) then i∗ϕ = √ (1 − ϕ)βση(η + ω) β(η + ω) = η √ 1 − ϕ β . suppose that βσ ̸= η(η + ω). if ϕ ≈ 1, then√ ∆ϕ ≈ |βσ − η(η + ω)|[1 + 2(1 − ϕ)βση(η + ω) [βσ − η(η + ω)]2 ] = |βσ − η(η + ω)| + 2(1 − ϕ)βση(η + ω) |βσ − η(η + ω)| and hence i∗ϕ ≈ (1 − ϕ)ση η(η + ω) − βσ if βσ < η(η + ω) while i∗ϕ ≈ (βσ − η(η + ω))2 + (1 − ϕ)ση(η + ω) (η + ω)[βσ − η(η + ω)] if βσ > η(η + ω), which are quite close to the disease levels in the case where ϕ = 1. the discussion also tells us that in order to develop a good control strategy, we should make βσ < η(η + ω). in this case, though the disease cannot be eradicated, the disease level can be very small and hence it cannot reach the economic injury level [15] (the lowest population density that will cause economic damage or the amount of injury which will justify the cost of using controls). we mention that we can use the same lyapunov functional to show the global attractivity of the endemic equilibrium of the following generalization of (1.2),  ds(t) dt = ϕσ − βs(t)f(i(t − τ)) − ηs(t), di(t) dt = σ(1 − ϕ) + βs(t)f(i(t − τ)) − (η + ω)i(t), (4.1) provided that an equilibrium (which is of course an endemic equilibrium) is guaranteed by the incidence function f. for example, f(x) = 1 1+αx in mccluskey [13]. here, we considered the case where f(x) = x for the simple reason that we could obtain the local stability of the endemic equilibrium. for a general a delayed plant disease model 37 incidence function f, it may not be easy to analyze the local stability. moreover, consider modifications of (4.1) with a distributed delay,  ds(t) dt = ϕσ − βs(t) ∫ τ 0 f(i(t − s))dk(s) − ηs(t), di(t) dt = σ(1 − ϕ) + βs(t) ∫ τ 0 f(i(t − s))dk(s) − (η + ω)i(t), where k : [0, τ] → r is nondecreasing and has bounded variation such that ∫ τ 0 dk(s) = k(τ) − k(0) = 1. same results can be obtained by replacing v+(φ2) with v̂+(φ2), where v̂+(φ2) = ∫ τ 0 α(s)g ( φ2(−s) i∗ϕ ) ds and α(h) = ∫ τ h dk(s) for h ∈ [0, τ]. we refer to mccluskey [12] for some guidance. references [1] e. beretta and y. takeuchi, global stability of an sir epidemic model with times, j. math. biol. 33(1995) 250–260. [2] m.s. chan and m.j. jeger, an analytical model of plant virus disease dynamics with roguing and replanting, j. appl. ecol. 31(1994) 413–427. [3] k.l. cooke, stability analysis for a vector disease model, rocky mount. j. math. 7(1979) 253–263. [4] m. de la sen, ravi p. agarwal, a. ibeas and s. alonso-quesada, on a generalized time-varying seir epidemic model with mixed point and distributed time-varying delays and combined regular and impulsive vaccination controls, advances in differential equations 2010 (2010), article id 281612, 42 pages. [5] o. diekmann, s.a. van gils, s.m. verduyn lunel and h.-o. walther, delay equations: functional-, complex-, and nonlinear analysis, springer-verlag, new york, 1995. [6] g. feng, g. yifu and z. pinbo, production and development of virus-free sweet potato in china, crop prot. 19(2000) 105–111. [7] s. fishman and r. marcus, a model for spread of plant disease with periodic removals, j. math. biol. 21(1984) 149–158. [8] j. hale and s.m. verduyn lunel, introduction to functional differential equations, springer-verlag, new york, 1993. [9] y. kuang, delay differential equations with applications in population dynamics, academic press, san diego, 1993. [10] v. lakshmikantham and s. leela, differential and integral inequalities: theory and applications, vol. i: ordinary differential equations, academic press, new york-london, 1969. [11] w. ma, y. takeuchi, t. hara and e. beretta, permanence of an sir epidemic model with distributed time delays, tohoku math. j. 54(2002) 581–591. [12] c.c. mccluskey, complete global stability for an sir epidemic model with delay–distributed or discrete, nonlinear anal. real world appl. 11(2010) 55–59. [13] c.c. mccluskey, global stability for an sir epidemic model with delay and nonlinear incidence, nonlinear anal. real world appl. 11(2010) 3106–3109. [14] s. soubeyrand, l. held, m. höhle and i. sache, modelling the spread in space and time of an airborne plant disease, j. roy. statist. soc. ser. c 57(2008) 253–272. [15] v.m. stern, r.f. smith, r. van den bosch and k.s. hagen, the integrated control concept, hilgardia, 29(1959) 81–101. [16] n. stollenwerk and k.m. briggs, master equation solution of a plant disease model, phys. lett. a 274(2000) 84–91. [17] y. takeuchi, w. ma and e. beretta, global asymptotic properties of a delay sir epidemic model with finite incubation times, nonlinear anal., 42(2000) 931–947. [18] h.r. thieme and j.a.p. heesterbeek, how to estimate the efficacy of periodic control of an infectious plant disease, math. biosci. 93(1989) 15–29. [19] f. van den bosch, m.j. jeger and c.a. gilligan, disease control and its selection for damaging plant virus strains in vegetatively propogated staple food crops; a theoretical assessment, proc. r. soc. b 274(2007) 11–18. [20] p. van den driessche, some epidemiological models with delays, in differential equations and applications to biology and to industry, pp. 507–520, world sci. publ., river edge, nj, 1996. 38 y. chen and c. zheng corresponding author. department of mathematics, wilfrid laurier university, waterloo, on n2l 3c5, canada e-mail address: ychen@wlu.ca department of applied mathematics, yuncheng university, yuncheng, shanxi 044000, p.r. china e-mail address: chongyang1894@163.com mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 1, number 2, june 2020, pp.150-180 https://doi.org/10.5206/mase/10644 selected topics on reaction-diffusion-advection models from spatial ecology king-yeung lam, shuang liu, and yuan lou abstract. we discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. topics include lokta-volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. open problems and conjectures are presented. parts of this survey are adopted from the materials in [89, 108, 109], and some very recent progress are also included. 1. introduction recent years have witnessed unprecedented progress of experimental technologies in the life sciences. the explosion of empirical results have rapidly generated massive sets of loosely structured data. the analytical methods from mathematics and statistics are required to synthesize the large data sets and extract insightful information from them. this trend continues to accelerate the development in mathematical biology. the study of mathematical biology usually includes two aspects. on the one hand, by introducing and analyzing mathematical models, researchers can elucidate and predict the basic mechanisms of underlying biological processes. on the other hand, deeper understanding of these mechanisms may drive the discovery of new mathematical problems, new analytical techniques, and even launch new research directions. most, if not all, areas of mathematics have found applications in mathematical biology. as an important mathematical area, nonlinear partial differential equations have been one of the most active research fields in the 21st century. it has also found new opportunities in mathematical biology in recent years. in [47] dr. avner friedman discussed some challenging problems arising from mathematical biology, and one of the major mathematical tools he used is nonlinear partial differential equations. the subject of partial differential equations concerns the changes of quantities in space and time. for biology, the importance of space is hardly a question, and a practical question is to quantify the effect of space in different biological scenarios. in this survey, we consider some issues in spatial ecology via reaction-diffusion models, focusing upon the effect of movement of species on population dynamics in spatially heterogeneous environments. an important ongoing trend in population dynamics is the integration with other directions of biosciences, such as evolution, epidemiology, cell biology, cancer research, etc. our main goal is to showcase the role of reaction-diffusion models by integrating across different research directions in mathematical biology, including spatial ecology, evolution and received by the editors 15 april, 2020; revised 15 may 2020; accepted 15 may 2020; published online 17 may 2020. 2010 mathematics subject classification. 35k57, 92d25, 92d40, 92d30, 37n25. key words and phrases. reaction-diffusion; advection; competition; evolution; dispersal; population dynamics. kyl and yl were supported in part by nsf grant dms-1853561. sl was partially supported by the outstanding innovative talents cultivation funded programs 2018 of renmin univertity of china and the nsfc grant no. 11571364. 150 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/10644 selected topics on reaction-diffusion-advection models 151 disease transmissions. we will illustrate that the quantitative analysis of some important questions in biosciences does bring interesting and novel mathematical problems, which in turn calls for the development of new mathematical tools. we will also formulate a number of potential research questions along the way. this paper is organized as follows. in section 2 we discuss several single species models, for which the dynamics have important implications for the invasion of exotic species. section 3 is devoted to the study of two-species competition models, with emphasis on the effects of dispersal and spatial heterogeneity on the outcome of competition for two similar species. in section 4 we investigate the competition models with directed movement and show how different dispersal (either random or biased) can bring dramatic changes to population dynamics. section 5 concerns the continuous trait models and evolution of dispersal. finally, in sections 6 and 7 we discuss some progress on dynamics of phytoplankton growth and disease transmissions, respectively. 2. single species models the study of mathematical models for single species has played a primary role in population dynamics. it is also crucial in analyzing the dynamics of multiple interacting species, e.g. on issues concerning the invasions of exotic species. in this section we will focus on two types of single species models and study the effect of varying the diffusion rate. 2.1. logistic model. in this subsection, we consider the logistic model  ∂tu = d∆u + u[m(x) −u] (x,t) ∈ ω × (0,∞), ∂u ∂ν = 0 (x,t) ∈ ∂ω × (0,∞), u(x, 0) = u0(x) x ∈ ω, (2.1) where ω is a bounded domain in rn with smooth boundary ∂ω and ν(x) denotes the unit outward normal vector at x ∈ ∂ω. here u(x,t) represents the density of the single species at location x and time t. parameter d > 0 is the diffusion rate and ∆ = ∑n i=1 ∂2 ∂x2 i is the usual laplace operator. the function m ∈ c2(ω) accounts for the local carrying capacity or the intrinsic growth rate of the species, which is assumed to be strictly positive in ω for the sake of clarity. the neumann boundary condition indicates that no individuals can move across ∂ω, i.e. the habitat is closed. we assume that the initial data u0 is non-negative and not identically zero. it is well known [12] that lim t→∞ u(x,t) = u∗(x,d) uniformly for x ∈ ω, where for each d > 0, the function u∗(·,d) is the unique positive solution of  d∆u + u(m−u) = 0 x ∈ ω, ∂u ∂ν = 0 x ∈ ∂ω. (2.2) if the spatial environment is homogeneous, i.e. m is a positive constant, then u∗(x,d) ≡ m is the unique positive solution of (2.2), which is independent of the diffusion rate d. if the spatial environment is heterogeneous, i.e. m = m(x) is a non-constant function, then u∗(x,d) depends non-trivially on d. in this case, a natural question is how the total biomass of the species at equilibrium, i.e. ∫ ω u∗(x,d) dx, depends on the diffusion rate d. the following property was observed in [105]: 152 k.-y. lam, s. liu, and y. lou lemma 2.1. if m is non-constant, then for any diffusion rate d > 0,∫ ω u∗ > ∫ ω m = lim d→0 ∫ ω u∗ = lim d→∞ ∫ ω u∗. lemma 2.1 means that (i) the heterogeneous environment can support a total biomass greater than the total carrying capacity of the environment, which is quite different from the homogeneous case; (ii) the total biomass, as a function of d, is non-monotone; it is maximized at some intermediate diffusion rate, and is minimized at d = 0 and d = ∞, respectively. examples were constructed in [98] to illustrate that this function can have two or more local maxima. lemma 2.1 implies that ∫ ω u∗/ ∫ ω m > 1 for all d > 0, and the lower bound 1 is optimal. for the upper bound of ∫ ω u∗/ ∫ ω m, ni raised the following question: conjecture 1 ([130]). there exists some constant c = c(n) depending only on n such that∫ ω u∗ /∫ ω m ≤ c(n), and c(1) = 3. conjecture 1 suggests that the supremum of the ratio of the biomass to the total carrying capacity depends only on the spatial dimension. for one-dimensional case, c(1) = 3 was proved in [6] and it turns out to be optimal. however, it is recently shown by inoue and kuto [74] that conjecture 1 fails for the higher-dimensional case even when ω is a multi-dimensional ball. that is, the nonlinear operator which maps m to u∗ is not bounded as a map from the positive cone of l1(ω) to itself. a related question is the optimal control of total biomass [38]: optimization problem. fix any δ ∈ (0, 1), and define the control set u := { m ∈ l∞(ω) : 0 ≤ m ≤ 1, ∫ ω m = δ|ω| } . determine those m∗ ∈ u which can maximize the total biomass ∫ ω u∗ in u. nagahara and yanagida [128] showed that, under a regularity assumption, the optimal m∗ is of “bang-bang” type, i.e. m∗ = χe for some measurable set e ⊂ ω, where χe denotes the characteristic function of e. more recently, mazari et al. [121] proved that the bang-bang property holds for all large enough diffusion rates. biologically, the set e can be viewed as the protected area, and the condition∫ ω m = δ|ω| means that the total resources are limited. the results in [121, 128] imply that under the limited resources, the way to maximize the total population size of species is to place all the resources evenly in some suitable subset e of ω. however, the characterization of the set e is a challenging problem. it seems that the problem has not been completely solved even in the one-dimensional case; see [79] for some progress on cylindrical domains. recent work [127] for the discrete patch model suggests that the set e is periodically fragmented; see also [123] for general fragmentation phenomenon. for the biased movement model with dispersal term ∇· ((1 + m)∇u), the optimal distribution of resources for maximizing the survival ability of a species was considered in [122]. they showed that the problem has “regular” solutions only when the domain is a ball and the optimal distribution can be characterized in this case; see [11] for the one-dimensional case. compared to (2.2), a more general model is  d∆u + r(x)u ( 1 − u k(x) ) = 0 x ∈ ω, ∂u ∂ν = 0 x ∈ ∂ω, (2.3) where r is the intrinsic growth rate and k is the carrying capacity of environment. is it possible to find general sufficient conditions for r,k such that the maximal biomass size of species can be reached at selected topics on reaction-diffusion-advection models 153 some intermediate diffusion rate? this question for (2.3) was proposed and studied by deangelis et al. [35]. we refer to [59] for some relevant developments in this regard, which shows that the total biomass can be a monotone increasing function of the diffusion rate for some choices of r and k. in the study of a predator-prey model [113], the following problem arises: is the maximum of the density in (2.2), i.e. maxx∈ω u ∗(x,d), monotone decreasing with respect to the diffusion rate d? some recent progress is obtained in [97], where the monotonicity was proved for several classes of function m. however, it remains open to show the monotonicity for general function m. in contrast, the minimum of the density, i.e. minx∈ω u ∗(x,d), is not necessarily monotone increasing in d; see [63]. we conjecture that the difference between the maximum and minimum values of the density, which measures the spatial variations of the density, is a monotone decreasing function of d. it is known that ∫ ω |∇u∗|2, which also measures the spatial variations of the density, is monotone decreasing with respect to d; see [98] for more details. 2.2. single species models in rivers. how do populations persist in streams when they are constantly subject to downstream drift? this problem is intensified when the habitat quality at the downstream end is very poor. once the species are washed downstream, the chances of survival will be greatly reduced. this problem, termed as the “drift paradox, has received considerable attention [65, 126]. speirs and gurney [144] pointed out that the action of dispersal can permit persistence in an advective environment, and they proposed the following reaction-diffusion-advection model:  ∂tu = d∂xxu−α∂xu + u(r −u) x ∈ (0,l), t > 0, d∂xu(0, t) −αu(0, t) = u(l,t) = 0 t > 0, u(x, 0) = u0(x) x ∈ (0,l), (2.4) where positive constants α and r are the advection rate and the growth rate of species, respectively. the interval [0,l] represents an idealized river, assuming that the upstream end x = 0 is a no-flux or closed boundary, while at the downstream end x = l the zero dirichlet boundary condition (also called lethal boundary condition) is imposed. biologically interpreted, the downstream end describes the junction of rivers and seas, as freshwater fishes cannot survive in the seawater. the persistence of species is equivalent to the instability of the trivial equilibrium, which is in turn characterized by the negativity of the principal eigenvalue of the following problem:{ d∂xxϕ−α∂xϕ + (r + λ)ϕ = 0 x ∈ (0,l), d∂xϕ(0) −αϕ(0) = ϕ(l) = 0. (2.5) by an explicit calculation, speirs and gurney [144] obtained the following necessary and sufficient condition for the persistence of species: d > α2 4r and l > l∗(d) := π − arctan (√ 4dr −α2/α ) √ 4dr −α2/2d . (2.6) set l := infd>0 l ∗(d). as illustrated in figure 1, condition (2.6) implies that (i) if l ≤ l, then the species cannot persist for any diffusion rate d. in this case, the equilibrium u = 0 is globally stable among all non-negative initial values; (ii) if l > l, then there exist 0 < d < d such that the species persists if and only if d ∈ (d,d). furthermore, if d 6∈ (d,d), then u = 0 is globally stable; if d ∈ (d,d) then (2.4) admits a unique positive steady state which is globally stable. biologically, the conditions for species survival are that the river has a minimal length and the diffusion rate of the species lies within a certain range: when the diffusion rate is small, the drift is so dominant that the species is washed downstream and cannot survive; when the diffusion rate is large, 154 k.-y. lam, s. liu, and y. lou figure 1. the diagram of l∗(d) illustrated by the black colored solid curve. the parameter region where the species persists is shaded in blue. the species is exposed too much to the lethal boundary at the downstream end and will also go to extinction. therefore, it was conjectured in [89, 110] that there is an intermediate diffusion rate d∗ which is the “best” strategy, i.e. once the diffusion rate of a species is d∗, no other species can replace it. this conjecture implies that if the majority of individuals in a population adopt the strategy d∗, then any invasion fails so long as the invading species is adopting a strategy different from d∗. in the evolutionary game theory, such d∗ is referred to as an evolutionary stable strategy [119]. see subsection 3.2 and conjecture 3 for further details. in (2.4), the downstream end is assumed to be the lethal boundary. recent work considering general boundary conditions [115, 148] showed that the species persists if and only if the velocity of flow drift is less than some threshold value. under some other boundary conditions, this threshold value turns out to be a monotone decreasing function of the diffusion rate, while this is not the case for (2.4). this leads to a natural evolutionary question: is the fast diffusion or slow diffusion more beneficial to the species in streams? we will discuss this question in subsection 3.2. some new persistence criteria has been found for general boundary conditions at the downstream in a most recent work [57], where the critical river length can be either monotone decreasing in the diffusion rate d or it can first decrease and then increase as d increases, depending on the loss rate at the downstream; see also [156] for related discussions. the diagram in figure 1 can be regarded as an extreme case of the new persistence criteria found in [57]. finally, we refer to [43, 78, 116, 117, 147, 151, 152] for related work on the persistence problem in rivers and river networks. 3. competing species models one basic question in spatial ecology is: how does the spatial heterogeneity of environment affect the invasion of exotic species and the competition outcome of native and exotic species? the term heterogeneity refers to the non-uniform distribution of various environmental conditions. for example, in the oceans, light intensity decreases with respect to the depth due to light absorption by phytoplankton and selected topics on reaction-diffusion-advection models 155 water [80], and thus is unevenly distributed. an exotic species can successfully invade often because it has some competitive advantage over the resident species. however, in the heterogeneous environment, with the help of diffusion alone, invasion is still possible even though the exotic species does not have any apparent competitive advantage. in this section, we consider some classical competition models to illustrate this phenomenon. 3.1. lotka-volterra competition models. we first consider the following lotka-volterra competition model in the homogeneous environment:   ∂tu = d∆u + u(a1 − b1u− c1v) (x,t) ∈ ω × (0,∞), ∂tu = d∆v + v(a2 − b2u− c2v) (x,t) ∈ ω × (0,∞), ∂u ∂ν = ∂v ∂ν = 0 (x,t) ∈ ∂ω × (0,∞), u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ ω, (3.1) where u(x,t) and v(x,t) are the population densities of two competing species with diffusion rates d and d > 0, parameters a1,a2 are their intrinsic growth rates, b1,c2 are the intra-specific competition coefficients, and b2,c1 are the inter-specific competition coefficients. we assume for the moment that ai,bi,ci, i = 1, 2, are positive constants, i.e. the spatial environment is homogeneous. if the coefficients satisfy the weak competition condition b1 b2 > a1 a2 > c1 c2 , (3.2) then (2.4) admits a unique positive steady state (u∗,v∗) given by u∗ = a1c2 −a2c1 b1c2 − b2c1 , v∗ = a2b1 −a1b2 b1c2 − b2c1 . (3.3) theorem 3.1 ([10]). suppose that the environment is homogeneous and (3.2) holds. then the positive steady state (u∗,v∗) is globally asymptotically stable among all non-negative and non-trivial initial conditions. theorem 3.1 shows that when the environment is homogeneous and the competition is weak, two competing species can always coexist regardless of the initial conditions. a natural question arises: does theorem 3.1 remain valid in spatially heterogeneous environment? obviously, when ai,bi,ci are non-constant functions in ω, (u ∗,v∗) given by (3.3) is generally nonconstant, and thus not necessarily a positive steady state of (3.1). however we may still ask: if condition (3.2) holds pointwisely on ω, does (3.1) admit a unique positive steady state which is globally asymptotically stable? surprisingly, even if (3.2) holds pointwisely in ω, it is possible that one of the two competing species eventually becomes extinct, regardless of initial conditions, provided that two competitors have proper diffusion rates. mathematically, for the homogeneous case, (3.1) can be viewed as an ordinary differential equation, which has a globally stable positive steady state for each x ∈ ω. in contrast, for the heterogeneous case, the global attractor of the partial differential equation model (3.1) could simply be a semi-trivial steady state, so that the density of one competitor stays bounded from below while the other tends to zero, as t → ∞. thus, the spatial heterogeneity of environment also plays a crucial role here. 156 k.-y. lam, s. liu, and y. lou for further discussions on the competitive exclusion in spatially heterogeneous environment, we consider the following simplified competition model:  ∂tu = d∆u + u[m(x) −u− bv] (x,t) ∈ ω × (0,∞), ∂tv = d∆v + v[m(x) − cu−v] (x,t) ∈ ω × (0,∞), ∂u ∂ν = ∂v ∂ν = 0 (x,t) ∈ ∂ω × (0,∞), u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ ω, (3.4) where m(x) represents the common resource of two competitors and the weak competition condition (3.2) is reduced to 0 < b,c < 1. we assume that m(x) is positive, smooth and non-constant to reflect the inhomogeneous distribution of resources across the habitat. then for d,d > 0, (3.4) has two semi-trivial states, (u∗, 0) and (0,v∗), where u∗ is the unique positive solution of (2.2), while v∗ is the unique positive solution of (2.2) with d replaced by d. for the spatially heterogeneous environment, the stability of semi-trivial states is a subtle question. for example, the stability of (u∗, 0) is determined by the sign of the principal eigenvalue λ1 of the problem   d∆ϕ + (m− cu∗)ϕ + λϕ = 0 x ∈ ω, ∂ϕ ∂ν = 0 x ∈ ∂ω, (3.5) where u∗(x,d) depends implicitly on d. it is well-known [12] that if λ1 > 0, then (u ∗, 0) is stable; if λ1 < 0, then (u ∗, 0) is unstable. when m is constant, under weak competition condition 0 < b,c < 1, direct calculations give λ1 < 0, i.e. (u ∗, 0) is always unstable for any d,d > 0. in contrast, for nonconstant m, the situation becomes complicated and interesting: first, the variational characterization of the principal eigenvalue [12] implies that λ1 is monotone increasing with respect to d, and it tends to minω(cu ∗ −m) as d approaches 0. notice from (2.2) that m−u∗ must change sign in ω. if c < 1 (in fact c ≤ 1 is sufficient), then minω(cu ∗ −m) < 0, which implies that λ1 < 0 for small d. for large d we have lim d→∞ λ1 = 1 |ω| ( c ∫ ω u∗ − ∫ ω m ) , which highlights the role played by the total biomass ∫ ω u∗ in the stability of (u∗, 0). to be more precise, define c∗ := inf d>0 ∫ ω m∫ ω u∗ . from subsection 2.1 we see that c∗ ∈ (0, 1), and if c < c∗, then (u∗, 0) is unstable for all d,d > 0. it is interesting that when c ∈ (c∗, 1), we can find some non-empty open set σ such that the semi-trivial solution (u∗, 0) is stable if and only of (d,d) ∈ σ, so that we can define σ := {(d,d) ∈ (0,∞) × (0,∞) : (u∗, 0) is stable} . (3.6) observe that for any c ∈ (c∗, 1), the set σ is always a subset of {(d,d) : 0 < d < d}. recently, he and ni [62] proved a surprising and insightful result: for a class of competition models, if the semi-trivial or positive solution is locally stable, then it must be globally stable. the following result, as a corollary of their general result, completely solves a conjecture proposed in [105, 107]: theorem 3.2. suppose that 0 < b ≤ 1, c ∈ (c∗, 1) and m is non-constant. if (d,d) ∈ σ, then (u∗, 0) is globally stable among all non-negative and non-trivial initial conditions; if (d,d) 6∈ σ and d < d, then (3.4) admits a unique positive steady state which is globally stable. selected topics on reaction-diffusion-advection models 157 theorem 3.2 shows that for some diffusion rates, competition exclusion takes place even if the weak competition condition is satisfied everywhere in ω. the case b = 1 and c∗ < c < 1 should be pointed out particularly. in this case, when the environment is homogeneous, i.e. m is constant, the equilibrium solution (0,m) is globally stable, so that the faster diffusing species excludes the slower one. in contrast, for the spatially heterogeneous environment, theorem 3.2 implies that for proper diffusion coefficients, the final outcome of the competition is reversed: the slower diffusing species excludes the faster one! furthermore, when b = 1 and c approaches 1 from below, the following result holds: theorem 3.3. assume that b = c = 1. if d < d, then (0,v∗) is unstable and (u∗, 0) is globally stable. conversely, if d > d, then (0,v∗) is globally stable. in fact, as c tends to 1 from below, the set σ increases monotonically and converges to the set {(d,d) : 0 < d < d} in the hausdorff sense. biologically, this implies that the slower diffusing species will replace the faster one in the spatially heterogeneous environment. this is the so-called evolution of slow dispersal. more precisely, if we consider the slower diffuser as a mutant among the faster diffusing population, then mutations for slower diffusion rate are selected, i.e. the diffusion rate of the “new resident” is smaller than the “previous resident”. through constant mutation and competition, the diffusion of new species thus becomes slower and slower. this phenomenon was first discovered by hastings [58], and theorem 3.3 was proved by dockery et al. [39]. moreover, it is shown in [71] that this result may fail for spatio-temporally varying environments; see also [104]. the above discussion illustrates that diffusion in a spatially heterogeneous environment has a dramatic effect on the dynamics of competition models, which is rather different from the case of a homogeneous environment. it should be noted that there are still many questions which remain open even for two competing species. we refer the interested readers to [23, 60, 61, 92, 129] for some related discussions. for general resource distributions, it is challenging to fully determine the stability of the semi-trivial state of (3.1); see [14, 70, 106] for examples of multiple reversals of the competition outcomes for two-species lotka-volterra competition models. a remaining problem is to find some conditions on the resource distribution, so that the local stability of two semi-trivial steady states of (3.1) can be accurately characterized. the answer may depend upon how to give some sufficient conditions ensuring that the total biomass of a single species as a function of diffusion rate has a unique local maximum (and thus also a global maximum). another interesting question is what happens if all the coefficients ai,bi,ci in (3.1) are non-constant, and the weak competition condition (3.2) still holds for all x ∈ ω. see the recent progress in [131] via construction of special lyapunov functions. for other population models, such as the predator-prey models, we can enquire similar issues: how do diffusion and spatial heterogeneity affect the dynamics of predator-prey models? for the competition models, under the weak competition condition, if the diffusion rate of a species lies within some proper ranges, the remaining average resources can be negative, which is closely related to the phenomenon that the total biomass of a single species can be greater than the total amount of resources, as discussed in subsection 2.1. therefore, the fast diffusing invader cannot invade successfully in this case as their available resources can be negative. similar consideration is applicable for the predator-prey models. however, the study of predator-prey models turns out to be more complicated [113]: understanding the relationship between the total biomass of species and the total amount of resources is far from being sufficient, and other issues are also involved, such as the dependence of the maximum density of species on the diffusion rate [97]. the study of reaction-diffusion models not only uncover some interesting biological phenomena, but also bring new and interesting mathematical problems. in the next two subsections, we continue on the theme of competition, but focus on the effect of biased movement of species in the spatially heterogeneous environment. 158 k.-y. lam, s. liu, and y. lou 3.2. competition in rivers. in this subsection we consider the following two-species competition model in a stream:  ∂tu = d∂xxu−α1∂xu + u(r −u−v) (x,t) ∈ (0,l) × (0,∞), ∂tv = d∂xxv −α2∂xv + v(r −u−v) (x,t) ∈ (0,l) × (0,∞), d∂xu−α1u = d∂xv −α2v = 0 (x,t) ∈{0,l}× (0,∞), u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ (0,l), (3.7) where α1 and α2 represent the advection rates of species u and v, respectively. when α1 = α2 = 0, it is shown in subsection 3.1 that the species with the smaller diffusion rate will drive the other species to extinction, provided that r is non-constant. what happens if α1,α2 6= 0? for the case when r is a positive constant (i.e. the environment is homogeneous) and α1 = α2 = α > 0, the following result was established in [115]: theorem 3.4. suppose that r > 0 is constant and α > 0, d > d. then the semi-trivial state (0,v∗) is globally stable, where v∗ is the unique positive solution of{ d∂xxv −α∂xv + v(r −v) = 0 x ∈ (0,l), d∂xv −αv = 0 x = 0,l. (3.8) theorem 3.4 shows that under the influence of passive drift, the species with faster diffusion will be favored, which is contrary to theorem 3.3 for model (3.4) in the non-advective environment. in the river, the “fate” of the two competitors is reversed: the slower diffusing species will be eventually wiped out! an intuitive reasoning is that the passive drift will push more individuals downstream to force a concentration of population at the downstream end, which causes mismatch of the population density with the resource distribution. faster diffusion can counterbalance such mismatch by sending more individuals upstream and therefore, it is competitively advantageous for species to adopt faster diffusion. when r is constant, the case α1 6= α2 and d = d was considered in [112], where it was proved that the smaller advection is always beneficial. a similar explanation is that under the drift, the species will move towards the downstream end, leading to the overcrowding and the overmatching of resources there, which in turn results in the extinction of the species with the stronger advection that can send more individuals to the downstream. the general case d 6= d and α1 6= α2 was addressed in [158], which showed that the fast diffusion and weak advection will be favored, consistent with the findings in [112, 115]. when r is non-constant, i.e. the spatial environment is heterogeneous, the dynamics of (3.7) turns out to be more complicated. the simplest situation might be when r is decreasing, for which we have the following conjecture: conjecture 2. suppose that r is positive and decreasing in (0,l). then there exists some α∗ > 0, independent of d and d, such that (i) for α ∈ (0,α∗), there exists d∗ = d∗(α) > 0 such that if d = d∗ and d 6= d∗, then the semi-trivial steady state (0,v∗) of (3.7) is globally stable; (ii) for α ≥ α∗, if d > d > 0, then (0,v∗) of (3.7) is globally stable. we observe that if r is non-constant and α = 0, the slower diffusion rate is always favored. this suggests that d∗(α) → 0 as α → 0. furthermore, we expect d∗(α) → +∞ as α → α∗−. we refer to [114, 157] for some recent developments for the case of decreasing r. the general case of r(x) is studied in [90], where sufficient conditions for the existence of evolutionarily stable strategies, or that selected topics on reaction-diffusion-advection models 159 of branching points, are found. we refer the readers to [36] for the definition of a branching point, and to subsection 4.2 for further discussions. for the speirs and gurney model (2.4), it was conjectured that there is an intermediate diffusion rate which is optimal for the evolution of dispersal. this seems to be in contrast to theorem 3.4, where the faster diffusion is favored. such discrepancy is due to the different boundary conditions at the downstream end x = l; see [110, 115, 148] for further results on the effect of boundary conditions. next, we impose the zero dirichlet boundary conditions at the downstream end and consider the following competition model:  ∂tu = d∂xxu−α∂xu + u(r −u−v) (x,t) ∈ (0,l) × (0,∞), ∂tv = d∂xxv −α∂xv + v(r −u−v) (x,t) ∈ (0,l) × (0,∞), d∂xu−αu = d∂xv −αv = 0 (x,t) ∈{0}× (0,∞), u = v = 0 (x,t) ∈{l}× (0,∞), u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ (0,l). (3.9) as shown in subsection 2.2, neither large nor small diffusion rate is beneficial to each species in this case. a natural conjecture is: conjecture 3. there exists some d∗(α) > 0 such that if d = d∗ and d 6= d∗, the semi-trivial steady state (0,v∗) of (3.9), whenever it exists, is globally stable. for the case of weak advection (α small), we conjecture that the approaches developed in [87, 88] may be applicable to yield the existence of d∗ for small α, satisfying d∗(α) → 0 as α → 0. however, the proof remains quite challenging for general α. recently it was shown in [155] that such diffusion rate d∗ exists and is unique for two-patch models: the semi-trivial steady state (0,v∗) is locally stable when d = d∗ and d 6= d∗. however, the global stability of (0,v∗) remains unclear. for multi-patch models, it remains open to determine the existence of d∗. 4. competition models with directed movement belgacem and cosner [7] argued that in the spatially heterogeneous environment, the species may have a tendency to move upward along the resource gradient, in addition to random diffusion. in this connection, they added an advection term to (2.1) and considered the reaction-diffusion-advection model   ∂tu = ∇· [d∇u−αu∇m] + u(m−u) (x,t) ∈ ω × (0,∞), ∂u ∂ν −αu ∂m ∂ν = 0 (x,t) ∈ ∂ω × (0,∞), u(x, 0) = u0(x) x ∈ ω. (4.1) here α ≥ 0 is a parameter which measures the tendency of the population to move upward along the gradient of the resource m(x). belgacem and cosner [7] and subsequent work [30] indicate that for a single species, the advection along the resource gradient can be beneficial to the persistence of the single species in convex habitats, but not necessary for non-convex ω. in particular, when m is positive somewhere, the species persists whenever the advection rate α is large. the biological explanation is that the species can utilize the large advection to concentrate at places with locally most abundant resources. 160 k.-y. lam, s. liu, and y. lou it is natural to enquire whether the advection along the resource gradient can confer competition advantage. to this end, cantrell et al. [15] proposed the following model:  ∂tu = ∇· [d∇u−αu∇m] + u(m−u−v) (x,t) ∈ ω × (0,∞), ∂tv = d∆v + v(m−u−v) (x,t) ∈ ω × (0,∞), d ∂u ∂ν −αu ∂m ∂ν = ∂v ∂ν = 0 (x,t) ∈ ∂ω × (0,∞), u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ ω. (4.2) when m is non-constant and ∫ ω m ≥ 0, (4.2) has two semi-trivial states, denoted by (u∗, 0) and (0,v∗), for every d,d > 0 and α ≥ 0 (see [30]), where u∗ is the unique positive steady state of (4.1). it is shown in [16] that if d = d and ω is convex, then the semi-trivial state (u∗, 0) is globally stable for small α > 0. this implies that for two species which differ only in their dispersal mechanisms, the weak advection can confer some competition advantages, provided that the underlying habitat is convex. however, for some non-convex ω, it is also proved in [16] that the opposite case may happen for proper m(x): the weak advection can actually put the species at a disadvantage. does the species have a competitive advantage if the advection is strong? the following result of [5] might seem surprising at the first look. it was first proved in [16] under the additional assumption that the set of critical points is of measure zero and has at least one strict local maximum point. theorem 4.1. suppose that m ∈ c2(ω) is positive and non-constant. then for α d ≥ 1 minω m , both semi-trivial steady states (u∗, 0) and (0,v∗) are unstable, and (4.2) has at least one stable coexistence state. theorem 4.1 implies that the two competing species are able to coexist when the advection is strong, which is in stark contrast with the case of weak advection. what is preventing the species u to exclude species v by strong advection? a series of work suggests that as α increases, species u resides only at the set m of local maximum points of m(x), and retreats from ω\m. if m is positive, then that leaves ample room for species v to survive in ω \ m, i.e. places with less resources [16, 26, 83, 84, 91]. this is proved in theorem 4.1. when m(x) changes sign, however, the overall leftover resources in ω \ m might be poor and u may competitively exclude v by strong advection. theorem 4.2 ([93]). suppose that m ∈ c2(ω) and it is sign-changing. (a) if ∫ ω\m m > 0, then for any d,d > 0, u and v coexist for large α. (b) if ∫ ω\m m < 0, then for some d,d > 0, u competitively excludes v for large α. theorem 4.1 also reveals that the two competitors modeled by (4.2) can achieve coexistence by different dispersal strategies including random diffusion and directed advection towards resources, which suggests a new mechanism for coexistence of two or more competing species. in [111] we apply the similar ideas to discuss the possibility of coexistence for three-species competition systems with the random diffusion and directed advection. the analysis of three-species models turns out to be much more difficult. an obvious challenge is that systems of three competing species, unlike two-species competition model, are not order-preserving [142]. as mentioned earlier, if the advection rate α is large, then the species u will concentrate near some local maximum points of m. further studies [84, 91] showed that the concentration positions are precisely those local maximum points of m where the function m − v∗ is also strictly positive, by establishing the uniform l∞ estimate and a new liouville-type result. we note that m is the common resource for both species u and v, and m − v∗ is the resource available for species u when the species v reaches its equilibrium. it is observed in [91, rmk. 1.4] that for some positive local selected topics on reaction-diffusion-advection models 161 figure 2. decomposition of the d-d parameter space into three regions: r1 and r2 are separated by the curve d = f(d). the regions r2 and r3 are separated by the line d = d. the bifurcation diagrams for each of these regions, with α as the bifurcation parameter, are given in figure 3. (first published in mem. amer. math. soc. 245 (spring 2017), published by the american mathematical society. @ 2017 american mathematical society.) maximum points of m, function m − v∗ can be negative, i.e. some apparently favorable places are actually dangerous ecological traps for the species u. this suggests that both distribution of resources and competitors should be taken into account when studying the directed movement of species along the resource gradient. 4.1. global bifurcation diagrams. the above results show that when the advection is weak, the two species cannot coexist, and the advection confers some competitive advantage for species u; when the advection is strong, the two species coexist for any diffusion rates. what happens for general advection rates? for instance, assuming d > d, when we increase the advection from weak to strong, how does that affect the dynamics of (4.2)? if α = 0, the result of dockery et al. [39] says that the semi-trivial state (0,v∗) is globally stable; if α is sufficiently large, theorem 4.1 shows that both (u∗, 0) and (0,v∗) are unstable and the coexistence happens. a possible conjecture is there exists some critical advection rate α∗ > 0 such that for α < α∗, (0,v∗) is globally stable, while for α > α∗, (4.2) has a unique positive steady state which is globally stable. however this conjecture fails in general and a more complicated bifurcation diagram of positive steady states occurs. to be more precise, averill et al. [5] showed that under some conditions on m(x), there exists a function f(x) : [0,∞) → [0,∞) satisfying f(0) = 0 and 0 < f(x) < x such that the d-d parameter space is divided into three regions r1,r2,r3, as shown in figure 2. using α as a bifurcation parameter, we see that the following three situations occur: (1) for d < f(d), there exists some critical value α1 > 0 such that as α < α1, the semi-trivial state (0,v∗) is stable but (u∗, 0) is unstable, while as α > α1, both (u ∗, 0) and (0,v∗) are unstable so that (4.2) has at least one positive steady state. by the bifurcation theory, there is a branch of positive steady states, denoted by γ1, which bifurcates from (0,v ∗) at α = α1 and can be extended to infinity in α. this is illustrated in figure 3(a). it is unclear whether (0,v∗) is 162 k.-y. lam, s. liu, and y. lou globally stable when α < α1, and whether the positive steady state is unique and globally stable when α > α1; (2) for f(d) < d < d, there exist three critical values 0 < α1 < α2 < α3 such that (i) as α < α1, (0,v∗) is stable but (u∗, 0) is unstable; (ii) as α2 < α < α3, (u ∗, 0) is stable but (0,v∗) is unstable; (iii) as α ∈ (α1,α2) ∪ (α3,∞), both (u∗, 0) and (0,v∗) are unstable so that (4.2) has at least one positive steady state. similarly, by the bifurcation theory, there is still a branch of positive steady states bifurcating from (0,v∗) at α = α1, which is denoted as γ2. the difference is that the branch γ2 cannot be extended to α = ∞, but is connected to (u∗, 0) at α = α2. interestingly, when α ∈ (α2,α3), the semi-trivial state (u∗, 0) is stable, which indicates that the advection can indeed be beneficial to the species in this case! when α > α3, the branch of a positive steady state, denoted as γ3, bifurcates from (u ∗, 0) at α = α3, and can be extended to α = ∞. we conjecture that as α < α1, (0,v∗) is globally stable; as α2 < α < α3, (u∗, 0) is globally stable; as α ∈ (α1,α2) ∪ (α3,∞), (4.2) admits a unique positive steady state which is globally stable. this is illustrated in figure 3(b). (3) for d > d, there exists some critical value α1 > 0 such that as α < α1, (u ∗, 0) is stable, while as α > α1, both (u ∗, 0) and (0,v∗) are unstable so that (4.2) has at least one positive steady state. similarly, a branch of positive steady states bifurcates from (u∗, 0) at α = α1, denoted as γ4, which can be extended to α = ∞. it is conjectured that (u∗, 0) is globally stable when α < α1, and the positive steady state is unique when α > α1. this is illustrated in figure 3(c). how, then, is the branch γ1 transformed to γ2 and γ3, and finally to γ4, as the diffusion rate d varies, with d being fixed? the result in [5] shows that in the critical situation d = f(d), the branch γ1 is connected to the semi-trivial state (u ∗, 0) at α = α∗ for some α∗ > 0. once d > f(d), α∗ is split into two different points α2 and α3, so that γ1 is split into two disjoint branches γ2 and γ3. as d approaches d from below, the bounded branch γ2 will approach α = 0 and vanishes; meanwhile γ3 approaches γ4. by these discussions, we explained that for the models with multiple parameters, it is possible to apply the bifurcation theory to describe the structure of positive steady states. of course, it remains challenging to completely characterize the global dynamics of (4.2), and many interesting mathematical questions, e.g. uniqueness and global stability, are left open [5]. one particular conjecture is that, for α sufficiently large, the positive steady state of the competition system (4.2) is unique and thus globally asymptotically stable. one way of proving it is to determine the precise limiting profile and show that each positive steady state is linearly stable, whenever it exists. one can then conclude the uniqueness and global asymptotic stability by the theory of monotone dynamical system [64]. this was carried out for the case when all critical points of m are non-degenerate, and that the value of m is constant on the set of all local maximum points [82, thm. 8.1]. the general case remains open. it is illustrated here that proper advection can confer some competition advantages. in next subsection, we will use evolutionary game theory to further show that proper advection can be an evolutionary stable strategy. 4.2. evolution of biased movement. in this subsection, we adopt the viewpoint of adaptive dynamics [36, 54] to consider the evolution of dispersal and the ecological impact of dispersal in spatially heterogeneous environments. an important concept in adaptive dynamics is evolutionarily stable strategy (ess), known also as evolutionarily steady strategy, which was introduced by maynard smith and price in the seminal paper [119]. a strategy is said to be evolutionarily stable if a population using it cannot be invaded by any small population using a different strategy. ess is a very general concept. in selected topics on reaction-diffusion-advection models 163 figure 3. bifurcation results from [5], where d,d are fixed and α is used as the bifurcation parameter. the three figures, from top to bottom, are the bifurcation diagrams concerning the parameter regions r1,r2, and r3 (respectively, d < f(d), f(d) < d < d and d > d) in figure 2. (first published in mem. amer. math. soc. 245 (spring 2017), published by the american mathematical society. @ 2017 american mathematical society.) this subsection we aim to connect it with the local stability of the semi-trivial steady states discussed earlier. an interesting question is what kind of advection rate will be evolutionarily stable, by regarding the rates of advection along the environmental gradient as movement strategies of the species. for this purpose, we consider the following model:   ∂tu = ∇· [d∇u−αu∇m] + u(m−u−v) (x,t) ∈ ω × (0,∞), ∂tv = ∇· [d∇v −βv∇m] + v(m−u−v) (x,t) ∈ ω × (0,∞), d ∂u ∂ν −αu ∂m ∂ν = d ∂v ∂ν −βv ∂m ∂ν = 0 (x,t) ∈ ∂ω × (0,∞), u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ ω, (4.3) where u and v are the densities of resident and mutant populations, and the two species differ only in their advection rates, given by the non-negative parameters α and β, respectively. 164 k.-y. lam, s. liu, and y. lou the mutant population v, which is assumed to be small initially, can invade successfully if and only if the semi-trivial steady state (u∗, 0) is unstable. here u∗ denotes the unique positive solution of  ∇· [d∇u−αu∇m] + u(m−u) = 0 x ∈ ω, d ∂u ∂ν −αu ∂m ∂ν = 0 x ∈ ∂ω. (4.4) a natural question is whether there is a critical value α∗ > 0 such that (u∗, 0) is a stable steady state of (4.3) whenever α = α∗ and β 6= α∗. we call such α∗, if it exists, an ess. to find such a magical strategy, we note that the stability of (u∗, 0) is determined by the principal eigenvalue, denoted by λ1, of the problem   ∇· [d∇ϕ−βϕ∇m] + ϕ(m−u∗) + λϕ = 0 x ∈ ω, d ∂ϕ ∂ν −βϕ ∂m ∂ν = 0 x ∈ ∂ω. (4.5) it is classical that the set of eigenvalues of (4.5) is real and bounded from below. the principal eigenvalue λ1 refers to the smallest eigenvalue. it is simple and corresponds to a strictly positive eigenfunction in ω; see [12, 26]. since u∗ is an implicit function of α, we regard λ1 = λ1(α,β). observe that λ1 = 0 for α = β, since the principal eigenfunction ϕ is simply a constant multiple of u∗ in such a case. it is well known that if λ1 > 0, then (u ∗, 0) is stable; otherwise, (u∗, 0) is unstable. hence if α∗ is an ess, then λ1(α ∗,β) ≥ 0 for all β, which together with λ1(α∗,α∗) = 0 implies that ∂λ1 ∂β ∣∣∣ α=β=α∗ = 0. (4.6) we call any strategy α∗ satisfying (4.6) an evolutionarily singular strategy, hence any ess is automatically an evolutionarily singular strategy. for the existence of evolutionarily singular strategies, we have the following result: theorem 4.3 ([87, thm. 2.2]). suppose that m ∈ c2(ω) is strictly positive in ω and maxω m/ minω m ≤ 3 + 2 √ 2. given any a > 1/ minω m, for all small positive d, there is exactly one evolutionarily singular strategy α∗ ∈ (0,da). the condition maxω m/ minω m ≤ 3 + 2 √ 2 is sharp, and theorem 4.3 may fail for general m [87, thm. 2.6]. when ω is one-dimensional, the evolutionarily singular strategy found in theorem 4.3 is also unique in the whole interval [0,∞), provided d > 0 is small enough [87, cor. 2.3]. we conjecture that the uniqueness holds for ω of higher dimensions as well. for intermediate or large d, however, the uniqueness of singular strategies remains open; see [20] for some recent progress. we conjecture that for any evolutionarily singular strategy α∗ = α∗(d), it holds that, for all d, min ω m < d α∗ < max ω m. (4.7) if this estimate were true, it suggests that a balanced combination of diffusion and advection might be a better movement strategy for populations. the following result shows that this is indeed the case: theorem 4.4 ([87, thm. 2.5]). suppose that ω is convex and m ∈ c2(ω) is strictly positive in ω. assume that ‖∇(ln m)‖l∞(ω) ≤ α0/diam(ω), where α0 ≈ 0.8 and diam(ω) is the diameter of ω. then for small d, the strategy α∗ given by theorem 4.3 is a local ess, i.e. there exists some small δ > 0 such that λ1(α ∗,β) > 0 for all β ∈ (α∗ −δ,α) ∪ (α,α∗ + δ). some other work also considered model (4.3) and related models. for instance, in [88] we discussed the evolution of diffusion rate in an advective environment. we refer to [20, 24, 25, 28, 29, 53, 55] for further discussions. selected topics on reaction-diffusion-advection models 165 can we decide whether the local ess α∗ given in theorem 4.3 is actually a global ess? or more generally, what can we say about the global structure of nodal set for the function λ1 = λ1(α,β)? it is clear that the nodal set of λ1(α,β) must include the diagonal line α = β, but otherwise little is known. for the reaction-diffusion models discussed in this paper, determining the complete structure of the nodal set for λ1(α,β) poses a challenging problem, both analytically and numerically [52]. to this end, some asymptotic behaviors of λ1 were studied in [26, 27, 99, 139] in order to have a better understanding for this problem. recently, the asymptotic behaviors for the principal eigenvalues of time-periodic parabolic operators were also studied, and we refer to [64, 69, 71, 101, 102, 103, 137]. these new developments of the related linear pde theory may open up a new venue to study the evolution of dispersal in spatio-temporally varying environments. the evolutionary dynamics, e.g. existence of ess, have various consequences on the two-species competition dynamics. this connection has been investigated in [13] for a broad class of model including reaction-diffusion equations and nonlocal diffusion equations. it was shown that frequently the ess playing species can competitively exclude the species playing a different but nearby strategy, regardless of the initial condition. in the jargon of adaptive dynamics, this result says that ess implies neighborhood invader strategy. this is quite unexpected, since an ess by its very definition only concerns the local stability of the semi-trivial steady state. the main tool in [13] is hadamard’s graph transform method. 5. continuous trait models regarding the advection rates as strategies of the species, in subsection 4.2 we investigated the course of evolution for two-species competition model (4.3). therein the strategies {α,β} form a discrete set, and the competition and mutation occur independently of each other. in this section, we consider population models with continuous traits, in which the competition and mutation can occur simultaneously. 5.1. mutation-selection model. to study the selection of random dispersal rate in multi-species competition, dockery et al. [39] considered the following model: ∂tui = ξi∆ui + ui  m(x) − k∑ j=1 uj   + k∑ j=1 mijuj, (5.1) where k interacting species, with densities ui(x,t), 1 ≤ i ≤ k, compete for a common resource m(x) and differ only in their diffusion rates ξi. these species are subject to mutation with switching rate mij from phenotype j to i. for k = 2 and mij = 0, a well-known result is that the slower diffusing species always win in the competition, but it remains open to determine the global dynamics for the case k ≥ 3. next, we extend the discrete set of strategies {ξi}ki=1 in (5.1) to a continuum set (ξ,ξ). by viewing the population u(x,ξ,t) as being structured by space, trait and time (i.e. u(·,ξ,t) gives the spatial distribution of the individuals playing strategy ξ at time t), we obtain the following mutation-selection model:   ∂tu = ξ∆xu + u [m(x) − û(x,t)] + �2∂ξξu for x ∈ ω,ξ ∈ (ξ,ξ), t > 0, ν(x) ·∇xu = 0 for x ∈ ∂ω,ξ ∈ (ξ,ξ), t > 0, ∂ξu = 0 for x ∈ ω,ξ ∈{ξ,ξ}, t > 0, u(x,ξ, 0) = u0(x,ξ) for x ∈ ω,ξ ∈ (ξ,ξ). (5.2) 166 k.-y. lam, s. liu, and y. lou here the parameters ξ,ξ, which are assumed to be positive, determine the range of continuous strategy ξ, and û(x,t) := ∫ ξ ξ u(x,ξ′, t) dξ′ represents the total population density at location x and time t, and the term �2∂ξξu accounts for the rare mutation, which is modeled by a diffusion with covariance √ 2�. we refer to [22] for a derivation of this type of equations from individual based on stochastic models. model (5.2) is an integro-pde model describing the diffusion, competition and mutation of a population with continuous traits. if ∫ ω m(x)dx > 0, then it is shown in [85] that (5.2) admits a unique positive steady state u�(x,ξ) for small �, which is locally asymptotically stable and satisfies  ξ∆xu + � 2∂ξξu + u [m(x) − û(x)] = 0 (x,ξ) ∈ ω × (ξ,ξ), ν(x) ·∇xu = 0 (x,ξ) ∈ ∂ω × (ξ,ξ), ∂ξu = 0 (x,ξ) ∈ ω ×{ξ,ξ}, (5.3) where û(x) = ∫ ξ ξ u(x,ξ′)dξ′. observe that when m is constant, i.e. m ≡ m0 for some positive constant m0, we have u� ≡ m0/(ξ − ξ) which is globally stable for any � > 0 [85]. this suggests that selection does not favor any particular strategy ξ ∈ [ξ,ξ], and the mutation �2∂ξξu has no effect on the population dynamics in the homogeneous environment. what happens in the spatially heterogeneous environment? the following result gives the answer: theorem 5.1 ([86]). suppose that ∫ ω m(x)dx > 0 and m is non-constant. then as � → 0,∥∥∥∥�2/3u�(x,ξ) −u∗(x,ξ)η∗ ( ξ − ξ �2/3 )∥∥∥∥ l∞(ω×(ξ,ξ)) → 0, where η∗(s) satisfying the following ordinary differential equation:{ η′′ + (a0 −a1s)η = 0 s > 0, η′(0) = η(+∞) = 0, ∫∞ 0 η(s) ds = 1, with some positive constants a0 and a1. in particular, we have as � → 0, u�(x,ξ) → u∗(x,ξ) ·δ(ξ − ξ) in distribution, where u∗(x,ξ) denotes the unique positive solution of (2.2) with d = ξ, and δ(·) is a dirac mass supported at 0. it can be verified that for any ξ0 ∈ [ξ,ξ], the limit profile u∗(x,ξ0) · δ(ξ − ξ0) is a solution of (5.3) with � = 0. theorem 5.1 says that only the steady state corresponding to the smallest diffusion rate is preserved by the perturbation of � = 0 to 0 < � � 1. theorem 5.1 implies that when the mutation rate is small, the species with the smallest diffusion rate ξ will eventually become the dominant phenotype by driving other phenotypes to extinction. this result connects with the case of two-species competition: the smallest available diffusion rate is selected. note from [85] that the steady state u� is the unique solution of (5.3) for small � > 0, and it is locally asymptotically stable. an unsolved problem here is the global stability of the unique positive steady state u�(x,ξ). selected topics on reaction-diffusion-advection models 167 ξ lo g (t im e ) 0.5 1 1.5 0 1 2 3 4 5 2 4 6 8 10 12 ξ lo g (t im e ) 0.5 1 1.5 0 1 2 3 4 5 2 4 6 8 10 12 figure 4. contour plot of ∫ u(x,ξ,t)dx as a function of ξ and time (log(time) for vertical axis) for a = 1 4 (left) and a = −1 4 (right). (first published in [w. hao et al., indiana univ. math. j. 68 (2019), 881-923.].) a closely related work is due to perthame and souganidis [140], where the following mutationselection model was considered:  ξ(z)∆xu + � 2∂zzu + u(m(x) − û) = 0 (x,z) ∈ ω × [0, 1], ν(x) ·∇xu = 0 (x,z) ∈ ∂ω × [0, 1], u(x, 0) = u(x, 1) x ∈ ω, (5.4) where ω is assumed to be convex and ξ(z) is assumed to be positive and periodic with unit period. for the solution u�(x,z) of (5.4), it is proved in [140] that u�(x,z) → u∗(x,ξ(z∗)) · δ(z − ξ(z∗)) in distribution as � → 0, where ξ(z∗) = min0≤z≤1 ξ(z). this implies that for rare mutations, the population concentrates on a single trait associated to the smallest diffusion rate, in agreement with the result for (5.3). see also [8, 76, 125] for some progress in this direction. 5.2. mutation-selection model with drift. similar to (5.2), we introduce the following integro-pde model in a stream:  ∂tu = ξ∂xxu−α∂xu + u [r(x) − û(x,t)] + �2∂ξξu for x ∈ (0,l),ξ ∈ (ξ,ξ), t > 0, (ξ∂xu−αu)(0,ξ,t) = (ξ∂xu−αu)(l,ξ,t) = 0 for ξ ∈ (ξ,ξ), t > 0, u(x,ξ,t) = u(x,ξ,t) = 0 for x ∈ (0,l) t > 0, u(x,ξ, 0) = u0(x,ξ) for x ∈ (0,l),ξ ∈ (ξ,ξ), (5.5) where α > 0 is the advection rate and û(x,t) := ∫ ξ ξ u(x,ξ′, t) dξ′ denotes the total population density at location x and time t. when r is a positive constant, from theorem 3.4 we conjecture that (i) the integro-pde (5.5) admits at most one positive steady state, denoted by u�(x,ξ), which is globally stable whenever it exists; (ii) as � → 0, u�(x,ξ) → u∗(x,ξ) · δ(ξ − ξ) in distribution sense, where u∗(x,ξ) is the unique positive solution of (3.8) with d = ξ. again, this predicts that the larger diffusion is selected as in theorem 3.4. to study the dynamics of (5.5) for non-constant r, we carried out some numerical simulations in [56], where the corresponding parameters were selected as follows: l = 1, α = 1, ξ = 0.5, ξ = 1.5, r(x) = e(1−a)x+ax 2 , � = 10−3. (5.6) we take initial conditions in the form of one dirac mass on the phenotypic space, and investigate their evolution for a = ±1 4 . the numerical results are presented in figure 4. (i) the numerical result in figure 4 (right) shows that when a = 1 4 , the steady state of (5.5) concentrates on the trait ξ∗ ≈ 1.1 in the limit of rare mutation, which suggests that the species 168 k.-y. lam, s. liu, and y. lou adopting the strategy ξ∗ persists and other species will disappear in the competition. this suggests that ξ∗ is an ess, which is in contrast with the assertion that “faster diffusion is more favorable” for the homogeneous environment. (ii) more interestingly, when a = −1 4 , figure 4 (left) shows that the steady state of (5.5) concentrates on two different traits, so that two species adopting different strategies form a coalition that is able to resist invasion by any of the remaining species! this phenomenon is called “evolutionary branching” in adaptive dynamics, and it is also clearly different from the homogeneous case. to explain these phenomena mathematically, we investigate the asymptotic behaviors of steady state u�(x,ξ) for (5.5) as � → 0. to this end, we first consider the two-species competition model (3.7) and denote λ = λ(ξ1,ξ2) as the principal eigenvalue of the problem{ ξ2∂xxϕ−α∂xϕ + [r −u∗(·,ξ1)] ϕ + λϕ = 0 x ∈ (0,l), ξ2∂xϕ−αϕ = 0 x = 0,l, where x 7→ u∗(x,ξ1) is the unique positive solution of (3.8) with d = ξ1. in the adaptive dynamics framework, λ(ξ1,ξ2) is called the invasion fitness, and an invader with trait ξ2 can (resp. cannot) invade an established trait ξ1 at equilibrium when rare if λ(ξ1,ξ2) < 0 (resp. λ(ξ1,ξ2) > 0). the limiting profiles of u�(x,ξ) depends critically on the behavior of λ(ξ1,ξ2) near ξ2 = ξ1. note that λ(ξ,ξ) ≡ 0 for all ξ > 0. we first consider the case of ∂ξ2λ(ξ,ξ) 6= 0, where the following result holds: theorem 5.2 ([56]). suppose that ∂ξ2λ(ξ,ξ) > 0 for all ξ ∈ (ξ,ξ), and ξ−ξ is sufficiently small. then any positive steady state u� of (5.5) satisfies as � → 0, u�(x,ξ) → δ(ξ − ξ) ·u∗(x,ξ) in distribution, where x 7→ u∗(x,ξ) is the unique positive solution of (3.8) with d = ξ. theorem 5.2 implies that for the case ∂ξ2λ(ξ,ξ) > 0, the slowest diffusion will be selected in the competition. similarly, we may conclude that if ∂ξ2λ(ξ,ξ) < 0 for all ξ ∈ (ξ,ξ), then the steady state u� concentrates on the trait ξ = ξ, so that the fastest diffusion will be selected, coinciding with the observation in theorem 3.4. these results show that if ∂ξ2λ 6= 0, it gives rise to a single diracconcentration at one of the two extreme traits, depending upon the sign of ∂ξ2λ. to study the case when ∂ξ2λ vanishes somewhere in (ξ,ξ), we introduce the convergence stable strategy ξ̂ (see [45]), which is characterized by the following relations: ∂ξ2λ(ξ̂, ξ̂) = 0 and d ds [∂ξ2λ(s,s)] ∣∣∣ s=ξ̂ > 0. (5.7) this leads to two generic cases: the convergence stable strategy ξ̂ is an ess if ∂2ξ2λ(ξ̂, ξ̂) > 0, and is a branching point (bp) if ∂2ξ2λ(ξ̂, ξ̂) < 0. in the case when the ess exists, there is concentration in the mutation-selection model. theorem 5.3. [56] suppose that ξ̂ is an ess and (5.7) holds. if ξ̂ ∈ (ξ,ξ) and ξ − ξ is sufficiently small, then as � → 0, any positive steady state u� of (5.5) satisfies u�(x,ξ) → u∗(x, ξ̂) ·δ(ξ − ξ̂) in distribution. theorem 5.3 indicates that if there is an ess in (ξ,ξ), then the phenotype adopting the ess dominates the competition in the limit of rare mutation. this corresponds to the numerical result in figure 4 (right) with ξ̂ ≈ 1.1. to further understand theorem 5.3, we consider a special case of (3.7) where r(x) = becx for some positive constants b and c. in this case, it is shown in [4] that the semi-trivial state (u∗(x,ξ1), 0) is globally stable when ξ1 = α/c and ξ2 6= α/c, i.e. the intermediate trait ξ̂ = α/c is an ess. selected topics on reaction-diffusion-advection models 169 we can offer some intuitive reasoning for this result by employing the theory of ideal free distribution [46]: when r(x) = becx, if ξ1 = α/c, it can be verified from (3.8) that u ∗(x,ξ1) ≡ becx ≡ r(x), so that species u attains the ideal free distribution, and thus α/c being an ess is natural. as c → 0, i.e. the environment tends to be homogeneous, we find ξ̂ = α/c approaches +∞, which is in agreement with the prediction of theorem 3.4. finally, in the neighborhood of a bp, we have the following result: theorem 5.4 ([56]). suppose that ξ̂ ∈ (ξ,ξ) is a bp and (5.7) holds. if ξ−ξ is sufficiently small, then as � → 0, any positive steady state u� of (5.5) satisfies u�(x,ξ) → δ(ξ − ξ) ·u1(x) + δ(ξ − ξ) ·u2(x) in distribution, where (u1,u2) is a positive solution of  ξ∂xxu1 −α∂xu1 + u1(r −u1 −u2) = 0 x ∈ (0,l), ξ∂xxu2 −α∂xu2 + u2(r −u1 −u2) = 0 x ∈ (0,l), ξ∂xu1 −αu1 = ξ∂xu2 −αu2 = 0 x = 0,l. (5.8) theorem 5.4 reveals a new phenomenon: in the neighborhood of a bp, no single trait can dominate the competition; instead, the two extreme traits, ξ and ξ, form a coalition that together dominates the competition such that any species with other traits cannot invade. to illustrate theorem 5.4, we consider the case r(x) = b1e c1x + b2e c2x in (3.7) with positive constants bi and ci, i = 1, 2. we observe that if ξ = α/c1 and ξ = α/c2, then (5.8) has the unique positive solution ui(x) = bie cix, i = 1, 2. again, in the framework of ideal free distribution, it is not difficult to understand the biological reasoning: in view of u1(x) + u2(x) = b1e c1x + b2e c2x = r(x), the total density of the two species can jointly reach an ideal free distribution, so that ξ = α/c1 and ξ = α/c2 potentially become a pair of ess. we refer to [17, 18, 19, 21, 53] for related work on the evolution of dispersal and ideal free distribution. an open problem is whether (5.5) or (5.8) has at most one positive solution, and if it exists, whether it is globally asymptotically stable. another challenging question is to determine the limit of the timedependent solutions uε(x,ξ,t) of (5.3) and (5.5) as � → 0. see [94, 132] and references therein for recent progress on the rigorous derivations of the canonical equations for the trait evolution in spatially structured mutation-selection models. 6. dynamics of phytoplankton besides the models discussed above, there are many other types of reaction-diffusion models in population dynamics. in this section, we briefly discuss some problems arising from the phytoplankton growth. phytoplankton are microscopic plant-like photosynthetic organisms drifting in lakes and oceans and are the foundation of the marine food chain. since they transport significant amounts of atmospheric carbon dioxide into the deep oceans, they play a crucial role in climate dynamics and have been one of the central topics in marine ecology. nutrients and light are the essential resources for the growth of phytoplankton. since most phytoplankton are heavier than water, they will sink into the bottom of the lakes or oceans where the light intensity is too weak for their growth. so how do these phytoplankton persist in the water columns? some biologists propose that biased movement, combining with water turbulence, can help phytoplankton diffuse to a position closer to the top of lakes or oceans, so that they can get access to sunlight. in this connection, a series of reaction-diffusion models including single and multiple phytoplankton species are introduced in [66, 67, 68] and the references therein to model the spatio-temporal dynamics of phytoplankton growth. 170 k.-y. lam, s. liu, and y. lou the following system of reaction-diffusion-advection equations was used by huisman et al. [68] to describe the population dynamics of two phytoplankton species:  ∂tu = d1∂xxu−α1ux + [g1(i(x,t)) −d1]u 0 < x < l, t > 0, ∂tv = d2∂xxv −α2vx + [g2(i(x,t)) −d2]v 0 < x < l, t > 0, d1∂xu(x,t) −α1u(x,t) = 0 x = 0,l, t > 0, d2∂xv(x,t) −α2v(x,t) = 0 x = 0,l, t > 0, u(x, 0) = u0(x) ≥, 6≡ 0, v(x, 0) = v0(x) ≥, 6≡ 0 0 ≤ x ≤ l. (6.1) here u(x,t),v(x,t) denote the population density of the phytoplankton species at depth x and time t, and d1,d2 > 0 are their diffusion rates. for i = 1, 2, αi ∈ r is the sinking (αi > 0) or buoyant (αi < 0) velocity, di > 0 is the death rate, and gi(i) represents the specific growth rate of phytoplankton species as a function of light intensity i, given by the lambert-beer law i(x,t) = i0 exp [ −k0x− ∫ x 0 (k1u(s,t) + k2v(s,t))ds ] , (6.2) where i0 > 0 is the incident light intensity, k0 > 0 is the background turbidity that summarizes light absorption by all non-phytoplankton components, and ki is the absorption coefficient of the i-th phytoplankton species. function gi(i) is smooth and satisfies gi(0) = 0 and g ′ i(i) > 0 for i ≥ 0. 6.1. single phytoplankton species. the single species model, i.e. v = 0 in (6.1), was first considered in [141] for the self-shading case and infinite long water column; see also [75, 81]. the authors [44] considered the global dynamics of the single species model when diffusion and drift rates are spatially dependent. in [42] the authors studied the effect of photoinhibition on the single phytoplankton species, and they found that, in contrast to the case of no photoinhibition, the model with photoinhibition possesses at least two positive steady states in certain parameter ranges. hsu et al. [73] studied the single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat, building upon some interesting recent findings of the principal eigenvalue of a 1-homogeneous positive compact operator. in [137, 138], the authors considered the effect of time-periodic light intensity at the surface. ma and ou [118] recently made an important finding that the biomass of the single species satisfies a comparison principle, even though the density itself does not obey such orders. in [72], we investigated a single phytoplankton model and obtained some necessary and sufficient conditions for the growth of phytoplankton, and the critical death rate, critical water column depth, critical sinking or buoyant coefficient and critical turbulent diffusion rate were studied respectively. one of the results is that the phytoplankton population persists if and only if the sinking velocity of phytoplankton is less than a critical value. there are many simplified assumptions in the model, e.g. the death rate is assumed to be a constant. an interesting issue is to consider the case when the death rate is non-constant in space and time. some preliminary results show that this situation is quite different from that of constant death rate. for instance, under some conditions phytoplankton can persist if and only if the sinking velocity stays within an intermediate range, rather than below a single critical value. 6.2. two phytoplankton species. the coexistence of two or multiple phytoplankton is an important issue. in contrast to single phytoplakton species, very few results exist for two or more phytoplankton species; see [40, 41, 124]. on the one hand, the coexistence of many species can often be observed in reality. on the other hand, the classical competition theory shows that only the most dominant phytoplankton persists. they seem to contradict each other. the reason is that the classical competition theory is generally established for ordinary differential equation models, i.e. it is assumed that the selected topics on reaction-diffusion-advection models 171 diffusion rates of phytoplankton are sufficiently large, so that only their average densities are considered; i.e. the water column is well mixed. a natural question is whether small diffusion can increase the possibility of coexistence. in this connection, we recently investigated the outcome of model (6.1)-(6.2) in [77] and established the following results: (i) if two phytoplankton differ only in their sinking velocities (d1 = d2,α1 6= α2,g1 = g2, d1 = d2), in [77] we showed that the phytoplankton with smaller sinking velocity has the competitive advantage. in terms of evolution, the mutation of phytoplankton can continually reduce their sinking velocities, so that the density of phytoplankton and water will get closer as the population evolves. in other words, as the environment permits, the natural selection may favor the phytoplankton whose density is lighter that of water in some circumstances. (ii) if two phytoplankton differ only in their diffusion rates (d1 6= d2,α1 = α2,g1 = g2, d1 = d2), it is shown in [77] that slower diffusion rate will be selected when buoyant, and in contrast, faster diffuser wins when they are sinking with large velocity. an underlying biological reasoning is that when the phytoplankton are buoyant, the slower diffuser are more likely to reach the top of the water column (i.e. without water turbulence), where the light intensity is the strongest, while when sinking with large velocity, faster diffusion can counterbalance the tendency to sink and provide individuals with better access to light. an important tool is the extension of the comparison principle in [118] for single-species models to two-species phytoplakton models as follows: theorem 6.1. suppose {(ui,vi)}i=1,2 are non-negative solutions of (6.1)-(6.2) such that∫ x 0 u1(s, 0) ds ≤, 6≡ ∫ x 0 u2(s, 0) ds and ∫ x 0 v1(s, 0) ds ≥, 6≡ ∫ x 0 v2(s, 0) ds hold for all x ∈ (0,l]. then for all x ∈ (0,l] and t > 0,∫ x 0 u1(s,t) ds < ∫ x 0 u2(s,t) ds and ∫ x 0 v1(s,t) ds > ∫ x 0 v2(s,t) ds. by theorem 6.1, system (6.1)-(6.2) is a strongly monotone dynamical system with respect to the order generated by the cone k := k1 × (−k1), where k1 := { φ ∈ c([0,l]; r) : ∫ x 0 φ(s) ds ≥ 0, ∀x ∈ (0,l] } . this in turn enables the application of the theory of strongly monotone dynamical system, which provides a powerful tool to investigate the global dynamics of two-species system (6.1)-(6.2). the above findings in part (ii) naturally lead to the following conjecture: conjecture 4. suppose that α1 = α2 := α, g1 = g2 and d1 = d2. there exist two critical sinking velocities αmin and αmax such that the slower diffuser wins if α < αmin, the faster diffuser wins if α > αmax, and two species coexist if α ∈ (αmin,αmax). 7. spatial dynamics of epidemic diseases the covid-19 pandemic has impacted or changed almost everyone’s life. there are many mysteries about the novel coronavirus, among which the effect and possible control strategies of the movement of individuals is an emerging question. the covid-19 pandemic has been spreading so fast, at least partially due to the movement of those infected individuals who are showing little or no symptoms. what is the general impact of movement on the persistence and extinction of a disease in spatially heterogeneous environment? we studied various susceptible-infected-susceptible (sis) models in the 172 k.-y. lam, s. liu, and y. lou heterogeneous environment [1, 2, 3], including the following reaction-diffusion model [2], where the susceptible and infected individuals move randomly and the disease transmission and recovery rates could be uneven across the space:  ∂ts −ds∆s = − βsi s + i + γi (x,t) ∈ ω × (0,∞), ∂ti −di∆i = βsi s + i −γi (x,t) ∈ ω × (0,∞), ∂s ∂ν = ∂i ∂ν = 0 (x,t) ∈ ∂ω × (0,∞), s(x, 0) = s0(x), i(x, 0) = i0(x) x ∈ ω, (7.1) where ω is a bounded domain in rn with smooth boundary ∂ω and ν denotes the unit outward normal vector on ∂ω. here s(x,t) and i(x,t) represent the density of susceptible and infected populations at location x and time t, respectively. parameters ds,di > 0 denote their diffusion rates. the functions β(x),γ(x) account for the rates of disease transmission and recovery, respectively. they are assumed to be positive and at least one of them is non-constant to reflect the spatial heterogeneity of the residing habitat. in [2], we defined the basic reproduction number r0 for the spatial sis model (7.1), which is consistent with the next generation approach for heterogeneous populations [37, 146, 150]. it is of practical significance to consider the effect of spatial heterogeneity of disease transmission and recovery rates on the basic reproduction number. for instance, in the study of dengue fever, it is shown in [149] that the infection risk may be underestimated if the spatially averaged parameters are used to compute the basic reproduction number for spatially heterogeneous infections. we proved that r0 is monotone decreasing with respect to the diffusion rate of the infected individuals. the basic reproduction number r0 characterizes the infection risk of disease and serves as the threshold value for the extinction and persistence of disease: namely, if r0 < 1, the unique disease-free equilibrium is globally stable, and if r0 > 1, the disease-free equilibrium is unstable and there is a unique endemic equilibrium. a standing open question is whether this unique endemic equilibrium is globally stable. interestingly, we also found in [2] that a disease may persist but it can be controlled by limiting the movement of the susceptible populations. to be more specific, if the spatial environment can be transformed to include low-risk sites where the recovery rates are greater than transmission rates (such as through vaccination and treatment), and the movement of the susceptible individuals can be restricted (such as through isolation or lock down), then it may be possible to control the disease by flattening the disease outbreak curve, i.e. decreasing the daily number of infected individuals. in recent years there have been many studies on sis and other types of disease transmission models in spatially and/or temporally varying environments; see [31, 32, 33, 34, 48, 49, 50, 51, 95, 96, 120, 133, 134, 135, 136, 154]. for instance, to study the effect of the movement of exposed individuals on disease outbreaks, the following seirs (susceptible-exposed-infected-recovered-susceptible) epidemic reaction-diffusion model was considered in [143]:  ∂ts = ds∆s − β(x)si s + i + e + r + αr (x,t) ∈ ω × (0,∞), ∂te = de∆e + β(x)si s + i + e + r −σe (x,t) ∈ ω × (0,∞), ∂ti = di∆i + σe −γ(x)i (x,t) ∈ ω × (0,∞), ∂tr = dr∆r + γ(x)i −αr (x,t) ∈ ω × (0,∞), ∂s ∂ν = ∂e ∂ν = ∂i ∂ν = ∂r ∂ν = 0 (x,t) ∈ ∂ω × (0,∞). (7.2) selected topics on reaction-diffusion-advection models 173 here we divide the individuals into four different compartments: susceptible (s), exposed (e), infectious (i), recovered (immune by vaccination, r). the susceptible individuals are infected by infectious individuals with a rate of β, and become exposed; exposed individuals become infectious with a rate σ; infected individuals are recovered with a rate γ; recovered individuals lose immunity and go back into the susceptible class with a rate of α. here s(x,t), e(x,t), i(x,t) and r(x,t) denote the density of susceptible, exposed, infected and recovered individuals at location x and time t, and ds, de, di, dr represent the diffusion rates for susceptible, exposed, infected and recovered populations, respectively. we assume that the disease transmission rate β(x) and recovery rate γ(x) are environmentally dependent and could be spatially heterogeneous. our results in [143] reveal that the travel of exposed individuals could have an important impact on the persistence of disease and the movement of recovered individuals may enhance the endemic. hence, further understanding of the behaviors of the exposed and recovered individuals could be important in designing effective disease control measures. in [100] we investigated the spatial sis model (7.1) with spatially heterogeneous and time-periodic coefficients and proved that the basic reproduction number r0 is non-decreasing with respect to the time-period t . in some scenario, this would imply that there is a critical period t∗ > 0 such that r0 < 1 for t < t∗ and r0 > 1 for t > t∗; i.e. increasing the period of disease transmissions and recovery will increase the chance of disease outbreak. this might suggest why the outbreak of some epidemic diseases are less frequent than others, e.g. annually vs bi-annually, as different epidemic diseases could have different transmission and recovery rates so that the threshold values t∗ are different. how do movement and spatial heterogeneity affect the competition among multiple strains? many diseases are present in multiple phenotypes or strains, e.g. the novel coronavirus has found three major strains which are prevalent in different continents. it is an important evolutionary problem to determine the environmental conditions under which the adaptability of a certain strain becomes dominant. bremermann and thieme [9] showed that for an sir model with one host and multiple pathogens, different pathogens will die out eventually except those that optimize the basic reproduction number. recent studies showed that for some non-autonomous multi-strain models, even if the transmission and recovery rates change time-periodically, these strains cannot coexist; i.e. the temporal heterogeneity may not increase the chance of the coexistence for multiple pathogens. tuncer and martcheva [145] and wu et al. [153] considered the following two-strain sis spatial model to address the question whether the presence of spatial structure would allow two strains to coexist, as the corresponding spatially homogeneous model generally leads to competitive exclusion:   ∂ts −ds∆s = − s(β1i1 + β2i2) s + i1 + i2 + γ1i1 + γ2i2 (x,t) ∈ ω × (0,∞), ∂tii −di,i∆ii = βisii s + i1 + i2 −γiii (x,t) ∈ ω × (0,∞), ∂s ∂ν = ∂ii ∂ν = 0 (x,t) ∈ ∂ω × (0,∞), s(x, 0) = s0(x), ii(x, 0) = i0,i(x) x ∈ ω, (7.3) where i = 1, 2. these two strains could be different in dispersal rates, transmission rates and recovery rates. for the two pathogens differing only in their diffusion rates, a conjecture is that the pathogen with the slower diffusion will drive the other one to extinction for proper transmission and recovery rates; see [153] for some partial results. for the two pathogens differing only in their recovery rates, we conjecture that the strain with the recovery rate of greater spatial variation will be selected eventually. the coexistence of multiple pathogens in spatially heterogeneous and temporally varying environment remains a promising open research direction. 174 k.-y. lam, s. liu, and y. lou acknowledgment. we thank the anonymous referee and grégoire nadin and lei zhang for their helpful comments. references [1] l. allen, b. bolker, y. lou, a. nevai, asymptotic profile of the steady states for an sis epidemic patch model, siam j. appl. math. 67 (2007), 1283-1309. 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[158] p. zhou, on a lotka-volterra competition system: diffusion vs advection, calc. var. partial differential equations 55 (2016): 137. 180 k.-y. lam, s. liu, and y. lou corresponding author. department of mathematics, the ohio state university, columbus, oh 43210, usa e-mail address: lam.184@math.ohio-state.edu institute for mathematical sciences, renmin university of china, beijing 100872, prc e-mail address: liushuangnqkg@ruc.edu.cn department of mathematics, the ohio state university, columbus, oh 43210, usa e-mail address: lou@math.ohio-state.edu 1. introduction 2. single species models 2.1. logistic model 2.2. single species models in rivers 3. competing species models 3.1. lotka-volterra competition models 3.2. competition in rivers 4. competition models with directed movement 4.1. global bifurcation diagrams 4.2. evolution of biased movement 5. continuous trait models 5.1. mutation-selection model 5.2. mutation-selection model with drift 6. dynamics of phytoplankton 6.1. single phytoplankton species 6.2. two phytoplankton species 7. spatial dynamics of epidemic diseases references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 3, number 2, 2022, pp.86-105 https://doi.org/10.5206/mase/14663 global analysis of a generalized viral infection cellular model with cell-to-cell transmission under therapy alexis nangue, paulin tiomo lemofouet, simon ndouvatama, and emmanuel kengne abstract. in virus dynamics, when a cell is infected, the number of virions outside the cells is reduced by one: this phenomenon is known as absorption effect. most mathematical in intra-host models neglects this phenomenon. virus-to-cell infection and direct cell-to-cell transmission are two fundamental modes whereby viruses can be propagated and transmitted. in this work, we propose a new virus dynamics model, which incorporates both modes and takes into account the absorption effect and treatment. first we show mathematically and biologically the well-posedness of our model preceded by the result on the existence and the uniqueness of the solutions. also, an explicit formula for the basic reproduction number r0 of the model is determined. by analyzing the characteristic equations we establish the local stability of the uninfected equilibrium and the infected equilibrium in terms of r0. the global behaviour of the model is investigated by constructing an appropriate lyapunov functional for uninfected equilibrium and by applying a geometric approach to the study of the infected equilibrium. numerical simulations are carried out, to confirm the obtained theoretical result in a particular case. 1. introduction mathematical modeling is one of the most coveted areas in which virus infection research is undertaken. it consists of modeling the evolution of the infection using tools, mainly differential equations. the modeling of infectious diseases is a tool that has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak, and to evaluate strategies to control an epidemic. in recent years, grandiose efforts have been devoted to the mathematical modeling of intra-host viral dynamics. mathematical models have been developed to describe the process of in vivo infection of many viruses. many viruses infect humans and cause different infectious diseases such as human immunodeficiency virus (hiv), hepatitis b virus (hbv), hepatitis c virus (hcv), ebola virus, zika virus, and nowadays new coronavirus (covid-19 virus). they are often transmitted in the body by two fundamentally distinct modes, either by virus-to-cell infection through the extracellular space or by cell-to-cell transmission involving direct cell-to-cell contact [7, 24, 26, 35, 43]. during both infection modes, a part of infected cells returns to the uninfected state by loss of all covalently closed circular dna (cccdna) from their nucleus [13, 20, 10]. to model viral infection dynamics, several mathematical models have been proposed and developed [3, 9]. most of these models are based on the assumption that healthy cells can only be infected by viruses, and so they consider only the virus-to-cell infection mode [1, 16, 29, 28, 30, 31]. authors in [23] consider a mathematical model that describes a viral infection of hiv-1 with both virus-to-cell and cell-to-cell transmission with other features. it should be noted that the total infection rate of received by the editors 2 february 2022; accepted 13 april 2022; published online 21 april 2022. 2010 mathematics subject classification. 92b99, 34d23, 92d25, 37c75. key words and phrases. absorption effect, cell-to-cell transmission, global solution, global stability, treatment, wellposedness. 86 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14663 global analysis of a generalized viral infection cellular model 87 uninfected cells that it has been considered is a particular case from the one we consider in this work. however, there are few virus dynamics models in literature with both modes of transmission [33, 34, 36] and taking into account the cure of infected cells. motivated by the mentioned biological and mathematical considerations above, [17] propose a virus dynamics model with two transmission modes, i.e., cell-to-cell and virus-to-cell transmission modes. we note that in virus dynamics model proposed by [16], the loss of pathogens due to the absorption into uninfected cells is ignored. in biology, it is natural that, when pathogens are absorbed into susceptible cells, the number of pathogens are reduced into the blood volume : this is called absorption effect. hence, some researchers (see, for example, [2, 27, 42, 14] and the references therein) have included the absorption effect into their models. we also find that the treatment [6] is not taken into account. so, to make this last model a little more realistic, motivated by the works in [16], in the present paper we are concerned with the effect of both virus-to-cell and cellto-cell transmissions with absorption and antiviral treatments on the global dynamics of a generalized infection viral model. the rest of this paper is organized as follows. in section 2, the mathematical model is constructed. section 3 deals with the existence, the positivity and boundedness of solutions of our model. in addition the threshold parameter r0 of model (2.2) is determined and the existence of the equilibria is discussed with respect to the value of r0. in section 4, local stability of the equilibria are completely discussed. in section 5, global stability of the equilibria is studied. the global behaviour of the model is investigated by constructing an appropriate lyapunov functional for uninfected equilibrium and by applying li-muldowney global stability-criterion to the infected equilibrium. numerical simulations are shown in section 6. finally, a brief discussion is presented in section 7. 2. formulation and description of the model the model studied in [16] is described by the following three differential system of equations:  dt dt = λ−dt −f(t,i,v )v −g(t,i)i + ρi, di dt = f(t,i,v )v + g(t,i)i − (a + ρ)i, dv dt = ki −µv, , (2.1) where t(t), i(t), and v (t) denote the concentrations of uninfected cells, infected cells, and free viruses at time t, respectively, λ is the recruitment rate of uninfected cells, ρ is the cure rate of infected cells, k is the production rate of free viruses by infected cells, and d, a, and µ are the death rates of uninfected cells, infected cells, and free viruses, respectively. in addition, healthy cells become infected either by free viruses at rate f(t,i,v )v or by direct contact with an infected cell at rate g(t,i)i. hence, the term f(t,i,v )v + g(t,i)i represents the total infection rate of uninfected cells. as we mentioned above, system (2.1) does not take into account the treatment and the absorption phenomenon. we consider in this paper the following virus dynamics model with therapy and absorption effect,   dt dt = λ−dt − (1 −η)f(t,i,v )v −g(t,i)i + ρi, di dt = (1 −η)f(t,i,v )v + g(t,i)i − (a + ρ)i, dv dt = (1 −ε)ki −µv − (1 −η)f(t,i,v )v, (2.2) where the term −(1 −η)f(t,i,v )v in the third equation represents the loss of pathogens due to the absorption into uninfected cells. in addition, the therapeutic effect in this model involved blocking virions production (referred to as drug effectiveness) and reducing new infections which, are described in fractions (1 −ε) and (1 −η), respectively. 88 a. nangue, p. t. lemofouet, s. ndouvatama, and e. kengne system (2.2) is subject to the initial conditions t(0) = t0, i(0) = i0, v (0) = v0 with t0 ≥ 0, i0 ≥ 0, v0 ≥ 0. (2.3) 3. relevant assumptions and preliminary results 3.1. relevant assumptions. the incidence function g for direct cell-to-cell transmission mode is assumed to be continuously differentiable in the interior of r2+ and satisfies the following properties : (h01): g(0,i) = 0 for all i ≥ 0, ∂g∂t (t,i) ≥ 0 for all t ≥ 0 and i ≥ 0(or g(t,i) is a strictly increasing function with respect to t when f ≡ 0). (h02): ∂g ∂i (t,i) ≤ 0 for all t ≥ 0 and i ≥ 0. in addition, the incidence function f for virus-to-cell infection mode, which denotes the average number of cells which are infected by each virus in unit time, is assumed to be continuously differentiable in the interior of r3+ and has the properties similar to those assumed in [14, 18] : (h1): f(0,i,v ) = 0; for all i ≥ 0 and v ≥ 0. (h2): ∂f ∂t (t,i,v ) ≥ 0; for all t > 0, i ≥ 0 and v ≥ 0. (h3): ∂f ∂i (t,i,v ) ≤ 0 and ∂f ∂v (t,i,v ) ≤ 0 for all t ≥ 0, i ≥ 0 and v ≥ 0. (h4): f(t,i,v ) + v ∂f ∂v (t,i,v ) ≥ 0 for all t > 0, i ≥ 0, and v ≥ 0. the significance of these assumptions is as follows : • (h01) means that the incidence rate by cell-to-cell transmission is equal to zero if there are no susceptible cells. this incidence rate is increasing when the numbers of infected cells are constant and the number of susceptible cells increases. biologically speaking, the greater the amount of susceptible cells, the greater the average number of cells infected by direct contact with an infected cell in the unit time. • (h02) means that the greater the amount of infected cells, the lower the average number of cells infected by direct contact in the unit time. • (h1) means that the incidence rate for virus-to-cell infection mode is equal to zero if there are no susceptible cells. • (h2) signifies that the incidence rate is increasing when the numbers of infected cells and viruses are constant and the number of susceptible cells increases. hence, the second hypothesis means; the more the amount of susceptible cells, the more the average number of cells which are infected by each virus in the unit time will occur. • the first assumption of (h3) means the more the amount of infected cells, the less the average number of cells which are infected by each virus in the unit time will be and the second assumption of (h3) means the more the amount of infected virus, the less the average number of cells which are infected by each virus in the unit time will be. • (h4) means that if the total number of cells is constant, the more the amount of virus is, then the more the number of cells which are infected in the unit time will be. therefore, the hypotheses summarized in assumptions (h01) (h4) are reasonable and consistent with the reality. for more informations concerning the biological significance of hypotheses (h01), (h02), (h1), (h2) and (h3) , we refer the readers to [15, 41]. furthermore, the five assumptions (h01), (h02), (h1), (h2) and (h3) are satisfied by most incidence rates existing in the literature. 3.2. positivity and boundedness. first of all we show that the solutions of system 2.2 with nonnegative initial conditions remain nonnegative and bounded for all t ≥ 0. let r3+ = {(t,i,v ) ∈ r3 : t ≥ 0, i ≥ 0, v ≥ 0}. it is well known by the fundamental theory of ordinary differential equations (uniqueness of solutions global analysis of a generalized viral infection cellular model 89 of cauchy problem), that system (2.2) has a unique local solution (t(t),i(t),v (t)) satisfying the initial conditions (2.3). we have the following results. theorem 3.1. let (t,i,v ) be a solution of the initial value problem (2.2), (2.3) on an interval [t0, t1) with t1 > t0 ≥ 0. assume that the initial data of the initial value problem (2.2), (2.3) satisfy t0 ≥ 0, i0 > 0, and v0 > 0. then t , i and v remain positive for all t ∈ [t0, t1). proof. we first prove that t(t) is positive for all t > 0. suppose t(t) is not always positive. let τ > 0 be the first time such that t(τ) = 0. by the first equation of (2.2) we have dt dt (τ) = λ + ρi(τ) > 0, provided t(τ) > 0, which implies t(t) < 0 for t ∈ (τ−�,τ), for sufficiently small � > 0. a contradiction. therefore t(t) is positive for all t > 0. now let us show that i(t) and v (t) are positive. call the variables xi. if there is an index i and a time t (t0 ≤ t < t1) for which xi(t) = 0, we consider t∗ be the infimum of all such t for any i. then the restriction of the solution to the interval [t0, t∗) is positive and xi(t∗) = 0 for a certain value of i. the second and third equation of system (2.2) can be written in the form: dxi(t) dt = −xifi(x1,x2,x3) + gi(x1,x2,x3), i ∈{2, 3} where f2(x1,x2,x3) = a + ρ, f3(x1,x2,x3) = µ + (1 −η)f(x1,x2,x3), g2(x1,x2,x3) = (1 −η)f(x1,x2,x3)x3 + g(x1,x2), g3(x1,x2,x3) = (1 −η)kx2 are. non-negative. as a consequence dxi(t) dt ≥−xifi(x1,x2,x3) and d dt (log xi) ≥−fi(x1,x2,x3) ≥−c, i ∈ 2, 3, for a positive constant c. to show this, we use the fact that the solution remains in a compact set. it follows that xi(t∗) ≥ xi(t0)e−c(t∗−t0) > 0, which is a contradiction and the theorem 3.1 is proven. � this means that the first quadrant r3+ is positively invariant with respect to system (2.2). the boundedness of the solutions is guaranteed by the following theorem. theorem 3.2. all solutions of (2.2) are uniformly bounded in the compact subset ω = ß (t,i,v ) ∈ r3+ : t ≤ λ δ , i ≤ λ δ , v ≤ (1 −ε)kλ δ ™ , where δ = min{a,d}. proof. let (t(t),i(t),v (t)) be any solution with nonnegative initial condition (t0,i0,v0). adding the first two equations of system (2.2), we obtain d dt (t + i) = λ−dt −ai, ≤ λ−δ(t + i). hence lim sup t→+∞ (t(t) + i(t)) ≤ λ δ . similarly, from the third equation of system (2.2) one has : dv dt ≤(1 −ε)k λ δ −µv −f(t,i,v )v, ≤(1 −ε)k λ δ −µv. 90 a. nangue, p. t. lemofouet, s. ndouvatama, and e. kengne hence, lim sup t→∞ (v (t)) ≤ (1 −ε)kλ δ . hence, all solutions of system (2.2) starting in r3+ are eventually confined in the region ω. this completes the proof. � remark 3.1. we would like to make the following remarks. (i) these two previous theorems show mathematically and biologically the well-posedness [22] of our model (2.2). (ii) from these two results, if the initial data satisfy the inequalities t0 + i0 ≤ λδ and v0 ≤ (1−ε)kλ δ then the whole solution most satisfy these inequalities. this means that we have identified an invariant subset and for biological considerations, we will study system (2.2) in the subset ω. 3.3. equilibra. 3.3.1. basic reproduction numbers and disease-free equilibrium. now, we discuss the existence of equilibria. by simple computation system (2.2) always has one uninfected equilibrium of the form e0 = (t0, 0, 0) with t0 = λ d . this will allow us to determine a threshold parameter to discuss the dynamic behaviour of the epidemic model. this later will be decisive for the rest of the work. this parameter is called the basic reproduction number and it measures the expected average number of new infected cells generated by a single virion in a completely healthy cell. according to the concept of next generation matrix in [8] and the computation of the basic reproduction number presented in [39], we can compute the basic reproduction number of system (2.2). we have the following result : proposition 3.3. the basic reproduction number of the model (2.2) is given as : r0 = (1 −ε)(1 −η)kf(λ d , 0, 0) + (µ + (1 −η)f(λ d , 0, 0))g(λ d , 0) (a + ρ)(µ + (1 −η)f(λ d , 0, 0)) which can be rewritten as r0 = r01 + r02 where r01 = (1 −ε)k(1 −η)f(λ d , 0, 0) (a + ρ)(µ + (1 −η)f(λ d , 0, 0)) and r02 = g(λ d , 0) a + ρ . proof. based on notations in [39], the nonnegative matrix f and the non-singular m-matrix v are given by : f = ç g(t0, 0) (1 −η)f(λ d , 0, 0) 0 0 å and v = ñ −(a + ρ) 0 (1 −ε)k −µ− (1 −η)f(λ d , 0, 0) é . the next generation matrix is given by f.v −1 = á − g(λ d , 0) a + ρ − (1 −ε)(1 −η)kf(λ d , 0, 0) (a + ρ) ( µ + (1 −η)f(λ d , 0, 0) ) − (1 −µ)f(λd , 0, 0) µ + (1 −µ)f(λ d , 0, 0) 0 0 ë , global analysis of a generalized viral infection cellular model 91 and the basic reproduction number of system (2.2) is defined as r0 = ρ(−f.v −1) where ρ(a) denotes the spectral radius of a matrix a. it follows that : r0 = (1 −ε)(1 −η)kf(λ d , 0, 0) (a + ρ) ( µ + (1 −η)f(λ d , 0, 0) ) + g(t0, 0) a + ρ , = (1 −ε)(1 −η)kf(λ d , 0, 0) + (µ + (1 −η)f(λ d , 0, 0))g(λ d , 0) (a + ρ)(µ + (1 −η)f(λ d , 0, 0)) . this completes the proof of theorem 3.3. � remark 3.2. according to [16], r01 is the basic reproduction number corresponding to virus-to-cell infection mode, whereas r02 is the basic reproduction number corresponding to cell-to-cell transmission mode. 3.3.2. infected equilibrium. proposition 3.4. if r0 > 1, then system (2.2) has a unique chronic infection equilibrium of the form e∗ = (t∗,i∗,v ∗) with t∗ ∈ (0,t0), i∗ > 0 and v ∗ > 0. proof. to find the other equilibrium of system (2.2), which is named the infected equilibrium, we solve the algebraic system   λ−dt − (1 −η)f(t,i,v )v −g(t,i)i + ρi = 0, (3.1) (1 −η)f(t,i,v )v + g(t,i)i − (a + ρ)i = 0, (3.2) (1 −ε)ki −µv − (1 −η)f(t,i,v )v = 0. (3.3) adding (3.2) and (3.3) one has : v = 1 µ [(1 −ε)k + g(t,i) − (a + ρ)]i. (3.4) reporting (3.4) into (3.2) yields : [(1 −ε)k + g(t,i) − (a + ρ)](1 −η)f(t,i,v ) + µg(t,i) −µ(a + ρ) = 0. (3.5) since i = 1 a (λ−dt) ≥ 0, we have t ≤ λ d . hence, there is no biological equilibrium when t > λ d . we define the function ψ on the interval [ 0,t0 ] by : ψ(t) = [(1 −ε)k + g(t,i) − (a + ρ)](1 −η)f(t,i,v ) + µg(t,i) −µ(a + ρ). we have ψ(0) = −µ(a + ρ) < 0. moreover, ψ(t0) =(1 −ε)k(1 −η)f(t0, 0, 0) + [µ + (1 −η)f(t0, 0, 0)]g(t0, 0), − (a + ρ) [ µ + (1 −η)f(t0, 0, 0) ] =(a + ρ) ( µ + (1 −η)f(t0, 0, 0) )ß (1 −ε)k(1 −η)f(t0, 0, 0) + [µ + (1 −η)f(t0, 0, 0)]g(t0, 0) (a + ρ)(µ + (1 −η)f(t0, 0, 0)) − 1 ™ , =(a + ρ) ( µ + (1 −η)f(t0, 0, 0) ) (r0 − 1) 92 a. nangue, p. t. lemofouet, s. ndouvatama, and e. kengne and dψ dt (t) =(1 −η) å ∂g ∂t (t,i) − d a . ∂g ∂i (t,i) ã f(t,i,v ) + (1 −η){ ∂f ∂t (t,i,v ) − d a ∂f ∂i (t,i,v ) + 1 µ [( ∂g ∂t (t,i) − d a . ∂g ∂i (t,i))i − d a ((1 −ε)k + g(t,i) − (a + ρ)]. ∂f ∂v (t,i,v )} [(1 −ε)k + g(t,i) − (a + ρ)] + µ å ∂g ∂t (t,i) − d a ∂g ∂i (t,i) ã , = å ∂g ∂t (t,i) − d a . ∂g ∂i (t,i) ã [µ + (1 −η)f(t,i,v )] + (1 −η) å ∂f ∂t (t,i,v ) − d a ∂f ∂i (t,i,v ) ã [(1 −ε)k + g(t,i) − (a + ρ)] + (1 −η) i µ å ( ∂g ∂t (t,i) − d a . ∂g ∂i (t,i)) ã [(1 −ε)k + g(t,i) − (a + ρ)] − (1 −η) d a 1 µ ∂f ∂v (t,i,v ) [(1 −ε)k + g(t,i) − (a + ρ)]2 . we deduce that dψ dt (t) > 0. hence, for r0 > 1, there exists a unique infected equilibrium e∗ = (t∗,i∗,v ∗) with t∗ ∈ ( 0, λ d ) , i∗ > 0 and v ∗ > 0. this completes the proof of proposition 3.4. � 4. local stability in this section, we discuss the local stability of the two equilibria of system (2.2). theorem 4.1. the disease-free equilibrium e0 is locally asymptotically stable if r0 < 1 and becomes unstable if r0 > 1. proof. the jacobian matrix of system (2.2) at the disease-free equilibrium e0 is given by : je0 = ñ −d −g(t0, 0) + ρ −(1 −η)f(t0, 0, 0) 0 g(t0, 0) − (a + ρ) (1 −η)f(t0, 0, 0) 0 (1 −ε)k −µ− (1 −η)f(t0, 0, 0) é . computing the characteristic equation of je0 , one has − (x + d)(x2 + a1x + a0) = 0 (4.1) where a1 = −g(t0, 0) + (a + ρ) + µ + (1 −η)f(t0, 0, 0), a0 = −g(t0, 0) [ µ + (1 −η)f(t0, 0, 0) ] + (a + ρ) [ µ + (1 −η)f(t0, 0, 0) ] −(1 −ε)(1 −η)kf(t0, 0, 0). a0 and a1 can also be written in the form : a1 = µ + (1 −η)f(t0, 0, 0) + (a + ρ)(1 −r02), and a0 = (a + ρ)(µ + (1 −η)f(t0, 0, 0))(1 −r0). global analysis of a generalized viral infection cellular model 93 since r0 < 1, it follows that a0 and a1 are positive. from the routh-hurwitz criteria [12] we know that all roots of x2 + a1x + a0 = 0 have negative real parts. thus all roots of (4.1) have negative real parts. therefore, the disease-free equilibrium e0 is locally asymptotically stable for r0 < 1 and unstable if r0 < 1. � next, we study the local stability of the chronic infection equilibrium e∗. note that the equilibrium e∗ does not exist if r0 < 1 and e∗ = e0 when r0 = 1. theorem 4.2. the chronic infection equilibrium e∗ is locally asymptotically stable if r0 > 1, ai∗ ∂g ∂t (t∗,i∗) > (1 −ε)kd and (h4) are satisfied. proof. the jacobian matrix of system (2.2) at the chronic infection equilibrium e∗ is given by je∗ = ñ a11 a12 a13 a21 a22 a23 a31 a32 a33 é where a11 = −d− (1 −η)v ∗ ∂f ∂t (e∗) − i∗ ∂g ∂t (t∗,i∗), a12 = −(1 −η)v ∗ ∂f ∂i (e∗) − i∗ ∂g ∂i (t∗,i∗) −g(t∗,i∗) + ρ, a13 = −(1 −η) å v ∗ ∂f ∂v (e∗) + f(e∗) ã , a21 = (1 −η)v ∗ ∂f ∂t (e∗) + i∗ ∂g ∂t (t∗,i∗), a22 = (1 −η)v ∗ ∂f ∂i (e∗) + i∗ ∂g ∂i (t∗,i∗) + g(t∗,i∗) − (a + ρ), a23 = (1 −η) å v ∗ ∂f ∂v (e∗) + f(e∗) ã , a31 = (1 −η)v ∗ ∂f ∂t (e∗), a32 = (1 −ε)k − (1 −η)v ∗ ∂f ∂t (e∗), a33 = −µ− (1 −η) å v ∗ ∂f ∂t (e∗) + f(e∗) ã . computing the characteristic equation of je∗, one has x3 + a2x + a1x + a0 = 0, (4.2) where a2 =d + a + ρ−g(t∗,i∗) + µ + (1 −η) å v ∗ ∂f ∂v (e∗) + f(e∗) ã + (1 −η)v ∗ ∂f ∂t (e∗) − (1 −η)v ∗ ∂f ∂i (e∗) + i∗ ∂g ∂t (t∗,i∗) − i∗ ∂g ∂i (t∗,i∗), a1 =d ï a + ρ−g(t∗,i∗) − i∗ ∂g ∂i (t∗,i∗) + (1 −η) å v ∗ ∂f ∂v (e∗) + f(e∗) ãò + ï µ + (1 −η) å v ∗ ∂f ∂v (e∗) + f(e∗) ãòï a + ρ−g(t∗,i∗) − i∗ ∂g ∂i (t∗,i∗) ò + (1 −η)(a + µ) ∂f ∂t (e∗) + (1 −η)ai∗ ∂f ∂t (e∗) − (1 −η)(1 + µ)v ∗ ∂f ∂i (e∗). 94 a. nangue, p. t. lemofouet, s. ndouvatama, and e. kengne and a0 =d å (a + ρ−g(t∗,i∗) − i∗ ∂g ∂i (t∗,i∗) ãï µ + (1 −η) å v ∗ ∂f ∂v (e∗) + f(e∗) ãò + (1 −η)av ∗i∗ ∂f ∂v (e∗) ∂g ∂t (t∗,i∗) −d(1 −ε)(1 −η)kv ∗ ∂f ∂v (e∗) −d(1 −ε)(1 −η)kf(e∗) + aµ å v ∗ ∂f ∂t (e∗) + i∗ ∂g ∂t (t∗,i∗) ã + ai∗ ∂g ∂t (t∗,i∗)f(e∗) −dµv ∗ ∂f ∂i (e∗). =d å a + ρ−g(t∗,i∗) − i∗ ∂g ∂i (t∗,i∗) ãï µ + (1 −η) å v ∗ ∂f ∂v (e∗) + f(e∗) ãò + (1 −η) å v ∗ ∂f ∂v (e∗) + f(e∗) ãå ai∗ ∂g ∂t (t∗,i∗) −d(1 −ε)k ã − (1 −η)dµv ∗ ∂f ∂i (e∗) + aµ å (1 −η)v ∗ ∂f ∂t (e∗) + i∗ ∂g ∂t (t∗,i∗) ã . since r0 > 1 and a + ρ−g(t∗,i∗) = 1 µ [(1 −ε)k + g(t∗,i∗) − (a + ρ)](1 −η)f(t∗,i∗,v ∗) = v ∗ i∗ (1 −η)f(t∗,i∗,v ∗) > 0, (4.3) we deduce that a0, a1, and a2 are positive. in fact, firstly, we deduce from (4.3) that a + ρ − g(t∗,i∗) > 0. furthermore, using assumptions (h01), (h01), (h2), (h3) and (h4), we obtain respectively, ∂g ∂t (t∗,i∗) ≥ 0, ∂g ∂i (t∗,i∗) ≤ 0, ∂f ∂t (e∗) ≥ 0, ∂f ∂i (e∗) ≤ 0 and v ∗ ∂f ∂v (e∗) + f(e∗) ≥ 0. this shows that a2 is positive. secondly, from the same arguments used to prove the positivity of a2, we deduce that a1 is positive. finally, in addition to assumptions (h01), (h01), (h2), (h3) and (h4), using the fact that ai∗ ∂g ∂t (t∗,i∗) −d(1 −ε)k > 0, we deduce that a0 is positive. moreover, a1a2 −a0 =da1 + (a + ρ + µ−g(t∗,i∗) − i∗ ∂g ∂i (t∗,i∗))a1 + (1 −η)å v ∗ ∂f ∂t (e∗)f(e∗) ã a1 + å (1 −η)(v ∗ ∂f ∂t (e∗)) + i∗ ∂g ∂t (t∗,i∗) ã a1 − (1 −η)v ∗ ∂f ∂i (e∗)a1 − (1 −η) å v ∗ ∂f ∂v (e∗) + f(e∗) ãå da + dρ−dg(t∗,i∗) −di∗ ∂g ∂i (t∗,i∗) + ai∗ ∂g ∂t (t∗,i∗) −d(1 −ε)k ã −dµ å a + ρ−g(t∗,i∗) − i∗ ∂g ∂i (t∗,i∗) − (1 −η)v ∗ ∂f ∂i (e∗) ã −aµ ï (1 −η)v ∗ ∂f ∂t (e∗) + i∗ ∂g ∂t (t∗,i∗) ò , global analysis of a generalized viral infection cellular model 95 =(d + µ)a1 + (1 −η) å v ∗ ∂f ∂v (e∗) + f(e∗) ãå a1 −ai∗ ∂g ∂t (t∗,i∗) ã (a + ρ −g(t∗,i∗) − i∗ ∂g ∂i (t∗,i∗)) å a1 −d(1 −η) å v ∗ ∂f ∂v (e∗) + f(e∗) ã −dµ ã − (1 −η)(a1 −dµ)v ∗ ∂f ∂i (e∗) + å (1 −η)v ∗ ∂f ∂i (e∗) + i∗ ∂g ∂t (t∗,i∗) ã (a1 −aµ) + d(1 −ε)(1 −η)k å v ∗ ∂f ∂v (e∗) + f(e∗) ã . one deduces that a1a2−a0 > 0. from the routh-hurwitz criteria in [12] we know that all roots of (4.2) have negative real parts. thus, the chronic infection equilibrium e∗ is locally asymptotically stable for r0 > 1. � 5. global stability in this section, we investigate the global stability of the disease-free equilibrium e0 and the chronic infection equilibrium e∗. for the global stability, we assume that a ≥ d. biologically, this assumption is often satisfied because a represents the death rate of infected cells and includes the possibility of death by bursting of infected cells. furthermore, this assumption is considered by many authors; see, for example, [16, 37, 32, 40]. particularly in [4], this condition means that the average life-time of infected cells 1 a is shorter than the average life-time of infected cells 1 d . therefore, we have the following result. theorem 5.1. if r0 < 1 and a ≥ d then the uninfected equilibrium e0 is globally asymptotically stable. proof. construct the following lyapunov functional l(t(t),i(t),v (t)) ≡ l(t) = i(t) + (1 −η)f(t0, 0, 0) µ + (1 −η)f(t0, 0, 0) v (t). calculating the time derivative of l(t) along the positive solution of system (2.2), we have dl dt = dl dt dt dt + dl di di dt + dl dv dv dt , = di dt + (1 −η)f(t0, 0, 0) µ + (1 −η)f(t0, 0, 0) dv dt , =(1 −η)f(t,i,v )v + g(t,i)i − (a + ρ)i + (1 −η)(1 −ε)kf(t0, 0, 0) µ + (1 −η)f(t0, 0, 0) i −µ (1 −η)f(t0, 0, 0) µ + (1 −η)f(t0, 0, 0) v − (1 −η)2f(t0, 0, 0) µ + (1 −η)f(t0, 0, 0) f(t,i,v )v, = ï f(t,i,v ) − (1 −η)f(t0, 0, 0) µ + (1 −η)f(t0, 0, 0) (µ + (1 −η)f(t,i,v )) ò v + (a + ρ)i ï (1 −η)(1 −ε)kf(t0, 0, 0) (a + ρ)(µ + (1 −η)f(t0, 0, 0)) + g(t,i) (a + ρ) − 1 ò . 96 a. nangue, p. t. lemofouet, s. ndouvatama, and e. kengne according to remark 3.1, we consider solutions for which t(t) ≤ λ d . using the expression of r0, one has dl dt ≤ ï f(t, 0, 0) − (1 −η)f(t0, 0, 0) µ + (1 −η)f(t0, 0, 0) (µ + (1 −η)f(t,i,v )) ò v + (a + ρ)i ï (1 −η)(1 −ε)kf(t0, 0, 0) + (µ + (1 −η)f(t0, 0, 0))g(t0, 0) (a + ρ)(µ + (1 −η)f(t0, 0, 0)) − 1 ò , ≤ µf(t, 0, 0) + f(t, 0, 0)f(t0, 0, 0) −µf(t0, 0, 0) −f(t0, 0, 0)f(t, 0, 0) µ + (1 −η)f(t0, 0, 0) + (a + ρ)i[r0 − 1], ≤ µ ( f(t, 0, 0) −f(t0, 0, 0) ) µ + (1 −η)f(t0, 0, 0) + (a + ρ)i[r0 − 1], ≤(a + ρ)i[r0 − 1]. consequently, dl dt ≤ 0 for r0 < 1. moreover, it is easy to show that the largest compact invariant set i in v ={(t,i,v ) / dl dt = 0}, ={(t,i,v ) / i = v = 0} is the singleton {e0}. by the lasalle invariance principle[19], the disease-free equilibrium e0 is globally asymptotically stable for r0 < 1. � we now investigate the global dynamics of system (2.2) when r0 > 1. firstly, we need the following lemma. lemma 5.2. if r0 > 1, then differential system (2.2) is uniformly persistent. proof. considering the notations as in theorem 4.5 of [11], we denote by x1 = int(r 3 +) the interior of r3+ and by x2 = bd(r 3 +) the boundary of r 3 +. since (t,i,v ) is bounded, there exists a compact set b of r3+ in which all the solutions of the differential system (2.2) initiated in r 3 + finally enter and stay there forever. let us denote by ω(x0) the omega limit set of the solution x = x(t,x0) of the system (2.2) (by the criterion of poincaré-bendixson, and the fact that the solutions of (2.2) remain in a compact, and the omega limit set always exists). we need to determine ω2 defined as in theorem 4.5 in [11] by ω2 = ⋃ y∈y2 ω(y) (5.1) with, y2 = {x ∈ x2 / φt(x) ∈ x2; ∀t > 0}. setting y2 = {(t,i,v )t ∈ bd(r3+) / i = v = 0}, one has ω2 = {e0 = (t0, 0, 0)} where t0 = λ d . thus, solutions initiated on the t-axis converge to e0, then e0 is an isolated recovering of ω2 (since e0 is an equilibrium of (2.2)) and secondly acyclic (because there is no non-trivial solution in bd(r3+) which links e0 to itself). finally, if it is shown that e0 is a weak repeller for x1, the proof will be achieved. by definition, e0 is a weak repeller for x1 if for each solution with initial data (t0,x0) ∈ j×x we have : lim t→+∞ sup d(x(t,x0),e 0) > 0. (5.2) global analysis of a generalized viral infection cellular model 97 inequality (5.2) holds if v s(e0) ∩ int(r3+) = ∅, (5.3) where v s(e0) denotes the stable manifold of e0. suppose that (5.2) does not hold for a solution x = x(t,x0) with initial data x0 ∈ x1. considering the fact that the closed positive orthant is positively invariant with respect to the system (2.2), then, lim t→+∞ sup d(x(t,x0),e0)) = lim t→+∞ inf d(x(t,x0),e0)) = 0. therefore, we will have lim t→+∞ x(t,x0) = e 0 which is clearly impossible if (5.3) is verified. it remains to show that (5.3) is valid in order to achieve a contradiction. for this, let us recall that the jacobian matrix associated with the system (2.2) in e0 is given by ∇f(e0) = ñ −d −g(t0, 0) + ρ −(1 −η)f(t0, 0, 0) 0 g(t0, 0) − (a + ρ) (1 −η)f(t0, 0, 0) 0 (1 −ε)k −µ− (1 −η)f(t0, 0, 0) é . the characteristic polynomial of ∇f(e0) is given by p∇f(e0)(x) = −(x + d) ( x2 + a1x + a0 ) . since the product of the real parts of the roots of the polynomial t(x) = x2 + a1x + a0 worth a0 = (a+ρ)(µ+ (1−η)f(t0, 0, 0))(1−r0) ≤ 0, then the point e0 is unstable for the system (2.2). this implies that the matrix ∇f(e0) defined previously has an eigenvalue with a positive real part denoted x+ and two others with negative real parts x 1 − and x 2 − which may or may not coincide with x 1 −. the eigenspace associated with the eigenvalue x1− is the vector space generated by the vector (1.0.0). if x1− 6= x2−, then the eigenspace associated with x2− has the structure (0,v2,v3) with v2 and v3 verifying the following equation :ç g(t0, 0) − (a + ρ) (1 −η)f(t0, 0, 0) (1 −ε)k −µ− (1 −η)f(t0, 0, 0) å . ç v2 v3 å = x2− ç v2 v3 å . (5.4) in the case of equality between x2− and x 1 −, we note that the squared matrix at the left side of (5.4) will be not diagonalizable. indeed if it was diagonalizable, it would be similar to the matrix d = ç x2− 0 0 x2− å , and this would imply thatç x2− 0 0 x2− å = ç g(t0, 0) − (a + ρ) (1 −η)f(t0, 0, 0) (1 −ε)k −µ− (1 −η)f(t0, 0, 0) å which is absurd. therefore, the structure of the eigenvector associated with x2− will have the structure (v1,v2,v3), where v1 satisfies the equation ∇f(e0). ñ v1 v2 v3 é = x2− ñ v1 v2 v3 é . in both cases (i.e., the cases x2− = x 1 − and x 2 − 6= x1−), we will always have (v2,v3) /∈ r2+. indeed, the matrix defined in (5.4) is a metzler matrix and irreducible. hence, the stability modulus of x+ of this metzler matrix will be an eigenvalue to which will correspond an eigenvector u of the positive orthant. the vector u being unique, this implies that (v1,v2,v3) is not contained in this positive orthant, i.e., (v1,v2,v3) /∈ r3+. therefore, v s(e0) ∩ int(r3+) = ∅ which completes the proof.. � 98 a. nangue, p. t. lemofouet, s. ndouvatama, and e. kengne remark 5.1. according to [5], we can deduce from lemma 5.2 the existence of a compact absorbing set in ω. next, we focus on the global stability of the chronic infection equilibrium e∗ by assuming that r0 > 1 and the incidence function f satisfies the hypothesis (h4). to prove the global stability of e∗, we apply the geometrical approach developed by li and muldowney [21]. theorem 5.3. assume that r0 > 1, (1−η) q1 < δ and (h4) hold, then the chronic infection equilibrium e∗ is globally asymptotically stable where q1 = 1 t ∫ t 0 å i(s) ∂f ∂t − i(s) ∂f ∂i −v (s) ∂f ∂v ã ds proof. the second additive compound matrix of the jacobian matrix j, of the system (2.2) is defined by j[2] = ñ j11 + j22 j23 −j13 j32 j11 + j33 j12 −j31 j21 j22 + j33 é = ñ c11 c12 c13 c21 c22 c23 c31 c32 c33 é where c11 = −(a + d + ρ) − i ∂g ∂t − (1 −η) å v ∂f ∂t + v ∂f ∂i ã + i ∂g ∂i + g, c12 = (1 −η) å v ∂f ∂v + f ã , c13 = (1 −η) å v ∂f ∂v + f ã , c21 = (1 −ε)k − (1 −η)v ∂f ∂i , c22 = − ï d + µ + (1 −η)v ∂f ∂t + i ∂g ∂t + (1 −η) å v ∂f ∂v + f ãò , c23 = ρ− (1 −η)v ∂f ∂i − i ∂g ∂i −g, c31 = (1 −η)v ∂f ∂t , c32 = (1 −η)v ∂f ∂t + i ∂g ∂t , c33 = − ï a + ρ + µ + (1 −η) å v ∂f ∂v + f ãò + (1 −η)v ∂f ∂i + i ∂g ∂i + g, and jij; i,j = 1, 2, 3 is the (k,l)th entry of the matrix j. we consider the matrix p(t,i,v ) = diag å 1, i v , i v ã which has inverse given by p−1 = diag å 1, v i , v i ã and the matrix pf which is obtained by replacing each entry pij of p by its derivative in the direction of the solution of system (2.2). thus pf = diag å 0, i′v −v ′i v 2 , i′v −v ′i v 2 ã global analysis of a generalized viral infection cellular model 99 with i′ = di dt and v ′ = dv dt . it follows that, pfp −1 = diag å 0, i′ i − v ′ v , i′ i − v ′ v ã . furthermore, one has b = pfp −1 + p(j[2]p−1 = ç b11 b12 b21 b22 å , where b11 = −(a + d + ρ) − i ∂g ∂t − (1 −η) å v ∂f ∂t + v ∂f ∂i ã + i ∂g ∂i + g, b12 = ä (1 −η)v i ä v ∂f ∂v + f ä (1 −η)v i ä v ∂f ∂v + f ää , b21 = öä (1 −ε)k − (1 −η)v ∂f ∂i ä i vä (1 −η)v ∂f ∂t ä i v è , b22 = ö − ä d + µ + (1 −η)v ∂f ∂t + i ∂g ∂t + (1 −η) ä v ∂f ∂v + f ää + ä i′ i − v ′ v ä ρ− (1 −η)v ∂f ∂i − i ∂g ∂i −g (1 −η)v ∂f ∂t + i ∂g ∂t − ä a + ρ + µ + (1 −η) ä v ∂f ∂v + f ää + (1 −η)v ∂f ∂i + i ∂g ∂i + g + ä i′ i − v ′ v äè . we define the norm on r3 as ‖(u,v,w)‖ = max{|u|, |v| + |w|} for all (u,v,w) ∈ r3 then the lozinskii measure µ with respect to the norm ‖ ·‖ can be estimated as follows (see [25]): µ(b) ≤ sup{g1,g2} (5.5) where g1 = µ1(b11) + ‖b12‖ et g2 = µ1(b22) + ‖b21‖. here, µ1 denotes the lozinskii measure with respect to the l1 vector norm, and ‖b12‖ and ‖b21‖ are matrix norms with respect to the l1 norm. moreover, we have µ1(b11) = −(a + d + ρ) − i ∂g ∂t − (1 −η) å v ∂f ∂t + v ∂f ∂i ã + i ∂g ∂i + g. (5.6) ‖b12‖ = (1 −η) v i å v ∂f ∂v + f ã . (5.7) according to the second equation of system (2.2), (5.7) becomes: ‖b12‖ = i′ i + (1 −η) v 2 i ∂f ∂v + a + ρ−g. µ1(b22) = i′ i − v ′ v −µ− (1 −η) å v ∂f ∂v + f ã + max{−d;−a}, (5.8) = i′ i − v ′ v −µ− (1 −η) å v ∂f ∂v + f ã − δ (5.9) and ‖b21‖ = ï (1 −ε)k − (1 −η)v ∂f ∂i + (1 −η)v ∂f ∂t ò i v . (5.10) hence, we obtain : g1 = i′ i −d + (1 −η)v 2 i ∂f ∂v − (1 −η) ä v ∂f ∂t + v ∂f ∂i ä + i ∂g ∂i − i ∂g ∂t , ≤ i ′ i − δ, since a ≥ d. (5.11) 100 a. nangue, p. t. lemofouet, s. ndouvatama, and e. kengne by using the first equation of (2.2), (5.8) and (5.10), one also has : g2 = i′ i − v ′ v −µ− (1 −η) å v ∂f ∂v + f ã − δ + ï (1 −ε)k − (1 −η)v ∂f ∂i + (1 −η)v ∂f ∂t ò i v , = i′ i − δ + (1 −η) å i ∂f ∂t − i ∂f ∂i −v ∂f ∂v ã . (5.12) from (5.5), (5.11) and (5.12), we get : µ(b) ≤ i′ i − δ + (1 −η) å i ∂f ∂t − i ∂f ∂i −v ∂f ∂v ã . from lemma 5.2 we know that the system (2.2) is uniformly persistent when r0 > 1. then there exists a compact absorbing set k ⊂ ω [5]. along each solution (t(t),i(t),v (t)) of (2.2) with x0 = (t(0),i(0),v (0)), we have 1 t ∫ t 0 (µ(b(x(s),x0))) ds ≤ (1 −η) 1 t ln å i(t) i0 ã −δ + 1 t ∫ t 0 å i(s) ∂f ∂t − i(s) ∂f ∂i −v (s) ∂f ∂v ã ds, which implies that lim sup t→∞ sup x0∈k 1 t ∫ t 0 (µ(b(x(s),x0))) ds ≤ −δ + lim sup t→∞ sup x0∈k (1 −η) 1 t ∫ t 0 å i(s) ∂f ∂t − i(s) ∂f ∂i −v (s) ∂f ∂v ã ds, ≤ −δ + (1 −η) q1, < 0. then, based on theorem 3.5 in [21], we deduce that the chronic infection equilibrium e∗ is globally asymptotically stable. this completes the proof of the theorem 5.3. � 6. applications and numerical simulations as an application of our theoretical results, we consider the system   dt dt = λ−dt − (1 −η)β1tv α0 + α1t + α2v + α3tv −β2ti + ρi, di dt = (1 −η)β1tv α0 + α1t + α2v + α3tv + β2ti − (a + ρ)i, dv dt = (1 −ε)ki −µv − (1 −η)β1tv α0 + α1t + α2v + α3tv , (6.1) which is a particular case of system (2.2) by letting f(t,i,v ) = β1t α0 + α1t + α2v + α3tv and g(t,i) = β2t where β1, β2, α1, α2, α3 and α4 are non negative constants. the functions g and f satisfy (h01), (h02) conditions and (h1), (h2), (h3) conditions, respectively. we have f(t,i,v ) + v ∂f ∂v (t,i,v ) = β1t(α0 + α1t) (α0 + α1t + α2v + α3tv )2 ≥ 0 global analysis of a generalized viral infection cellular model 101 and we conclude that assumption (h4) is also satisfied. other state variables and parameters are the same as in model (2.2). the threshold number, r0 takes the following form r0 = ((1 −η)(1 −ε)kβ1λd + (µ(α0d + α1λ) + (1 −η)β1λ) β2λ d(a + ρ) (µ(α0d + α1λ) + (1 −η)β1λ) . firstly, we simulate the model (6.1) by using the following parameter values : λ = 10 cells mm−3 day−1 [15], η = 0.4 , β1 = 0.000024 mm 3virion−1day−1 [15], ρ = 0.01 [38], d = 0.02day−1[15]; ε = 0.5, k = 600 virions cell−1day−1, µ = 3 day−1 [15], α0 = 1, α1 = 0.1, α2 = 0.01, α3 = 0.00001, β2 = 0.0001 [38], a = 0.5 day−1 [15]. according to the values of these parameters, r0 = 0.1257 < 1, which means that r0 satisfies the conditions mentioned in theorem 4.1 and theorem 5.1. this implies that the diseasefree equilibrium e0 = (500, 0, 0) is globally asymptotically stable. furthermore, numerical simulation shown in figure 1 confirms the result. 0 100 200 300 400 500 600 700 800 900 1000 time t [in days] 300 320 340 360 380 400 420 440 460 480 500 u n in fe ct e d c e lls t (t ) (a) 0 100 200 300 400 500 600 700 800 900 1000 time t [in days] -10 0 10 20 30 40 50 60 70 80 in fe ct e d c e lls i (t ) (b) 0 100 200 300 400 500 600 700 800 900 1000 time t [in days] -1000 0 1000 2000 3000 4000 5000 6000 v ir u s lo a d v (t ) (c) 0 80 2000 4000 60 500 v (t ) 6000 400 phase diagram i(t) 8000 40 300 t(t) 10000 20020 100 0 0 (d) figure 1. time evolutions of model (2.2) with initial values (300; 100; 15). e0 is globally asymptotically stable in figure 1 (a), the proliferation of uninfected cells reaches its equilibrium value at λ d and t(t) converges to λ d = 500 whereas in figure 1 (b) and in figure 1 (c), infected cells i(t) and viral load v (t) 102 a. nangue, p. t. lemofouet, s. ndouvatama, and e. kengne converge to zero. secondly, we choose β1 = 0.000024 mm 3virion−1day−1 [15] and the other parameter values are the same as above. the reason to just modify the parameter β1 is based on the fact that r0 is an increasing function with respect to β1. by calculating, we r0 = 1.2571 > 1, which means that it satisfies the conditions mentioned in theorem 4.1 and theorem 5.3. this implies that the chronic infection equilibrium e∗ = (405, 12.10, 1275) is globally asymptotically stable. in addition to this, numerical simulations shown in figure 2 confirms the result. 0 100 200 300 400 500 600 700 800 900 1000 time t [in days] 240 260 280 300 320 340 360 380 400 420 440 u n in fe ct e d c e lls t (t ) (a) 0 100 200 300 400 500 600 700 800 900 1000 time t [in days] 0 10 20 30 40 50 60 70 80 90 100 in fe ct e d c e lls i (t ) (b) 0 100 200 300 400 500 600 700 800 900 1000 time t [in days] 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 v ir u s lo a d v (t ) (c) 0 100 2000 4000 450 v (t ) 6000 phase diagram 400 i(t) 8000 50 t(t) 10000 350 300 0 250 (d) figure 2. time evolutions of model (2.2) with initial values (300; 75; 15). e∗ is globally asymptotically stable in figure 2, we observe damped oscillations at the beginning of the simulation. infection will not become totally extinct, but a considerable reduction of the viral load and infected cells will be observed. particularly, in figure 2 (a), the number of uninfected cells t(t) decrease rapidly and then increase slightly until equilibrium is reached whereas in figure 2 (b), infected cells i(t) do not tend to zero as t increases and in figure 2 (c) viruses persist in the presence of treatment leading to the system going to an endemic equilibrium. global analysis of a generalized viral infection cellular model 103 7. conclusion in this study, we have proposed and studied a virus dynamic model with a generalized functional response, treatment and absorption effect. by analyzing the characteristic equations of model (2.2) at the disease-free equilibrium point, it has been completely established that the disease-free equilibrium is locally asymptotically stable if the basic reproduction number r0 is less than or equal to one (r0 ≤ 1). the local stability result of the chronic infection equilibrium of model (2.2) is shown in theorem 4.2. if the basic reproduction number is greater than one (r0 > 1), then the disease-free equilibrium is unstable and the chronic infection equilibrium is locally asymptotically stable. the global behaviour of the model is investigated by constructing an appropriate lyapunov functional for disease-free equilibrium and by applying li-muldowney global stability-criterion to the chronic infection equilibrium. numerical simulations are carried out, performed in matlab, to confirm obtained theoretical result in a particular case. furthermore, for model (2.2), we found that the basic reproduction number is less than that of a model without absorption effect. from the above discussion, it can be seen that there is a positive effect on eliminating viruses from the blood vessel than the model without absorption effect. finally, it should be noted that most of the results contained in this work extend and complete the results of the works in [14] and [17]. it would be interesting to incorporate time delay into the current model. also, taking into account random phenomena could be a serious issue. these two challenges will be the concerns of future investigation. references [1] a. abdon and f. d. g. emile, on the mathematical analysis of ebola hemorrhagic fever : deathly infection disease in west african countries, biomed research international 2014 (2014), 1–7. 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[43] p. zhong, l. m. agosto, j. b. munro, and w. mothes, cell-to-cell transmission of viruses, curr. opin. virol. 3 (2013), no. 1, 44–50. https://doi.org/10.1007/978-981-13-6581-22 https://doi.org/10.1007/978-981-13-6581-22 https://doi.org/10.1007/s10884-017-9622-2 https://doi.org/10.1142/s1793524512600121 global analysis of a generalized viral infection cellular model 105 higher teachers’ training college, university of maroua, p.o.box 55, maroua, cameroon email address: alexnanga02@gmail.com higher teachers’ training college, university of maroua, p.o.box 55, maroua, cameroon email address: plemofouettiomo@yahoo.fr higher teachers’ training college, university of maroua, p.o.box 55, maroua, cameroon email address: simonvotsomafils@gmail.com corresponding author, school of physics and electronic information engineering, zhejiang normal university, jinhua 321004, china email address: ekengne6@zjnu.edu.cn 1. introduction 2. formulation and description of the model 3. relevant assumptions and preliminary results 3.1. relevant assumptions 3.2. positivity and boundedness 3.3. equilibra 4. local stability 5. global stability 6. applications and numerical simulations 7. conclusion references mathematics in applied sciences and engineering https://doi.org/10.5206/mase/8120 volume 1, number 1, march 2020, pp.1-15 https://ojs.lib.uwo.ca/mase existence and metastability of non-constant steady states in a keller-segel model with density-suppressed motility peng xia, yazhou han, jicheng tao, and manjun ma abstract. we are concerned with stationary solutions of a keller-segel model with density-suppressed motility and without cell proliferation. we establish the existence and the analytical approximation of non-constant stationary solutions by applying the phase plane analysis and bifurcation analysis. we show that the one-step solutions is stable and two or more-step solutions are always unstable. then we further show that two or more-step solutions possess metastability. our analytical results are corroborated by numerical simulations of the underlying system. 1. introduction stripe pattern formation was observed in the experiment of the engineered e. coli strains with the behavior of density suppressing motility in an isolated apparatus (see [9]), which showed that spatio-temporal patterns could be driven by a “self-trapping” mechanism besides diffusion-driven and chemotaxis-driven instabilities [10]. in order to describe the essential features of the stripe pattern formation driven by the density-suppressed motility, in [1] authors proposed the following model  ut = ∆(r(v)u) + σu(1 −u), x ∈ ω, t > 0, vt = d∆v + ηu−βv, x ∈ ω, t > 0, ∂u ∂ν = ∂v ∂ν = 0, x ∈ ∂ω, t > 0, u(x, 0) = u0(x),v(x, 0) = v0(x), x ∈ ω, (1.1) where the domain ω ⊂ rn,n ≥ 1 is bounded and has a smooth boundary ∂ω, ν is the outward unit normal vector on ∂ω; u(x,t) and v(x,t) denote the densities of e. coli cells and chemical substance acyl-homoserine lactone (ahl), respectively; the chemical substance ahl is produced by e. coli cells with a rate η > 0, degraded with a rate β > 0, and diffused with a rate d. e. coli cells have a logistic growth with an intrinsic rate σ ≥ 0 saturated at the normalized density 1 and a non-random diffusion with the diffusion rate r(v). the motility function r(v) decreases as the density of ahl increases, i.e., dr dv < 0, which implies the suppressing effect of ahl concentration on cell’s motility. when σ > 0, as far as we know, there have been the following study to (1.1). the existence of global classical solutions and the stability of constant steady state for ω ⊂ r2 were investigated in [2]. received by the editors 1 july 2019; revised 27 september 2019; accepted 27 september 2019; published online 30 september 2019. 2000 mathematics subject classification. 35k55, 35k45, 35k57, 35k50, 92c15, 92c17. key words and phrases. keller-segel model, density-suppressed motility, metastability, nonconstant steady states. manjun ma and peng xia were supported by the national natural science foundation of china (no. 11671359). yazhou han was supported by the provincial natural science foundation of zhejiang (no. ly18a010013)). jicheng tao was supported by the provincial natural science foundation of zhejiang (no. ly16a010009)). 1 2 peng xia, yazhou han, jicheng tao, and manjun ma the global existence of classical solutions was recently extended to higher dimensions (n ≥ 3) under appropriate conditions in [14]. in [12] the authors studied the dynamics of interface of discontinuity of solutions when r(v) is a piecewise constant function. for the case where σ = 0, global classical solutions in two dimensions and global weak solutions in three dimensions were established in [13] by supposing that r(v) has a positive lower and upper bounds. in [15], the authors studied the global existence of classical solutions, the stability of constant steady states and the existence of non-constant solutions in any dimensions for the motility function r(v) given by r(v) = c0/v p, p > 0, c0 > 0 and small enough. (1.2) specifically, the reference [15] dealt with the model as follows:{ ut = ∇· ( r(v) ( ∇u− p v u∇v )) , x ∈ ω, t > 0, vt = d∆v + ηu−βv, x ∈ ω, t > 0 (1.3) with the following boundary condition and initial value{ ∂u ∂ν = ∂v ∂ν = 0, x ∈ ∂ω, t > 0, u(x, 0) = u0(x),v(x, 0) = v0(x), x ∈ ω, (1.4) where the initial functions u0(x) ≥ 0 and v0(x) > 0 are smooth. the first equation of (1.3) is also in the form of ut = ∇· (r(v)∇u + r′(v)u∇v). obviously, the system (1.3) is a special form of original keller-segel model [3]{ ut = ∇· (γ(v)∇u−χ(v)u∇v) , x ∈ ω, t > 0, vt = d∆v + η(v)u−β(v)v, x ∈ ω, t > 0 (1.5) with χ(v) = (l− 1)γ′(v), γ′(v) < 0, l ∈ [0, 1). (1.6) it is seen that the case of l = 0 corresponds to the model (1.3) when both η(v) and β(v) are positive constants. for the biological interpretation of l = 0 and l ∈ (0, 1) as well as other more details about (1.3), we refer interested readers to [15]. the keller-segel models [3] have been extensively investigated for a cell aggregation phenomenon and the global existence and boundedness of solutions in various forms, for example, see [4, 5, 6, 16, 17]. the aim of this paper is to establish the existence of non-constant steady states of (1.3)-(1.4) by using a different method from [15] and to derive conditions for their stability and metastability in one-dimensional space. for some different models the metastability was discussed, for example, see [7, 8, 11]. throughout the paper, we assume (1.2) is true. our presentation is structured as follows. in section 2 we give some properties of the negative laplace operator used later, the interval and the number of unstable modes and the sufficient conditions for the instability of constant steady state, and the expression of the most unstable mode. in section 3 we establish the existence of non-constant steady states by applying the phase plane analysis and the third-order approximate expression of the local bifurcations. in section 4, we analyze the stability and metastability of the stationary solutions with small amplitudes. full numerical solutions of the original system are also carried out to corroborate the results of our analytical analysis. finally, in section 5 we conclude our work and bring some forward problems for further study. existence and metastability of non-constant steady states 3 2. preliminaries in this section we give some results which will play an important role in our later discussion. we first present one known property of the negative laplace operator −∆ in the interval [0, l], where l is a positive real number. the eigenvalue problem  −φ′′ = λφ, x ∈ (0, l), φ′ = 0 at x = 0, l, (2.7) has a sequence of simple eigenvalues λj = (jπ/l) 2, j = 0, 1, 2, · · ·, (2.8) whose corresponding eigenfunctions are φj(x) = { 1, j = 0, cos(jπx/l), j > 0. (2.9) the steady states of (1.3)-(1.4) with n = 1 satisfy the elliptical boundary-value problem  ( r(v) ( u′ − p v uv′ ))′ = 0, x ∈ (0, l), dv′′ + ηu−βv = 0, x ∈ (0, l), u′(0) = u′(l) = 0, v′(0) = v′(l) = 0. (2.10) obviously, system(1.3) possesses one conserved quantity m = 1 l ∫ l 0 u(x,t)dx = 1 l ∫ l 0 u(x, 0)dx, (2.11) where m is an implicit positive parameter. then (2.10) has the constant solution (m, η β m). naturally, we first study its stability to establish the unstable mode band and its relationship with the system parameters. lemma 2.1. if there exists j ∈ n satisfies 0 < ( πj l )2 < λ∗, λ∗ = β d (p− 1), (2.12) then (i) the constant steady state solution of (1.3) is linearly unstable. (ii) the number of the unstable fourier modes jmax is equal to the greatest j satisfying (2.12), i.e., jmax = l √ λ∗ π − 1 if l √ λ∗ π is a positive integer; otherwise, jmax = [ l √ λ∗ π ] . (iii) the most unstable mode ju = [ l √ λu π ] or ju = [ l √ λu π ] + 1, where λu > √ r(ηm β )d(√ r(ηm β ) + √ d )2 λ∗; furthermore, the wave number √ λu is monotone increasing in β if 1 < p ≤ 2; when p > 2, √ λu is monotone decreasing if β ∈ [β∗, +∞) or is monotone increasing if β ∈ (0,β∗), where β∗ = ηmc −1 p 0 ( (p + 2) √ d p− 2 )2 p . here [·] denotes the integer part. 4 peng xia, yazhou han, jicheng tao, and manjun ma proof. let u = m + u(x,t), v = ηm β + v (x,t). then substitute this into (1.3) to obtain the linearized system of u and v   ut = r( ηm β )u′′ + r′( ηm β )mv ′′, x ∈ ω, t > 0, vt = dv ′′ −βv + ηu, x ∈ ω, t > 0, ux = vx = 0,x = 0, l. (2.13) it is well know that the jth mode cos(πjx l ) grows at the exponential function exp(µt), where µ is the larger eigenvalue of the matrix a(λ) = ( −λr(ηm β ) −r′(ηm β )mλ η −dλ−β ) , λ = ( πj l )2. this gives the characteristic equation of µ µ2 − traµ + deta = 0, (2.14) where tra = − ( r( ηm β ) + d ) λ−β < 0, det a = −dr( ηm β )λ(λ∗ −λ). thus, the discriminant of (2.14) is d =(tra)2 − 4deta = [( r( ηm β ) + d ) λ + β ]2 + 4dr( ηm β )λ(λ∗ −λ) = ( r( ηm β ) −d )2 λ2 + 2 ( r( ηm β ) + d ) βλ + β2 + 4dr( ηm β )λλ∗ ≥ ( r( ηm β ) −d )2 λ2 − 2 ( r( ηm β ) −d ) βλ + β2 = [( r( ηm β ) −d ) λ−β ]2 ≥ 0. (2.15) by this, we have µ = tra + √ d 2 . (2.16) and µ > 0 if and only if deta < 0, i.e., 0 < λ < λ∗, which implies that on a bounded domain [0, l] there is a finite set of unstable modes j satisfies 0 < j < l √ λ∗ π . then (i) and (ii) are proved. next, we prove (iii). we regard µ as a continuous function of λ and assume that the maximum of µ(λ) attains at λu. to find the value of λu we have to solve the equation dµ dλ = 0. by (2.16), we introduce a new function h(µ) = 2µ− tra, hence dµ dλ = 0 is equivalent to dh dλ = − tra dλ = r( ηm β ) + d. (2.17) applying (2.15) and (2.16), we have h2 = (tra)2 −4deta. differentiation of this expression by λ yields 2hhλ = 2tra(tra)λ − 4(deta)λ. existence and metastability of non-constant steady states 5 then taking the square of both sides and substituting the expression of h2 and dh dλ leads to 4 [( (r( ηm β ) + d ) λ + β)2 + 4dr( ηm β )λ(λ∗ −µ) ]( r( ηm β ) + d )2 = [ 2 ( (r( ηm β ) + d ) λ + β) ( r( ηm β ) + d ) + 4dr( ηm β )(λ∗ − 2λ) ]2 . (2.18) to simplify the expression we let τ = r(ηm β ) + d dr(ηm β ) . then (2.18) can be rewritten as( τ2dr( ηm β ) − 4 ) λ2 + 2λ(τβ + 2λ∗) −λ∗(λ∗ + τβ) = 0, which has a unique positive root λu = −2(τβ + 2λ∗) + √ (4(τβ + 2λ∗)2 + 4(τ2dr(ηm β ) − 4)λ∗(λ∗ + τβ))) 2(dr(ηm β )τ2 − 4) . it is easy to check that d( λuλ∗ ) dλ∗ < 0 so that λu λ∗ > lim λ∗→+∞ λu λ∗ = 1 2 + τ √ dr(ηm β ) = √ r(ηm β )d(√ r(ηm β ) + √ d )2 . then (iii) follows. � 3. existence and analytical approximation 3.1. existence of non-constant steady states. this subsection is devoted to the discussion of the existence of nonconstant solutions to the stationary system (2.10) by using the method of phase plane analysis. to this end, we apply the first equation of (2.10) with the given boundary conditions to get u = v0 r(v) . (3.19) applying the conserved quantity m, we know that the integration constant v0 is determined by v0 = lmc0∫ l 0 vpdx > 0. (3.20) integrating the second equation of (2.10) from 0 to l and using the boundary conditions yield an additional information for v(x) ∫ l 0 vdx = lmη β . (3.21) combining (3.20) with (3.21), we conclude that value range of v0 is 0 < v0 ≤ r( ηm β )m when p > 1. (3.22) the maximum v0 = r( ηm β )m corresponds to the positive constant steady state (m, η β m). in the cases where the steady states are not constant, v0 are gotten by solving (3.20). substituting (3.19) into the the second equation of (2.10) leads to the single second-order equation dvxx + ηv0 c0 vp −βv = 0, (3.23) 6 peng xia, yazhou han, jicheng tao, and manjun ma with the boundary conditions vx(0) = vx(l) = 0. (3.24) in [18] the authors established the conditions for the existence and the attractivity of positive nonconstant solutions to (3.23). let vx = ω and regard v0(v) in (3.19) as a given constant. then (3.23) is equivalent to the first-order system  vx = w, dwx = β(v −g(v)), g(v) = ηv0 βr(v) = ηv0 βc0 vp, (3.25) which is a hamiltonian system if letting the variable x act as the time. its hamiltonian function is h(w,v) = 1 2 w2 + ηv0 d(p + 1)c0 vp+1 − β 2d v2. thus system (3.25) has only two types of fixed points, i.e., saddles and centers. it is easy to observe that (3.25) has two or three fixed points denoted by (vk, 0) satisfying v 1−p k = ηv0 βc0 , (3.26) and the eigenvalues ρ of the fixed point (vk, 0) solve the equation dρ2 = β(1 −g′(v̄k) = β ( 1 − ηv0 βc0 pv p−1 k ) . (3.27) the type and the number of the fixed points explicitly depend on the values of the parameter p. by a straightforward computation, we have the proposition below. proposition 3.1. the following statements are true: (a) for all p > 0, system (3.25) has always a fixed point v0 = (0, 0). it is a saddle if p > 1; while it is a non-differentiable point if 0 < p < 1. (b) suppose that p > 1 is a integer or p−1 = s q , where s and q are coprime positive integers. then we have (b1) if p is odd or s is even, then system (3.25) has two nontrivial fixed points: v1 = (( βc0 ηv0 ) 1 p−1 , 0 ) and v2 = ( − ( βc0 ηv0 ) 1 p−1 , 0 ) , and both of them are centers. (b2) if p is even or s is odd, then system (3.25) has one nontrivial fixed point v1 = (( βc0 ηv0 ) 1 p−1 , 0 ) , and it is a center. (c) assume that 0 < p < 1 and 1 −p = s q , where s,q are coprime integers. then we have (c1) if s is even, then system (3.25) has two nontrivial fixed points: v3 = (( ηv0 βc0 ) 1 1−p , 0 ) and v4 = ( − ( ηv0 βc0 ) 1 1−p , 0 ) ; moreover, they are saddles. (c2) if s is odd, then system (3.25) has one nontrivial fixed point v3 = (( ηv0 βc0 ) 1 1−p , 0 ) , and it is a saddle. (d) if p = 1, then there is one nontrivial fixed point (m, η β m) which is a node. for the boundary value problem (3.23)-(3.24), fixed points are its spatially homogeneous solutions (i.e., constant solutions). even if there are more than one fixed points, only one of them satisfies (3.20) and thus it is a solution of (3.23)-(3.24). obviously, corresponding to this constant solution, we have vk = η β m for all p > 0 and v0 = mr( ηm β ). existence and metastability of non-constant steady states 7 (a) (b) (c) (d) figure 1. typical phase portraits for (3.25) with d = 1,β = 1,η = 1 and m = 2. (a) p = 2; (b) p = 3; (c) p = 7/3; (d) p = 8/3. to obtain spatially positive and inhomogeneous solutions (i.e., non-constant positive solutions) of (3.23)-(3.24) , it is well known to require that the trajectory of (3.25) on the phase plane meets two conditions: (i) it begins and ends at the line ω = 0 to satisfy the boundary value conditions; (ii) the transition time x between this two points is equal to l. then, based on the results of bendixson and poincaré, only some of the periodic trajectories circling the center can be candidates. therefore, by proposition 3.1, (3.23)-(3.24) (equivalent to (2.10)) has no non-constant positive solutions when 0 < p ≤ 1. next we discuss the case where p > 1. notice of (b) in proposition 3.1, there are two types of phase portraits for four different values of p, which are shown in fig.1. let v∗ ∈ [v0,v1] and (v∗, 0) is the point where the trajectory touches the v−axis. we use l(v∗) to represent the length of a half circle which ends at (v∗, 0). if v∗ approaches v0, then the corresponding orbit approaches a homoclinic, then lim v∗→v0 l(v∗) = +∞. if v∗ approaches v1, then by the linearized system of (3.25) around the center v1, the length of the half circle is lim v∗→v1 l(v∗) = π√ β d (g′(v1) − 1) = π√ β d (p− 1) def = l∗. thus, when the interval length l > l∗, there is at least one non-constant steady state in (3.23)-(3.24). through the above discuss, we conclude the following result on solutions to (2.10). theorem 3.2. if 0 < p ≤ 1, then (2.10) has only constant positive solutions. let the positive parameters d,β be fixed, and assume that l > l∗. then (2.10) has at least one non-constant solution provided that p > 1. 8 peng xia, yazhou han, jicheng tao, and manjun ma remark 3.1. theorem 3.2 is just the result of proposition 3.1 and theorem 3.3 with n = 1 in [15]. this result implies that the occurrence of non-constant solutions of (2.10) requires β d to be large for some fixed p. this can be done by giving a sufficiently small diffusion rate d and sufficiently large degradation rate β of the chemical substance ahl. 3.2. analytical expression of local bifurcation. next under the condition that p > 1, we shall establish the third-order approximations of non-constant solutions with small amplitudes of (2.10) (equivalent to (3.23)-(3.24)). the above phase plane analysis shows that non-constant solutions bifurcate from the spatially homogenous solution (m, η β m) having at least one unstable mode. if we treat β as a bifurcation parameter, then, by theorem 3.2, the bifurcation points are β0 = dλj p− 1 = d(πj l )2 p− 1 def = β j 0,j = 1, 2, 3, · · ·, (3.28) where, for two given positive integers j1 and j2, d(πj1 l )2 p− 1 6= d(πj2 l )2 p− 1 if j1 6= j2. (3.29) notice of (2.12), from (3.28) it follows that β10 is the smallest bifurcation point βmin. let (u ∗,v∗) be a non-constant solutions of (2.10). then v∗ solves (3.23)-(3.24)). we now make an asymptotic analysis for v∗ with a small amplitude by assuming  v∗ = ηm β + εv1 + ε 2v2 + ε 3v3 + · · ·, v0 r(v) = u∗ = m + εu1 + ε 2u2 + ε 3u3 + · · ·, β = β0 + εβ1 + ε 2β2 + ε 3β3 + · · · (3.30) with β0 = β j 0,β1 = β j 1 · · · ,j = 1, 2, · · · and un = n∑ j=1 anj cos ( πjx l ) , n = 1, 2, · · ·. here the positive parameter 0 < ε � 1 keeps the parameter β is in the small neighborhood of the bifurcation location β0 so that the corresponding small amplitude solution (u ∗,v∗) can be bifurcated at this location from the constant steady state (m, η β m). substituting (3.30) into (3.23)-(3.24), under the condition (3.29) we have β1 = 0 and{ u1 = c(j) cos( πjx l ), v1 = cos( πjx l ), (3.31) where c(j) = λjd + β0 η = pdλj (p− 1)η > 0 (3.32) as well as β2 = β30 (p 2 + 3p) 12(ηm)2 > 0, (3.33) and { u2 = d1(j) cos( 2πjx l ), v2 = d2(j) cos( 2πjx l ), (3.34) where   d1(j) = 4dλj+β0 η d2(j) = (4p−3)dλj (p−1)η d2(j), d2(j) = − 2r′( ηm β0 )c(j)+r′′( ηm β0 )m 4r( ηm β0 ) ( (4p−3)dλj (p−1)η ) +4r′( ηm β0 )m . (3.35) existence and metastability of non-constant steady states 9 0 5 10 15 20 space x 13.7 13.75 13.8 13.85 v (x , t) t = 100000 0 5 10 15 20 space x 12.49 12.495 12.5 12.505 12.51 12.515 a p p ro x im a ti o n o f v 0 5 10 15 20 space x 13.65 13.7 13.75 13.8 v (x , t) t = 20000 0 5 10 15 20 space x 12.49 12.495 12.5 12.505 12.51 12.515 a p p ro x im a ti o n o f v figure 2. non-constant steady states to system (1.3) in the interval (0, 20) for m = 5 and different wave modes. system parameters are taken as d = 1,c0 = 0.01,p = 2,η = 1 and β = 0.4. left: the numerical solutions obtained by integrating the full system (1.3) with the initial condition (u0(x),v0(x)) = ( m + rand(1), ηm β + rand(1) ) . right: the analytical solutions (3.36) obtained by our asymptotic analysis with ε = 0.01 . the top line is the three-step pattern at t = 105, i.e., j = 3. the bottom line is the four-step pattern at t = 2 × 104, i.e., j = 4. by (3.30), (3.31) and (3.34), the third-order approximation of non-constant solutions with small amplitudes to system (2.10) reads{ u∗ = m + ε pdλj (p−1)η cos( πjx l ) + ε2d1(j) cos( 2πjx l ) + o(ε3), v∗ = ηm β + ε cos(πjx l ) + ε2d2(j) cos( 2πjx l ) + o(ε3), (3.36) where d1(j) and d2(j) are defined in (3.35). fig.2 compares the long-time numerical steady states (see left panels) with the prediction (3.36) from our asymptotic analysis (see right panels) for the bifurcations with the principal wave modes j = 3 and j = 4, respectively. for the sake of brevity, only the numerical results of the solution component v are presented here. as it can be noticed from the figure, there is a qualitative agreement between the full numerics and the analytical prediction. the variation in amplitude originates from omitting higher order terms in the analysis. therefore, we shall use the expression in (3.36) to analyze the stability of (u∗,v∗) by estimating the sign of the principal eigenvalue. 4. stability and metastability this section is devoted to the analysis of stability and metastability for non-constant steady states with small amplitudes. 4.1. stability analysis. following the method used in [19], we first indicate the relationship between the solution (u∗,v∗) and its bifurcation location β j 0 by relabelling (u ∗,v∗) as (u∗j,v ∗ j ). for system 10 peng xia, yazhou han, jicheng tao, and manjun ma (1.3)-(1.4) with n = 1, we set { u = u∗j + ϕ(x)e γt, v = v∗j + ψ(x)e γt. (4.37) substitution of (4.37) into (1.3)-(1.4) yields a linear system  r(v∗j )ϕ ′′ + r′(v∗j )u ∗ jψ ′′ + 2r′(v∗j )v ∗ j ′ϕ′ + q1ψ ′ + q2ψ + q3ϕ = γϕ, x ∈ (0, l), dψ′′ + ηϕ−βψ = γψ, x ∈ (0, l), ϕ′(0) = ϕ′(l) = 0, ψ′(0) = ψ′(l) = 0, (4.38) where q1 = 2r ′′(v∗j )v ∗ j ′ u∗j + 2r ′(v∗j )v ∗ j ′ , (4.39) q2 = r ′′′(v∗j )(v ∗ j ′ )2u∗j + r ′′(v∗j )(v ∗ j ) ′′u∗j + 2r ′′(v∗j )v ∗ j ′ u∗j ′ + r′(v∗j )u ∗ j ′′ , q3 = r ′′(v∗j )(v ∗ j ′ )2 + r′(v∗j )v ∗ j ′′ . then substituting the asymptotic expansions of u∗j , v ∗ j and β in (3.30) and  γ = γ j 0 + �γ j 1 + � 2γ j 2 + · · · , ϕ = ϕ0 + �ϕ1 + � 2ϕ2 + · · · , ψ = ψ0 + �ψ1 + � 2ψ2 + · · · , into (4.38) and equating the o(1) terms lead to  r(ηm β0 )ϕ′′0 + r ′(ηm β0 )mψ′′0 = γ0ϕ0, x ∈ (0, l), dψ′′0 + ηϕ0 −β0ψ0 = γ0ψ0, x ∈ (0, l), ϕ′0(0) = ϕ ′ 0(l) = 0, ψ′0(0) = ψ ′ 0(l) = 0. (4.40) the sign of γ0 determines the stability of the stationary solution (u ∗ j,v ∗ j ). to solve the eigenvalue problem (4.40) for γ0, in view of (2.7)-(2.9) we replace (ϕ ′′ 0,ψ ′′ 0 ) with −λm(ϕ0,ψ0) for some integer m ≥ 0. thus, the characteristic equation is γ20 + σγ0 + τ = 0 (4.41) with σ = ( λmr( ηm β j 0 ) + dλm + β j 0 ) , τ = λm(p− 1) c0(β j 0) p (ηm)p ( βm0 −β j 0 ) . it is easy to see that β j 0 6= βmin when j 6= 1, and then there exists a integer m = 1 such that τ < 0. hence (4.41) has a positive root γ0. we now have the following result. proposition 4.1. for system (1.3)-(1.4) with n = 1, non-constant steady state (u∗j,v ∗ j ) is unstable if j ≥ 2. in other words, the stable non-constant steady state with small-amplitude is always located on the first bifurcation. next, we look for the sufficient condition of stability of the first bifurcation. obviously, when j = 1, the principal eigenvalue of (4.41) is γ0 = 0 corresponding to m = 1 with the eigenvector (ϕ0,ψ0) = ( pdπ2 (p− 1)l2 cos( πx l ), cos( πx l ) ) . (4.42) existence and metastability of non-constant steady states 11 thus, we have to find the value of γ1. now equating the o(ε) terms in (4.38) gives  r(ηm β0 )ϕ′′1 − r′( ηm β0 )mη d ϕ1 + r′( ηm β0 )mβ0 d ψ1 = γ1 ( ϕ0 − r′( ηm β0 )m d ψ0 ) −g1, x ∈ (0, l), dψ′′1 + ηϕ1 −β0ψ1 = γ1ψ0, x ∈ (0, l), ϕ′1(0) = ϕ ′ 1(l) = 0, ψ′1(0) = ψ ′ 1(l) = 0, (4.43) where g1 =r ′( ηm β0 ) (v1ϕ ′′ 0 + u1ψ ′′ 0 + +2v ′ 1ϕ ′ 0 + 2u ′ 1ψ ′ 0 + u ′′ 1ψ0 + v ′′ 1ϕ0) + r′′( ηm β0 )m (v1ψ ′′ 0 + 2v ′ 1ψ ′ 0 + v ′′ 1ψ0) . simplifying g1 by applying (3.20) and (4.42), we have g1 = −2 ( 2r′( ηm β0 )c(1)λ1 + r ′′( ηm β0 )mλ1 ) cos( 2πx l ). (4.44) since the solution to the adjoint system of the homogeneous system corresponding to (4.43) is{ ϕ = c(1) cos(πx l ), ψ = cos(πx l ), where c(1) = pd r′(ηm β0 )m < 0, (4.45) by solvability condition of (4.43), we have γ1 = ∫ l 0 g1ϕdx∫ l 0 [ ϕ0ϕ− r′( ηm β0 )m d ψ0ϕ + ψ0ψ ] dx = 0. thus, we have to compute the value of γ2. in view of (4.44), we set the particular solution of (4.43) as{ ϕ1 = d3(1) cos( 2πx l ), ψ1 = d4(1) cos( 2πx l ). (4.46) substitution of this into (4.43) yields  d3(1) = (4p−3)dλ1 (p−1)η d4(1), d4(1) = − 2r′( ηm β0 )c(1)+r′′( ηm β0 )m 2r( ηm β0 ) ( (4p−3)dλ1 (p−1)η ) +2r′( ηm β0 )m . (4.47) by equating the o(ε2) terms in (4.38), we have  r(ηm β0 )ϕ′′2 − r′( ηm β0 )mη d ϕ2 + r′( ηm β0 )mβ0 d ψ2 = γ2(ϕ0 − r′( ηm β0 )m d ψ0) − r′( ηm β0 )m d ψ0β2 −g2, x ∈ (0, l), dψ′′2 + ηϕ2 −β0ψ2 = γ2ψ0 + ψ0β2, x ∈ (0, l), ϕ′2(0) = ϕ ′ 2(l) = 0, ψ′2(0) = ψ ′ 2(l) = 0, (4.48) where the explicit expression of g2 is omitted here since it is too cumbersome. using the the solvability condition once more, we obtain γ2 = (p− 1)2β2 1 2 (c(1)c(1) − (p− 1)) , (4.49) 12 peng xia, yazhou han, jicheng tao, and manjun ma where c(1) and c(1) are defined in (3.32) and (4.45), respectively. thus, by this, (3.33) and proposition 4.1, we have the following theorem. theorem 4.2. let the positive parameters c0,d,η and l be fixed. assume that p > 1, β > d p− 1 (π l )2 . (4.50) then, for any positive constant m the first bifurcation (i.e., (u∗1,v ∗ 1 )) of (2.10) is supercritical and linearly stable. all other bifurcations (i.e., (u∗j,v ∗ j ),j ≥ 2) are also supercritical but unstable. taking into account proposition 4.1 and theorem 4.2, we give the bifurcation diagram in fig.3. (a) (b) 0 0.05 0.1 0.15 0.2 0.25 0.3 β 0 50 100 150 200 250 300 v m a x ,m in 0.02 0.022 0.024 0.026 0.028 0.03 β 100 150 200 250 300 v m a x ,m in figure 3. bifurcation diagram of vmax,min as a function of β. the mean density is set as m = 5. other parameters are taken as d = 1,η = 1,p = 2 and l = 20. the blue and thick curve shows the constant solution v = ηm β . the black and thin curves correspond to bifurcations. solid curves correspond to stable solutions, dashed to unstable solutions. the constant solution loses its stability at the first bifurcation point where the non-constant solution with the principal wave mode 1 appears. (b) is an enlargement of the first bifurcation point from (a). 4.2. metastability of multi-step solutions. solutions located on the jth bifurcation possess the principal wave mode j. hence we call them j-step solutions. as obtained in the previous sections, under the condition (4.50) the one-step solution is stable and multi-step ones are unstable. by the equation (4.41), we know that the principal eigenvalues of multi-step solutions correspond to m = 1, that is, γ0(j, l) = −σ(j) + √ σ(j)2 − 4τ(j) 2 > 0, j = 2, 3, · · ·, (4.51) where σ(j) = ( λ1r( ηm β j 0 ) + dλ1 + β j 0 ) = ( λ1r( ηm β j 0 ) + dλ1 + dλj p− 1 ) , τ(j) = λ1(p− 1)r( ηm β j 0 ) ( β10 −β j 0 ) = λ1dr( ηm β j 0 ) (λ1 −λj) , λj = ( πj l )2 . through a simple analysis of (4.51), we know that γ0(j, l) is an increasing function of j for some fixed l, and if the wave mode j is fixed, then γ0(j, l) is decreasing with the increase of length l; when the length l is sufficiently large, it is obvious that, for all j the principal eigenvalue γ0 is close to zero. these existence and metastability of non-constant steady states 13 dependency are also explicitly shown in fig.4. thus, two or more-step solutions have metastability; moreover, the less steps the stationary solution has, the more stable it is. 10 20 30 40 50 l 0 0.2 0.4 0.6 0.8 1 γ 0 ×10 -8 j=2 j=3 j=4 j=5 figure 4. examples of dependence of principal eigenvalues γ0 on l and j for 2-5-step solutions. the mean density is m = 5 and the interval length is l = 50. system parameters are set as d = 0.1,c0 = 0.01,η = 1 and p = 2. we now give a numerical example to demonstrate our theoretical results. system parameters are taken as c0 = 0.1,p = 2,d = 0.5,β = 0.4 and η = 0.4. the interval length is chosen as l = 20. the mean density is set as m = 0.5. by a computation, we have that (1) the upper bound of unstable modes is λ∗ = 0.8; (2) the number of unstable fourier modes is jmax = 6; (3) the most unstable mode is j = 3 corresponding to λu = 0.2492. fig.5 demonstrates all of above analytical results obtained by applying lemma2.1, theorem 3.2, theorem 4.2 and our metastability analysis. the initial data is a random perturbation of the spatially homogeneous background (m, ηm β ) = (0.5, 0.5). as observed in fig.5, the most unstable three-step solution appears first at about t = 60. its left step disappears at about t = 1.1 × 103, and then a two-step solution develops. the two-step solution finally becomes a stable one-step solution at about t = 1.1×105. hence, the three-step solution persists for about 103 time units and the two-step solution exits for about 106 time units. when we vary system parameters, especially increase the length of interval, we do observe that every transient period of duration will become longer, which is not shown here. since such non-constant steady states that stays almost unchanged for a rather long period may not be distinguishable from true stable steady sates, we call them metastable steady states. 5. conclusion in this work, in one-dimensional space we obtain the conditions for the existence of non-constant steady states of (1.3) by using the phase plane analysis. then relying on the bifurcation analysis by treating the decay rate of chemical substance as a bifurcation parameter, we derive the thirdorder approximate expression of non-constant steady states with small amplitudes. furthermore, the stability of one-step solutions and the metastability of two or more-step solutions are established. the analytical results are corroborated by the numerical computation as well as by numerical simulations of the underlying keller-segel system. the metastable states are the phenomenon that can be seen in experiments. thus the establishment of metastability of stationary solutions will provide theoretical 14 peng xia, yazhou han, jicheng tao, and manjun ma 0 5 10 15 20 space x 0.54 0.545 0.55 0.555 u , v t = 1 0 5 10 15 20 space x 0.54 0.545 0.55 0.555 0.56 u , v t = 50 0 5 10 15 20 space x 0.52 0.54 0.56 0.58 u , v t = 60 0 5 10 15 20 space x 0 0.5 1 1.5 2 2.5 u , v t = 300 0 5 10 15 20 space x 0 1 2 3 u , v t = 1000 0 5 10 15 20 space x 0 1 2 3 4 u , v t = 1100 0 5 10 15 20 space x 0 1 2 3 4 u , v t = 30000 0 5 10 15 20 space x 0 1 2 3 4 u , v t = 110000 0 5 10 15 20 space x 0 2 4 6 8 u , v t = 115000 figure 5. example of metastable steady states in (1.3). the initial data is (u0(x),v0(x)) = (m + rand(1), ηm β + rand(1)). the blue and solid curve is u(x). the red and dashed curve is v(x). basis for experimental study related to the model (1.3). the analytical results show that e. coli cells and chemical substance ahl, under the condition that other environmental factors remain unchanged, will be inhomogenously distributed when the diffusion rate of e. coli cells is sufficiently small and the degradation rate of ahl is sufficiently large (see theorem 3.2 ), and their state always appears to be stable when the spatial domain is large enough (see (4.51)). this makes good biological sense. however, how to explain the formation of metastability and how to understand mergings and dissolvings of steps of non-constant steady states are not explored here. this is an interesting problem. another challenging possibility is to consider the metastability of (1.3) in two or higher dimensional spaces. acknowledgement. we would like to thank the anonymous reviewers for their valuable comments and suggestions which greatly improved the exposure of this manuscript. existence and metastability of non-constant steady states 15 references [1] x. fu, l.-h. tang, c. liu, j. -d. huang, t. hwa, p. lenz, stripe formation in bacterial systems with density -suppressed motility, physical review letters 108 (2012): 198102. 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[19] m. j. ma, c.h. ou, and z.a. wang. stationary solutions of a volume filling chemotaxis model with logistic growth and their stability, siam j. appl. math. 72 (2012), 740-766. department of mathematics, school of sciences, zhejiang sci-tech university, hangzhou, zhejiang, 310018, china e-mail address: 624598686@qq.com department of mathematics, college of science, china jiliang university, hangzhou, zhejiang 310018, china e-mail address: taoer558@163.com department of mathematics, college of science, china jiliang university, hangzhou, zhejiang 310018, china e-mail address: 02a0802031@cjlu.edu.cn corresponding author. department of mathematics, school of sciences, zhejiang sci-tech university, hangzhou, zhejiang, 310018, china e-mail address: mjunm9@zstu.edu.cn mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase online first, pp.1-20 https://doi.org/10.5206/mase/15145 long-time behavior of a nonlocal dispersal logistic model with seasonal succession zhenzhen li and binxiang dai abstract. this paper is devoted to a nonlocal dispersal logistic model with seasonal succession in one-dimensional bounded habitat, where the seasonal succession accounts for the effect of two different seasons. firstly, we provide the persistence-extinction criterion for the species, which is different from that for local diffusion model. then we show the asymptotic profile of the time-periodic positive solution as the species persists in long run. 1. introduction the nonlocal diffusion as a long range process can well describe some natural phenomena in many situations (andreu-vaillo et al. [1], fife [9]). recently, nonlocal diffusion equations have attracted much attention and have been used to simulate different dispersal phenomena in material science (bates [2]), neurology (sun et at. [25]), population ecology (hutson et al. [13], kao et al.[15]). especially, the spectral properties of nonlocal dispersal operators and the essential differences between them and local dispersal operators are studied in coville [5], coville et al. [6], garćıa-melián and rossi [10], shen and zhang [22] and sun, yang and li [25]. a widely used nonlocal diffusion operator has the form (j ∗u−u)(t,x) := ∫ r j(x−y)u(t,y)dy −u(t,x), which can capture the factors of ‘long-range dispersal’ as well as ‘short-range dispersal’. time-varying environmental conditions are important for the growth and survival of species. seasonal forces in nature are a common cause of environmental change, affecting not only the growth of species but also the composition of communities [7, 8]. the growth of species is actually driven by both external and internal dynamics. for instance, in temperate lakes, phytoplankton and zooplankton grow during the warmer months and may die or lie dormant during the winter. this phenomenon is termed as seasonal succession. in the present paper, we are concerned with the nonlocal dispersal logistic model with seasonal succession as follows:  ut = −δu, iω < t ≤ (i + ρ)ω, l1 ≤ x ≤ l2, ut = d( ∫ l2 l1 j(x−y)u(t,y)dy −u) + u(a− bu), (i + ρ)ω < t ≤ (i + 1)ω, l1 ≤ x ≤ l2, u(0,x) = u0(x), x ∈ [l1, l2], (1.1) received by the editors 27 october 2022; accepted 10 december 2022; published online 14 december 2022. 2020 mathematics subject classification. primary 35k57; 92d25; secondary 35b40. key words and phrases. nonlocal dispersal; seasonal succession; persistence-extinction. z. li was supported by the fundamental research funds for the central universities of central south university (no. 2020zzts040) and b. dai was supported by the national natural science foundation of china (no. 11871475). 1 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15145 2 z. li and b. dai where u(t,x) is the population density of a species at time t and location x in the one-dimensional bounded habitat [l1, l2] ⊂ r. all parameters δ,a,b and d are positive constants. the kernel function j : r → r is assumed to satisfy (j): j ∈ c(r) ∩l∞(r) is nonnegative, even, j(0) > 0 and ∫ r j(x)dx = 1. here the parameter d stands for the diffusion rate of the species. let j(x − y) be the probability distribution of the species jumping from location y to location x, then ∫ r j(x−y)u(t,y)dy represents the rate where individuals are arriving at location x from all other places and −u(t,x) = − ∫ r j(x − y)u(t,x)dy is the rate at which they are leaving location x to travel to all other sites. in such model, the integral operator ∫ l2 l1 j(x−y)u(t,y)dy−u can be viewed as the nonlocal dispersal counterpart of the elliptic operator uxx with homogeneous dirichlet type boundary condition. the initial function u0(x) is nonnegative continuous function. here and in what follows, unless specified otherwise, we always take i ∈ z+ = {0, 1, 2, · · ·}. in (1.1), it is assumed that the species u undergoes two different seasons: the bad season and the good season. in the bad season: iω < t < (i + ρ)ω, for instance, from winter to spring, the species can not get enough food to feed themselves and its density are declining exponentially. during this season, the population has no ability to move in space. in the good season (for instance, from summer and autumn): (i + ρ)ω < t ≤ (i + 1)ω, we assume that the spatiotemporal distribution of the species u are governed by the classical nonlocal dispersal logistic equation. parameters ω and 1 − ρ represent the period of seasonal succession and the duration of the good season, respectively. in fact, if we define the time-periodic finctions d(t) = { 0, t ∈ (iω, (i + ρ)ω], d, t ∈ ((i + ρ)ω, (i + 1)ω], ā(t) = { −δ, t ∈ (iω, (i + ρ)ω], a, t ∈ ((i + ρ)ω, (i + 1)ω], b̄(t) = { 0, t ∈ (iω, (i + ρ)ω], b, t ∈ ((i + ρ)ω, (i + 1)ω], (1.2) then model (1.1) can be rewritten as{ ut = d(t)(j ∗u−u)(t,x) + u(t,x)(ā(t) − b̄(t)u(t,x)), t > 0, l1 ≤ x ≤ l2, u(0,x) = u0(x), x ∈ [l1, l2], (1.3) which is a nonlocal dispersal piecewise smooth time-periodic system. the models with seasonal succession have been investigated by several authors. ignoring the spatial evolution of the involved species, the effects of seasonal succession on the dynamics of population can be analysed by ode models, see [14, 16] and references therein. there are also some investigations on it by the numerical method, see, e.g. [12, 20]. in [14], hsu and zhao first considered the single species model with seasonal succession:  zt = −δ, iω < t ≤ (i + ρ)ω, zt = z(a− bz), (i + ρ)ω < t ≤ (i + 1)ω, z(0) = z0 ∈ r+ := [0,∞), (1.4) where z(t) denotes the population density of a species at time t. they showed the threshold dynamics of model (1.4): when a(1 − ρ) − δρ ≤ 0, the unique solution of (1.4) converges to zero among all nonnegative initial value, while when a(1 −ρ) − δρ > 0, it converges to the unique positive ω-periodic solution of (1.4) for all positive initial value. a nonlocal dispersal logistic model with seasonal succession 3 taking spatial factor into account, peng and zhao [18] investigated the following local diffusion model with seasonal succession:  ut = −δu, iω < t ≤ (i + ρ)ω, x ∈ (l1, l2), ut −duxx = u(a− bu), (i + ρ)ω < t ≤ (i + 1)ω, x ∈ (l1, l2), u(t, l1) = u(t, l2) = 0, t ≥ 0, u(0,x) = u0(x) ≥ 0, x ∈ (l1, l2), (1.5) where the parameter d stands for the intensity of random diffusion. the positive constants ω,ρ,δ,a,b have the same biological interpretations as in (1.1), and the initial function u0 ∈ c2([l1, l2]). denote by λl1 the principal eigenvalue of the eigenvalue problem  ϕt = −δϕ + λϕ, iω < t ≤ (i + ρ)ω, x ∈ (l1, l2), ϕt −dϕxx = aϕ + λϕ, (i + ρ)ω < t ≤ (i + 1)ω, x ∈ (l1, l2), ϕ > 0, (i + ρ)ω < t ≤ (i + 1)ω, x ∈ (l1, l2), ϕ(t, l1) = ϕ(t, l2) = 0, t ≥ 0, ϕ(t,x) = ϕ(t + ω,x), x ∈ (l1, l2). one can calculate exactly that λl1 = (1 − ρ)( π2d (l2−l1)2 − a) + ρδ. by the consequence of [27, theorem 2.3.4], peng and zhao [18] has showed that, the solution of (1.5) converges to zero among all nonnegative initial value if λl1 ≥ 0, while when λl1 < 0, it converges to the unique positive ω-periodic solution of (1.5) for all nonnegative and not identically zero initial value. specially, we can observe that (i) if (1 − ρ)a − ρδ > 0, then the solution of (1.5) converges to the unique positive ω-periodic solution of (1.5) for all nonnegative and not identically zero initial value; (ii) if (1−ρ)a−ρδ < 0, then there exists a critical value l̂ such that the solution of (1.5) converges to the unique positive ω-periodic solution of (1.5) for all nonnegative and not identically zero initial value if and only if l2 − l1 > l̂. the dynamics of the time-periodic nonlocal dispersal logistic equation have been studied by many authors (see [19, 24, 21, 23]). in [19], rawal and shen studied the eigenvalue problems of time-periodic nonlocal dispersal operator, and then showed that the existence of positive periodic solution relies on the sign of principal eigenvalue of a linearized eigenvalue problem. sun et al. [24] considered a timeperiodic nonlocal dispersal logistic equation in spatial degenerate environment. shen and vo [21] and su et al. [23] have studied the asymptotic profiles of the generalised principal eigenvalue of time-periodic nonlocal dispersal operators under dirichlet type boundary conditions and neumann type boundary conditions, respectively. the models considered in the above mentioned work are all smooth periodic systems. the purpose of current paper is to study the dynamical properties of nonlocal dispersal model (1.1). clearly, system (1.1) is in time-periodic environment and the dispersal term and reaction term are both discontinuous and periodic in t caused by the seasonal succession. note that, by general semigroup theory (see [17]), (1.1) has a unique local solution u(t, ·; u0) with initial value u(0, ·; u0) = u0 ∈ c([l1, l2]), which is continuous in t. if u0 is nonnegative over [l1, l2], then by a comparison argument, u(t, ·; u0) exists and is nonnegative for all t > 0 (see lemma 2.2). next, we have the following theorem on the long time behavior of model (1.1). theorem 1.1. assume that (j) holds and −∞ < l1 < l2 < +∞. let u(t, ·; u0) be the unique solution to (1.1) with the initial value u0(x) ∈ c([l1, l2]), where u0(x) is nonnegative and not identically zero. then the following statements are true: 4 z. li and b. dai (1) if (1−ρ)a−ρδ > (1−ρ)d, then lim n→∞ u(t + nω,x; u0) = u ∗ (l1,l2) (t,x) in c([0,ω]× [l1, l2]), where u∗ (l1,l2) (t,x) is the unique ω-periodic positive solution of  ut = −δu, iω < t ≤ (i + ρ)ω, x ∈ [l1, l2], ut = d ∫ l2 l1 j(x−y)u(t,y)dy −du(t,x) + u(a− bu), (i + ρ)ω < t ≤ (i + 1)ω, x ∈ [l1, l2], u(t,x) = u(t + ω,x), t ≥ 0, x ∈ [l1, l2]; (1.6) (2) if 0 < (1−ρ)a−ρδ ≤ (1−ρ)d, then there exists a unique `∗ > 0 such that lim n→∞ u(t+nω,x; u0) = u∗ (l1,l2) (t,x) in c([0,ω] × [l1, l2]) if and only if l2 − l1 > `∗; (3) if (1−ρ)a−ρδ ≤ 0, then 0 is the unique nonnegative solution of (1.6), and lim t→∞ u(t,x; u0) = 0 uniformly for x ∈ [l1, l2]. theorem 1.1 shows a complete classification on all possible long time behavior of system (1.1) with the assumption (j). the criteria governing persistence and extinction of the species show that: (i) when the duration of the bad season is too long (namely, ρ is close to 1), or the season is too bad (for example, bad weather and food shortages contributes to the large death rate δ) such that (1−ρ)a−ρδ ≤ 0, then the species will die out eventually regardless the initial population size; (ii) if the bad season is not long, or the food resource a is not small such that ρδ < (1−ρ)a ≤ (1−ρ)d + ρδ, then both persistence and extinction are determined by the range of the habitat of the species; (iii) when the good season is very long (i.e., ρ is close to 0), or the species has enough food such that (1 −ρ)(a−d) −ρδ > 0, then the species can persist for long time, which is different from that for the local diffusion model (1.5). the following conclusion concerns the asymptotic profile of the ω-periodic positive solution u∗ (l1,l2) of (1.6). theorem 1.2. assume that (j) holds. if (1 − ρ)a − ρδ > 0, then there exists ˆ̀ > 0 such that λ1(−l(l1,l2)) < 0 for every interval (l1, l2) with l2 − l1 > ˆ̀ and hence (1.6) admits a unique positive ω-periodic solution u∗ (l1,l2) (t,x). moreover, lim −l1,l2→+∞ u∗(l1,l2)(t,x) = z ∗(t) in cloc([0,ω] ×r), where z∗(t) is the unique ω-periodic positive solution of the following equation  zt = −δz, iω < t ≤ (i + ρ)ω, zt = z(a− bz), (i + ρ)ω < t ≤ (i + 1)ω, z(t + ω) = z(t), t ≥ 0. (1.7) the rest part of this paper is organized as follows. sections 2 are devoted to the global existence and uniqueness of solution of (1.1). in section 3, we then study the long-time dynamical behavior of system (1.1) based on the results for the time-periodic eigenvalue problem and time periodic upperlower solutions. we also show some numerical simulations and discussion in section 4 and the final section, respectively. 2. well-posedness in this section, we show the existence and uniqueness of the global solution of (1.1). before the statement of well-posedness of solution to (1.1), we provide a maximum principle. a nonlocal dispersal logistic model with seasonal succession 5 lemma 2.1 (maximum principle). let m be a positive integer. assume that (j) holds and −∞ < l1 < l2 < +∞. suppose that v,vt ∈ c([0,mω] × [l1, l2]),c ∈ l∞([0,mω] × [l1, l2]) and  vt ≥−δv, iω < t ≤ (i + ρ)ω, l1 ≤ x ≤ l2, vt ≥ d ∫ l2 l1 j(x−y)v(t,y)dy −dv(t,x) + c(t,x)v, (i + ρ)ω < t ≤ (i + 1)ω, l1 ≤ x ≤ l2, v(0,x) ≥ 0, x ∈ [l1, l2], (2.1) where i = 0, 1, · · · ,m − 1. then v(t,x) ≥ 0 for (t,x) ∈ [0,mω] × [l1, l2]. moreover, if v(0,x) 6≡ 0 in [l1, l2], then v(t,x) > 0 for (t,x) ∈ (ρω,mω] × (l1, l2); if v(0,x) > 0 in (l1, l2), then v(t,x) > 0 for (t,x) ∈ (0,mω] × (l1, l2). proof. let v (t,x) = ektv(t,x). then v (0, ·) ≥ 0 and v (t,x) satisfies  vt ≥ p0v (t,x), iω < t ≤ (i + ρ)ω, x ∈ [l1, l2], vt ≥ d ∫ l2 l1 j(x−y)v (t,y)dy + p1(t,x)v (t,x), (i + ρ)ω < t ≤ (i + 1)ω, x ∈ [l1, l2], (2.2) where p0 = k − δ, p1(t,x) = k + c(t,x) −d. due to the boundedness of c, there exists k > 0 such that p0 > 0 and inf t∈[0,mω],x∈[l1,l2] p1(t,x) > 0. we now claim that v (t,x) ≥ 0 in [0,mω] × [l1, l2]. let p1,0 = supt∈[0,mω],t∈[l1,l2] p1(t,x) and t0 = min { mω, 1 2(p0+d+p1,0) } . in the following, we will show that the claim holds for t ∈ (0,t0],x ∈ [l1, l2]. assume to the contrary that vinf := inft∈(0,t0),x∈[l1,l2] v (t,x) < 0. then there exists (t0,x0) ∈ (0,t0] × [l1, l2] such that vinf = v (t0,x0) < 0. notice that there are tn ∈ (0, t0] and xn ∈ [l1, l2] such that v (tn,xn) → vinf as n →∞. we only need to consider the following two cases. case 1. t0 ∈ (i0ω, (i0 + ρ)ω] for some i0 ∈{0, 1, · · · ,m− 1}. in this case, tn ∈ (i0ω,t0] for large n. then it follows from (2.2) that v (tn,xn) −v (0,xn) = i0−1∑ i=0 (∫ (i+ρ)ω iω vtdt + ∫ (i+1)ω (i+ρ)ω vtdt ) + ∫ tn i0ω vtdt ≥ i0−1∑ i=0 ∫ (i+ρ)ω iω p0v (t,xn)dt + ∫ tn i0ω p0v (t,xn)dt + i0−1∑ i=0 ∫ (i+1)ω (i+ρ)ω [ d ∫ l2 l1 j(xn −y)v (t,y)dy + p1(t,xn)v (t,xn) ] dt ≥ ∫ tn 0 p0vinf dt + d ∫ tn 0 ∫ l2 l1 j(xn −y)vinf dydt + ∫ tn 0 p1,0vinf dt ≥ tn(p0 + d + p1,0)vinf ≥ t0(p0 + d + p1,0)vinf for large n. recall that v (0,xn) ≥ 0 for n = 0, 1, 2, · · · . thus we have v (tn,xn) ≥ t0(p0 + d + p1,0)vinf 6 z. li and b. dai for large n. taking the limit as n →∞, it holds that vinf ≥ t0(p0 + d + p1,0)vinf ≥ 1 2 vinf, which is a contradiction. case 2. t0 ∈ ((i0 + ρ)ω, (i0 + 1)ω] for some i0 ∈{0, 1, · · · ,m− 1}. similarly, we can also derive a contradiction since v (tn,xn) −v (0,xn) = i0−1∑ i=0 (∫ (i+ρ)ω iω vtdt + ∫ (i+1)ω (i+ρ)ω vtdt ) + ∫ (i0+ρ)ω i0ω vtdt + ∫ tn (i0+ρ)ω vtdt ≥ tn(p0 + d + p1,0)vinf ≥ t0(p0 + d + p1,0)vinf for large n. therefore, v (t,x) ≥ 0 for (t,x) ∈ (0,t0] × [l1, l2] and then v(t, ·) ≥ 0 for t ∈ [0,t0]. if t0 = mω, then v(t,x) ≥ 0 in [0,mω]× [l1, l2] follows directly; while if t0 < mω, we can repeat the above process by replacing v (0, ·) and (0,t0] as v (t0, ·) and (t0,mω]. obviously, this process can be repeated in finite many times, and consequently, v(t, ·) ≥ 0 for t ∈ [0,mω]. now we assume that v(0,x) 6≡ 0 in [l1, l2]. to finish the proof, it suffices to prove that v > 0 in (ρω,ω] × (l1, l2). suppose that there exists a point (t∗,x∗) ∈ (ρω,ω] × (l1, l2) such that v (t∗,x∗) = 0. first, we prove that v (t∗,x) = 0 for x ∈ (l1, l2). otherwise, we can find x̃ ∈ [l1, l2] ∩∂{x ∈ (l1, l2) : v (t∗,x) > 0}. then v (t∗, x̃) = 0 and it follows from (2.2) that 0 ≥ vt(t∗, x̃) ≥ d ∫ l2 l1 j(x̃−y)v (t∗,y)dy > 0, by assumption (j). this is impossible, and hence v (t∗,x) = 0 for x ∈ (l1, l2). thus, we can derive from (2.2) that for x ∈ [l1, l2] −v (0,x) = v (t∗,x) −v (0,x) = ∫ ρω 0 vtdt + ∫ t∗ ρω vtdt ≥ p0 ∫ ρω 0 v (t,x)dt + d ∫ t∗ ρω ∫ l2 l1 j(x−y)v (t,y)dydt + ∫ t∗ ρω p1(t,x)v (t,x)dt ≥ 0. this means that v(0,x) ≡ 0 in [l1, l2], which is a contradiction. � lemma 2.2 (existence and uniqueness). assume that (j) holds and −∞ < l1 < l2 < +∞. then for any nonnegative and bounded initial value u0(x) ∈ c([l1, l2]), problem (1.1) admits a unique global solution u ∈ c1,0((iω, (i + ρ)ω] × [l1, l2]) ∩ c1,0(((i + ρ)ω, (i + 1)ω] × [l1, l2]) for i ∈ z+. moreover, u(t,x) > 0 for t > 0 and x ∈ (l1, l2), if u0(x) > 0 in (l1, l2). proof. at first, we set û = e−δtu0(x) for t ∈ [0,ρω]. then û ∈ c1,0((0,ρω] × [l1, l2]) satisfies{ ût = −δû, 0 < t ≤ ρω, l1 ≤ x ≤ l2, û(0,x) = u0(x), l1 ≤ x ≤ l2. a nonlocal dispersal logistic model with seasonal succession 7 consider the following problem  ut = d ∫ l2 l1 j(x−y)u(t,y)dy −du(t,x) + u(a− bu), ρω < t ≤ ω, x ∈ (l1, l2), u(ρω,x) = e−δρωu0(x), x ∈ [l1, l2]. (2.3) then one can apply the banach’s fixed theorem and comparison argument (see [1]) to conclude that (2.3) has a unique solution ū(t,x) ∈ c1,0((ρω,ω] × [l1, l2]). moreover, by the maximum principle and comparison argument, we have that 0 < ū(t,x) ≤ max { a b , max −h0≤x≤h0 u0(x) } for t ∈ (ρω,ω],x ∈ (l1, l2). define u(t,x) = { û(t,x) in [0,ρω] × [l1, l2], ū(t,x) in [ρω,ω] × [l1, l2]. we have that u ∈ û ∈ c1,0((0,ρω] × [l1, l2]) ∩c1,0((ρω,ω] × [l1, l2]). based on the above obtained function u, we let u1(t,x) = e −δ(t−ω)u(ω,x) for ω ≤ t ≤ (1 + ρ)ω. then u1 ∈ c1,0((ω, (1 + ρ)ω] × [l1, l2]) satisfies{ u1,t = −δu1, ω < t ≤ (1 + ρ)ω, l1 ≤ x ≤ l2, u1(ω,x) = u(ω,x), l1 ≤ x ≤ l2. likewise, the nonlocal dispersal problem  ut = d ∫ l2 l1 j(x−y)u(t,y)dy −du(t,x) + u(a− bu), (1 + ρ)ω < t ≤ 2ω, x ∈ (l1, l2), u((1 + ρ)ω,x) = e−δρωu(ω,x), x ∈ [l1, l2] has a unique solution ū1 ∈ c1,0(((1 + ρ)ω, 2ω] × [l1, l2]), in which 0 < ū1(t,x) ≤ max { a b , max −h0≤x≤h0 u0(x) } for t ∈ ((1 + ρ)ω, 2ω],x ∈ (l1, l2). define u(t,x) =   u(t,x) in [0,ω] × [l1, l2], u1(t,x) in [ω, (1 + ρ)ω] × [l1, l2], ū1(t,x) in [(1 + ρ)ω, 2ω] × [l1, l2]. then it holds that u ∈ c1,0((iω, (i + ρ)ω] × [l1, l2]) ∩c1,0(((i + ρ)ω, (i + 1)ω] × [l1, l2]) for i = 0, 1. by repeating the above procedure, we therefore obtain the existence and uniqueness of the solution (u,g,h) of (1.1). � 3. global dynamics in this subsection, we first establish the periodic upper-lower solutions method for model (1.1). using this method, we can consider the long time behavior of model (1.1). 8 z. li and b. dai 3.1. the method of periodic upper-lower solutions. following hess [11], we can define the upperlower solutions of (1.6) as follows. definition 3.1. a bounded and continuous function ũ(t,x) is called an upper-solution of (1.6) if ũ(t,x) ∈ c1,0((iω, (i + ρ)ω] × [l1, l2]) ∩c1,0(((i + ρ)ω, (i + 1)ω] × [l1, l2]) satisfies  ũt ≥−δũ, iω < t ≤ (i + ρ)ω, l1 ≤ x ≤ l2, ũt ≥ d ∫ l2 l1 j(x−y)ũ(t,y)dy −dũ(t,x) + ũ(a− bũ), (i + ρ)ω < t ≤ (i + 1)ω, l1 ≤ x ≤ l2, ũ(0,x) ≥ ũ(ω,x), x ∈ [l1, l2]. (3.1) for i ∈ z+. meanwhile, the function û(t,x) ∈ c1,0((iω, (i+ρ)ω]×[l1, l2])∩c1,0(((i+ρ)ω, (i+1)ω]×[l1, l2]) is called a lower-solution of (1.6) if the inequalities in (3.1) are reversed. similarly, we can define the upper-solution (resp. lower-solution) of (1.1) by replacing the inequality ũ(0,x) ≥ ũ(ω,x) in (3.1) as ũ(0,x) ≥ ũ0(x) (resp. û(0,x) ≤ û0(x)). we say that a pair of upper-lower solution ũ and û are ordered if ũ(t,x) ≥ û(t,x) in [0, +∞) × [l1, l2]. using the semigroup theory, we have the following result. lemma 3.1. let d(t), ā(t) and b̄(t) be defined as in (1.2). assume that u(t,x) is bounded for (t,x) ∈ [0, +∞] × [l1, l2]. then u(t,x) is a solution of (1.1) if and only if u(t,x) = u(0,x) + ∫ t 0 [ d(s) (∫ l2 l1 j(x−y)u(s,y)dy −u(s,x) ) + u(s,x)[ā(s) − b̄(s)u(s,x)] ] ds, t > 0, x ∈ [l1, l2]. (3.2) proof. it follows from the semigroup method [17], we have that u(t,x) = e−tu(0,x) + ∫ t 0 e−(t−s) [ d(s) (∫ l2 l1 j(x−y)u(s,y)dy −u(s,x) ) + u(s,x) + u(s,x)[ā(s) − b̄(s)u(s,x)] ] ds, (3.3) which implies ∫ t 0 u(s,x)ds = (1 −e−t)u(0,x) + i1[u](t,x), (3.4) where i1[u](t, x) = ∫ t 0 ∫ s 0 e −(s−z) [ d(z) (∫ l2 l1 j(x − y)u(z, y)dy − u(z, x) ) + u(z, x)[1 + ā(z) − b̄(z)u(z, x)] ] dzds = ∫ t 0 ∫ t z e −(s−z) [ d(z) (∫ l2 l1 j(x − y)u(z, y)dy − u(z, x) ) + u(z, x)[1 + ā(z) − b̄(z)u(z, x)] ] dsdz + ∫ t 0 (1 − es−t) [ d(s) (∫ l2 l1 j(x − y)u(s, y)dy − u(s, x) ) + u(s, x)[1 + ā(s) − b̄(s)u(s, x)] ] ds. therefore, (3.2) can be derived from (3.3) and (3.4). on the other hand, if u satisfies (3.2), then we can also show that (3.3) holds. � similarly, we have the following result for (1.6). a nonlocal dispersal logistic model with seasonal succession 9 lemma 3.2. assume that u(t,x) is bounded for (t,x) ∈ [0, +∞]× [l1, l2]. then u(t,x) is a solution of (1.6) if and only if  u(t,x) = u(0,x) + ∫ t 0 [ d(s) (∫ l2 l1 j(x−y)u(s,y)dy −u(s,x) ) + u(s,x)[ā(s) − b̄(s)u(s,x)] ] ds, t > 0, x ∈ [l1, l2], u(t,x) = u(t + ω,x), t ≥ 0, x ∈ [l1, l2]. (3.5) corollary 3.3. assume that u(t,x) is bounded for (t,x) ∈ [0, +∞] × [l1, l2]. then u(t,x) is a solution of (1.6) if and only if  u(t,x) = e−ctu(0,x) + ∫ t 0 e−c(t−s) [ d(s) (∫ l2 l1 j(x−y)u(s,y)dy −u(s,x) ) + u(s,x)[c + ā(s) − b̄(s)u(s,x)] ] ds, t > 0, x ∈ [l1, l2], u(t,x) = u(t + ω,x), t ≥ 0, x ∈ [l1, l2], (3.6) where c is a constant. the following conclusion establishes a method of periodic upper-lower solutions. theorem 3.4. assume that (j) holds and u0(x) ∈ c([l1, l2]) is bounded. let u(t,x) be the unique solution to (1.1) and ũ(t,x), û(t,x) be a pair of ordered and bounded upper-lower solutions to (1.6) satisfying û(0,x) ≤ u0(x) ≤ ũ(0,x) on [l1, l2]. then the time periodic problem (1.6) admits a minimal solution u ∈ c1,0((iω, (i + ρ)ω] × [l1, l2]) ∩ c1,0(((i + ρ)ω, (i + 1)ω] × [l1, l2]) and a maximal solution u ∈ c1,0((iω, (i + ρ)ω] × [l1, l2]) ∩c1,0(((i + ρ)ω, (i + 1)ω] × [l1, l2]) (i ∈ z+) satisfying û(t,x) ≤ u(t,x) ≤ lim n→∞ u(t + nω,x) ≤ lim n→∞ u(t + nω,x) ≤ u(t,x) ≤ ũ(t,x) for t ≥ 0,x ∈ [l1, l2]. proof. for notational convenience, denote q = (0, +∞) × [l1, l2] and j ∗u(t,x) = ∫ l2 l1 j(x−y)u(t,y)dy for u ∈ c(q). set i = [ −‖û‖l∞(q) −‖ũ‖l∞(q),‖û‖l∞(q) + ‖ũ‖l∞(q) ] . at first we take k > 1 such that for u ∈ i, u[ā(t)−b̄u] +ku and ā(t)u−2(‖û‖l∞(q) +‖ũ‖l∞(q))b̄(t)u+ ku are both increasing with respect to u. define l[u](t,x) = u(t,x) −e−ktu(0,x) − ∫ t 0 e−k(t−s)d(s) [j ∗u(s,x) −u(s,x)] ds and f(t,u) = u(t,x)[ā(t) − b̄(t)u(t,x)] + ku(t,x). we construct two iterations sequences by the following linear nonlocal evolution equations{ l[un](t,x) = ∫ t 0 e−k(t−s)f(s,un−1(s,x))ds, (t,x) ∈ q, un(0,x) = un−1(ω,x), x ∈ [l1, l2] (3.7) and { l[un](t,x) = ∫ t 0 e−k(t−s)f(s,un−1(s,x))ds, (t,x) ∈ q, un(0,x) = un−1(ω,x), x ∈ [l1, l2], (3.8) 10 z. li and b. dai where n ≥ 1,u0(t,x) = ũ(t,x) and u0(t,x) = û(t,x). we can check that a sufficiently large constant is an upper-solution of (3.7) (resp. (3.8)) since k > 1. then an application of banach’s fixed point theorem and comparison principle yields that the linear initial value problem (3.7) (resp. (3.8)) has a unique bounded global solution un(t,x) (resp. un(t,x)) for any n ≥ 1. we complete the proof of this theorem by the following four steps. step 1. the sequences {un}∞n=1 and {un}∞n=1 satisfy û(t,x) ≤ un(t,x) ≤ un+1(t,x) ≤ u(t + nω,x) ≤ un+1(t,x) ≤ un(t,x) ≤ ũ(t,x), (t,x) ∈ q, (3.9) for n ≥ 1. since ũ(t,x) is a bounded upper-solution of (1.6) and ũ(0,x) ≥ u0(x), we see that ũ(t,x) is also a bounded upper-solution of (1.1). then by lemma 2.1, we have ũ(t,x) ≥ u(t,x) in q. it follows from (3.7) and corollary 3.3 that u1(t,x) satisfies  u1t (t,x) = d(t) [ j ∗u1(t,x) −u1(t,x) ] + ũ(t,x)[ā(t) − b̄(t)ũ(t,x)] + k[ũ(t,x) −u1(t,x)], (t,x) ∈ q, u1(0,x) = ũ(ω,x), x ∈ [l1, l2]. (3.10) set w1(t,x) = u0(t,x) −u1(t,x) = ũ(t,x) −u1(t,x). since{ ũt(t,x) ≥ d(t) [j ∗ ũ(t,x) − ũ(t,x)] + ũ(t,x)[ā(t) − b̄(t)ũ(t,x)], (t,x) ∈ q, ũ(0,x) ≥ ũ(ω,x), x ∈ [l1, l2], there holds that { w1t (t,x) ≥ d(t) [ j ∗w1(t,x) −w1(t,x) ] −kw1(t,x), (t,x) ∈ q, w1(0,x) ≥ 0, x ∈ [l1, l2], which together with lemma 2.1 implies that w1(t,x) ≥ 0 and so u1(t,x) ≤ u0(t,x) = ũ(t,x) for (t,x) ∈ q. by a similar manner for lower-solution û(t,x), we have u(t,x) ≥ û(t,x) and u1(t,x) ≥ u0(t,x) = û(t,x) for (t,x) ∈ q. now we let w2(t,x) = u1(t,x) − u1(t,x). in view of û(t,x) ≤ u(t,x) ≤ ũ(t,x) in q, by (3.7) and (3.8), we have w2(0,x) = ũ(ω,x) − û(ω,x) ≥ 0 and w2t (t,x) = d(t) [ j ∗w2(t,x) −w2(t,x) ] −kw2(t,x) + ā(t)[ũ(t,x) − û(t,x)] − b̄(t)[ũ2(t,x) − û2(t,x)] + k[ũ(t,x) − û(t,x)] ≥ d(t) [ j ∗w2(t,x) −w2(t,x) ] −kw2(t,x) in q, where the conditions satisfied by k are used here. it follows from lemma 2.1 that w2(t,x) ≥ 0 and hence u1(t,x) ≥ u1(t,x) in q. next, we show that u1(t,x) ≤ u(t + ω) ≤ u1(t,x) in q. let w3(t,x) = u1(t,x) −u(t + ω,x). notice that u(t + ω,x) satisfies ut(t + ω,x) = d(t + ω) [j ∗ ũ(t + ω,x) − ũ(t + ω,x)] + ũ(t + ω,x)[ā(t + ω) − b̄(t + ω)ũ(t + ω,x)] = d(t) [j ∗ ũ(t + ω,x) − ũ(t + ω,x)] + ũ(t + ω,x)[ā(t) − b̄(t)ũ(t + ω,x)] in q. (3.11) a nonlocal dispersal logistic model with seasonal succession 11 combining (3.10) and (3.11), there holds that w3(0,x) = u1(0,x) −u(ω,x) = ũ(ω,x) −u(ω,x) ≥ 0 for x ∈ [l1, l2] and w3t (t,x) = d(t) [ j ∗w3(t,x) −w3(t,x) ] + ũ(t,x) [ ā(t) − b̄(t)ũ(t,x) ] + k [ ũ(t,x) −u1(t,x) ] −u(t + ω) [ ā(t) − b̄(t)u(t + ω,x) ] = d(t) [ j ∗w3(t,x) −w3(t,x) ] + [ ā(t) − b̄(t) ( u1(t,x) + u(t + ω,x) )] w3(t,x) + ā(t) [ ũ(t,x) −u1(t,x) ] − b̄(t) [ ũ(t,x) + u1(t,x) ][ ũ(t,x) −u1(t,x) ] + k [ ũ(t,x) −u1(t,x) ] . since ũ(t,x) ≥ u1(t,x) ≥ u1(t,x) ≥ û(t,x), by the condition satisfied by k, we see that ā(t) [ ũ(t,x) −u1(t,x) ] − b̄(t) [ ũ(t,x) + u1(t,x) ][ ũ(t,x) −u1(t,x) ] + k [ ũ(t,x) −u1(t,x) ] ≥ 0 in q, which leads to that w3t (t,x) ≥ d(t) [ j ∗w3(t,x) −w3(t,x) ] + [ ā(t) − b̄(t) ( u1(t,x) + u(t + ω,x) )] w3(t,x) in q. due to the boundedness of u1(t,x) and u(t + ω,x), we can derive from lemma 2.1 that w3(t,x) ≥ 0 and then u1(t,x) ≥ u(t + ω,x) in q. similarly, we also have u1(t,x) ≤ u(t + ω,x) in q. therefore, the following inequalities are true: û(t,x) ≤ u1(t,x) ≤ u(t + ω,x) ≤ u1(t,x) ≤ ũ(t,x), (t,x) ∈ q. an induction argument implies the monotone property (3.9) immediately. since {un} and {un} monotonically bounded sequences, there exist two bounded function u(t,x) and u(t,x) such that lim n→∞ un(t,x) = u(t,x) and lim n→∞ un(t,x) = u(t,x) and u(t,x) ≤ lim n→∞ u(t + nω,x) ≤ lim n→∞ u(t + nω,x) ≤ u(t,x) for each (t,x) ∈ q. thus from the dominated convergence theorem, we obtain that u(t,x) and u(t,x) are bounded solutions of the initial value problem{ ut(t,x) = d(t) [j ∗u(t,x) −u(t,x)] + u(t,x)[ā(t) − b̄(t)u(t,x)], (t,x) ∈ q, u(0,x) = u(ω,x), x ∈ [l1, l2]. step 2. we prove that u(t,x),u(t,x) ∈ c1,0((iω, (i+ρ)ω]×[l1, l2])∩c1,0(((i+ρ)ω, (i+ 1)ω]×[l1, l2]) for all i ∈ z+. since u(t,x) = u(0,x) + ∫ t 0 [ d(s) [ j ∗u(s,x) −u(s,x) ] + u(s,x) [ ā(s) − b̄(s)u(s,x) ]] ds, it holds that u(t + ε,x) −u(t,x) = ∫ t+ε t [ d(s) [ j ∗u(s,x) −u(s,x) ] + u(s,x) [ ā(s) − b̄(s)u(s,x) ]] ds for each fixed (t,x) ∈ q, where |ε| > 0 is sufficiently small. then, we have |u(t + ε,x) −u(t,x)| ≤ ∫ t+ε t ∣∣∣[d(s)[j ∗u(s,x) −u(s,x)] + u(s,x)[ā(s) − b̄(s)u(s,x)]]∣∣∣ ds ≤ c|ε|, where c > 0 is a constant independent of ε. this means that u(t,x) is continuous in t ∈ [0, +∞). the continuity of u(t,x) in x ∈ [l1, l2] follows from the argument in [1]. 12 z. li and b. dai for any t0 ∈ (0, +∞), there must exist a unique i0 ∈ z+ such that either t0 ∈ (i0ω, (i0 + ρ)ω], or t0 ∈ ((i0 + ρ)ω, (i0 + 1)ω]. when t0, t0 + ε ∈ (i0ω, (i0 + ρ)ω], we see that lim ε→0 u(t + ε,x) −u(t,x) ε = lim ε→0 1 ε ∫ t+ε t [ − δu(s,x) ] ds = −δ lim ε→0 u(t + θε,x) (0 < θ < 1) = −δu(t,x). when t0, t0 + ε ∈ ((i0 + ρ)ω, (i0 + 1)ω], we have lim ε→0 u(t + ε,x) −u(t,x) ε = lim ε→0 1 ε ∫ t+ε t [ d [ j ∗u(s,x) −u(s,x) ] + u(s,x) [ a− bu(s,x) ]] ds = lim ε→0 d [ j ∗u(t + θε,x) −u(t + θε,x) ] + u(t + θε,x) [ a− bu(t + θε,x) ] (0 < θ < 1) = d [ j ∗u(t,x) −u(t,x) ] + u(t,x) [ a− bu(t,x) ] . hence, u(t,x) ∈ c1,0((iω, (i + ρ)ω] × [l1, l2]) ∩c1,0(((i + ρ)ω, (i + 1)ω] × [l1, l2]) for all i ≥ 0 due to the arbitrariness of t0. the proof for u(t,x) is similar. step 3. we prove that u(t,x) = u(t + ω,x) and u(t,x) = u(t + ω,x) for all t ≥ 0. let v(t,x) = u(t + ω,x) −u(t,x). note that d(t), ā(t) and b̄(t) are all ω−periodic in t. then vt(t,x) = d(t) [∫ ω j(x−y)v(t,y)dy −v(t,x) ] + ā(t)v(t,x) − b̄(t)[u(t + ω,x) + u(t,x)]v(t,x). (3.12) since v(0,x) = u(ω,x) −u(0,x) = 0, the uniqueness of solution of initial value problem (3.12) implies that v(t,x) ≡ 0 in q, equivalently, u(t,x) is ω-periodic in t. similarly, we can also prove that u(t,x) = u(t + ω,x), and omit the details here. step 4. we show the maximality of u(t,x) and minimality of u(t,x). notice that every ω-periodic solution u∗(t,x) of (1.6) satisfies û(t,x) ≤ u∗(t,x) ≤ ũ(t,x). meanwhile, u∗(t,x) is a lower-solution as well as a upper-solution of (1.6). by choosing ũ and u∗ as a pair of upper-lower solutions to (1.6), there holds that u∗(t,x) ≤ un(t,x) ≤ ũ(t,x) and hence u∗(t,x) ≤ u(t,x) ≤ ũ(t,x). on the other hand, if we take u∗ and û as a pair of upper-lower solutions to (1.6), then û(t,x) ≤ u(t,x) ≤ u∗(t,x). � 3.2. proofs of theorems 1.1 and 1.2. in this subsection, we complete the proof of theorems 1.1 and 1.2. linearizing model (1.1) at zero, we obtain the time-periodic eigenvalue problem{ vt + δv = λv, iω < t ≤ (i + ρ)ω, l1 ≤ x ≤ l2, vt −d(j ∗v −v)(t,x) + av = λv, (i + ρ)ω < t ≤ (i + 1)ω, l1 ≤ x ≤ l2. (3.13) it is well known (see, e.g., [3, 5, 6]) that the time independent eigenvalue equation d(j ∗φ−φ)(t,x) + aφ(x) = −σφ(x), x ∈ [l1, l2]. (3.14) admits a principal eigenvalue σ1, which satisfies σ1 < d−a. moreover, from [4, proposition 3.4], we see that proposition 3.5. assume that (j) holds and −∞ < l1 < l2 < +∞. then the following hold true: (1) σ1 is strictly decreasing and continuous in ` := l2 − l1; (2) liml2−l1→+∞σ1 = −a; (3) liml2−l1→0+ σ1 = d−a. a nonlocal dispersal logistic model with seasonal succession 13 let φ1(x) be the positive eigenfunction of (3.14) associated with σ1. by defining σ(t) = { δ, t ∈ (iω, (i + ρ)ω], σ1, t ∈ ((i + ρ)ω, (i + 1)ω], we see that d(t) [j ∗φ1(x) −φ1(x)] + ā(t)φ1(x) = −σ(t)φ1(x), ∀t ∈ r,x ∈ ω. set ϕ(t,x) = exp [( (1 −ρ)σ1 + ρδ ) t− ∫ t 0 σ(s)ds ] φ1(x). (3.15) then there holds that  ϕt + δϕ = [(1 −ρ)σ1 + ρδ]ϕ, iω < t ≤ (i + ρ)ω, l1 ≤ x ≤ l2, ϕt −d(j ∗ϕ−ϕ)(t,x) + aϕ = [(1 −ρ)σ1 + ρδ]ϕ, (i + ρ)ω < t ≤ (i + 1)ω, l1 ≤ x ≤ l2, ϕ(t,x) = ϕ(t + ω,x), t ≥ 0, l1 ≤ x ≤ l2. this means that (1 −ρ)σ1 + ρδ is a eigenvalue of (3.13) with the positive eigenfunction ϕ(t,x). in the proof of theorems 1.1, we will show that (1−ρ)σ1 + ρδ serves as a threshold which determines whether the species can persist. proof of theorem 1.1. let λ1 = (1 −ρ)σ1 + ρδ. we first consider two cases on the sign of λ1. case 1. suppose that λ1 < 0. one can easily check that a sufficiently large positive constant m is a upper-solution of (1.1) as well as the upper-solution of (1.6). following the comparison argument in theorem 3.4, we see that for any (t,x) ∈ [0,ω] × [l1, l2], u(t + nω,x; m) is non-increasing with respect to n. then the function u+(t,x) := lim n→∞ u(t + nω,x; m), (t,x) ∈ [0,ω] × [l1, l2] is well-defined and upper semi-continuous. on the other hand, let ϕ(t,x) be defined as in (3.15). for any 0 < ε � 1, by λ1 < 0, we see that εϕ is a lower-solution of (1.1) as well as a lower-solution of (1.6). again, by the comparison argument, u(t + nω,x; εϕ(0,x)) is non-decreasing as n increases for any (t,x) ∈ [0,ω] × [l1, l2]. thus, the function u−(t,x) := lim n→∞ u(t + nω,x; εϕ), (t,x) ∈ [0,ω] × [l1, l2] is well-defined and lower semi-continuous. obviously, u− ≤ u+. next, we show u−(t,x) ≡ u+(t,x). for this purpose, we define γn := inf { ln α : 1 α u(t + nω,x; m) ≤ u(t + nω,x; εϕ(0,x)) ≤ αu(t + nω,x; m), (t,x) ∈ [0,ω] × [l1, l2] } . since the sequence {u(· + nω, ·; εϕ(0, ·))}n is non-decreasing and {u(· + nω, ·; m)}n is non-increasing, u(· + nω, ·; εϕ(0, ·)) and u(· + nω, ·; m) will be closer to each other when n decreases. consequently, {γn}n is a non-increasing sequence, and then the limit γ∗ := limn→∞γn exists. if γ∗ > 0, then by the comparison argument, we can construct some α∗ > 1 and 0 < σ � 1 such that 1α∗u(· + nω, ·; m) ≤ u(· + nω, ·; εϕ(0, ·)) ≤ α∗u(· + nω, ·; m) and ln α∗ < γn − σ for sufficiently large n. this causes a contradiction with the definition of γ∗. hence, γ∗ = 0 and the equality u + = u− follows. notice that u+ is upper semi-continuous and u− is lower semi-continuous. then u∗ := u+ is continuous and inf [0,ω]×[l1,l2] u∗ > 0. using dini’s theorem, we have limn→∞u(· + nω, ·; m) = limn→∞u(· + nω, ·; εϕ(0, ·)) = u∗ uniformly for (t,x) ∈ [0,ω] × [l1, l2]. this also means that u(t + ω,x; u∗(0,x)) = lim n→∞ u(t + ω,x; u(nω, ·; m)) = lim n→∞ u(t + (n + 1)ω,x; m) = u∗(t,x). 14 z. li and b. dai this is, u(t,x; u∗(0,x)) is ω-periodic in t. the existence of time periodic positive solution of (1.6) is established. by the above contraction argument, we can obtain the existence of the solutions of (1.6). the uniqueness follows directly from theorem 3.4 and the above argument. to emphasize the dependence of u∗(t,x) on l1, l2, denote by u ∗ (l1,l2) (t,x) the unique time periodic positive solution of (1.6). since εϕ(t,x) ≤ u(t,x; u0) ≤ m, (t,x) ∈ [0,ω] × [l1, l2] for 0 < ε � 1 and m � 1, the above contraction argument also implies that lim n→∞ u(t + nω,x; u0) = u ∗ (l1,l2) (t,x) uniformly in c([0,ω] × [l1, l2]). the global stability of u∗(l1,l2) can also be inferred from theorem 3.4 and the uniqueness of the solutions of (1.6). case 2. suppose that λ1 ≥ 0. at first, we show the nonexistence of positive solution of (1.6). by way of contradiction, suppose that v∗ is a positive solution of (1.6). then we can choose � > 0 small enough such that �ϕ < v∗ in [0, +∞) × [l1, l2]. there holds that 0 ≤ λ1ϕ(t,x) = ∂tϕ(t,x) −d(t) [∫ l2 l1 j(x−y)ϕ(t,y)dy −ϕ(t,x) ] − ā(t)ϕ(t,x) ≤ ∂tϕ(t,x) −d(t) [∫ l2 l1 j(x−y)ϕ(t,y)dy −ϕ(t,x) ] − ā(t)ϕ(t,x) + b̄(t)ϕ2(t,x) in [0, +∞) × [l1, l2], which means ϕ is an upper-solution of (1.6). it follows from the comparison argument that v∗ ≤ ϕ in [0, +∞) × [l1, l2]. this is a contradiction. hence the equation (1.6) admits no positive solution. since λ1 ≥ 0, a simple calculation gives that for large e > 0, eϕ(t,x) and 0 are a pair of upper-lower solutions of (1.1) as well as a pair of upper-lower solutions of (1.6). it then follows from theorem 3.4 that the time periodic problem (1.6) admits a minimal solution u and a maximal solution u satisfying 0 ≤ u(t,x) ≤ lim n→∞ u(t + nω,x; u0) ≤ lim n→∞ u(t + nω,x; u0) ≤ u(t,x) ≤ eϕ(t,x) in [0, +∞) × [l1, l2]. the nonexistence of positive solution to (1.6) implies that u = u = 0. thus, the solution u(t,x; u0) of (1.1) converges to 0 point by point. since u,u ∈ c([0, +∞) × [l1, l2]) and the sequences constructed in (3.9) are monotone, we have lim t→∞ u(t,x; u0) = 0 uniformly for x ∈ [l1, l2] by dini’s theorem. from the above two cases, one can obtain that the sign of (1 −ρ)σ1 + ρδ can completely determine the long time behavior of the species. therefore, the conclusions of theorem 1.1 follows from the above argument and proposition 3.5. � we further discuss the behaviors of the positive ω-periodic solution to (1.6). look at the ode system (1.4). it is known from [14, theorem 2.1] that (1.4) admits a unique positive ω-periodic solution z∗ satisfying the equation (1.7), if and only if (1 −ρ)a−ρδ > 0, where z∗ ∈ c1((iω, (i + ρ)ω]) ∩c1(((i + ρ)ω, (i + 1)ω]) is bounded. moreover, if (1 −ρ)a−ρδ ≤ 0, then the solution z(t; z0) of (1.4) converges to 0 for all z0 ∈ [0, +∞) as t → +∞, while if (1−ρ)a−ρδ > 0, then limn→∞(z(t + nω; z0)−z∗(t)) = 0 in c([0,ω]) for all z0 ∈ (0, +∞). using this fact, we can prove theorem 1.2. proof of theorem 1.2. since (1 −ρ)a−ρδ > 0, by proposition 3.5, there holds that lim l2−l1→+∞ [ (1 −ρ)σ1 + ρδ ] = ρδ − (1 −ρ)a < 0, a nonlocal dispersal logistic model with seasonal succession 15 and thus there exists a large ˆ̀> 0 such that (1 −ρ)σ1 + ρδ < 0 as l2 − l1 > ˆ̀. the existence and uniqueness of u∗ (l1,l2) (t,x) follow from theorem 1.1. note that z∗(t) satisfies  zt = −δz, 0 < t ≤ ρω, zt = z(a− bz), ρω < t ≤ ω, z(0) = z(ω). then z∗(t) = e−δtz∗(0) for 0 ≤ t ≤ ρω. meanwhile, u∗ (l1,l2) (t,x) = e−δtu∗ (l1,l2) (0,x) for 0 ≤ t ≤ ρω and l1 ≤ x ≤ l2. we make an assertion that for each 0 < � � 1, there exists `� ≥ ˆ̀ > 0 such that for each l1 ∈ (−∞,−`�) and l2 ∈ (`�, +∞), z∗(t) − � ≤ u∗(l1,l2)(t,x) ≤ z ∗(t) + �, (t,x) ∈ [ρω,ω] × [l1, l2]. (3.16) only the proof for the lower bound will be given here since that for the upper bound is similar. clearly, min[0,ω] z ∗(t) > 0. in fact, if there is t0 > 0 such that z ∗(t0) = 0, then z ∗(t) = 0 for all t ≥ t0, which is impossible as z∗ is periodic in t. set 0 < � � 1. then there exists η(�) > 0 such that ẑ(t) := (1 −η)z∗(t) ≥ z∗(t) − � > 0, t ∈ [ρω,ω]. observe that ẑt(t) −d [∫ l2 l1 j(x−y)ẑ(t)dy − ẑ(t) ] − ẑ(t) [ a− bẑ(t) ] = (1 −η)z∗t (t) − (1 −η)z ∗(t)d [∫ l2 l1 j(x−y)dy − 1 ] − (1 −η)z∗(t) [ a− bz∗(t) ] − ẑ(t) [ a− bẑ(t) ] + (1 −η)z∗(t) [ a− bz∗(t) ] = −d(1 −η)z∗(t) [∫ l2 l1 j(x−y)dy − 1 ] − bη(1 −η)z∗2(t), ρω < t ≤ ω, l1 ≤ x ≤ l2. denote el(t,x) := d(1 −η)z∗(t) [∫ l −l j(x−y)dy − 1 ] , (t,x) ∈ (ρω,ω] ×r. since j ∈ c(r) ∩l∞(r) is nonnegative and lim l→+∞ ∫ l −l j(x)dx = 1, we know that el is non-decreasing with respect to l > 0 and continuous, bounded for all (l, t,x) ∈ (0, +∞) × (ρω,ω] × r. it then follows from dini’s theorem that el(t,x) converges to zero locally uniformly in (ρω,ω]×r as l → +∞. hence, there exists `� ≥ ˆ̀> 0 such that for each l1 ∈ (−∞,−`�), l2 ∈ (`�, +∞), the following inequality holds ẑ(t) −d [∫ l2 l1 j(x−y)ẑ(t)dy − ẑ(t) ] − ẑ(t) [ a− bẑ(t) ] < 0, (t,x) ∈ (ρω,ω] × [l1, l2]. (3.17) it suffices to prove that for each l1 ∈ (−∞,−`�), l2 ∈ (`�, +∞), we have ẑ(t) ≤ u∗(l1,l2)(t,x) in [ρω,ω] × [l1, l2]. to this end, we fix any l1 ∈ (−∞,−`�), l2 ∈ (`�, +∞) and set β∗ = inf { β > 0 : ẑ(t) ≤ βu∗(l1,l2)(t,x) for all (t,x) ∈ (ρω,ω) × [l1, l2] } . we see that β∗ is well-defined and positive since min [ρω,ω]×[l1,l2] u∗ (l1,l2) (t,x) > 0 and ẑ(t) is bounded. it follows from the continuity of u∗ (l1,l2) and ẑ that ẑ(t) ≤ β∗u∗(l1,l2)(t,x) for all (t,x) ∈ (ρω,ω)× [l1, l2]. in particular, there must exist (t0,x0) ∈ (ρω,ω) × [l1, l2] such that ẑ(t0) = β∗u∗(l1,l2)(t0,x0). 16 z. li and b. dai when β∗ ≤ 1, the lower bound in (3.16) holds immediately. on the contrary, suppose that β∗ > 1. let w(t,x) = ẑ(t) −β∗u∗(l1,l2)(t,x). then by (3.17) and the equation satisfied by u ∗ (l1,l2) (t,x), a simple calculation yields that wt(t,x) < d [∫ l2 l1 j(x−y)w(t,y)dy −w(t,x) ] + ẑ(t) [ a− bẑ(t) ] −β∗u∗(l1,l2)(t,x)[a− bu ∗ (l1,l2) (t,x)] for (t,x) ∈ (ρω,ω)×[l1, l2]. however, by the definition of β∗, we have that wt(t0,x0) = 0. this together with ∫ l2 l1 j(x0 −y)w(t0,y)dy −w(t0,x0) ≤ 0 leads to that 0 = wt(t0,x0) ≤ ẑ(t0) [ a− bẑ(t0) ] −β∗u∗(l1,l2)(t0,x0)[a− bu ∗ (l1,l2) (t0,x0)] < ẑ(t0) [ a− bẑ(t0) ] −β∗u∗(l1,l2)(t0,x0)[a− bβ∗u ∗ (l1,l2) (t0,x0)] = 0, which is a contradiction. consequently, β∗ ≤ 1 and so ẑ(t) ≤ u∗(l1,l2)(t,x) for all (t,x) ∈ (ρω,ω)×[l1, l2]. in fact, the domain (ρω,ω) × [l1, l2] can be extended to [ρω,ω] × [l1, l2] since ẑ(t) and u∗(l1,l2)(t,x) are both continuous and bounded. hence, (3.16) holds true and so lim−l1,l2→+∞u ∗ (l1,l2) (t,x) = z∗(t) in cloc([ρω,ω] ×r). on the other hand, when t ∈ [0,ρω], it holds that z∗(t) = e−δtz∗(0) = e−δtz∗(ω) and u∗ (l1,l2) (t,x) = e−δtu∗ (l1,l2) (0,x) = e−δtu∗ (l1,l2) (ω,x) for 0 ≤ t ≤ ρω and l1 ≤ x ≤ l2. this means that lim −l1,l2→+∞ u∗(l1,l2)(t,x) = z ∗(t) in cloc([0,ρω] ×r). as a result, lim −l1,l2→+∞ u∗(l1,l2)(t,x) = z ∗(t) in cloc([0,ω] ×r). the proof is completed. � 4. simulations in this section, we present the simulations to illustrate some of our results. referring to [26], we choose the form of j to be a simple laplace kernel: j(x) = 1 2d e− |x| d with d = 20. consider the following parameter sets: (p1) δ = 0.2,d = 0.6,a = 1.2,b = 0.6,ρ = 0.6,ω = 1; (p2) δ = 0.2,d = 1,a = 1.2,b = 0.6,ρ = 0.6,ω = 1; (p3) δ = 0.8,d = 0.6,a = 1.2,b = 0.6,ρ = 0.6,ω = 1; and the initial condition (ic) u0(x) = cos( πx l ),x ∈ (−l, l). clearly, the parameter set (p1) satisfies the condition in theorem 1.1 (1). then fig. 1 shows that when the domain length l := 2l = 0.4, the solution of (1.1) satisfying (p1) and (ic) converges to a spatially nonhomogeneous positive periodic solution. this is consistent with the conclusion of theorem 1.1 (1). the parameter set (p2) satisfies the condition in theorem 1.1 (2). then fig. 2 shows that when the domain length l = 2l = 0.4, the solution of (1.1) satisfying (p2) and (ic) converges to a spatially nonhomogeneous positive periodic solution, but when l = 2l = 8, the solution of (1.1) with the same parameters and initial condition converges to zero. this is consistent with the conclusion of theorem 1.1 (2). a nonlocal dispersal logistic model with seasonal succession 17 figure 1. numerical simulations of (1.1) with parameter set (p1) and initial condition (ic), where l = 0.2. figure 2. numerical simulations of (1.1) with parameter set (p2) and initial condition (ic). left: l = 0.2; right: l = 4. the parameter set (p3) satisfies the condition in theorem 1.1 (3). then fig. 3 shows that when the domain length l = 2l = 8, the solution of (1.1) satisfying (p3) and (ic) converges to zero. this is consistent with the conclusion of theorem 1.1 (3). 5. discussion in this paper, we mainly examine a nonlocal dispersal logistic model with seasonal succession subject to dirichlet type boundary condition. in section 3, in order to study the long time behavior of the solutions to (1.1), we establish a method of time periodic upper-lower solutions, and show that the sign of the eigenvalue (1 − ρ)σ1 + ρδ of the linearized operator can completely determine the asymptotic behavior of the solutions to (1.1). meanwhile, we see that the ω-periodic positive solution corresponding to the nonlocal dispersal model (1.1) behaves like the ω-periodic positive solution corresponding to the ode model (1.4) when the range of the habitat tends to the entire space r. 18 z. li and b. dai figure 3. numerical simulations of (1.1) with parameter set (p3) and initial condition (ic), where l = 4. in the following, we give some remarks on a nonlocal dispersal logistic model under neumann type boundary condition, which is associated with model (1.1), that is,  ut = −δu, iω < t ≤ (i + ρ)ω, l1 ≤ x ≤ l2, ut = d ∫ l2 l1 j(x−y) ( u(t,y) −u(t,x) ) dy + u(a− bu), (i + ρ)ω < t ≤ (i + 1)ω, l1 ≤ x ≤ l2, u(0,x) = u0(x) ≥ 0, x ∈ [l1, l2], (5.1) the kernel function j : r → r is assumed to satisfy (j). the integral operator ∫ l2 l1 j(x−y) ( u(t,y) − u(t,x) ) dy describes diffusion processes, where ∫ l2 l1 j(x−y)u(t,y)dy is the rate at which individuals are arriving at position x from all other places and ∫ l2 l1 j(x−y)u(t,x)dy is the rate at which they are leaving location x to travel to all other sites. since diffusion takes places only in [l1, l2] and individuals may not enter or leave the domain [l1, l2], we call it neumann type boundary condition. linearizing system (5.1) at u = 0, we obtain the time-periodic operator: l̃(l1,l2)[v](t,x) =   −vt − δv, t ∈ (iω, (i + ρ)ω], x ∈ [l1, l2], −vt + d ∫ l2 l1 j(x−y) ( u(t,y) −u(t,x) ) dy + av, t ∈ ((i + ρ)ω, (i + 1)ω], x ∈ [l1, l2]. (5.2) a easy calculation yields that λ1 = δρ−a(1−ρ) is a eigenvalue of −l̃(l1,l2) with a positive eigenfunction. moreover, one can also derive as in theorem 1.1 that theorem 5.1. assume that (j) holds and −∞ < l1 < l2 < +∞. let u(t, ·; u0) be the unique solution to (5.2) with the initial value u0(x) ∈ c([l1, l2]), where u0(x) is bounded, nonnegative and not identically zero. the following statements are true: (1) if δρ−a(1 −ρ) < 0, then lim n→∞ u(t + nω,x; u0) = z ∗(t) in c([0,ω] × [l1, l2]), where z∗(t) is the unique positive ω-solution to (1.7); (2) if δρ−a(1 −ρ) ≥ 0, then lim t→∞ u(t,x; u0) = 0 uniformly for x ∈ [l1, l2]. form above discussion, we can conclude that in spatial homogeneous environment, the nonlocal diffusion model with seasonal succession under neumann boundary condition has the same dynamical a nonlocal dispersal logistic model with seasonal succession 19 behavior as that of the corresponding ode model. when the environment is spatially dependent (e.g., the parameter a in model (1.1) or (5.1) is replaced by a spatially dependent function a(x)), one can also obtain the existence of a eigenvalue of the linearized operator associated with a positive eigenfunction under the additional compact support condition on the kernel function j(x). likewise, the sign of such obtained eigenvalue can completely determine the dynamical behavior of model (1.1) or (5.1). however, in spatially heterogeneous environment, the criteria governing persistence and extinction of the species becomes more difficult to be obtained (see [21, 23] for details). we slao remark that if the term a−bu in (1.1) (resp. 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[27] x.-q. zhao, dynamical systems in population biology, springer-verlag, new york, 2003. school of mathematics and statistics, hnp-lama, central south university, changsha, hunan, 410083, pr china email address: zhzhli@csu.edu.cn corresponding author, school of mathematics and statistics, hnp-lama, central south university, changsha, hunan, 410083, pr china email address: bxdai@csu.edu.cn 1. introduction 2. well-posedness 3. global dynamics 3.1. the method of periodic upper-lower solutions 3.2. proofs of theorems 1.1 and 1.2 4. simulations 5. discussion references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase online first, pp.1-27 https://doi.org/10.5206/mase/14918 global stability of a hcv dynamics model with cellular proliferation and delay alexis nangue, armel willy fokam tacteu, and ayouba guedlai abstract. in this work, we propose and investigate a delay cell population model of hepatitis c virus (hcv) infection with cellular proliferation, absorption effect, and a non-linear incidence function. first of all, we prove the existence of the local solutions of the model, followed by the existence of the global solutions and the positivity. moreover, we determine the infection free equilibrium and the basic reproduction rate r0, which is a threshold number in mathematical epidemiology. then we prove the existence and uniqueness of the infection persist equilibrium. we also proceed to study the local and global stability of this equilibrium. we show that if r0 < 1, the infection free equilibrium is globally asymptotically stable, which means that the disease will disappear and if r0 > 1, we have a unique infection persist equilibrium that is globally asymptotically stable under some conditions. finally, we perform some numerical simulations to illustrate the obtained theoretical results. 1. introduction hepatitis c is a disease caused by hcv, which is a virus that attacks liver cells, causing them to become inflamed. the world health organization (who) estimates in [29] that globally, 71 million people are chronic carriers of hepatitis c infection and that in 2016, around 399,000 people died from it, most often as a result of cirrhosis or hepatocellular carcinoma (primary liver cancer). this organization is leading actions to reduce the number of new cases of viral hepatitis by 90% and the number of deaths associated with this disease by 65% by 2030. due to the severity of hepatitis c infection, it is necessary to develop the tools that help to understand this disease. it is for this reason that several mathematical models have been developed to better understand the dynamics of the hepatitis c virus within the liver itself [1, 2, 7, 18, 20, 19, 17]. eric avila vales et al. in [2] studied an intra-host delay model, which is a pioneer work which inspired the work in the present paper. indeed, we note that in their model, the loss of pathogens due to the absorption that we can find in [10, 30, 23] in uninfected cells is ignored. when a pathogen enters an uninfected cell, the number of pathogens in the blood decreases by one. this is called the absorption effect (see, for example, [4]). to place the model on a more solid biological basis, we use the saturated infection rate (saturated infection rate found in [25, 30]) and cellular proliferation effect. these three aspects added to the model studied in [2] make the model we are studying a more realistic model in the biological sense. we note that in most intra-host models of virus dynamics, the loss of pathogens due to the absorption into uninfected cells is ignored. in biology, it is natural that, when pathogens are absorbed into susceptible cells, the numbers of pathogens are reduced in the blood volume which : this phenomenon is called absorption effect. hence, some researchers (see, for example, [4, 16, 30] and the references therein) have included the absorption effect into their models. in several of models with or without delay, the process of cellular infection by free virus particles is typically modelled by the mass action principle, that is to say, the infection rate is assumed to occur at a rate received by the editors 30 april 2022; accepted 25 august 2022; published online september 10, 2022. 2020 mathematics subject classification. 34a05, 34a06, 34a34, 34d20, 34d23, 37n25. key words and phrases. absorption effect, cellular proliferation, delay, global stability, hepatitis c virus, wellposedness. 1 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14918 2 a. nangue, a. w. fokam tacteu, and a. guedlai proportional to the product of the concentration of virus particles and uninfected target cells. this principle is insufficient to describe the cellular infection process in detail, and some non-linear infection rates have been proposed. li and ma [14], song and neumann [25] considered a virus dynamics model with monod functional response bxv 1+αv . regoes et al.[21] and song and neumann [26] considered a virus dynamics model with the non-linear infection rates bx (v k )p / ( 1 + (v k )p) and bxv q (1 + αvp) , where p, q, k > 0 are constants, respectively. recently, huang et al. [12] considered a class of models of viral infections with a non-linear infection rate and two discrete intracellular delays, and assumed that the infection rate is given by a general non-linear function of uninfected target cells and free virus particles f(x,v), which satisfies certain conditions. the rest of this paper is organized as follows. section 2 gives a description of the newly constructed model. section 3 deals with the existence, the positivity and boundedness of solutions of our model. in section 4, the threshold parameter r0(τ) of our dde model (2.1) is derived and the existence of the equilibria are discussed in relation to the value of r0(τ). section 5 and section 6 show the local and global stability of the infection free equilibrium and infection persist equilibrium respectively. additionally, some numerical results are displayed this supports the obtained theoretical results. finally, a brief discussion and some possible future ideas are presented in section 7. 2. model construction in this section, motivated by what has been said previously, we propose an hcv infection model with time delay, absorption effect and monod functional response, taking into account the proliferation of both exposed cells and hcv infected hepatocytes :  dh(t) dt = λ + rhh(t) ( 1 − h(t) + i(t) k ) −µh(t) − βh(t)v (t) 1 + av (t) di(t) dt = βe−τmh(t− τ)v (t− τ) 1 + av (t− τ) + rii(t) ( 1 − h(t) + i(t) k ) −αi(t) dv (t) dt = ηi(t) −γv (t) − βh(t)v (t) 1 + av (t) (2.1) with initial conditions h(θ) = ϕ1(θ) ≥ 0, i(θ) = ϕ2(θ) ≥ 0, v (θ) = ϕ3(θ), −τ ≤ θ ≤ 0, (2.2) where ϕ = (ϕ1,ϕ2,ϕ3) ∈c([−τ, 0],r3+) which is the banach space of continuous functions ϕ : [−τ, 0] −→r3+ = { (h,i,v ) ∈ r3 |h ≥ 0, i ≥ 0, v ≥ 0 } with norm ‖ϕ‖ = sup −τ≤θ≤0 {|ϕ1|, |ϕ2|, |ϕ3|}. the model (2.1) is a modification of model (2) studied in [2] and later in [1]. the features of the latter is as follows : h(t), i(t) and v (t) denote the concentration of uninfected hepatocytes (or target cells), infected hepatocytes and free virus, respectively. all parameters are assumed to be positive constants. here, target cells are generated at a constant rate λ and die at a rate µ per uninfected hepatocyte. these hepatocytes are infected at rate β per target cell per virion. infected cells die at rate α per cell by cytopathic effects. because of the viral burden on the virus-infected cells, we assume that µ ≥ α. in other words, we assume that the average life-time of infected cells ( 1 α ) is shorter than the average life-time of uninfected cells ( 1 α ) [5, 24]. the proliferation of infected and uninfected hepatocytes due to mitotic division obeys to a logistic growth. the mitotic proliferation of uninfected hepatocytes is described by rhh(t) [1 − (h(t) + i(t))/k], and mitotic transmission occurs at a rate rii(t) [1 − (h(t) + i(t))/k], which is the mitotic division of infected hepatocytes. it should be mentioned global stability of a hcv dynamics model with cellular proliferation and delay 3 that the model (2) in [2] has ri = rh. uninfected and infected hepatocytes grow at the constant rate rh and ri respectively , and k is the maximal number of total hepatocyte population proliferation. infected cells produce virions at an average rate µ per infected cell, and γ is the clearance rate of virus particles. the population of virions decreases due to the infection at a rate βh(t)v (t)/[1 + av (t)] : this is absorption phenomenon. it should be noted that according to [20], to have a physiologically realistic model, in an uninfected liver when k is reached, liver size should no longer increase i.e. λ ≤ µk. we assume that the contacts between viruses and uninfected target cells are given by an infection rate βh(t)v (t)/[1 + av (t)], it is reasonable for us to assume that the infection has a maximal rate of β a . the parameter τ accounts for the time between viral entry into a target cell and the production of new virus particles. the recruitment of virus producing cells at time t is given by the number of cells that were newly infected at time t − τ and are still alive at time t. here, m is assumed to be a constant death rate for infected but not yet virus-producing hepatocytes. thus, the probability of surviving the time period from t− τ to t is e−mτ . 3. wellposedness in this section, we show that our model (2.1) is mathematically and biologically well posed. theorem 3.1. all solutions of system (2.1) with initial conditions (2.2), where h(0) > 0, i(0) > 0, v (0) > 0, are positive and under the initial conditions (2.2), the solution (h(t), i(t), v(t)) of model (2.1) is existent and unique. moreover, for any positive solution (h(t), i(t), v(t)) of system (2.1) we have : lim sup t→+∞ h(t) ≤ h0 = [ (rh −µ) + ( (rh −µ)2 + 4rhλk )1/2] k 2rh , the existence of constants mi > 0 and mv > 0 such that i(t) < mi, v (t) < mv . proof. firstly, we deal with the fact that r3+ is positively invariant with respect to the dde model system (2.1). we prove the positivity by contradiction. suppose h(t) is not always positive. then, let t0 > 0 be the first time such that h(t0) = 0. from the first equation of system (2.1), dh(t0) dt = λ > 0. by our hypothesis this means that h(t) < 0 for t ∈ (t − ε,t0), where ε is an arbitrary small positive constant. implying that exist t′0 < t0 such that h(t ′ 0) = 0 : this is a contradiction because we take t0 as the first value which h(t0) = 0 . it follows that t(t) is always positive. we now show that i(t) > 0 for all t > 0. by considering the second equation of system (2.1), one has : i(t) = i(0) exp ( −αt + ∫ t 0 ri ( 1 − h(u) + i(u) k ) du ) + exp [ −αt + ∫ t 0 ri ( 1 − h(u) + i(u) k ) du ] × ∫ t 0 [ βe−τmh(u− τ)v (u− τ) 1 + av (u− τ) exp ( −αu + ∫u 0 ri ( 1 − h(θ) + i(θ) k ) dθ )] du. (3.1) u − τ ∈ [−τ, 0] since u ∈ [0,τ] and furthermore by assumption for t ∈ [−τ, 0] we have : h(t) > 0, i(t) > 0, v (t) > 0, h(0) > 0, i(0) > 0 and v (0) > 0. we deduce that∫ t 0 [ βe−τmh(u− τ)v (u− τ) 1 + av (u− τ) exp ( −αu + ∫ u 0 ri ( 1 − h(θ) + i(θ) k ) dθ )] du is positive for all t ∈ [0,τ] and hence i(t) > 0 for all t ∈ [0,τ]. let us show by recurrence on n that i(t) > 0 for all t ∈ [nτ, (n + 1)τ]. let pn, the proposition : ∀n ∈ n i(t) > 0 ∀t ∈ [nτ, (n + 1)τ]. a) for n = 0, p0 is verified. b) suppose that for n ∈ n, i(t) > 0 for all t ∈ [nτ, (n + 1)τ] and show that i(t) > 0 for all t ∈ [(n+1)τ, (n+2)τ]. according to (3.1), for u ∈ [(n+1)τ, (n+2)τ], we have u−τ ∈ [nτ, (n+1)τ]. 4 a. nangue, a. w. fokam tacteu, and a. guedlai by recurrence assumption, the term∫ t 0 [ βe−τmh(u− τ)v (u− τ) 1 + av (u− τ) exp ( −αu + ∫ u 0 ri ( 1 − h(θ) + i(θ) k ) dθ )] du is positive. therefore i is positive on [(n + 1)τ, (n + 2)τ] and consequently pn+1 is true. hence i(t) > 0 for all t > 0. we finally show that v (t) > 0 for all t > 0. since i(t) > 0 for all t > 0 and η > 0, we have : dv (t) dt ≥ ( −γ − βh(t) 1 + av (t) ) v (t). integrating the previous expression on [0, t], we obtain : v (t) ≥ v0 exp (∫ t 0 ( −γ − βh(u) 1 + av (u) ) du ) , which ensures that v (t) > 0 for all t > 0. thus deduce that r3+ is positively invariant with respect to model (2.1). secondly, the existence and uniqueness of the solution (h(t),i(t),v (t)) can be easily proved by using the following theorems(theorem 2.1 and theorem 2.2 page 19 in [13]). finally, let us show that the solution (h(t),i(t),v (t)) is uniformly bounded. for any positive solution of system (2.1), we have lim sup t→+∞ h(t) ≤ h0 = [ (rh −µ) + ( (rh −µ)2 + 4 rhλ k )1/2] k 2rh since the first equation of (2.1) yields dh(t) dt ≤ λ− (µ−rh)h(t) − rh k h2(t). then there is a t1 > 0 such that for any sufficiently small ε > 0 one has h(t) < h0 + ε for t > t1. now let for t ≥ 0, define u(t) as below u(t) = h(t) + i(t) + β ∫ t t−τ e−m(t−s) h(s)v (s) 1 + av (s) ds. (3.2) taking the derivation of the previous expression along the solution, collecting and simplifying some terms, we obtain, for t ≥ 0, du(t) dt = λ + ((h(t) + i(t)) (rh + ri)) ( 1 − h(t) + i(t) k ) −µh(t) −αi(t) −(rhi(t) + rih(t)) ( 1 − h(t) + i(t) k ) −mβ ∫ t t−τ e−m(t−s) h(s)v (s) 1 + av (s) ds = λ + ( rh + ri 4 ) k − ( rh + ri k )( h(t) + i(t) − k 2 )2 −µh(t) −αi(t) −(rhi(t) + rih(t)) ( 1 − h(t) + i(t) k ) −mβ ∫ t t−τ e−m(t−s) h(s)v (s) 1 + av (s) ds, ≤ λ + ( rh + ri 4 ) k −µh(t) −αi(t) −mβ ∫ t t−τ e−m(t−s) h(s)v (s) 1 + av (s) ds ≤ λ + ( rh + ri 4 ) k − bu(t), global stability of a hcv dynamics model with cellular proliferation and delay 5 where b = min{µ,α,m}. it follows that lim sup t→+∞ u(t) ≤ 4λ + (rh + ri)k 4b , that is, there exist t2 > 0 and m1 > 0 such that u(t) < m1 for t > t2. then i(t) has an upper bound mi. it follows from the third equation of system (2.1), that, for t ≥ 0 dv (t) dt ≤ ηi(t) −γv (t), from which we have that lim sup t−→+∞ v (t) ≤ η γ 4λ + (rh + ri)k 4b . then there exists mv > 0 such that v (t) < mv for t > t2. this completes the proof of theorem 3.1 . � remark 3.1. from theorem 3.1, one has that the solution of initial value problem (2.1), (2.2) enters the region γ = { (h,i,v ) ∈ r3+|0 ≤ h(t) ≤ h0, 0 ≤ i(t) ≤ mi, 0 ≤ v (t) ≤ mv } . hence γ, of biological interest, positively-invariant under the flow induced by the problem (2.1), (2.2). now, we determine the equilibria of model (2.1). 4. equilibria and basic reproduction number 4.1. infection free equilibrium and basic reproduction number. model (2.1) always has the uninfected equilibrium e0 = (h0, 0, 0). by using similar techniques in [9] and [27], we obtain the basic reproduction number (spectral radius of next generation matrix) for model (2.1) as r0(τ) = 1 α [ ri ( 1 − h0 k ) + ηβe−τmh0 γ + βh0 ] . here, r0(τ) is the average number infected cells produced by one infected hepatocyte after introducing infected hepatocyte into fully susceptible hepatocyte population, that plays a crucial role in the dynamics. generally, the basic reproduction number r0(τ) helps us to decide whether viruses clean out with time or not. e0 = (h0, 0, 0) is the trivial equilibrium of model (2.1). to find the other equilibrium of (2.1), we solve the following algebraic system :  λ + rhh ( 1 − h + i k ) −µh − βhv 1 + av = 0, (4.1) βe−τmhv 1 + av + rii ( 1 − h + i k ) −αi = 0, (4.2) ηi −γv − βhv 1 + av = 0. (4.3) 4.2. infection persist equilibrium. 6 a. nangue, a. w. fokam tacteu, and a. guedlai 4.2.1. existence of an infection persist equilibrium. proposition 4.1. the system (2.1) possesses an infection persist equilibrium denoted as e1 = (h1,i1,v1) if r0(τ) > 1. proof. from (4.3), one has i1 = γ η v1 + βh1v1 η(1 + av1) . (4.4) reporting (4.4) into (4.1) yields the following second degree algebraic equation in h1 : − ( rh k + rhβv1 kη(1 + av1) ) h21 + ( rh −µ− rh k γ η v1 − βv1 1 + av1 ) h1 + λ = 0. (4.5) the discriminant of the algebraic equation (4.5) is given by : ∆ = ( rh −µ− rh k γ η v1 − βv1 1 + av1 )2 + 4λ ( rh k + rhβv1 kη(1 + av1) ) > 0. (4.6) thus, equation (4.5) has a unique positive root known as : h1 = ( rh −µ− rh k γ η v1 − βv1 1 + av1 ) + √ ∆ 2 ( rh k + rhβv1 kη(1 + av1) ) . we can once again denote h1 = f(v1). substituting (4.4) into (4.2), one gets: βe−τmh1v1 1 + av1 + ri ( γ η v1 + βh1v1 η(1 + av1) )( 1 − h1 + γ η v1 + βh1v1 η(1+av1) k ) −α ( γ η v1 + βh1v1 η(1 + av1) ) = 0, which is equivalent to βe−τmh1 1 + av1 + ri ( γ η + βh1 η(1 + av1) )( 1 − h1 + γ η v1 + βh1v1 η(1+av1) k ) −α ( γ η + βh1 η(1 + av1) ) = 0, since v1 > 0. taking into account the fact that h1 = f(v1) one has βe−τmf(v1) 1 + av1 + ri ( γ η + βf(v1) η(1 + av1) )1 − f(v1) + γηv1 + βf(v1)v1η(1+av1) k  −α(γ η + βf(v1) η(1 + av1) ) = 0. on the other hand we also have − ( rh k + rhβv1 kη(1 + av1) ) f(v1) 2 + ( rh −µ− rh k γ η v1 − βv1 1 + av1 ) f(v1) + λ = 0. (4.7) let f(v1) = βe−τmf(v1) 1 + av1 + ri ( γ η + βf(v1) η(1+av1) )1 − f(v1) + γηv1 + βf(v1)v1η(1+av1) k   −α ( γ η + βf(v1) η(1+av1) ) . (4.8) obviously, f is continuous on [0; +∞[. we have, for v1 = 0, f(0) = βe−τmf(0) + ri ( γ η + βf(0) η )( 1 − f(0) k ) −α ( γ η + βf(0) η ) , global stability of a hcv dynamics model with cellular proliferation and delay 7 with f(0) = (rh −µ) + √ (rh −µ) 2 + 4λ rh k 2 (rh k ) = h0. we finally obtain f(0) = α ( γ η + βh0 η )[ ηβe−τmh0 α(γ + βh0) + ri α ( 1 − h0 k ) − 1 ] which is equivalent to : f(0) = α ( γ η + βh0 η ) (r0 − 1) . (4.9) r0(τ) > 1, implies f(0) > 0. from 0 < h1 = f(v1) ≤ c, with c > 0 according to theorem 3.1, we have lim v1→+∞ βe−τmf(v1) 1 + av1 = 0 since lim v1→+∞ βe−τmc 1 + av1 = 0 and, consequently lim v1→+∞ −α ( γ η + βf(v1) η(1 + av1) ) = − αγ η and lim v1→+∞ ri ( γ η + βf(v1) η(1 + av1) )1 − f(v1) + γηv1 + βf(v1)v1η(1+av1) k   = −∞. therefore lim v1→+∞ f(v1) = −∞. the intermediate value theorem ensures the existence of v1 > 0 such that f(v1) = 0. the existence of v1 also ensures the existence of h1 and i1. therefore the infection persist equilibrium e1 = (h1,i1,v1) exists. � now let’s take a look at uniqueness. 4.2.2. uniqueness of the infection persist equilibrium. proposition 4.2. let λ = [4λ + (rh + ri)k]/4b. if e −τm > α/η and λ/k ≤ 1/2 then the infection persist equilibrium e1 = (h1,i1,v1) is unique when it exists. proof. from equation (4.8), one has f ′(v1) = ( β 1 + av1 ( e−τm − α η ) + riβ η(1 + av1)  1 − f(v1) + γηv1 + βf(v1)v1η(1+av1) k   + a ) f′(v1) + aβf(v1) (1 + av1)2 ( α η −e−τm ) − ri k ( γ η + βf(v1) η(1 + av1) )( γ η + βf(v1) η(1 + av1)2 ) − riaβf(v1) η(1 + av1)2  1 − f(v1) + γηv1 + βf(v1)v1η(1+av1)2 k   , where a = − ri k ( γ η + βf(v1) η(1 + av1) )( 1 + βv1 η(1 + av1) ) . the expression of a can be rewritten in the following form : a = riβ η(1 + av1)  1 − f(v1) + γηv1 + βf(v1)v1η(1+av1) k  − riγ ηk − riβ η(1 + av1) . 8 a. nangue, a. w. fokam tacteu, and a. guedlai thus f ′(v1) = [ β 1 + av1 ( e−τm − α η ) + riβ η(1+av1)  1 − 2f(v1) + 2γη v1 + 2βf(v1)v1η(1+av1) k   − riγ ηk ] f′(v1) − riaβf(v1) η(1+av1)2  1 − f(v1) + γηv1 + βf(v1)v1η(1+av1) k   + aβf(v1) (1 + av1)2 ( α η −e−τm ) − ri k ( γ η + βf(v1) η(1+av1) )( γ η + βf(v1) η(1+av1)2 ) . dividing equation (4.7) by f(v1), we obtain − ( rh k + rhβv1 kη(1 + av1) ) f(v1) + ( rh −µ− rh k γ η v1 − βv1 1 + av1 ) + λ f(v1) = 0. (4.10) using the implicit differentiation we get from 4.10 : f′(v1) = ( rhγ ηk + β (1 + av1)2 + rhβf(v1) kη(1 + av1)2 )( rh k + λ f(v1)2 + rhβv1 kη(1 + av1) )−1 . we deduce from the latter that f ′(v1) = − β (1 + av1)2 ( e−τm − α η )( rhγ ηk + β (1+av1)2 + rhβf(v1) kη(1+av1)2 )( rh k + λ f(v1)2 + rhβv1 kη(1+av1) )−1 − riβ η(1+av1)  1 − 2f(v1) + 2γη v1 + 2βf(v1)v1η(1+av1) k   × ( rhγ ηk + β (1+av1)2 + rhβf(v1) kη(1+av1)2 )( rh k + λ f(v1)2 + rhβv1 kη(1+av1) )−1 + riγ kη ( rhγ ηk + β (1+av1)2 + rhβf(v1) kη(1+av1)2 )( rh k + λ f(v1)2 + rhβv1 kη(1+av1) )−1 − riaβf(v1) η(1 + av1)  1 − f(v1) + γηv1 + βf(v1)v1η(1+av1) k   + aβf(v1) (1 + av1)2 ( α η −e−τm ) −ri k ( γ η + βf(v1) η(1+av1) )( γ η + βf(v1) η(1+av1)2 ) . the terms aβf(v1) (1 + av1)2 ( α η −e−τm ) , −riβ η(1 + av1)  1 − 2f(v1) + 2γη v1 + 2βf(v1)v1η(1+av1) k  (rhγ ηk + β (1 + av1)2 + rhβf(v1) kη(1 + av1)2 ) × ( rh k + λ f(v1)2 + rhβv1 kη(1 + av1) )−1 and − β (1 + av1)2 ( e−τm − α η )( rhγ ηk + β (1 + av1)2 + rhβf(v1) kη(1 + av1)2 )( rh k + λ f(v1)2 + rhβv1 kη(1 + av1) )−1 global stability of a hcv dynamics model with cellular proliferation and delay 9 are negative since α η −e−τm ≤ 0 and λ k ≤ 1 2 . once more we deduce that f ′(v1) ≤ riγ kη ( rhγ kη + rhβf(v1) (1 + av1)2kη + rhβf(v1) (1 + av1)kη + rhβ 2f2(v1) (1 + av1)3γkη )(rh k )−1 − riaβf(v1) η(1 + av1)2  1 − f(v1) + γηv1 + βf(v1)v1η(1+av1) k  − ri k ( γ η + βf(v1) η(1 + av1) ) × ( γ η + βf(v1) η(1 + av1)2 ) , since ( rhγ ηk + β (1 + av1)2 + rhβf(v1) kη(1 + av1)2 )( rh k + λ f(v1)2 + rhβv1 kη(1 + av1) )−1 ≤ ( rhγ kη + rhβf(v1) (1 + av1)2kη + rhβf(v1) (1 + av1)kη + rhβ 2f2(v1) (1 + av1)3γkη )(rh k )−1 . thus f ′(v1) ≤ ri k ( γ η + βf(v1) η(1 + av1) )( γ η + βf(v1) η(1 + av1)2 ) − ri k ( γ η + βf(v1) η(1 + av1) )( γ η + βf(v1) η(1 + av1)2 ) − riaβf(v1) η(1 + av1)2  1 − f(v1) + γηv1 + βf(v1)v1η(1+av1) k   , f ′(v1) ≤ − riaβf(v1) η(1 + av1)2  1 − f(v1) + γηv1 + βf(v1)v1η(1+av1) k   . therefore f ′(v1) < 0. the fact that f is strictly decreasing function allows us to state that v1 is unique. uniqueness of v1 implies those of h1 and i1. therefore one can concludes that e1 = (h1,i1,v1) is unique. � 5. asymptotic stability analysis of the infection free equilibrium the aim of this section is to study the local and global stability of the infection free equilibrium. 5.1. local stability analysis of e0. the following result gives conditions for equilibrium e0 to be locally asymptotically stable. proposition 5.1. if r0(τ) < 1, then the infection free equilibrium e0 of system (2.1) is locally asymptotically stable. proof. the characteristic equation associated to the jacobian matrix at the infection free equilibrium, e0 = (h0, 0, 0) is given by the following determinant : pj (e0)(x) = ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ( rh −µ− 2rhk h0 ) −x − rh k h0 −βh0 0 ( ri − rik h0 −α ) −x βe−τ(m+x)h0 0 η (−γ −βh0) −x ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ = 0. (5.1) 10 a. nangue, a. w. fokam tacteu, and a. guedlai computation of the determinant (5.1) yields : pj (e0)(x) = ( x − ( rh −µ− 2 rh k h0 ))( x2 + (ri k h0 + γ + α−ri + βh0 ) x ) + ( −riγ −riβh0 + αγ + αβh0 + ri k h0γ + ri k h20β −ηβe (m+x)τh0 ) = 0. since rh ( 1 − h0 k ) = µ− λ h0 , the first factor of the characteristic equation pj (e0)(x) = 0 is x = ( rh −µ− 2 rh k h0 ) −µ = − ( λ h0 + rhh0 k ) , which have a negative eigenvalue. the other two eigenvalues satisfy the following transcendental polynomial x2 + a2x + a3 + b3(τ)e −xτ = 0, (5.2) where a2 = ri rh λ h0 − ri rh µ + λ + α + βh0, a3 = −riγ ( 1 − h0 k ) −riβh0 ( 1 − h0 k ) + α (γ + βh0) and b3(τ) = −ηβe−mτh0. when τ = 0, (5.2) yields x2 + a2x + a3 + b3(0) = 0. (5.3) note that a2 > 0 as − rirh µ + α > 0 since ri ≤ rh and µ ≤ α, and a3 + b3(0) = −α(γ + βh0) [ ri α ( 1 − h0 k ) + ηβh0 γ + βh0 − 1 ] = −α(γ + βh0) [r0(0) − 1] . if r0(0) < 1 then a3 +b3(0) > 0. it follows that for τ = 0, according to routh-hurwitz criteria [8, 3], infection free equilibrium e0 = (h0, 0, 0) is locally asymptotically stable. now, let us consider the distribution of the roots of (5.2) when τ > 0. assume that, x = ωi, (ω > 0) is a solution of (5.2). the substitution of x = ωi, (ω > 0) into (5.2) yields : −ω2 + iωa2 + a3 + b3(τ) cos ωτ − ib3(τ) sin ωτ = 0, then separating in real and imaginary parts the previous equation, we obtain{ a3 −ω2 = −b3(τ) cos ωτ a2ω = b3(τ) sin ωτ. squaring and adding the last two equations, we get{ (a3 −ω2)2 = b23(τ) cos2 ωτ, a22ω 2 = b23(τ) sin 2 ωτ, and simplifications yields ω4 + (a22 − 2a3)ω 2 + (a23 − b 2 3(τ)) = 0. furthermore, if ω2 = z; a = a22 − 2a3; b(τ) = a 2 3 − b 2 3(τ), global stability of a hcv dynamics model with cellular proliferation and delay 11 we obtain equation f(z) = z2 + az + b(τ) = 0, (5.4) where a = (γ + βh0) 2 + ( riλ rhh0 − riµ rh + α )2 > 0 and b(τ) = a23 − b 2 3(τ) = (a3 − b3(τ)) (a3 + b3(τ)) . we know that a3 + b3(τ) = −α(γ + βh0)(r0(τ) − 1). thus, b(τ) = α(γ + βh0) 2(1 −r0(τ)) ( riλ rhh0 − riµ rh + α + µβe −τmh0 γ+βh0 ) . since − rih rh + α > 0, if r0(τ) < 1, then b(τ) > 0. now as a > 0, b(τ) > 0 and ω > 0, then f(z) > 0 for any z > 0 which contradicts f(z) = 0. this show that characteristic equation (5.4) does not have pure imaginary roots when r0(τ) < 1. now, let us show that equation (5.2) has all its roots with real negative part when r0(τ) < 1. let p(x) = x2 + a2x + a3 et q(x) = b3(τ). and a = −ri ( 1 − h0 k ) + α et p = γ + βh0, thus : a2 = a + p et a3 = ap, and p(x) = x + (a + p)x + ap. let us verify if the four conditions of theorem 1 p.187 [6] are satisfied. 1) using the routh-hurwitz criteria [8, 3], if p(x) = 0, it follows that if a + p = a2 > 0 and ap = a3(γ + βh0) ( −ri ( 1 − h0 k ) + α ) > 0, then the real part of x 2: re(x) < 0. here the first condition holds. 2) for 0 ≤ y < +∞, we have: p(−iy) = −y2 − i(a + p)y + ap p(−iy) = −y2 + i(a + p)y + ap = p(iy) q(−iy) = ηβh0e−mτ = q(iy). and the second condition is satisfied. 12 a. nangue, a. w. fokam tacteu, and a. guedlai 3) for 0 ≤ y < +∞ p(iy) = −y2 + (a + p)iy + ap |p(iy)|2 = (ap−y)2 + y(a + p)2 = (a2 + y2)(p2 + y2), thus |p(iy)| ≥ ap. note that for r0(τ) < 1, ap−ηβh0emτ = (γ + βh0) ( −ri ( 1 − h0 k ) + α− ηβh0e −mτ γ + βh0 ) = (γ + βh0) α (1 −r0(τ)) > 0. thus ap > ηβh0e −mτ. therefore ap > |q(iy)|, and it follows that |p(iy)| > |q(iy)|. 4) the last condition holds since: |q(x)| |p(x)| = ηβh0e −mτ x2 + (a + p)x + ap and lim |x|→+∞ ∣∣∣∣q(x)p(x) ∣∣∣∣ = 0. finally, if r0(τ) < 1, the infection free equilibrium e0 of system (2.1) is locally asymptotically stable. this completes the proof. � 5.2. global stability analysis of e0. for biological models and virus dynamics models in particular, it is interesting to study the stability of positive equilibria. all hepatocytes populations must persist. it is also necessary for all hepatocyte population to be present initially. therefore, a genuine concept of global stability for positive equilibrium points in biological models is that every model solution that starts in the positive orthant r3+ must remain there for all finite values of t and converge to the equilibrium when t tends to ∞. in this section, applying lyapunov functionals as in vargas-de-leon [28], we consider the global stability of the infection free equilibrium e0. theorem 5.2. assume that the condition rh = (γ + βh0)rie τm/γ holds. if r0(τ) < 1, then the infection free equilibrium e0 = (h0, 0, 0) of system (2.1) is globally asymptotically stable in r 3 +. proof. define the lyapunov functional u(t) = emτri ∫ h(t) h0 η −h0 η dη + emτrhi(t) + rhβh0v (t) γ + βh0 +rhβ ∫ τ 0 h(t−ω)v (t−ω) 1 + cv (t−ω) dω. (5.5) global stability of a hcv dynamics model with cellular proliferation and delay 13 u is defined and continuous for any positive solution (h(t),i(t),v (t)) of system (2.1). let us calculate the derivative of u(t) along a positive solution of (2.1). we have: du(t) dt = emτri (h −h0) h ḣ(t) + rhe τmi̇(t) + riβh0 γ + βh0 v̇ (t) +βrh d dt ∫ τ 0 h(t−ω)v (t−ω) 1 + av (t−ω) dω. since u = t−ω and βrh d dt ∫ τ 0 h(t−ω)v (t−ω) 1 + av (t−ω) dω = −βrh d du ∫ t−τ t h(u)v (u) 1 + av (u) du, it follows that du(t) dt = emτri (h −h0) h ( λ + rh ( 1 − h + i k ) −µh − βhv 1 + av ) + rhβh(t− τ)v (t− τ) 1 + av (t− τ) + rhe mτrii ( 1 − h + i k ) −rhαemτi + rhβh0ηi γ + βh0 − rhβh0γv γ + βh0 − rhβ 2h0hv (γ + βh0)(1 + av ) − rhβh(t− τ)v (t− τ) 1 + av (t− τ) + rhβhv 1 + av . using the fact that rh −µ = rhh0 k − λ h0 , we get du(t) dt = emτri (h −h0) h ( λ + rh k h0h − λ h0 h − rh k ih − βhv 1 + av ) +emτirhri ( 1 − h0 k ) −emτirhri ( h −h0 k ) −rhemτri i2 k −rhγαemτi + rhβh0ηi γ + βh0 − rhβh0γv γ + βh0 − rhβ 2h0hv (γ + βh0)(1 + av ) + rhβhv 1 + av ; = − rhri h0 emτ ((h −h0) + i) 2 + emτrhαi(r0(τ) − 1) −λemτ ri(h −h0)2 hh0 + βh0v ( rie mτ 1 + av − rhγ γ + βh0 ) + βhv 1 + av ( −riemτ + rhγ γ + βh0 ) . therefore, du(t) dt ≤ emτri (h−h0)2 hh0 − rh k rie mτ ((h −h0) + i) 2 + emτrhαi(r0(τ) − 1) (5.6) since rhγ γ + βh0 = rie mτ. thus du(t) dt ≤ 0 since r0(τ) < 1.furthermore, du(t) dt = 0 if and only if h(t) = h0, i(t) = 0 and v (t) = 0. therefore, the largest compact invariant set in { (h(t),i(t),v (t)) / du(t) dt = 0 } when r0(τ) ≤ 1 is e0 = (h0, 0, 0), where e0 is the infection free equilibrium. this shows that lim t−→∞ (h(t),i(t),v (t)) = (h0, 0, 0). by the lyapunov-lasalle invariance theorem for delay differential systems (theorem 2.5.3 in [13]), this implies that e0 is globally asymptotically stable in the interior of r 3 +. � 14 a. nangue, a. w. fokam tacteu, and a. guedlai 5.3. numerical results. in this section, we present some numerical simulations which validate our theoretical results. to explore system (2.1) and illustrate the stability of infection free equilibrium solution, we consider the set of following parameters value : τ = 5; rh = 0, 05; ri = 0.0428; m = 0, 021; k = 1200 ; a = 0, 001; β = 9, 2419.10−7; µ = 0, 02; γ = 0, 02; α = 0, 021; λ = 20; η = 0, 2 . (5.7) we obtain figure 1. 0 100 200 300 400 500 600 700 800 900 t 0 200 400 600 800 1000 1200 (a) 0 80 100 200 60 1200 v (t ) 300 phase space i(t) 400 40 1000 h(t) 500 20 800 0 600 (b) figure 1. dynamical behaviour of system (2.1) with the set of parameter values (5.7) we have r0(5) = 0.5513 < 1 and the infection free equilibrium e0 = (1140.8; 0; 0) is globally asymptotically stable. the graph in (a) shows the time series of the solutions with constant initial conditions. the graph in (b) shows the trajectory in the phase diagram of system 2.1, which illustrates the stability of the uninfected e0 with the history functions ϕ1(θ) = 800, ϕ2(θ) = 80, ϕ3(θ) = 300 (first trajectory); ϕ1(θ) = 750, ϕ2(θ) = 75, ϕ3(θ) = 250 (second trajectory); ϕ1(θ) = 700, ϕ2(θ) = 70, ϕ3(θ) = 200 (third trajectory). 6. asymptotic stability analysis of the infection persist equilibrium the aim of this section is to study the local and global stability of the infection persist equilibrium for system (2.1). 6.1. local stability analysis of e1. we now study the local stability behaviour of the infection persist equilibrium e1 when r0(τ) > 1 for system (2.1). thus, linearizing system (2.1) at the infection persist equilibrium e1 = (h1,i1,v1), we obtain that the associated transcendental characteristic equation is given by global stability of a hcv dynamics model with cellular proliferation and delay 15 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ − λ h1 − rh k h1 −x −rhk h1 − βh1 (1+av1)2 βe−(m+x)τ 1+av1 − ri k i1 −βe −τmh1v1 (1+av1)i1 − ri k i1 −x βe −(m+x)τh1 (1+av1)2 − βv1 1+av1 η −γ − βh1 (1+av1)2 −x ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ = 0. when we take into account the identities rh −µ = − λ h1 + rh k h1 + rh k i1 + βv1 1 + av1 , ri −α = − βe−τmh1v1 (1 + av1)i1 + ri k h1 + ri k i1, the characteristic equation reduces to x3 + a2(τ)x 2 + a1(τ)x + [ b1(τ)x + b2(τ) ] e−xτ + a0(τ) = 0. (6.1) where a2(τ) = λ h1 + rh k h1 + βe−mτh1v1 (1 + av1)i1 + ri k i1 + γ + βh1 (1 + av1)2 , a1(τ) = βe−τmh1v1 (1 + av1)i1 γ + ri k γi1 + riβh1i1 (1 + av1)2 + β2e−τmh21v1 (1 + av1)3i1 + ( λ h1 + rh k h1 ) × ( βe−τmh1v1 (1 + av1)i1 + rh k i1 + βh1 (1 + av1)2 + γ ) − rh k h1i1 − β2h1v1 (1 + av1)3 , a0(τ) = ( λ h1 + rh k h1 )( βe−mτh1v1 (1 + av1)i1 γ + ri k γi1 + riβh1i1 k(1 + av1)2 + β2e−τmh21v1 (1 + av1)3i1 ) + rh k h1 ( − ri k γi1 − riβh1i1 k(1 + av1)2 ) + βh1 (1 + av1)2 × ( − riηi1 k − riβv1i1 k(1 + av1) − β2e−τmh1v 2 1 (1 + av1)2i1 ) , b1(τ) = − ηβe−(m+x)τh1 (1 + av1)2 + rh k h1 βe−(m+x)τv1 (1 + av1) , b2(τ) = − ( λ h1 + rh k h1 + rh k i1 ) ηβe−(m+x)τh1 (1 + av1)2 + rhβe −(m+x)τh1v1 k(1 + av1) γ − βh1 (1 + av1)2 ηβe−(m+x)τv1 (1 + av1) . define p(λ,τ) = x3 + a2(τ)x 2 + a1(τ)x + a0(τ) and q(λ,τ) = b1(τ)x + b2(τ). 16 a. nangue, a. w. fokam tacteu, and a. guedlai when τ = 0, we have from (6.1) that p(λ, 0) + q(λ, 0) = x3 + a2(0)x 2 + (a1(0) + b1(0))x + b2(0) + a0(0) = 0. (6.2) by the routh-hurwitz criterion the conditions for the real part of x to be negative are a2(0) > 0, a0(0) + b2(0) > 0 and q = a2(0)(a1(0) + b1(0)) − (a0(0) + b1(0)) > 0. in our case a2(0) > 0. in other hand, we have a1(0) + b1(0) = βh1v1 (1 + av1)i1 γ av1 1 + av1 + ri k γi1 + riβh1i1 (1 + av1)2 + rh k h21 β (1 + av1)2 + ( λ h1 + rh k h1 )( βh1v1 (1 + av1)i1 + γ ) + λ h1 ri k i1 + rhh1i1 k (rh −ri) −rhh1 ( 1 − h1 + i1 k ) β (1 + av1)2 + βh1 (1 + av1)2 µ = aβh1v 2 1 γ (1 + av1)2i1 + βh1 (1 + av1)2 ( µ + rii1 −rh ( 1 − 2h1 + i1 k )) + ( λ h1 + rh k h1 )( βh1v1 (1 + av1)i1 + γ ) + λ h1 ri k i1 + rhh1i1 k (rh −ri) + riγi1 k , since ηβh1 (1 + av1)3i1 = βh1v1γ (1 + av1)2i1 + β2h21v1 (1 + av1)3i1 and − β2h1v1 (1 + av1)3 = − βλ (1 + av1)2 −rhh1 ( 1 − h1 + i1 k ) β (1 + av1)2 + βh1 (1 + av1)2 µ. it follows that if µ + rii1 −rh ( 1 − 2h1 + i1 k ) > 0, (6.3) then a1(0) + b1(0) > 0. (6.4) global stability of a hcv dynamics model with cellular proliferation and delay 17 furthermore, using (4.3) we obtain a0(0) + b2(0) = ( λ h1 + rh k h1 )( 1 − 1 1 + av1 ) βh1v1 (1 + av1)i1 γ + λ h1 ( ri k γi1 + riβh1i1 k(1 + av1)2 ) + rhβh1v1 k(1 + av1) γ + ηβ2h1v1 (1 + av1)3 − riβh1i1 k(1 + av1)2 ( γ i1 v1 + βh1v1 1 + av1i1 ) − riβ 2h1v1i1 k(1 + av1)3 − β3h21v 2 1 (1 + av1)4i1 , = ( λ h1 + rh k h1 )( 1 − 1 1 + av1 ) βh1v1 (1 + av1)i1 γ + λ h1 ( ri k γi1 + riβh1i1 k(1 + av1)2 ) + βh1v1γ k(1 + av1) × ( rh − ri 1 + av1 ) − riβ 2h21v1 k(1 + av1)3 − riβ 2h1v1i1 k(1 + av1)3 − β3h21v 2 1 (1 + av1)4i1 + β2h1v1 (1 + av1)3 ( γ i1 v1 + βh1v1 (1 + av1)i1 ) = ( λ h1 + rh k h1 )( 1 − 1 1 + av1 ) βh1v1 (1 + av1)i1 γ + λ h1 ( ri k γi1 + riβh1i1 k(1 + av1)2 ) + βh1v1 k(1 + av1) γ× ( rh − ri 1 + av1 ) − riβ 2h21v1 k(1 + av1)3 − riβ 2h1v1i1 k(1 + av1)3 + β2h1v 2 1 (1 + av1)3i1 γ, = ( λ h1 + rh k h1 ) aβh1v 2 1 (1 + av1)2i1 γ + βh1v1 k(1 + av1) γ ( rh − ri 1 + av1 ) + λ h1 ( ri k γi1 + riβh1i1 k(1 + av1)2 ) + riβ 2h1v1 (1 + av1)3 ( 1 − h1 + i1 k ) + β2h1v1 (1 + av1)3 ( v1 i1 −ri ) . finally, if ri < v1 i1 , then a0(0) + b2(0) > 0. thus, if τ = 0, by the routh-hurwitz criterion, we have the following theorem : theorem 6.1. assume r0(0) > 1, if q > 0 and ri < v1i1 then the unique infection persist equilibrium e1 = (h1,i1,v1) is locally asymptotically stable as τ = 0. 18 a. nangue, a. w. fokam tacteu, and a. guedlai now we are going to check if it is possible to have a complex root with a positive real part for τ > 0, assuming h(1) : a0 + b0 > 0, a2(a1 + b1) − (a0 + b0) > 0. note that x = 0 is not root of the equation because a0 + b2 > 0. suppose then that x = iω, (ω > 0), is a root of equation given by: p(x,τ) + q(x,τ)e−xτ = 0 (6.5) with p(x,τ) = x3 + a2(τ)x 2 + a1(τ) + a0(τ), q(x,τ) = b1(τ)x + b2(τ) as x = iω is a root of equation (6.5), we have p(iω,τ) + q(iω,τ)e−iωτ = 0 and −iω3 −a2(τ)ω2 + ia1(τ)ω + a0(τ) + (b1(τ)iω + b2(τ)) e−iωτ = 0. and it follows that −iω3 −a2(τ)ω2 + ia1(τ)ω + a0(τ) + (ib1(τ)ω + b2(τ)) (cos ωτ − i sin ωτ) = 0 we obtain : −iω3 −a2(τ)ω2 + ia1(τ)ω + a0(τ) + ib1(τ)ω cos ωτ + b2(τ) cos ωτ + b1(τ)ω cos ωτ − ib2(τ) sin ωτ = 0. consequently, we have the following relation :{ a0(τ) −a2(τ)ω2 = −b2(τ) cos ωτ − b1(τ)ω sin ωτ (6.6) a1(τ)ω −ω3 = −b1(τ)ω cos ωτ + b2(τ) sin ωτ. (6.7) multiplying (6.6) by −b1(τ)ω and (6.7) by b2(τ), we get : (−a0(τ) + a2(τ)ω2)b1(τ)ω + (a1(τ)ω −ω3)b2(τ) b22(τ) + b 2 1(τ)ω 2 = sin ωτ. therefore sin ωτ = (a2(τ)b1(τ) − b2(τ))ω3 + (a1(τ)b2(τ) −a0(τ)b1(τ))ω b22(τ) + b 2 1(τ)ω 2 . similarly, multiplying (6.6) by b2(τ) and (6.7) by b1(τ)ω, we obtain: cos ωτ = b1(τ)ω 4 + [a2(τ)b2(τ) −a1(τ)b1(τ)]ω2 −a0(τ)b2(τ) b22(τ) + b 2 1(τ)ω 2 . moreover for x = iω, we get : p(iω,τ) = −iω3 −a2(τ)ω2 + ia1(τ)ω + a0(τ), and q(iω,τ) = ib1(τ)ω + b2(τ). therefore one has : p(iω,τ) q(iω,τ) = i(a2(τ)b1(τ) − b2(τ))ω3 + (a1(τ)b2(τ) −a0(τ)b1(τ)ω) b22(τ) + b 2 1(τ)ω 2 − ( b1(τ)ω 4 + ( a2(τ)b2(τ) −a1(τ)b1(τ) ) ω2 −a0(τ)b2(τ) ) b22(τ) + b 2 1(τ)ω 2 . it follows that : sin(ωτ) = im ( q(iω,τ) p(iω,τ) ) global stability of a hcv dynamics model with cellular proliferation and delay 19 and cos(ωτ) = −re ( q(iω,τ) p(iω,τ) ) . thus from the fact that sin2(ωτ) + cos2(ωτ) = 1 we have |q(iω,τ)|2 |p(iω,τ)|2 = 1 and |q(iω,τ)|2 = |q(iω,τ)|2. we conclude that ω is a positive root of the equation |p(iω,τ)|2 −|q(iω,τ)|2 = 0. furthermore |p(x,τ)|2 = ω6 + a22(τ) − 2a1(τ)ω 4 + a21(τ) − 2a0(τ)a2(τ)ω 2 + a20(τ) and |q(x,τ)|2 = b22(τ) + b 2 1(τ)ω 2. we get ω6 −aω4 + bω2 + c = 0 (6.8) where a = a2(τ) − 2a1(τ), b = a21(τ) − 2a0(τ)a2(τ) − b21(τ), c = a20(τ) − b22(τ). let z = ω2 , we obtain z3 + az2 + bz + c = 0 (6.9) suppose that (6.9) has at least one positive root. let z0 be the smallest value of its roots. then (6.8) has the root ω0 = √ z0 and from (6.7) we get the value of τ associated with ω0 such that x = ωi is a pure imaginary root of (6.5). this value of τ is given by: τ0 = 1 ω0 arccos [ b2(a2ω 2 0 −a0) + b1ω0(ω30 −a1ω0) b22 + b 2 1ω 2 0 ] . ultimately we can summarize what precedes in the following result. thus according to theorem 2.4 p.48 [22], we have : theorem 6.2. suppose (h(1)) is verified, (1) if c ≥ 0 and λ = a2 − 3b < 0, then the roots of (6.5) have a negative real part for all τ ≥ 0, therefore the infection persist equilibrium e1(h1,i1,v1) is locally asymptotically stable. (2) if c < 0 or c ≥ 0, z1 > 0 and z31 + az21 + bz1 + c ≤ 0, then all the roots of the equation (6.5) have a negative real part when τ ∈ [0,τ0], and therefore the infection persist equilibrium e1(h1,i1,v1) is locally asymptotic stable in [0,τ0]. 6.2. global stability analysis of e1. theorem 6.3. denote ε = µ−rh + rh k (h1 + i1), λ = 4λ + (rh + ri)k 4b , et ω = ηλ γ . assume rh = rie mτ, ε > 0 and k > β3h1v 2 1 λ(1 + aω) 4εη2(1 + av1)4 + aβλv1 ηi1(1 + av1) , then the unique infection persist equilibriume1 = (h1,i1,v1) for system (2.1) is globally asymptotically stable for any τ ≥ 0. 20 a. nangue, a. w. fokam tacteu, and a. guedlai proof. define a lyapunov functional for e1 l(t) = l̃(t) + βh1v1 1 + av1 l+(t), (6.10) where l̃(t) = rie mτ ∫ h h1 η −h1 η dη + rhe mτ ∫ i i1 η − i1 η dη + ri emτβh1v1 ηi1(1 + av1) ∫ v v1 ( 1 − v1(1 + η) η(1 + av1) ) dη and l+(t) = rh ∫ τ 0 ( h(t− τ)v (t− τ)(1 + av1) h1v1(1 + av (t− τ)) − 1 − ln h(t−ω)v (t− τ)(1 + av ) h1v1(1 + av (t− τ)) ) dω. we have : dl+(t) dt = d dt rh ∫ τ 0 ( h(t−ω)v (t−ω)(1 + av1) h1v1(1 + av (t−ω)) − 1 − ln h(t−ω)v (t−ω)(1 + av ) h1v1(1 + av (t−ω)) ) dω. denote u = t−ω, we obtain : dl+(t) dt = −rh du dt d du ∫ t−τ t ( h(u)v (u)(1 + av1) h1v1(1 + av (u)) − 1 − ln h(u)v (u)(1 + av ) h1v1(1 + av (u)) ) dω = −rh [ h(u)v (u)(1 + av1) h1v1(1 + av (u)) − 1 − ln h(u)v (u)(1 + av ) h1v1(1 + av (u)) ]t−τ ω=t , = − rhh(t− τ)v (t− τ)(1 + av1) h1v1(1 + av (t− τ)) + rh ln h(t− τ)v (t− τ)(1 + av ) h1v1(1 + av (t− τ)) + rhhv (1 + av1) h1v1(1 + av ) −rh ln hv (1 + av1) h1v1(1 + av ) , = − rhh(t− τ)v (t− τ)(1 + av1) h1v1(1 + av (t− τ)) + rh ln i1h(t− τ)v (t− τ)(1 + av ) ih1v1(1 + av (t− τ)) , + rhhv (1 + av1) h1v1(1 + av ) + rh ln h1 h + rh ln iv1(1 + av ) i1v (1 + av1) . besides, a direct calculation yields : dl̃(t) dt = rie mτ (h −h1) h ḣ + rhe mτ (i − i1) i i̇ + rie mτ βh1v1 ηi1(1 + av1) ( 1 − v1(1 + av ) v (1 + av1) ) v̇ , = (h −h1)riemτ ( λ h − rh k (h + i) − βv 1 + av + rh −µ ) +rhe mτ (i − i1) ( βe−τh(t− τ)v (t− τ) i(1 + av (t− τ)) − ri k (h + i) + ri −α ) +rie mτ βh1v1 ηi1(1 + av1) ( 1 − v1(1 + av ) v (1 + av1) )( ηi −γv − βhv 1 + av ) . hence using (4.1), (4.2) and (4.3) we get : global stability of a hcv dynamics model with cellular proliferation and delay 21 dl̃(t) dt = rie mτ (h −h1) ( λ h − rh k (h + i) − βv 1 + av1 − λ h1 + rh k (h1 + i1) + βv1 1 + av1 ) + rhe mτ (i − i1) ( βe−mτh(t− τ)v (t− τ) i(1 + av (t− τ)) − ri k (h + i) + ri k (h1 + i1) − βe−mτh1(t− τ)v1(t− τ) i1(1 + av1) ) + rie mτβh1v1 ηi1(1 + av1) × ( 1 − v1(1 + av ) v (1 + av1) )( (ηiv1 − i1v ) v1 + βv h1 (1 + av1) − βv h (1 + av ) ) . cancelling identical terms with opposite signs and collecting terms yields consecutively dl̃(t) dt = rie mτ (h −h1) ( −λ(h −h1)2 hh1 − rh k [(h −h1) + (i − i1)] −β ( v 1 + av − v1 1 + av )) + rh(i − i1) ( βh(t− τ)v (t− τ) i(1 + av (t− τ)) − βh1v1 i1(1 + av1) − ri k emτ ((h −h1) + (i − i1)) ) + r1e mτβh1v1 ηi1(1 + av1) × ( 1 − v1(1 + av ) v (1 + av1) )( η(iv1 − i1v ) v1 + βv h1 (1 + av1) − βv h (1 + av ) ) = −λriemτ (h −h1)2 hh1 + ( − rh k rie mτ (h −h1)2 − rh k rie mτ (h −h1)(i − i1) − rirh k emτ (h −h1)(i − i1) − rirh k emτ (i − i1)2 ) + βhivi 1 + av1 ( −emτ rhhv (1 + av1) h1v1(1 + av ) + rhh(t− τ)v (t− τ)(1 + av1) h1v1(1 + av (t− τ)) ) + βh1v1 1 + av1 ( −rhemτiv1(1 + av ) i1v (1 + av ) − rhi1h(t− τ)v (t− τ)(1 + av1) h1iv1(1 + av (t− τ)) ) + βh1v1 1 + av1 ( rie mτ h h1 + rie mτ v (1 + av1) v1(1 + av ) −riemτ v v1 + rie mτ (1 + av ) 1 + av1 ) − rie mτβ2h1v1 η(1 + av1)3 (v −v1)(h −h1) + rie mτaβ2h1hv1(v −v1)2 ηi1(1 + av1)3(1 + av ) . 22 a. nangue, a. w. fokam tacteu, and a. guedlai = −λriemτ (h −h1)2 hh1 − rhri k emτ ( (h −h1) + (i − i1) )2 + βh1v1 1 + av1 ( −emτ rihv (1 + av1) h1v1(1 + av ) + rhh(t− τ)v (t− τ)(1 + av1) h1v1(1 + av (t− τ)) ) + βh1v1 1 + av1 ( emτriv (1 + av1) v1(1 + av ) −emτri v v1 + emτri(1 + av ) 1 + av1 −emτri ) + βh1v1 1 + av1 ( 3rie mτ −riemτ h1 h − emτriv1(1 + av )i i1v (1 + av1) − rhi1h(t− τ)v (t− τ)(1 + av1) h1iv1(1 + av (t− τ)) ) − rie mτβ2h1v1 η(1 + av1)3 (v −v1)(h −h1) + rie mτaβ2h1hv1(v −v1)2 ηi1(1 + av1)3(1 + av ) + rie mτβh1v1 1 + av1 ( h1 h + h h1 − 2 ) . additionally, h1 h + h h1 − 2 = (h −h1)2 hh1 , thus, dl̃(t) dt = rie mτ ( −λ + βh1v1 1 + av1 ) (h −h1)2 hh1 − rhri k emτ ( (h −h1) + (i − i1) )2 + βh1v1 1 + av1 ( − emτrihv (1 + av1) h1v1(1 + av ) + rhh(t− τ)v (t− τ)(1 + av1) h1v1(1 + av (t− τ)) ) + βh1v1 1 + av1 ( − rie mτv (1 + av1)i i1v1(1 + av ) − rhi1h(t− τ)v (t− τ)(1 + av1)i h1iv1(1 + av (t− τ)) + 3rie mτ −riemτ h h1 ) + rie mτ βh1v1 1 + av1 ( 1 − v1(1 + av ) v (1 + av1) )( v (1 + av1) v1(1 + av ) − v v1 ) − r1e mτβ2h1v1 η(1 + av1)3 (v −v1)(h −h1) + r1e mτaβ2h1hv1(v −v1)2 ηi1(1 + av1)3(1 + av ) . the fact that λ− βh1v1 1 + av1 = (µ−rh)h1 + rhh1 k (h1 + i1) yields dl̃(t) dt = −riemτ ε h (h −h1)2 − rhri k emτ ((h −h1) + (i − i1)) 2 + βh1v1 1 + av1 ( −emτ r1hv (1 + av1) h1v1(1 + av ) + rhh(t− τ)v (t− τ)(1 + av1) h1v1(1 + av (t− τ)) ) global stability of a hcv dynamics model with cellular proliferation and delay 23 + βh1v1 1 + av1 ( 3emτri −emτri h1 h − emτr1iv1(1 + av ) i1v (1 + av1) − rhi1h(t− τ)v (t− τ)(1 + av1) h1iv1(1 + av (t− τ)) ) − krie mτβh1 (1 + av )(1 + av1)2 (v −v1)2 −ri emτβ2h1v1 η(1 + av1)3 (v −v1)(h −h1) + rie mτaβ2h1hv1(v −v1)2 ηi1(1 + av1)3(1 + av ) . it follows that dl̃(t) dt ≤ −riemτ ε h ( (h −h1) + β2hh1v1 2η(1 + av1)3 (v −v1) )2 + rie mτβh1 (1 + av )(1 + av1)2 ( −k + β3h1v 2 1 λ(1 + aω) 4εη2(1 + av1)4 + aβλv1 ηi1(1 + av1) ) (v −v1)2 + βh1v1 1 + av1 ( −emτ rihv (1 + av1) h1v1(1 + av ) + rhh(t− τ)v (t− τ)(1 + av1) h1iv1(1 + av (t− τ)) ) + βh1v1 1 + av1 ( 3emτri −emτri h1 h − emτr1iv1(1 + av ) i1v (1 + av1) − rhi1h(t− τ)v (t− τ)(1 + av1) h1iv1(1 + av (t− τ)) ) − rirh k emτ ( (h −h1) + (i − i1) )2 . according to (6.10), we have: dl dt = dl̃ dt + βh1v1 1 + av1 dl+ dt . thus, dl(t) dt ≤− rhri k emτ ((h −h1) + (i − i1)) 2 −− rirh k emτ ( (h −h1) + (i − i1) )2 −riemτ ε h ( (h −h1) + β2hh1v1 2η(1 + av1)3 (v −v1) )2 + rie mτβh1 (1 + av )(1 + av1)2 × ( −k + β3h1v 2 1 λ(1 + aω) 4εη2(1 + av1)4 + aβλv1 ηi1(1 + av1) ) (v −v1)2 − rie mτβh1v1 (1 + av1) ( h1 h − 1 − ln h1 h ) − rie mτβh1v1 (1 + av1) ( iv1(1 + av ) i1v (1 + av1) − 1 − ln iv1(1 + av ) i1v (1 + av1) ) −riemτ βh1v1 1 + av1 ( h(t− τ)v (t− τ)(1 + av1) h1iv1(1 + av (t− τ)) − 1 − ln h(t− τ)v (t− τ)(1 + av ) h1iv1(1 + av (t− τ)) ) . 24 a. nangue, a. w. fokam tacteu, and a. guedlai the function x 7→ x− 1 − ln x is positive on ]0; +∞[, therefore, since −k + β3h1v 2 1 λ(1 + aω) 4εη2(1 + av1)4 + aβλv1 ηi1(1 + av1) ≤ 0 and ε > 0 we deduce that dl(t) dt ≤ 0. furthermore, dl(t) dt = 0 if and only if h(t) = h(t − τ) = h1, v (t) = v (t − τ) = v1 and i(t) = i1. therefore, the largest compact invariant set m is the singleton {e1}, where e1 is the infection persist equilibrium. this shows that lim t−→∞ (h,i,v ) = (h1,i1,v1). by the lyapunov-lasalle invariance theorem for delay differential systems (theorem 2.5.3 in [13]), this implies that e1 is globally asymptotically stable in the interior of r3+. � 6.3. numerical results. in this section, we present some numerical simulations which validate our theoretical results. to explore system (2.1) and illustrate the stability of infection persist equilibrium solution, we consider the set of following parameter value : τ = 10; β = 0.0027; rh = 0.01; m = 0.02; ri = 0.0061; λ = 5; µ = 0.02; a = 0.001; η = 0.5 ; γ = 2.1; α = 0.05 ; ε = 0.0116 > 0; λ = 550; e0(379.7959; 0; 0); e1 = (156.9218; 38.0708; 7.5522) k = 1200 > β3h1v 2 1 λ(1+aω) 4εη2(1+av1)4 + aβλv1 ηi1(1+av1) = 8.0643. (6.11) for these parameter values, calculation by the formula of basic reproduction number leads to r0 = 2.0729 > 1; and thus, the infection persist equilibrium e1 = (156.9218; 38.0708; 7.5522) is globally asymptotically stable. this conclusion is demonstrated in figure 2). the graph in (a) shows the time series of the solutions with constant initial conditions. the graph in (b) shows the trajectory in the phase diagram of system 2.1, which illustrates the stability of the infected e1 with the history functions ϕ1(θ) = 120, ϕ2(θ) = 80, ϕ3(θ) = 20 (first trajectory); ϕ1(θ) = 210, ϕ2(θ) = 120, ϕ3(θ) = 25 (second trajectory); ϕ1(θ) = 330, ϕ2(θ) = 200, ϕ3(θ) = 30 (third trajectory). 7. conclusion in order to better understand the dynamics of hcv viral infection, this paper presents a mathematical study on the global dynamics of improved intra-host hcv models based on models in [2] and [1]. in this work we have established results about the local and global stability of equilibria known as infection free equilibrium and infection persist equilibrium. we can conclude that the stability of infection free equilibrium is completely determined by the value of the basic reproductive number r0(τ). if r0(τ) < 1, then the infection free equilibrium will be asymptomatically stable and unstable if r0(τ) > 1. for the infection persist equilibrium, we established conditions to ensure the local stability. we have needed another conditions to ensure the global stability for this equilibrium. most of the results obtained in the present work generalize, in the same framework, those obtained by eric avila vales et al. in [1] in the sense that in [1] the proliferation rates of infected and uninfected hepatocytes are identical and the global stability of a hcv dynamics model with cellular proliferation and delay 25 0 100 200 300 400 500 600 700 800 900 1000 t 0 50 100 150 200 250 300 350 (a) 0 300 20 400 40 v (t ) 200 phase space 60 i(t) h(t) 80 200100 0 0 (b) figure 2. dynamical behaviour of system (2.1) with the set of parameter values (6.11) absorption phenomenon was absent. this work can be developed in many ways as follows: (i) by taking into consideration the effect of several time discrete delays that may occur during the infection process, diffusion of cells and virions or other biological processes. for example, the following model can be study:  ∂h(x,t) ∂t = dh∆h(x,t) + λ + rhh(x,t) ( 1 − h(x,t) + i(x,t) k ) −µh(x,t) − βh(x,t)v (x,t) 1 + av (x,t) , ∂i(x,t) ∂t = di∆i(x,t) + βe−τ1m1h(x,t− τ1)v (x,t− τ1) 1 + av (x,t− τ1) +rii(x,t) ( 1 − h(x,t) + i(x,t) k ) −αi(x,t)v, ∂v (x,t) ∂t = dv ∆v (x,t) + ηe −τ2m2i(x,t− τ2) −γv (x,t) − βh(x,t)v (x,t) 1 + av (x,t) , where the parameter τ1 accounts for the time between viral entry into a target cell and the production of new virus particles, after which the cells produce virus at per capita rate ηe−τ2m2 with a delay of τ2. the constant m1 is assumed to be the death rate for cells that are infected but not yet producing virus, so that e−τ1m1 is the probability of surviving from time t− τ2 to time t. likewise the constant m2 is assumed to be the birth rate of the virions so that e −τ2m2 is the probability of producing a fraction of η virions from time t− τ2 to time t. and dh, di and dv give the rates at which the target cells, the infected cells and the virus particles diffuse respectively; (ii) by fitting the model with real data and finding a better estimation for the values of parameters. (iii) using a more general incidence rate as a function with certain desired properties, as considered in [11] or considering distributed delays in the equations for the infected cells and the virus particles, as is discussed in [15]. (iv) by investigating hcv co-infections with other viruses. 26 a. nangue, a. w. fokam tacteu, and a. guedlai disclosure statement no potential conflict of interest was reported by the authors. references [1] e. avila-vales, a. canul-pech, g. e. garćıa-almeida, and a. g. c. pérez, global stability of a delay virus dynamics model with mitotic transmission and cure rate, mathematical modelling and analysis of infectious diseases (k hattaf and h dutta, eds.), vol. 302, (2020) springer, pp. 93–136. [2] e. avila-vales, n. chan-chi, g. e .garćıa-almeida, and c. v. leon, stability and hopf bifurcation in a delayed viral infection model with mitosis transmission, applied mathematics and computation 259 (2015), 293–312. 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[30] r. xu, global dynamics of a delayed hiv-1 infection model with absorption and saturation infection, int. j. biomath. 5 (2012), no. 3, id 1260012 (13 pages). corresponding author, higher teachers’ training college , university of maroua, cameroon, p.o.box 55 maroua current address: same email address: alexnanga02@gmail.com higher teachers’ training college , university of maroua, cameroon, p.o.box 55 maroua current address: university of maroua, faculty of sciences, department of mathematics and computer science, cameroon, p.o. box 814 maroua email address: tacteufokam@gmail.com higher teachers’ training college , university of maroua, cameroon, p.o.box 55 maroua current address: university of maroua, faculty of sciences, department of mathematics and computer science, cameroon, p.o. box 814 maroua email address: guedlaiayoubagg@gmail.com 1. introduction 2. model construction 3. wellposedness 4. equilibria and basic reproduction number 4.1. infection free equilibrium and basic reproduction number 4.2. infection persist equilibrium 5. asymptotic stability analysis of the infection free equilibrium 5.1. local stability analysis of e0 5.2. global stability analysis of e0 5.3. numerical results 6. asymptotic stability analysis of the infection persist equilibrium 6.1. local stability analysis of e1 6.2. global stability analysis of e1 6.3. numerical results 7. conclusion disclosure statement references mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 4, number 1, march 2023, pp.12-29 https://doi.org/10.5206/mase/15505 a novel ivgtt model including interstitial insulin jun jin, jiaxu li∗, rui xu, lei yu, and zhen jin∗ abstract. minimal model (mm) is the top-scoring model for assessing physiological characteristics to diagnose the potential or onset of type 2 diabetes mellitus (t2dm) through the intravenous glucose tolerance test (ivgtt) for the past four decades. nevertheless it has been arguable that mm method either overestimates glucose effectiveness (ge) or underestimates insulin sensitivity (is) in some cases by both biologists through in vivo experiments and mathematicians by analysis and/or simulations. we propose a novel model including the interstitial insulin according to physiology and adapted from the well accepted sturis’ model for the glucose-insulin metabolic system suitable to the ivgtt setting. our model consistently overcomes the aforementioned defects in a subgroup of subjects. in addition, the variable x for insulin action in mm might be appropriately interpreted as an increment of insulin in the interstitial space in response to the bolus stimulus, rather than being proportional to the interstitial insulin as believed. 1. introduction quantitatively assessing physiological characteristics, e.g., insulin sensitivity (is) and glucose effectiveness (ge), is essential to diagnose the progression and/or onset of type 2 diabetes mellitus (t2dm) and develop drug for treatment. the gold standard for the assessment of is is hyperinsulinemic euglycemic glucose clamp test originally proposed by defronzo et al.[11], which is direct and accurate, and widely accepted [32]. but it is very invasive, expensive, time consuming, and the subjects suffer. a much less invasive protocol as a result is the intravenous glucose tolerance test (ivgtt). the ivgtt data sampled at the time marks, for example, -10, -1, 0, 2, 3, 5, 7, 10, 13, 17, 21, 25, 30, 35, 40, 45, 50, 60, 70, 80, 90, 100, 120, 160 and 180 min after a rapid bolus intravenous glucose infusion, is used to estimate the parameter values of a differential equation model so that the aforementioned physiological characteristics can be assessed through these parameters. many such models have been proposed at least as early as 1975 [28, 4, 5, 15, 33, 36, 27], but the top-scoring model widely used in experiments in both laboratory and clinic research is the minimal model (mm) developed by bergman, cobelli, and their colleagues a few years later [4, 5] for the past 40 years [2]. commercial software include minmod, minmod millennium [34, 7], and a software implemented in mlab [21] by civilized software inc. (silver spring, md 20906) are available. mm has been used by many biologists in their experiments [20]. the mm is given by bergman et al. [4]:{ g′(t) = −[sg + x(t)]g(t) + sggb, x′(t) = −p2x(t) + p3[i(t) − ib]+, (1.1) received by the editors 15 november 2022; accepted 13 february 2023; published online 20 february 2023. 2020 mathematics subject classification. primary 54c40, 14e20; secondary 46e25, 20c20. key words and phrases. diabetes, ivgtt, minimal model, remote insulin, time delay. ∗corresponding authors. rx, ly and zj were supported in part by nsfc grant 12271317, 61803242, and 12231012, respectively. zj was supported in part by key r&d project in shanxi province grant 201803d31032. 12 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15505 a novel ivgtt model including interstitial insulin 13 with the initial condition g(0) = g0 and x(0) = 0, where g(t) and i(t) are respectively plasma glucose and insulin concentration, sg is the ge index, the positive parameter p2 and p3 are used to calculate the is index si = p3/p2, and g0 is the initial value of the plasma glucose concentration value. [i(t)−ib]+ = i(t)−ib if i(t)−ib ≥ 0; [i(t)−ib]+ = 0 if i(t)−ib < 0, where gb and ib denote the basal glucose and insulin level, respectively. the state variable x with initial condition x(0) = 0 was called insulin action in the original paper [4] and later was believed to be proportional to remote insulin (i.e. insulin in interstitial compartment) by, for example, pacini and bergman [34], ader et al. [1], bergman et al. [3] and, even recently, bergman [2]. while along with the resounding success has been achieved by mm, several drawbacks and limitations in mm have been pointed out by both biologists and applied mathematicians [13, 39, 9, 15, 32, 37, 18, 14, 23]. finegood and tzur [13] pointed out that the estimation of sg through mm may be not correct, later caumo and cobelli [8] elucidated that the inaccuracy is due to “the second hidden compartment”. cobelli et al. [9] furthermore showed that mm overestimates sg and underestimates si. patarrao et al. [37] believed that “many limitations of minimal model analysis stem from the fact that the model oversimplifies the physiology of glucose homeostasis”. recently, ha et al. [18], fosam et al. [14] and koh et al. [23] showcased that mm systematically underestimates si in african american females comparing with non-hispanic white females, which unveiled the paradox in predicting the risks of t2dm in different ethnic groups. in particular, ha et al. [18] pointed out that the presence of a strong first insulin secretion phase leads to an underestimation of insulin sensitivity. after the creation of mm, deeper understanding for the action of interstitial insulin on blood sugar removal became much clearer. for example, an endothelial barrier delays the transcapillary transport of insulin from plasma to interstitial space [42] and capillary endothelium poses a barrier that delays the onset of muscle insulin action [49]. factors that impede insulin access to muscle could contribute to increase insulin resistance [49]. in this paper, we propose a novel ivgtt model taking interstitial insulin as an explicit state variable based on the recent advanced physiology and the formulation from sturis’ model in [46]. we demonstrate that the aforementioned drawbacks could be rectified in some subgroup of animal subjects. we organize this paper as follows. we present the model formulation in details in next section, followed by a section showing this model is well posed. then in section 4, we utilize latin hypercube sampling method to generate pure random parameter values in the parameter space and then analyze the correlations of the profiles from mm, the new model and the nine available ivgtt data. we also elucidate that the new model would not produce ruinous dynamics when the model parameter values are within physiological ranges. we will discuss our findings in the last section. 2. formulation of the novel ivgtt model adapted from a physiological model a well known model describes the general glucose-insulin metabolism was formulated by sturis et al. [46], which contains three state variables for the plasma glucose (g), the plasma insulin (ip) and the interstitial insulin (ii) and three auxiliary variables mimicking time delays. through this model, sturis et al. [46] successfully elucidated and well accepted that the ultradian oscillation of insulin secretion in physiological settings is intrinsically caused by time delays and the transfer of insulin between the plasma compartment and interstitial compartment. a number of models in this area have been stemmed from sturis’ model, particularly the metabolic feedback loop involving more organs [47], with explicit time delays [26, 24], and consequent attempts for algorithms of the artificial pancreas for type 1 diabetes (t1dm) by wu et al. [50], huang et al. [19], kissler et al. [22] and song et al. [44]. these, in addition to physiological observation by prigeon et al. [39], allow us to confidently adapt the insulin in the interstitial compartment as an independent state variable as in sturis’ model and the explicit time 14 j. jin, j. li, r. xu, l. yu, and z. jin delay in the model in [26] to formulate a novel model suitable for the ivgtt environment for our aforementioned aims. figure 1. diagram of the model (2.1). refer to the model diagram in figure 1 for the glucose and insulin regulations, we denote g(t), ip(t) and ii(t) as the concentrations of the plasma glucose, plasma insulin and interstitial insulin at time t, respectively. according to the procedure of ivgtt, we assume that the glucose bolus is infused into the vein in the time interval [0, t0], where t0 = 2 or 3, and then our model is defined in [t0,∞). insulin is secreted from pancreatic β-cells. summarized by straub and sharp [45], glucose-stimulated biphasic insulin secretion in the ivgtt setting includes the katp channel-dependent pathways, and katp channel-independent pathways, respectively. the first phase of insulin secretion is resulted from the exocytosis of immediately releasable β-cell granules followed by the second phase of due to the katp channel-independent pathways acting in synergy with the katp channel-dependent pathway. in the same way as bergman et al. [4], de gaetano and arino [15], li et al. [25] and panunzi et al. [36], we model the first phase quick insulin release by the initial condition ip(t0) = ip0 taken at the time mark t0. the subsequent long lasting and a larger amount insulin release in the second phase exhibits an explicit time delay in the same manner as in the physiological setting. together we mimic the slower secretion through liver to plasma compartment by σf(g(t− τ)) with time delay τ > 0 and the maximum insulin secretion σ > 0. the insulin transfer relates to the biological action of insulin in the slowly equilibrating interstitial space, obeying a passive diffusion process determined by the difference between the concentration levels in the plasma and interstitial compartments, for which, we adapt the structure in sturis’ model for insulin transfer between the plasma and interstitial space is described by the term e (ip(t) − ii(t)) with the diffusion transfer rate e > 0 between the two compartment. different from the saturating responses of both insulin-dependent and insulin-independent glucose uptakes in the environment of the daily life [46, 26], under the three-hour short clinic setting of ivgtt, the response of β-cells to the bolus glucose injection into the vein is to release the insulin in the readily-releasablepool abruptly followed by a large amount secretion [38, 10]. thus, taking the same modeling approach a novel ivgtt model including interstitial insulin 15 as mm and most of the subsequent work [15, 16, 35, 27, 40, 41], we also assume the hepatic glucose production (hgp) to be a constant (b > 0), the insulin-independent glucose utilization to be a linearly dependent term sgg(t), and the insulin-dependent glucose removal in interstitial space to be a term of bi-linear dependence, sig(t)ii(t), where sg stands for the ge index and si stands for the is index. the bolus glucose is infused in at t = 0 min and takes t0 minutes, typically t0 = 2 or 3, to complete. thus our model is given as follows  g′(t) = b−sgg(t) −sig(t)ii(t), i′i(t) = e (ip(t) − ii(t)) −diiii(t), i′p(t) = σf(g(t− τ)) −e (ip(t) − ii(t)) −dpiip(t), (2.1) with initial condition g(t) = φ(t) > 0,φ ∈ c([t0 −τ,t0]), ip(t0) = ip0 > 0,ii(t0) = ii0 > 0, where ip0 is the plasma insulin data at t0, and ii0 is the corresponding interstitial insulin concentration. the initial function φ(t) = { gb, for t ∈ [t0 − τ, 0), gb + t −1 0 (g0 −gb)t, for t ∈ [0, t0], (2.2) where g0 is the glucose data at the time mark t0 when the bolus glucose infusion is completed. according to [17] and [26], the term f(g(t − τ)) = (g(t − τ))γ/(αγ + (g(t − τ))γ) models the insulin secretion stimulated by glucose g with time delay τ > 0, half saturation α > 0, and γ > 1. positive constant parameters dpi and dii stand for the rates of insulin degradation in plasma and interstitial space, respectively. the model (2.1) is a generalization of the model in [27], which strengthens the stability properties for the models in [36] and [35]. it is easy to show that the model (2.1) assumes a unique positive equilibrium e0. in physiological observation, the dynamics of glucose, plasma insulin and interstitial insulin will return to their basal level after the test (180 min). we can therefore assume that the unique equilibrium point is at the basal level, that is, e0 = (gb,ipb,iib) = (gb,ib,m1ib), where m1 = e/(e + dii). this allows us to further express three model parameters in terms of other parameters, b = (sg + sim1ib)gb, σ = e(1 −m1 + dpi)ib/f(gb), dii = e(1 −m1)/m1, (2.3) which reduces the total number of parameters to be estimated from ten to seven. notice that 0 < m1 = e/(e + dii) < 1. the relation iib = m1ib is in agreement with that the interstitial insulin concentration is a fraction of the concentration of the plasma insulin at equilibrium. the fraction is shown to be about m1 ≈ 0.60 according to [39, 42, 43]. so we assume throughout this paper that iib < ipb. (2.4) remark 2.1. in mm, the initial condition of the insulin action x(0) = 0. this could hint x(t) might be interpreted as the increment of the interstitial insulin caused by the stimulus of abrupt glucose increase. we will discuss this in the last section in more details. in model (2.1), the initial condition of the interstitial insulin concentration ii0 is positive but a fraction of the plasma insulin at the basal state. remark 2.2. clearly the best way to verify the assumption of the state variable x in mm is through a carefully designed in-vivo or in human experiment by clinicians and biologists. however this had been 16 j. jin, j. li, r. xu, l. yu, and z. jin overlooked in the past. we utilize the economic in-silico approach to perform the verification through a delay differential equation (dde) model and its analysis. 3. global stability of the equilibrium of the novel model we first ensure that the model (2.1) manifests ideal qualitative behaviours. by lemma 1 in [27], the proof of the following proposition is straightforward. proposition 3.1. all solutions of model (2.1) exist for all t > t0 and are strictly positive and bounded, that is, 0 < ip(t) + ii(t) ≤ mi . = max{ip(t0) + ii(t0), σ/ min{dpi,dii}} , for t > t0, and 0 < g(t) ≤ gm = max{g(t0), b/sg} , for t > t0. now we show that the equilibrium e0 is globally asymptotically stable by employing a suitable lyapunov function. theorem 3.2. if there exist positive constants a1,b1 and c1 such that the following assumptions (h1) and (h2) hold: (h1) τ < min { a1r + a1 k 2 − b1 l2 b1c , e + dpi − l2 c + d } , (h2) [ c1(e + dii) − b1dτ + a1k 2 ][ e + dpi − τ(c + d) − l 2 ] − (b1 + c1) 2 4b1 e 2 > 0, where l = σ (γ + 1)2 4γα ( γ − 1 γ + 1 )γ−1 γ and r = sg + siiib,k = sigm , c = rl/2,d = kl/2 and gm is the maximum value of g(t), then the equilibrium eb of the system (2.1) is globally asymptotically stable. proof. assume that the condition (h1) and (h2) are satisfied. let (g(t),ip(t),ii(t)) be any positive solution. let v1(t) = a1 2 (g(t) −gb)2 + b1 2 (ip(t) − ipb)2 + c1 2 (ii(t) − iib)2, and v2(t) = b1c ∫ t t−τ ∫ t z (g(s) −gb)2dsdz + b1d ∫ t t−τ ∫ t z (ii(s) − iib)2dsdz. where a1,b1,c1 > 0. define v (t) = v1(t) + v2(t). (3.1) we shall show that dv (t)/dt < 0 along any positive solution of the system (2.1) and thus the unique equilibrium point is globally asymptotically stable. we leave the details in appendix a. � in section 4, we will apply theorem 3.2 to show the global stability in seven of the nine ivgtt experiments by determining the coefficients of the liapunov function. remark 3.1. if the parameters e = 0 and dii = 0, then the system (2.1) is reduced to the model (1) in [27], then the conditions (h1) and (h2) become a novel ivgtt model including interstitial insulin 17 (h1)′ τ < min { 2a1r + a1k − b1l b1rl , 2dpi − l l(r + k) } , (h2)′ (a1 − b1lτ) [2dpi − τl(r + k) − l] > 0. comparing with the conditions in theorem 3 in [27], the two theorems might not imply each other. 4. studies of the time courses x(t) and ii(t) with ivgtt data 4.1. data. we obtained nine individual data sets in ivgtt from literature [3, 34, 15, 36, 21] (see table 1). we respectively fit mm (1.1) and the model (2.1) with these data to estimate the model parameters and then compare the profiles {x(tk)} from mm (1.1), the insulin profiles {ip(tk)} and {ii(tk)} from the model (2.1), and the ivgtt the data {i(tk)} for each subject and analyze their correlations, where {tk}, k = 1, 2, 3, · · · , are the time marks at which the data were sampled. we compare the indices of the is and ge from mm (1.1) with that from the model (2.1) and then we observed noticeable improvement for a subgroup of subjects (dogs) in [3] labeled by fig32a, fig32b and fig32c in this paper. 4.2. method of generating the time courses. we first obtain the original parameter values and is and ge indices si and sg of mm from the report for the subject 2 by minmod millennium [6], and from [3] for the subject fig32a, fig32b and fig32c. to obtain the estimated indices si and sg for the other subjects (6, 7, 8, 27, mlabex) that are not available in the literature, we implemented mm (1.1) in python, referenced from the details of the commercial ivgtt mm software implemented in mlab by civilized software inc., silver spring, md 20906 [21]. we implemented the model (2.1) in python. then, we employ latin hypercube sampling (lhs) method [31] to estimate the parameter values for both models. all the parameter values are shown in table 3 and the corresponding time courses are demonstrated in figure 2 for the subject 2 and figure 3 for the other subjects. lhs is an effective sampling technique for generating a nearly random sample of the parameter values from a multidimensional parameter space. the property of the stratified sampling ensures that lhs requires fewer samples to represent the real distribution than simple random sampling [29]. to ensure the randomness in parameter search by lhs, we search the parameter space for sg,si,p2 and p3 in the logarithmic manner for mm, while the searches of other parameter values are linear for our model (2.1). among these samplings of parameter sets, we choose the parameter set that minimizes the deviation between the model profile and the ivgtt data characterized by the coefficient of determination (rsquared): r2 = 1 − ∑n i=1 (yexp(ti) −ysim(ti)) 2∑n i=1 (yexp(ti) −yexp) 2 where yexp(ti) and ysim(ti) are the ivgtt data and simulated profiles at time ti, respectively. to determine the parameter values of mm for the other subjects (6, 7, 8, 27 and mlabex), the initial value of the glucose level g0 is also taken as a model parameter in addition to the three model parameters, sg,p2 and p3. so the parameter space is given by θmm = {sg,p2,p3,g0} , where the ranges of the parameter values are obtained from [7] listed in table 2. the initial value of x is set to be zero. linear interpolation of the sampled insulin data is performed and applied as the input i(t). the basal insulin level is taken as the insulin data at the time mark 180 min according to mlab’s implementation. then we utilize lhs method to generate 1,000,000 independent sets of random parameter values. then, the set of parameter values with the best fitting effect for the glucose data is taken as the parameter values from which the profiles g(t) and x(t) are determined. 18 j. jin, j. li, r. xu, l. yu, and z. jin table 1. nine ivgtt data. the data of subject 2 is from [34], subjects 6, 7 and 8 are from [15], subject 27 is from [36], subject labeled by fig32a, fig32b and fig32c are from [3], and subject labeled by mlabex is from [21]. s u b j 2 s u b j 6 s u b j 7 s u b j 8 s u b j 2 7 f ig 3 2 a f ig 3 2 b f ig 3 2 c m l a b e x m in g ( m g / d l) i ( p m ) g ( m g / d l) i ( p m ) g ( m g / d l) i ( p m ) g ( m g / d l) i ( p m ) g ( m g / d l) i ( p m ) g ( m g / d l) i ( p m ) g ( m g / d l) i ( p m ) g ( m g / d l) i ( p m ) g ( m g / d l) i ( p m ) 0 9 2 .0 0 1 1 .0 0 8 7 .7 4 6 7 .9 2 8 7 .2 1 3 8 .5 7 7 7 .9 9 5 7 .9 0 8 6 .4 7 4 4 .0 0 9 3 .3 0 2 2 .7 6 6 6 .5 9 5 .8 6 1 0 1 .4 3 2 8 .6 0 9 3 .0 0 3 .0 0 1 3 0 0 .0 1 5 0 .0 1 2 3 5 0 .0 0 2 6 .0 0 2 2 5 .4 7 4 1 3 .2 1 2 9 9 .3 7 1 7 9 .4 5 2 2 6 .4 2 1 0 3 1 .4 0 3 4 5 .9 0 1 0 3 6 .0 0 2 7 8 .5 2 9 9 .8 2 2 6 4 .9 2 7 7 .3 4 3 0 0 .0 0 1 1 2 .0 6 3 2 7 3 .5 1 1 1 0 .3 4 2 8 1 .3 8 1 2 5 .4 9 5 6 3 .0 0 1 5 5 .0 0 4 2 8 7 .0 0 1 3 0 .0 0 2 1 4 .1 5 4 1 0 .3 8 2 5 9 .9 6 1 0 3 .9 8 2 2 8 .9 3 9 1 5 .7 0 2 7 5 .6 4 1 0 6 7 .0 0 2 6 6 .3 5 8 4 .6 7 2 1 6 .2 3 7 8 .5 2 2 7 4 .2 3 8 4 .0 5 5 2 3 4 .8 5 8 5 .2 6 4 0 2 .0 0 1 7 3 .0 0 6 2 5 1 .0 0 8 5 .0 0 2 0 3 .7 7 3 0 5 .6 6 2 5 3 .2 5 9 9 .7 9 2 0 3 .7 7 7 5 9 .7 0 2 6 3 .0 3 9 1 4 .0 0 2 2 4 .1 1 7 6 .5 2 1 9 9 .0 5 5 5 .0 8 2 6 6 .3 6 9 6 .3 0 7 3 3 7 .0 0 1 2 2 .0 0 8 2 4 0 .0 0 5 1 .0 0 2 0 0 .0 0 2 8 6 .7 9 2 4 4 .0 3 9 3 .9 2 2 0 1 .2 6 7 7 2 .3 0 2 4 1 .4 1 4 1 5 .0 0 1 9 5 .4 7 7 3 .0 3 1 8 4 .0 1 5 6 .8 4 2 3 9 .9 1 9 1 .0 5 1 0 2 1 6 .0 0 4 9 .0 0 1 9 5 .2 8 2 3 4 .9 1 2 2 5 .5 8 1 0 4 .8 2 1 9 6 .2 3 6 4 6 .5 0 2 2 8 .8 0 4 5 5 .0 0 1 5 8 .2 3 7 9 .4 7 1 6 8 .2 6 6 5 .0 4 2 2 7 .7 4 9 5 .1 4 3 0 3 .0 0 8 3 .0 0 1 2 2 1 1 .0 0 4 5 .0 0 1 9 2 .4 5 3 1 7 .9 2 2 2 3 .9 0 7 7 .1 5 1 8 3 .6 5 6 6 9 .2 0 2 2 7 .9 0 4 0 4 .0 0 1 5 1 .0 7 7 1 .3 2 1 5 3 .9 4 5 0 .9 8 2 1 6 .2 9 1 1 2 .0 6 1 4 2 0 5 .0 0 4 1 .0 0 2 1 8 .8 9 2 1 6 .0 0 1 3 3 .1 7 4 6 .8 3 2 0 6 .2 7 9 2 .2 2 1 5 1 7 4 .5 3 2 7 8 .3 0 2 0 3 .7 7 8 8 .8 9 1 7 3 .5 8 5 1 3 .2 0 1 4 2 .4 8 6 0 .3 5 2 4 4 .0 0 6 3 .0 0 1 6 1 9 6 .0 0 3 5 .0 0 1 1 6 .7 1 3 1 .6 7 1 9 3 .3 8 1 0 3 .9 0 1 7 1 2 6 .7 3 4 3 .3 6 1 8 2 0 8 .9 8 3 4 4 .0 0 1 9 1 9 2 .0 0 3 0 .0 0 9 0 .9 3 2 2 .9 5 1 7 6 .9 2 9 9 .8 1 2 0 1 5 8 .4 9 2 3 8 .6 8 1 8 8 .6 8 9 5 .6 0 1 4 8 .4 3 5 0 8 .2 0 1 1 5 .2 7 3 3 .4 0 2 0 4 .0 0 4 4 .0 0 2 1 1 9 9 .9 7 2 8 2 .0 0 2 2 1 7 2 .0 0 3 0 .0 0 8 3 .7 7 2 0 .0 6 1 6 4 .7 4 8 9 .3 0 2 4 1 9 2 .7 7 2 3 2 .0 0 2 5 1 5 0 .0 0 2 5 0 .0 0 1 7 0 .2 3 7 9 .6 6 1 2 3 .2 7 4 4 0 .3 0 8 6 .6 3 1 8 .3 4 9 8 .0 9 2 5 .7 8 1 4 8 .9 9 8 5 .2 1 2 7 1 6 3 .0 0 2 7 .0 0 3 0 1 3 1 .1 3 2 3 3 .9 6 1 5 0 .9 4 9 7 .2 7 1 1 5 .7 2 3 2 7 .0 0 1 7 5 .6 5 2 9 4 .0 0 8 8 .7 8 2 4 .2 3 8 3 .0 5 2 0 .5 1 1 4 2 .5 0 6 7 .7 0 1 3 7 .0 0 2 8 .0 0 3 2 1 4 2 .0 0 3 0 .0 0 3 5 1 1 8 .8 7 2 0 3 .7 7 1 3 4 .1 7 8 6 .3 7 1 0 0 .6 3 2 8 6 .8 0 1 6 3 .9 4 1 9 3 .0 0 7 2 .3 2 1 4 .6 5 4 0 1 1 5 .0 9 1 5 3 .7 7 9 5 .6 0 2 2 6 .4 0 1 5 7 .6 4 2 2 7 .0 0 9 0 .9 3 1 9 .6 5 6 6 .5 9 1 3 .4 8 1 1 5 .2 4 3 9 .6 9 4 2 1 2 4 .0 0 2 2 .0 0 4 5 1 4 9 .5 3 2 1 0 .0 0 6 1 .5 8 8 .2 0 1 1 0 .0 0 2 0 .0 0 5 0 1 0 6 .6 0 1 6 9 .8 1 1 0 1 .4 7 4 4 .4 4 8 5 .5 3 1 6 6 .0 0 1 4 7 .7 3 1 8 8 .0 0 9 8 .8 1 1 4 .4 9 5 9 .4 3 8 .2 0 1 0 5 .8 4 2 6 .8 5 5 2 1 0 5 .0 0 1 5 .0 0 5 5 5 9 .4 3 8 .2 0 6 0 9 3 .4 0 1 1 5 .0 9 8 9 .7 3 2 4 .3 2 7 5 .4 7 1 4 8 .4 0 1 3 2 .4 1 1 1 6 .0 0 1 0 9 .5 5 1 2 .9 4 6 0 .8 6 8 .2 0 9 8 .9 0 8 .2 7 8 7 .0 0 1 0 .0 0 6 2 9 2 .0 0 1 5 .0 0 7 0 1 0 8 .9 9 1 9 4 .0 0 1 0 4 .5 4 1 1 .7 6 6 3 .7 2 6 .4 5 9 6 .7 0 1 3 .5 0 7 2 8 4 .0 0 1 1 .0 0 7 5 7 8 .0 0 4 .0 0 8 0 8 2 .0 8 1 1 1 .3 2 8 5 .5 3 3 3 .5 4 7 2 .9 6 1 1 8 .2 0 9 7 .2 8 1 5 4 .0 0 1 0 4 .5 4 1 2 .9 4 6 5 .1 6 4 .6 9 9 2 .3 4 2 8 .6 8 8 2 7 7 .0 0 1 0 .0 0 9 0 9 3 .6 8 9 5 .0 0 9 9 .5 2 1 9 .4 1 6 6 .5 9 5 .2 7 9 5 .8 6 6 .3 7 8 2 .0 0 2 .0 0 9 2 8 2 .0 0 8 .0 0 1 0 0 7 7 .3 6 5 3 .7 7 8 5 .5 3 2 9 .3 5 8 9 .1 8 7 2 .0 0 1 0 5 .9 7 1 1 .7 6 6 5 .8 7 4 .1 0 9 4 .3 7 5 .7 4 1 0 2 8 1 .0 0 1 1 .0 0 1 1 0 9 9 .5 2 1 5 .2 9 6 5 .1 6 3 .5 2 9 4 .3 2 1 5 .0 6 1 2 0 8 3 .0 2 4 6 .2 3 8 8 .0 5 3 7 .7 4 7 7 .9 9 6 7 .9 0 8 4 .6 7 5 0 .0 0 9 4 .5 1 1 1 .1 8 6 5 .8 7 4 .1 0 9 4 .9 7 2 1 .4 6 8 5 .0 0 6 .0 0 1 2 2 8 2 .0 0 7 .0 0 1 4 0 8 3 .0 2 5 8 .4 9 8 7 .2 1 3 1 .0 3 8 0 .5 0 4 2 .8 0 7 9 .2 7 3 8 .0 0 1 0 0 .2 4 1 8 .8 2 6 3 .7 2 6 .4 5 9 5 .5 7 6 .7 2 1 4 2 8 2 .0 0 8 .0 0 1 5 0 8 1 .0 0 1 .0 0 1 6 0 8 2 .0 8 6 4 .1 5 8 6 .3 7 3 3 .5 4 7 7 .9 9 6 0 .4 0 7 2 .0 6 3 6 .0 0 1 0 3 .1 0 1 1 .7 6 6 5 .1 6 5 .8 6 9 6 .1 8 2 0 .1 0 1 6 2 8 5 .0 0 8 .0 0 1 8 0 8 5 .8 5 5 5 .6 6 8 7 .2 1 4 6 .9 6 8 0 .5 0 5 7 .9 0 7 2 .0 6 3 3 .0 0 1 0 3 .0 0 1 1 .7 6 6 7 .3 0 5 .2 7 9 7 .4 9 8 .8 7 8 6 .0 0 1 .0 0 1 8 2 9 0 .0 0 7 .0 0 a novel ivgtt model including interstitial insulin 19 table 2. model parameter ranges. for mm (1.1), all the parameter ranges of mm are obtained from [7]. for model (2.1), the ranges of sg and si are set slightly larger than that in [7] and the ranges of other parameter values are cited in the table. mm value range model (2.1) value range source sg (1.2 × 10−3, 4.5 × 10−2) sg (1 × 10−4, 9.9 × 10−2) [15, 7, 27] p2 (1.3 × 10−3, 2.0 × 10−1) si (1 × 10−5, 9.9 × 10−3) [15, 7] p3 (5.4 × 10−7, 8.0 × 10−5) e (0.075, 0.25) [46] g0 (150, 400) τ (3, 25) [15, 27] dpi (0, 0.2) [26] α (100, 200) [48] γ (2, 5) [48] table 3. parameter values and is and ge indices si and sg of mm and model (2.1). subj 2 subj 6 subj 7 subj 8 subj 27 fig32a fig32b fig32c mlabex mm by mlab: si (×10−4) 7.46 0.63 4.02 0.382 0.27 9.10 5.40 1.00 4.21 sg (×10−2) 0.95 1.40 1.85 2.36 1.33 7.70 4.70 5.20 1.86 g0 246.19 224.42 281.15 238.11 264.00 320.00 249.00 324.00 399.71 p2 (×10−2) 3.86 11.31 5.26 6.77 2.48 13.10 6.20 2.80 7.72 p3 (×10−5) 2.87 0.71 2.12 0.26 0.07 12.00 3.40 0.30 3.25 model (2.1): si (×10−4) 0.65 0.32 2.82 0.60 0.95 13.18 9.35 4.21 4.63 sg (×10−2) 4.63 2.54 2.04 3.99 0.02 1.91 3.06 2.06 4.50 e 0.10 0.09 0.23 0.10 0.16 0.14 0.23 0.12 0.16 τ 6.06 6.34 9.30 8.72 15.70 3.33 5.31 6.51 21.49 dpi (×10−1) 1.58 1.54 1.33 1.39 0.89 1.64 1.48 0.99 0.40 α 102.20 135.95 106.00 118.34 124.00 177.00 164.27 181.93 140.93 γ 3.08 3.38 2.70 2.15 3.98 2.80 3.00 3.19 2.94 in the initial condition function (2.2) of the model (2.1), t0 = 2 or 3 (min) according to the experiments, and the basal levels of the glucose (gb) and insulin (ib) are determined by averaging the data at the time mark t = 0 and the last data point so that the measure errors for basal concentrations are reduced. again we apply lhs method to produce 10,000 independent sets of random values in the parameter space θ(2.1) = {sg,si,τ,e,dpi,α,γ} , and fit the glucose data for g and plasma insulin data for ip to estimate the values of the seven parameters within the ranges given in table 2. 4.3. results. we take the subject 2 as an example to show the detailed comparison while summarize the comparisons of other subjects at the last. for the subject 2, applying theorem 3.2 we find a set of coefficients of the liapunov function v (t) =(g(t) −gb)2 + (ip(t) − ipb)2 + (ii(t) − iib)2 20 j. jin, j. li, r. xu, l. yu, and z. jin + 0.0017 ∫ t t−τ ∫ t z (g(s) −gb)2dsdz + 6.8414 × 10−4 ∫ t t−τ ∫ t z (ii(s) − iib)2dsdz that ensures the global stability of the equilibrium e0, that is, both glucose and insulin concentrations will return their basal levels. we plot the ivgtt data, the profiles from mm, and the profiles from the model (2.1) in figure 2. the dynamics of x(t) is multiplied by 4000 to fit into the scale of the window. denote the values of the model profiles x, ii, ip using the parameter values in table 3, and the ivgtt plasma insulin data at the sampling time marks tk by {x(tk)}, {ii(tk)}, {ip(tk)} and {i(tk)}, k = 1, 2, 3, · · · , respectively. simple correlation analysis discloses that {ip(tk)} and {i(tk)}, {ii(tk)} and {i(tk)} are strongly correlated (0.989 and 0.909), but, on the contrast, {x(tk)} is not correlated with {ii(tk)}, nor {i(tk)} with the correlation coefficients merely 0.520 and 0.219, respectively. 0 5 0 1 0 0 1 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 glu cos e (m g/d l) t ( m i n ) g ( t ) g ( t ) g l u c o s e 0 3 0 6 0 9 0 1 2 0 1 5 0 i p ( t ) i i ( t ) x ( t ) p l a s m a i n s u l i n ins uli n ( mu /m l) m o d e l ( 2 . 1 ) : d a t a : m o d e l m m : figure 2. profiles generated by mm and model (2.1) for the subject 2. for six out of the other eight subjects, given the parameter values in table 3, we are able to construct liapunov functions in theorem 3.2. the coefficients of liapunov function defined by (3.1) are shown in table 4 and hence the equilibrium e0 is globally asymptotically stable. we plot the ivgtt data and all model solution curves of these eight subjects in figure 3 in smaller size of plots. table 4. coefficients of liapunov functions in theorem 3.2 for the subject 2, 7 and 8 in [27], the subjects in fig. 32a, 32b, 32c in [3], and the example in [21]. subj 7 subj 8 fig32a fig32b fig32c mlabex a1/2 5 4 2 3 4 3 b1/2 3 1 2 1 3 2 c1/2 4 3 4 5 5 1 b1c 0.0132 0.0084 0.0056 0.0038 0.0078 7.6824 × 10−4 b1d 0.042 0.0028 0.068 0.028 0.0384 0.0044 the correlation analyses shown in table 5 on the model profiles for the other eight subjects using the parameter values in table 3 estimated in section 4.2 consistently reveal that the plasma insulin profiles a novel ivgtt model including interstitial insulin 21 table 5. correlation coefficients between insulin concentrations in different compartments from mm (1.1) and the model (2.1). subj 6 subj 7 subj 8 subj 27 fig32a fig32b fig32c mlabex {ip(tk), i(tk)} 0.958 0.939 0.929 0.961 0.967 0.960 0.975 0.984 {ii(tk), i(tk)} 0.976 0.900 0.974 0.955 0.938 0.968 0.959 0.977 {ip(tk), ii(tk)} 0.986 0.984 0.977 0.984 0.991 0.991 0.990 0.996 {x(tk), i(tk)} 0.609 0.445 0.426 -0.112 0.576 0.526 0.146 0.492 {x(tk), ii(tk)} 0.895 0.740 0.533 0.398 0.919 0.804 0.627 0.787 {ip(tk)} from the model (2.1) are strongly correlated to the ivgtt plasma insulin data {i(tk)}. the interstitial insulin profile {ii(tk)} by model (2.1) is also strongly correlated to the plasma insulin data, although slightly weaker than the correlation between the plasma insulin {ip(tk)} and the ivgtt data {i(tk)}. without any surprise, the model profile {ip(tk)} and {ii(tk)} are strongly correlated as well. these analyses for the subject 2 and other eight subjects can be supported by the known physiological fact that interstitial insulin level is fractional of plasma insulin level, roughly about 60%, when at the basal state [39, 42, 43]. these reasonably indicate that the time course ii(t) from the model (2.1) may indeed stand for the interstitial insulin concentration. on the other hand, neither of the nine profiles {x(tk)} of the auxiliary variable x in mm (1.1) show strong positive correlation to the sampled ivgtt plasma insulin data {i(tk)}. on the other hand, interestingly, the correlation coefficient between x and ii are greater than 0.8 for subject 6, fig32a, fig32b, three of the total nine subjects. but the three subjects inconsistently belong to different subgroups. the above analysis indicates that, even though x is not proportional to or consistently and strongly correlated to interstitial insulin, certain weak correlation between x and interstitial insulin still exists. in addition, observing that the initial condition x(0) = 0 and x(t) stabilizes at 0, perhaps, x(t) could be appropriately interpreted as a proportional increment of the interstitial insulin when the β-cells respond to the stimulus during ivgtt. carefully examining the parameter values of the nine subjects in table 3, it can be seen by comparison that the estimations by the model (2.1) is consistently overcome the aforementioned drawbacks of mm for the (dog) subject fig32a, fig32b and fig32c [3]. for other (human) subjects, no consistent result is observed. ha et al. [18] pointed out that the presence of a strong first insulin secretion phase could result in an underestimation of insulin sensitivity through a set of data generated by a putative model. unfortunately we could not repeat their findings by our model (2.1). in a short summary, the above results manifest that a) the dynamics of ip(t) and ii(t) produced by the model (2.1) reflect the plasma and interstitial insulin levels during the ivgtt duration. b) in mm, the variable x(t) might be the increment of the interstitial insulin, which is not correlated to the interstitial insulin dynamics. instead, it is more suitable to be considered as the proportional increment of the interstitial insulin from the basal level in responding to the bolus glucose stimulus. c) numerical studies for a subgroups (dog subjects) evidenced the potentiality of the model (2.1) to overcome the mm’s limitation – overestimating sg and/or underestimating si. 4.4. physiological meaningful parameter values of e and τ would not destabilize the equilibrium. in this section, we consider the robustness of the model (2.1) in physiological applications. 22 j. jin, j. li, r. xu, l. yu, and z. jin figure 3. profiles generated by mm and model (2.1) for the other eight subjects. a novel ivgtt model including interstitial insulin 23 under certain conditions, we have shown that the equilibrium e0 of our model (2.1) is globally asymptotically stable. it is well known that larger time delays destabilize a system. we take the explicit time delay τ and the transfer rate e between the two insulin compartments as bifurcation parameters and investigate whether their changes would undermine the stability of e0. as results, omitting the routine but lengthy mathematical treatments, we obtained that, in addition to a natural physiological assumption iib < ipb (basal interstitial insulin level is lower than the basal plasma insulin level), a necessary condition for e0 to be unstable is sg < (γ(1 −f(gb)) − 1)siiib. (4.1) for the parameter values of seven out of the nine subjects in table 3 estimated from the ivgtt data, we successfully constructed liapunov functions (refer to table 4 for the coefficients) and thus the equilibrium point e0 is globally asymptotically stable. so we focus on the subject 27 and 6 now. the parameter values of both subjects do satisfy (4.1). however our intensive numerical simulations cannot detect a bifurcation value when e varies from 0 to large, but we did detect a hopf bifurcation value for τ at τ0 ≈ 242.9 for subject 27 as shown in figure 4, which is very much out of the physiological ranges and the instability would not happen. 230 235 240 245 250 255 60 70 80 90 100 20 30 40 50 60 70 taug(t) i p (t ) figure 4. hopf bifurcation of subject 27 when τ varies. limit cycles bifurcated from a hopf bifurcation when τ changes from 0 to large. the bifurcation point is at τ0 ≈ 242.9. 5. discussions as the rapid increase of diabetic population in the world, research on the metabolic regulation in glucose and insulin becomes more and more pressing. determining is and ge are critical to find the progression pathways of t2dm and drug developments for t2dm. ivgtt is an appropriate protocol to determine these physiological characters less invasive and relatively accurate. however, the utility of the ivgtt for evaluating the essential physiological characteristics, insulin sensitivity and glucose effectiveness, has been challenged due to the minimal model [23]. a carefully formulated mathematical model for ivgtt setting is important to accomplish the assessment. one possible reason causing the 24 j. jin, j. li, r. xu, l. yu, and z. jin drawbacks may be due to limited knowledge in biology regarding the interstitial insulin when the model was formulated, which resulted in a flaw in model structure. till 2002, straub and sharp [45] pointed out that the glucose-stimulated biphasic insulin secretion in the ivgtt setting includes at lease two different pathways, the katp channel-dependent pathways and katp channel-independent pathways, which was not considered in the model formulations in mm. in this paper we formulate a reasonable model (2.1) to describe ivgtt based on physiological observations and the well accepted sturis’ model for the glucose-insulin regulation. our intensive numerical studies reveal that the parameter estimations of si and sg by our model show improvements of mm’s limitations for the dog subjects in the available data. even though the number of the available data is limited, the consistency of the estimation in the entire subgroup may bring to light the right way of the model formulation. we analytically justified that the model (2.1) is well posed and the unique positive equilibrium point is globally asymptotically stable under certain conditions, which precludes the model (2.1) from the invalidity encountered by the extended mm by pacini and bergman [34] as pointed by de gaetano and arino [15]. seven of the nine (≈ 78%) ivgtt data used in this paper do satisfy such conditions. bifurcation analysis shows that the equilibrium point is stable when the values of the key parameter e and τ are within physiological ranges, which makes applications at ease. in addition, we also observed followings through our analyses. a) as known, in daily life (also called free living), the interstitial insulin and the plasma insulin keep at a balanced stable state [39, 42, 43]. in ivgtt, the balance is contravened for about 100 minutes due to the readily released insulin stored in β-cell granules into the peripheral triggered by the bolus glucose injection into the vein. b) the interstitial insulin level can be close to the level of the plasma insulin when the second phase insulin release occurs in the first 30 minute but hardly overpasses the plasma insulin level. the balance is gradually restored. c) when the plasma insulin exhibits the second peak caused by the second phase secretion, the interstitial insulin also acts like a bump but not a peak. in such case, the difference between the plasma insulin and interstitial insulin is larger. mm directly models the insulin action. when the insulin actions on adipocytes, skeletal muscle and liver in obesity are impaired, or decreased in the pathways of glucose uptake and metabolism, the glucose removal is eventually reduced [10]. our analysis manifests that the variable x for insulin action in mm could be understood as a quantity proportional to the increment of the interstitial insulin, rather than proportional to nor strongly correlated to the interstitial insulin. it has been shown that hostile environmental factors promote the development of t2dm. marunaka [30] found that the level of ph (potential of hydrogen) in the interstitial fluid is a critical factor responsible for the occurrence of insulin resistance as ph level affects the binding affinity of insulin to its receptors. even under mild metabolic disorder conditions with blood ph is kept constant within a normal range (7.35 − 7.45), the interstitial fluid ph would be lower than a normal level due to the less ph-buffering molecules in interstitial fluids than the powerful ph-buffering molecules in blood (e.g., hemoglobin and albumin), which leads to insulin resistance [30]. quantifying the impact of ph on insulin resistance in detail by mathematical models could be significant in this area. other environmental factors promote the development of t2dm can be found in the review by dendup et al. 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and d. h. wasserman, capillary endothelial insulin transport: the rate-limiting step for insulinstimulated glucose uptake, endocrinology. 163 (2022), bqab252. 50. z. wu, c. k. chui, g. s. hong, and s. chang, physiological analysis on oscillatory behavior of glucose–insulin regulation by model with delays, j. theor. biol. 280 (2011), 1–9. a novel ivgtt model including interstitial insulin 27 appendix a. details of the proof of theorem 3.2 to calculate the derivation of v (t) = v1(t) + v2(t) along any positive solutions of system (2.1), we first have dv1(t) dt = a1(g(t) −gb)[b−sgg(t) −sig(t)ii(t)] + b1(ip(t) − ipb) [σf(g(t− τ)) −e (ip(t) − ii(t)) −dpiip(t)] + c1(ii(t) − iib) [e (ip(t) − ii(t)) −diiii(t)] = a1(g(t) −gb)[−sg(g(t) −gb) −sig(t)(ii(t) − iib) −siiib(g(t) −gb)] + b1σ(ip(t) − ipb) (f(g(t− τ)) −f(gb)) − b1(ip(t) − ipb) [(e + dpi) (ip(t) − ipb) −e(ii(t) − iib)] + c1(ii(t) − iib) [e(ip(t) − ipb) − (e + dii) (ii(t) − iib)] . (a.1) according to the mean value theorem, we have from (a.1) that dv1(t) dt = a1(g(t) −gb)[−(sg + siiib)(g(t) −gb) −sig(t)(ii(t) − iib)] + b1σ(ip(t) − ipb)f′(g(ξ))(g(t− τ) −gb) − b1(ip(t) − ipb) [(e + dpi) (ip(t) − ipb) −e(ii(t) − iib)] + c1(ii(t) − iib) [e(ip(t) − ipb) − (e + dii) (ii(t) − iib)] = − a1(sg + siiib)(g(t) −gb)2 −a1sig(t)(g(t) −gb)(ii(t) − iib) − b1(e + dpi)(ip(t) − ipb)2 + (b1 + c1)e(ip(t) − ipb)(ii(t) − iib) − c1(e + dii)(ii(t) − iib)2 + b1σ(ip(t) − ipb)f′(g(ξ))[(g(t− τ) −g(t)) + (g(t) −gb)], (a.2) where g(ξ) is between g(t−τ) and gb. because g is bounded and α and γ are positive constants, we get that f′(g(ξ)) is bounded. similarly, setting r = sg + siiib,k = sigm and l = σ (γ + 1)2 4γα ( γ − 1 γ + 1 ) γ−1 γ , where gm and l are the maximum values of g(t) and function of σf ′(g(ξ)) for g > 0, we obtain from (a.2) that dv1(t) dt ≤−a1r(g(t) −gb)2 −a1k | (g(t) −gb)(ii(t) − iib) | −b1(e + dpi)(ip(t) − ipb)2 + (b1 + c1)e(ip(t) − ipb)(ii(t) − iib) −c1(e + dii)(ii(t) − iib)2 + b1l | (ip(t) − ipb)(g(t− τ) −g(t)) | + b1l | (ip(t) − ipb)(g(t) −gb) | . (a.3) furthermore, since xy ≤ 1 2 (x2 + y2) for all x,y ≥ 0, we obtain that |(ip(t) − ipb)(g(t− τ) −g(t))| = ∣∣∣∣(ip(t) − ipb) ∫ t t−τ g′(s)ds ∣∣∣∣ = ∣∣∣∣ ∫ t t−τ [−sg(g(s) −gb) −sig(s)(ii(s) − iib) −siiib(g(s) −gb)](ip(t) − ipb)ds ∣∣∣∣ = ∣∣∣∣ ∫ t t−τ [(sg + siiib)(g(s) −gb)(ip(t) − ipb) + sig(s)(ii(s) − iib)(ip(t) − ipb)]ds ∣∣∣∣ 28 j. jin, j. li, r. xu, l. yu, and z. jin ≤ 1 2 (sg + siiib) (∫ t t−τ (g(s) −gb)2ds + τ(ip(t) − ipb)2 ) + 1 2 sigm (∫ t t−τ (ii(s) − iib)2ds + τ(ip(t) − ipb)2 ) = 1 2 [ r ∫ t t−τ (g(s) −gb)2ds + k ∫ t t−τ (ii(s) − iib)2ds + τ(r + k)(ip(t) − ipb)2 ] . (a.4) it follows from (a.3) and (a.4), we derive that dv1(t) dt ≤ −a1r(g(t) −gb)2 −a1 k 2 [ (g(t) −gb)2 + (ii(t) − iib)2 ] | −b1(e + dpi)(ip(t) − ipb)2 + (b1 + c1)e(ip(t) − ipb)(ii(t) − iib) −c1(e + dii)(ii(t) − iib)2 + b1 l 2 [ (ip(t) − ipb)2 + (g(t) −gb)2 ] + b1 lr 2 ∫ t t−τ (g(s) −gb)2ds + b1 lk 2 ∫ t t−τ (ii(s) − iib)2ds + b1τ l(r + k) 2 (ip(t) − ipb)2 = − ( a1r + a1 k 2 −b1 l 2 ) (g(t) −gb)2 −b1 [ e + dpi − τ(c + d) − l 2 ] (ip(t) − ipb)2 − [ c1(e + dii) + a1 k 2 ] (ii(t) − iib)2 + (b1 + c1)e(ip(t) − ipb)(ii(t) − iib) + b1c ∫ t t−τ (g(s) −gb)2ds + b1d ∫ t t−τ (ii(s) − iib)2ds, (a.5) where c = lr 2 ,d = lk 2 . second, for v2(t), it follows dv dt ≤ − ( a1r + a1 k 2 −b1 l 2 ) (g(t) −gb)2 −b1 [ e + dpi − τ(c + d) − l 2 ] (ip(t) − ipb)2 − [ c1(e + dii) + a1 k 2 ] (ii(t) − iib)2 + (b1 + c1)e(ip(t) − ipb)(ii(t) − iib) + b1cτ(g(t) −gb)2 + b1dτ(ii(t) − iib)2 = − ( a1r + a1 k 2 −b1 l 2 −b1cτ ) (g(t) −gb)2 −b1 [ e + dpi − τ(c + d) − l 2 ] (ip(t) − ipb)2 − [ c1(e + dii) + a1 k 2 −b1dτ ] (ii(t) − iib)2 + (b1 + c1)e(ip(t) − ipb)(ii(t) − iib). a novel ivgtt model including interstitial insulin 29 clearly, if the conditions (h1) and (h2) hold, then dv dt < 0. therefore, the steady state of eb is globally asymptotically stable according to the lyapunov theorem. this completes the proof. � jun jin, complex systems research center, shanxi university, taiyuan, shanxi, 030006, p. r. china. jiaxu li (corresponding author), department of mathematics, university of louisville, on, usa. email address: jiaxu.li@louisville.edu rui xu, complex systems research center, shanxi university, taiyuan, shanxi, 030006, p. r. china. lei yu, shanxi key laboratory of mathematical techniques and big data analysis on disease control and prevention, shanxi university, taiyuan, shanxi 030006, china. zhen jin (corresponding author), complex systems research center, shanxi university, taiyuan, shanxi, 030006, p. r. china; shanxi key laboratory of mathematical techniques and big data analysis on disease control and prevention, shanxi university, taiyuan, shanxi 030006, china. email address: jinzhn@263.net 1. introduction 2. formulation of the novel ivgtt model adapted from a physiological model 3. global stability of the equilibrium of the novel model 4. studies of the time courses x(t) and ii(t) with ivgtt data 4.1. data 4.2. method of generating the time courses 4.3. results 4.4. physiological meaningful parameter values of e and would not destabilize the equilibrium 5. discussions acknowledgements references appendix a. details of the proof of theorem 3.2 mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase volume 3, number 2, 2022, pp.119-149 https://doi.org/10.5206/mase/14612 a mathematical model of quorum quenching in biofilms and its potential role as an adjuvant for antibiotic treatment viktoria freingruber, christina kuttler, hermann j eberl, and maryam ghasemi abstract. we extend a previously presented mesoscopic (i.e. colony scale) mathematical model of the reaction of bacterial biofilms to antibiotics. in that earlier model, exposure to antibiotics evokes two responses: inactivation as the antibiotics kill the bacteria, and induction of a quorum sensing based stress response mechanism upon exposure to small sublethal dosages. to this model we add now quorum quenching as an adjuvant to antibiotic therapy. quorum quenchers are modeled as enzymes that degrade the quorum sensing signal concentration. the resulting model is a quasilinear system of seven reaction-diffusion equations for the following dependent variables: the volume fractions of up-regulated (protected), down-regulated (unprotected) and inert (inactive) biomass [particulate substances], and the concentrations of a growth promoting nutrient, antibiotics, quorum sensing signals, and quorum quenchers [dissolved substances]. the biomass fractions are subject to two nonlinear diffusion effects: (i) degeneracy, as in the porous medium equation, where biomass vanishes, and (ii) a super-diffusion singularity as it attains its theoretically possible maximum. we study this model in numerical simulations. our simulations suggest that for maximum efficacy quorum quenchers should be applied early on before quorum sensing induction in the biofilm can take place, and that an antibiotic strategy that by itself might not be successful can be notably improved upon if paired with quorum quenchers as an adjuvant. 1. introduction bacterial biofilms are groups of microorganisms that attach to an immersed surface called substratum and are enclosed in a self-produced extracellular polymeric substance (eps). this gel-like surrounding protects biofilms against eradication which makes them beneficial in various fields such as waste water treatment [53]. on the other hand in medicine and food processing where bacterial proliferation can cause serious problems such as bacterial infections, failure of medical implants or downstream proliferation of pathogens, it is important to control bacterial biofilms [33, 38, 41]. a feature of biofilms that distinguishes them from planktonic cells is their resistance against chemical and mechanical washout [48]. thus, finding a way to eradicate biofilms as effectively as possible is an active research area. various reasons have been suggested for the increased resistance of biofilm bacteria against antibiotics, such as its heterogeneous structure that allows the formation of microenvironments with different growth conditions within a colony, protection by eps that limits penetration of antibiotics such that inner received by the editors 18 january 2022; accepted 8 june 2022; published online 16 june 2022. 2020 mathematics subject classification. 92d25, 35k65, 65n08, 34c60. key words and phrases. antibiotics, biofilm, mathematical model, quorum quenching, quorum sensing. parts of this study were carried out when the authors participated various roles in the thematic program “emerging challenges in mathematical biology” of the fields institute for research in mathematical sciences in toronto. the support received is greatly acknowledged. hje also acknowledges the support received from the natural sciences and engineering research council of canada (nserc) through discovery grant rgpin 2019-05003, and research tools and infrastructure grant rti-2016-00080. 119 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14612 120 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi layers of the biofilm can not be reached and remain active after treatment, and formation of persister cells that are extremely resistant to antibiotics [1, 5, 38, 48, 50]. heterogeneity in and of biofilms occurs due to diffusion gradients that drive life in biofilms, and quorum sensing (qs). qs is a type of cell-cell communication, used to coordinate gene expression and group behavior [27]. bacteria produce and sense signalling substances; when a critical signal concentration is reached, the cells become induced, and undergo changes in gene expressions. however, induction does not always occur homogeneously. depending on environmental conditions and physical properties of biofilms like their thickness, the inner region of the biofilms might be up-regulated whereas the outer layers are downregulated. the signal substance is also called ”autoinducer” because of a positive feedback loop in the underlying gene regulatory system upregulating its own production. our focus in this study is on gramnegative bacteria [55], which often use acyl homoserine lactones (ahl) as signal molecules. besides such spatial heterogeneity there are other ways, by which qs can influence the biofilm’s resistance against antibiotics, namely regulation of virulence factors, and control of eps production [31]. furthermore, it has been shown experimentally that a low concentration of antibiotics can increase the qs activity and make the biofilms even more resistant [3, 46]. in this case the antibiotic acts as a stressor and qs, being used as a stress response mechanism, changes the biofilm’s behavior [28, 44], for example qs leads to more cooperation between cells which can induce resistance against antibiotics. one strategy to increase the susceptibility of biofilms is to disturb the qs mechanism. disruption of qs can be done in several ways such as inhibiting qs signal production, blocking the qs receptors of cells, or accelerating the degradation of qs signals. the last possibility is also referred to as ”quorum quenching” (qq) [34], while the former two will be referred to as ”quorum sensing inhibition” later in this paper. qq molecules appear naturally and are used by microbial species to gain an advantage in competitive environments. concerning an ahl-based qs system these molecules can be categorised mainly into two distinct groups of ahl-degrading molecules, the acyl-homoserine lactonase (ahllactonase) and the acyl-homoserine lactone acylase (ahl-acylase) [11]. ahl-lactonase degrades ahl by hydrolysing its lactone bond and ahl-acylase degrades ahl by hydrolysing the amide linkage between the fatty acid chain and the homoserine lactone moiety [11]. several mathematical models have been suggested using different mathematical concepts to describe biofilm development. the model that we use for our study is formulated in the framework that was originally proposed in [15]. the underlying model, which is able to reflect the ecological-mechanical duality of biofilms, was extended later to consider qs and biofilm response to antibiotics [17, 20, 25, 31]. the biofilm growth model that we use as a foundation of our work, is a nonlinear density dependent reaction-diffusion model for biomass, which is coupled with a semi-linear reaction-diffusion equation for nutrients. the nonlinearity of the biomass equation stems from two interacting nonlinear diffusion effects: (i) porous medium degeneracy as the dependent variable biomass volume fraction approaches zero, and (ii) super-diffusion singularity as this dependent variable in the diffusion coefficient approaches the known maximum density. the interplay of these two nonlinear effects assures that the solution of the biomass equation is bounded by the maximum cell density regardless of growth activity [35], and that spatial expansion of the biofilm does not take place if there is enough space for new biomass to be accumulated. moreover, it shows that interfaces between the biofilm and the liquid phase are not stationary and change over time. eventually neighbouring colonies can combine into a bigger colony, in which case their interfaces may merge and dissolve. a model of quorum quenching in biofilms. 121 mechanisms of resistance of biofilms to antibiotics have been studied extensively by proposing several mathematical models and approaches [5, 8, 9, 42, 51]. however, the focus in these studies is on physical protection. to understand the mechanism of qs in biofilms and investigate how biofilm properties and environmental conditions influence the time of qs induction, some studies propose mathematical models [6, 7, 24, 29, 57]. other modelling studies focus on examples of qs induction changing biofilm behavior, rather than on qs induction itself [20, 25, 57]. in [31], the authors studied the effect of qs on biofilm response to exposure to antibiotics. they suggested a model which accounts for the stress response mechanism and showed through computer simulations how a low, sub-lethal concentration of antibiotics can upregulate the qs activity as a stress response and thus increase the biofilm’s resistance. using qq to assist in the eradication of biofilms is quite a new strategy, and the range of mathematical models is still very limited. in [54] a quorum sensing inhibition model was introduced. the authors in that study expanded the quorum sensing models introduced in [39] and [56] and developed a complex ode model considering the whole ahl production process and all three possibilities of quorum sensing inhibition. it is shown in [23] that quorum quenching can improve the efficiency of quorum sensing inhibition and vice versa. however, their simulations show that when the therapy strategies are not combined, quorum sensing inhibitors can reduce the signal molecule by 35% and qq can reduce it by almost 100%. to the best of our knowledge, the effects of quorum quenching or quorum sensing inhibition on a bacterial community has only been studied in [54]. the authors use a complex multiscale approach to capture the mechanisms involved. our main objective is to introduce a model that captures the interplay between qs signals and qq enzymes on the population level, and its role in the temporal and spatial behavior of biofims, as well as the stress response mechanism. for this purpose, we develop further the spatio-temporal model that was proposed in [31] and account for the effect of qq on qs disruption. the resulting model will include three biomass volume fractions: down-regulated active, up-regulated active and inert (inactive) biomass, and nutrients, antibiotics, ahl and qq concentration as dissolved substrates. due to the complexity of the suggested model, we will identify the key parameters that affect the interaction of qs and qq, and study the efficiency of qq interference with biofilm stress response to sublethal antibiotics concentrations by computer simulations. 2. mathematical model 2.1. model assumptions. we develop further a mathematical model that was introduced in [31] to study the qs stress response mechanism of biofilms that are exposed to a sublethal small doses of antibiotics. new in the developed model is to include also quorum quenching (qq) to investigate how disturbing qs can affect the resistance of a biofilm to antibiotics. the model is based on the following assumptions: (1) the computational domain ω is divided into two regions: (1) the aqueous phase, ω1(t), in which the total biomass density, m is zero, i.e. ω1(t) = {(x,y) ∈ ω ⊂ r2 : m(x,y,t) = 0}; (2) the region of biofilms, ω2(t), with positive density of biomass ω2(t) = {(x,y) ∈ ω ⊂ r2 : m(x,y,t) > 0} that is surrounded by the liquid phase [15, 16]. the regions ω1(t) and ω2(t) are separated by the biofilm/water interface γ(t) = ∂ω1(t)∩∂ω̄2(t) which is not stationary and might change as the biofilm grows. 122 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi (2) biofilm growth is controlled by a dissolved nutrient, which is transported in the liquid phase by fickian diffusion. following [49], we assume that in the biofilm itself (ω2) the diffusion coefficient is smaller than in the aqueous phase, due to the increased diffusive resistance of eps. we also assume that the biofilm decays naturally. new biomass is produced by cell division. in this process nutrients are consumed at a rate proportional to the rate at which biomass is produced. the bacterial growth rate is proportional to the local biomass density and depends on the local nutrient concentration in a nonlinear fashion: if nutrient is available in abundance, 0th order kinetics (i.e. constant rate) apply [i.e. we have a saturation effect], if nutrient becomes limited the growth rate is proportional to the available nutrient concentration, i.e. we are in the 1st order reaction regime. the transition between both regimes is modeled in the usual manner by monod kinetics. eps is not explicitly modeled but subsumed in the biomass fraction, as is common in biofilm modeling. this corresponds to the assumption that the eps-to-cell ratio is constant, cf. [40]. biomass is assumed to not spread notably if locally space is available to accommodate new cells, but if the volume fraction occupied by biomass (active or inert) approaches the maximum cell density (one after non-dimensionalization) spatial movement of biomass takes places. both these effects are modeled as a single density dependent diffusion mechanism. (3) quorum sensing. bacterial cells can secret and sense ahl to communicate with each other. we define two distinct active types of biomass in the sense of ahl concentration: downregulated biomass and up-regulated biomass. when the local ahl concentration passes a critical threshold value, changes in gene expressions occur and down-regulated cells convert into up-regulated cells. back transformation from the upto the down-regulated state occurs if the ahl concentration locally drops below the critical threshold [20, 25]. down-regulated cells produce ahl at lower rates than up-regulated cells (by about one order of magnitude), see [22] for an experimental study for the bacterial species pseudomonas putida. ahl signals are subject to abiotic decay at a constant rate that depends on environmental conditions. upregulated cells are assumed to be more resistant to antibiotics than down-regulated cells (as shown in an experimental study, [37]), and are assumed to have a slightly slower growth rate due to the resources required to maintain increased resistance against antibiotics, as such processes are metabolically costly [45]. in the presence of antibiotics as stressor, more ahl is produced due to a stress response mechanism. ahl is dissolved and transported by diffusion in the surrounding aqueous phase and in the biofilm, there however at a reduced rate. (4) antibiotics are modeled as dissolved substrates which diffuse in the surrounding liquid and in the biofilm, however at different rates. they remove both types of active biomass (i.e. upregulated and down-regulated), but as per our assumption up-regulated biomass is killed slower, i.e. it is more resistant. beyond the above-mentioned improved resistance of up-regulated cells further effects like an increased biofilm production (see [10]) and by that a better physical protection against antibiotics may play a role. active cells that are killed by antibiotics become inert (i.e. still occupy some volume), which we include in the model as third biomass volume fraction. we assume that antibiotics are degraded in the action against bacteria, and also underlie natural abiotic decay following the assumption made by [13, 17]. (5) the production of ahl is counteracted by quorum quenching, which consequently delays and damps the qs up-regulation. we assume that qq molecules are a dissolved substrate and added to the system externally through the top boundary. they react with the signal molecules a model of quorum quenching in biofilms. 123 and inactivate them. in our model, we introduce quorum quenchers that act like enzymes degrading the qs signal. even for the ahl type qs signals, there are many different sources of enzymatic degradation known, not only bacteria but also eukaryotes [32]. in this reaction they are catalysts and not degraded themselves. qq molecules diffuse in the liquid and biofilm regions at different rates. as the simplest possible assumption they loose viability abiotically, at a constant rate, potential further biotic processes are left out here. vis-a-vis the underlying model of [31] the last assumption on quorum quenching is newly added in this study. 2.2. governing equation. according to the assumptions and based on the basic biofilm growth model introduced originally in [15], the mathematical model is formulated as a system of differential mass balance equations for the fractions of space occupied by the bacterial biomass types (down-regulated a, up-regulated b, inert i), and the concentrations of growth-limiting nutrient substrate n, ahl signal molecule s, antibiotics c, and quorum quenchers q over the spatial domain ω ⊂ r2. the biomass densities are then i × mmax, a × mmax, b × mmax where mmax[gm−3] is the maximum biomass density, in terms of mass cod (chemical oxygen demand) per unit volume. all these together give the governing equation as:  ∂i ∂t = ∇(d(m)∇i) + βa cn1a kn1c + c n1 + βb cn1b kn1c + c n1︸ ︷︷ ︸ inactivation of active biomass ∂a ∂t = ∇(d(m)∇a) + µa na kn + n︸ ︷︷ ︸ growth of down-regulated biomass − βa cn1a kn1c + c n1︸ ︷︷ ︸ inactivation by antibiotics + ψ τn2b τn2 + sn2︸ ︷︷ ︸ downregulation −ω sn2a τn2 + sn2︸ ︷︷ ︸ upregulation − kaa︸︷︷︸ natural decay ∂b ∂t = ∇(d(m)∇b) + µb nb kn + n︸ ︷︷ ︸ growth of up-regulated biomass − βb cn1b kn1c + c n1︸ ︷︷ ︸ inactivation by antibiotics − ψ τn2b τn2 + sn 2︸ ︷︷ ︸ downregulation + ω sn2a τn2 + sn2︸ ︷︷ ︸ upregulation − kbb︸︷︷︸ natural decay ∂n ∂t = ∇(dn (m)∇n) −νa na kn + n −νb nb kn + n︸ ︷︷ ︸ nutrient uptake ——continued on next page (2.1) 124 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi   ∂c ∂t = ∇(dc(m)∇c) − δa cn1a kn1c + c n1 − δb cn1b kn1c + c n1︸ ︷︷ ︸ antibiotic degradation − θc︸︷︷︸ abiotic decay ∂s ∂t = ∇(ds(m)∇s) + σ0(a + b)︸ ︷︷ ︸ base level signal production + µs(a + b) c ḱc + c︸ ︷︷ ︸ increased signal production in response to antibiotics + σs sn2 τn2 + sn2 b︸ ︷︷ ︸ increased signal production by up-regulated cells − νqq s kq + s︸ ︷︷ ︸ qs-qq interaction − γss︸︷︷︸ abiotic decay ∂q ∂t = ∇(dq(m)∇q) − γqq︸︷︷︸ abiotic decay where all variables are explained in the table 1: here we use m := i + a + b for the total volume table 1. description of the variables in the model (2.2) variable definition dimension i inert biomass volume fraction − a down-regulated biomass volume fraction − b up-regulated biomass volume fraction − n nutrient concentration gm−3 c antibiotic concentration gm−3 s signal molecule concentration nm q concentration of quorum quencher gm−3 fraction occupied by biomass. as per the previous studies [15, 20, 25, 31] etc., the diffusion coefficient for all biomass fractions is nonlinearly density dependent and is defined as d(m) = d m α (1−m)β [m 2d−1]. in d(m), the parameter d [m2d−1] denotes the biomass motility coefficient which is positive and much smaller than the diffusion coefficients of dissolved substrates in liquid. the nonlinear effects represented by d(m) are: (i) a porous medium degeneracy, i.e. d(m) vanishes as m ≈ 0 and (ii) a super diffusion singularity as m approaches unity. the porous medium degeneracy, mα, guarantees the finite speed for biofilm/water interface propagation if the biomass density is small, 0 < m � 1, and it is also responsible for the formation of a sharp interface between the biofilm and the surrounding liquid. the second effect (ii) at 0 � m < 1 enforces the solution to be bounded by unity as it was shown by efendiev et al. [35]. this is counteracted by the degeneracy as m = 0 at the interface. consequently, m squeezes in the biofilm region and approaches its maximum value 1. hence, the interaction of both a model of quorum quenching in biofilms. 125 non-linear diffusion effects with the growth term is needed to describe spatial biomass spreading [16]. it is known that in models of this type, as in the porous medium equation, the biomass gradients at the interface between ω1 and ω2 can blow up and that accordingly regions with m ≈ 0 and m ≈ 1 can be very close together. all reaction terms in the model (2.2) are described in detail in [31] except for the interaction between qs and qq. the reaction of quorum sensing molecules with quorum quenching molecules is modeled with michaelis-menten kinetics [12, 52, 54] in which νq is the maximum qs-qq reaction rate, and kq is the michaelis constant for qs-qq reaction. it is assumed that signal molecules do not have any effect on qq acting as enzymes, and decay of qq is described by a constant rate. the computational domain to study the mathematical model (2.2) in our simulation experiments is a rectangle of size ω = [0,l]×[0,h]. we assume dissolved substrates nutrient, antibiotics, and quorum quenchers are added through the top boundary of the domain y = h and ahl are removed through this segment. thus, a robin condition is posed for dissolved substrates at y = h. we also assume the biofilm colonies are formed on a substratum at the bottom boundary, y = 0 which is impermeable to biomass and dissolved substrates. at the lateral boundaries, x = 0 and x = l, a symmetry boundary condition, i.e. the homogeneous neumann condition, is applied for all dependent variables. this allows us to view the domain as a part of a continuously repeating (and repeatedly symmetrically mirrored) larger domain. at the top boundary, y = h, we pose a homogeneous neumann condition for the biomass fraction. thus the boundary conditions on domain ω = [0,l] × [0,h] are defined as:   at x = 0, l and y = 0 : ∂ni = ∂na = ∂nb = 0, ∂nn = ∂nc = ∂ns = ∂nq = 0, at y=h: ∂ni = ∂na = ∂nb = 0, n + λ∂nn = n∞, c + λ∂nc = c∞, s + λ∂ns = 0, q + λ∂nq = q∞ (2.2) where c∞, n∞ and q∞ are the antibiotics, nutrient and qq bulk concentration; we assumed that the bulk concentration for ahl is 0, which forces a flux of signals out of the system there. ∂n denotes the outward normal derivative. the parameter λ [mm] can be interpreted as an (externally enforced) concentration boundary layer thickness. this concentration boundary layer is linked to the convective contribution of external bulk flow to substrate supply and removal. according to [18] a small bulk flow velocity implies a thick concentration boundary layer, while a thin concentration boundary layer represents fast bulk flow, see also [14]. hence, 1/λ [mm−1] is a measure for the mass transfer from the external bulk phase into the computational domain. we refer to appendix a for existence of a bounded solution for this model. it closely follows the approach taken for similar models [19]. it is based on a regularisation of the density dependent biomass diffusion coefficient to overcome the challenges posed by the degeneracy at m = 0 and the singularity at m = 1. 126 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi 3. numerical methods and simulation setting for the numerical treatment and further result discussions, the whole system is non-dimen-sionalized with choices x̃ = x/l, t̃ = tµa for the independent variables, where l is a characteristic length scale of the computational domain and 1 µa is the characteristic time scale for growth of biomass species a. the concentration variables n, c, s and q are non-dimensionalized as ñ = n n∞ , c̃ = c c0 , s̃ = s τ , and q̃ = q q∞ where n∞ and q∞ are the bulk concentrations for nutrient and qq respectively and c0 = δa µa . note that the volume fractions i, a and b are already defined as dimensionless variables. note that, for the sake of simplicity and easier biological and physical interpretation, we will make our choices of parameters based on the dimensional values and describe simulation results in terms of those and drop the ”tilde” from our notation. for the detailed description of the nondimensionalization procedure we refer to [31]. for the spatial discretization, we introduce a uniform grid of n×m grid cells over the domain ω, and discretize the partial differential equations using a finite volume method where fluxes across grid cell boundaries are obtained from arithmetic averaging. upon introducing a lexicographical grid ordering one obtains the discrete-in-space, continuous-in-time system of 7 ·n ·m ordinary differential equations (see appendix b for the details).   di dt = dii + r1i a + r2i b + bi da dt = daa + r1aa + r2ab + ba db dt = dbb + r1b a + r2b b + bb dn dt = dnn + r1n a + r2n b + bn dc dt = dcc + r1c a + r2c b + r3c c + bc ds dt = dss + r1s a + r2s b + r3s s + bs dq dt = dqq + r1qq + bq (3.1) the matrices di,a,b,n,c,s,q are block matrices of size nm × nm that depend on the dependent variables i,a,b. they are symmetric, and weakly diagonally dominant with non-positive main diagonals and non-negative off-diagonals and contain the spatial derivative terms of each equation. the matrices r1,2(i,a,b,n,c,s) , r3(c,s) , and r1(q) are diagonal matrices of size nm ×nm which contain the reaction terms of each equation. the vectors bi,a,b,n,c,s,q are of size nm and contain contributions from the imposed boundary conditions for each biomass species and dissolved substrates. by the posed boundary conditions for biomass species the entries of bi,a,b are zero. for substrates due to the robin boundary conditions at the top, the entries bn,c,s,q are zero for all grid cells (i,j) with j < m, and for j = m they are obtained from the boundary conditions. the introduced model in (2.2) is a stiff problem due to the different time scales between the substrate and biomass equations. this can be exacerbated by non-linear diffusion effects if the biomass approaches unity somewhere. however, it is proved in appendix a that the biomass density remains below unity, i.e. the singularity is not attained. this allows using a high order time adaptive, error controlled numerical a model of quorum quenching in biofilms. 127 figure 1. the geometry of initial inoculation ω2(0) of substratum by biofilm. in our simulations initially the colonies are down-regulated, i.e. entirely of type a. method. for the time integration of the resulting ode system, the embedded rosenbrock-wanner method ros3prl [43] is used which was previously proposed and used to solve similar problems [29, 30]. initially, three down-regulated semispherical colonies are placed equidistantly on the substratum, cf. figure 1. the initial value of the nutrient n is 1; the initial values for i,b,c,s,q are set at zero. the simulations stop when a specified simulation time is reached. the parameters and their values used in computer simulations are summarized in table 2. for a better interpretation of the computer simulations of the model, we will provide two-dimensional visualizations of the simulations and define the following quantitative lumped measures: u(t) = ∫ ω u(x,y,t)dxdy where u = a,b,s,q represents dimensionless quantities. these output parameters give the total amount of active biomass fraction and dissolved substrates in the considered computational domain. further quantities of interest will be calculated from those. 4. numerical simulations the derivation of model (2.2) in section 2.2 was based on converting the assumptions detailed in section 2.1 into mathematical language. a first qualitative validation of the model formulation is to confirm that simulations of the model give results that agree with what one could deduct directly in a straightforward manner from the assumptions in simple thought experiments, i.e. to ascertain that the model assumptions and the model output are compatible and that no unforeseen effects are observed in simple scenarios. we will conduct several such model validation simulations in the next sections 4.1.1-4.1.4. after that, in section 4.2 we will conduct more targeted simulation experiments. we focus here on the role of the parameters of the quorum quenching process that is the new addition of the present study. more specifically, we focus on the parameters that encode the environmental conditions that are under the control of an experimenter. the focus here is to obtain a better understanding of the role that quorum quenching might play as an adjuvant to antibiotic therapy. per our assumptions we expect that increasing the amount of quorum quenchers and the rate at which they are supplied reduces the quorum 128 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi table 2. model parameters for system (2.2) used for computer simulations parameter symbol value dimension source growth rate of a µa 6 d −1 [20] growth rate of b µb 4 d −1 assumed nutrient monod half saturation kn 4 gm −3 [20] antibiotics monod half saturation kc 0.034 gm −3 assumed antibiotics monod half saturation for ahl ḱc 0.034 gm −3 assumed up regulation rate ω 2.5 d−1 assumed down regulation rate ψ 2.5 d−1 assumed decay rate of a by antibiotics βa 30 d −1 assumed decay rate of b by antibiotics βb 3 d −1 assumed substrate uptake rate for a and b νa,b 10 4 gm−3d−1 assumed decay rate of ahl γs 0.12 d −1 [21] threshold of ahl τ 10, 20 nm [20] ahl production induced by antibiotics µs 55000 nmd −1 [20] basic production rate of ahl σ0 5500 nmd −1 [20] production rate of ahl after qs induction σs 55000 nmd −1 assumed antibiotics degradation rate (a) δa 4 × 103 gm−3d−1 assumed antibiotics degradation rate (b) δb 4 × 103 gm−3d−1 assumed decay rate of antibiotics θ 0.001 d−1 assumed biomass motility coefficient d 10−12 m2d−1 [16] exponent of hill function for removal by antibiotics n1 2.5 − assumed degree of polymerisation n2 2.5 − [25] biofilm/water diffusivity ratio of each substrate ρn,c,s 0.1 − assumed nutrient and antibiotics diffusion coefficient in water d0n,c 10 −4 m2d−1 [20] ahl diffusion coefficient in water d0s,q 0.00007758 m 2d−1 [20] lysis rate ka,b 0.1 d −1 assumed concentration boundary layer thickness λ 0.5 mm assumed michaelis constant for qs-qq reaction kq 100 nm [12, 52, 54] max. reaction rate qs-qq νq 8640 nm [12, 52, 54] qq concentration in bulk q∞ 10 nm − decay rate of qq γq 0.12 d −1 assumed sensing signal and thus the bacteria’s protection mechanism against antibiotics, so that a given amount of antibiotics administered will become more effective. what is not a priori clear is whether there are limitations to this, i.e. whether from a certain point on further increase of quenchers supplied does not lead to further noticeable additional gain. it is also not clear a priori whether there is a threshold that must be exceeded for quorum quenchers to be effective. 4.1. model validation experiments. 4.1.1. illustrative simulation. to illustrate the interplay of qs and qq, the results of an exemplary simulation are shown in fig.2. the bulk concentration of qq is in one case set to q∞ = 10[gm −3], in the other one no qq are added (base case scenario). the qs induction threshold is chosen as τ = 10[nm]. antibiotics are added at t = 10. qq is added from the beginning, t = 0. both are turned off at the end of simulation. before adding antibiotics, the total value of active biomass, a(t) + b(t), is the same in the system with and without qq. however with qq the biofilm is mostly down-regulated, a model of quorum quenching in biofilms. 129 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 5 10 15 20 25 30 a (t )+ b (t ) t w/o qq w qq 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 5 10 15 20 25 30 a (t ) t w/o qq w qq 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 5 10 15 20 25 30 b (t ) t w/o qq w qq 0 2 4 6 8 10 12 0 5 10 15 20 25 30 s (t ) t w/o qq w qq figure 2. biomass volume fractions a(t), b(t), signal concentration s(t) in simulations with and without qq. antibiotics are administered at t = 10, qq at t = 0 in one case, and absent in the other. q∞ = 10 [nm], c∞ = 2 [gm −3]. s = 1 is the qs induction threshold. because the ahl concentration does not exceed the induction threshold. this means the biofilm is less resistant so that it can be removed over the simulation time. without qq the biofilm is mostly up-regulated at the time antibiotics are added. we observe in fig.2 that bluethe ahl concentration increases upon the onset of treatment, however, in the case with qq it is always below the threshold. this, of course, is what one expects, providing a first qualitative validation of our model formulation that reflects correctly the assumptions on which it was built. visualizations of the spatial structure of relative fraction of active biomass, r := a+b i+a+b , and concentration of ahl at t = 12 are given in fig.3. as qq prevents qs upregulation, conversion of biomass of type a to b does not occur and antibiotics can kill bacterial cells effectively which results in the development of homogeneous biofilm with near zero density of active cells, see fig.3(a). however, without qq, qs up-regulation occurs before adding antibiotics and the biofilm consists of more upregulated biomass which is more resistant to antibiotics. in the case without qq added, we observe a clear heterogeneous biomass distribution. active biomass is largest in the inner region of the biofilm. there are two possible reasons for this: (i) upregulation initiates from the inner region of biofilm, which means that the bacteria there are more resistant to antibiotics. (ii) antibiotics diffuse into the biofilm from the aqueous surrounding and decay as they inactivate biomass. this leads to lower antibiotics concentrations in the inner regions as a consequence of the interplay of diffusion and reaction. the spatial distribution of ahl concentration is given in fig.3(c). although the ahl concentration is low, in the down-regulated regime, we observe a clear ahl gradient in s from inside out. for the parameter set at hand qq is effective in that it reduces the signal to the down-regulated range. it is interesting to point out that we have here a counter diffusion phenomenon: signals are produced in the inner layer 130 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi (a) (b) (c) figure 3. relative fraction of active biomass, r := a+b i+a+b with qq (a) and without (b), at t = 12. ahl distribution with qq (c). and diffuse toward the aqueous phase, whereas quorum quenchers that neutralised the signal diffuse in the opposing direction. however, since there is no inactivation of quorum quenchers by interaction with signal or biomass, after t = 12 quorum quenchers have been in the system long enough to have completely penetrated the biofilm for our parameter set and no noteworthy gradients of q are observed. the heterogeneity in the biofilm therefore is controlled by almost fickian diffusion (of the signal) and therefore can be explained directly by maximum principles. 4.1.2. biofilm growth and disinfection versus timing of exposure to quorum quenching. an important aspect of biofilm control strategies is the timing such that best performance is achieved. in the case of qq based strategies, this means maximum qs disruption. to investigate this question, we add a specific amount of qq, 10[gm−3], at three different times: at the beginning (before adding antibiotics), at t = 10 (together with the antibiotics), and t = 15 (after adding antibiotics). the results are reported in fig.4. adding qq initially keeps the concentration of ahl below the qs threshold and upregulation does not occur before adding antibiotics. at the time of exposure to antibiotics, upregulation takes place due to the stress response mechanism but at the end of the simulation interval only a very small amount of bacterial cells remains active. the values of active biomass species in the two other cases are larger because the biofilm is already up-regulated when the treatment starts and in our parameter regime, qq is not strong enough to reduce the ahl concentration below the threshold. noteworthy here is that there is no significant difference between the results of adding qq at t = 10 and t = 15 at the end of simulation (t = 20). this finding shows that over the considered time interval and by the amount of applied qq, adding qq early on, before the biofilm can develop, is the most effective approach in terms of fast removal of bacterial cells. for a better understanding of the importance of timing of exposure to qq, we added qq at four different times and recorded the value of up-regulated biomass at t = 10, results are shown in table (3). the later qq is added, the more up-regulated biomass is present at the time of treatment. 4.1.3. the effect of qq on periodic administration of antibiotics. in medical and industrial settings, administering antibiotics periodically is a realistic and practically implementable, but not necessarily an optimal biofilm control strategy. important is here how long the periods of exposure are, and how long the periods between them during which no antibiotics are supplied. if the latter is too long the bacteria might be fully recovering, if it is too short the second dose might be ineffective. a model of quorum quenching in biofilms. 131 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 5 10 15 20 a (t )+ b (t ) t w/o qq qq at t=0 qq at t=10 qq at t=15 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 5 10 15 20 a (t ) t w/o qq qq at t=0 qq at t=10 qq at t=15 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 5 10 15 20 b (t ) t w/o qq qq at t=0 qq at t=10 qq at t=15 0 2 4 6 8 10 12 0 5 10 15 20 s (t ) t w/o qq qq at t=0 qq at t=10 qq at t=15 figure 4. biomass volume fraction a(t), b(t), ahl s(t) with and without qq. antibiotics are added at t = 10 and qq at three differentent times t = 0, t = 10, and t = 15. q∞ = 10 [gm −3], c∞ = 2 [gm −3]. s = 1 is the qs induction threshold. table 3. amount of up-regulated biomass b at t = 10, in dependence of the time qq are administered. time of adding qq b(t)|t=10 0 0.1468066116e-02 2 0.1541007139e-02 5 0.6240219094e-02 10 0.7814844603e-01 we already showed above that adding qq early on can keep the biofilm down-regulated and susceptible to antibiotics. we explore now whether qq can improve an otherwise inefficient way of antibiotics administration. for this purpose we assumed that a limited amount of qq, 10[gm−3], is added at t = 0 and antibiotics are added at t = 10 periodically (with period 5). antibiotics are on during the fist quarter of a treatment period and off over the remaining three quarters, whereas qq is continuously added. the total amount of active biomass and ahl are plotted in fig.5. we observe that without qq, ahl exceeds the induction threshold and the biofilm consists mostly of up-regulated biomass, which is more resistant but grows slower than down-regulated cells would. after adding the antibiotics the biomass volume fraction decreases only slightly and it will recover after antibiotics supply is turned off. at the beginning of the second cycle, even more biomass is available that cannot be removed during the 132 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 5 10 15 20 25 30 a (t )+ b (t ) t w/o qq w qq 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 5 10 15 20 25 30 a (t ) t w/o qq w qq 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 5 10 15 20 25 30 b (t ) t w/o qq w qq 0 2 4 6 8 10 12 14 16 0 5 10 15 20 25 30 s (t ) t w/o qq w qq figure 5. biomass volume fractions a(t) and b(t), ahl s(t) with and without qq. antibiotics are administered at t = 10 periodically and qq is added continuously starting at t = 0. q∞ = 10 [nm], c∞ = 2 [gm −3]. s = 1 is the qs induction threshold. following treatment cycles. upon adding qq at t = 0, qs up-regulation is prohibited before starting the treatment. thus, down-regulated biofilm is treated by antibiotics which is more susceptible and its value is reduced significantly after the first cycle of treatment. the given time for re-establishment of the biofilm is not large enough so total value of active biomass volume fractions decreases at the end of each cycle. at the beginning of treatment, the concentration of ahl increases instantaneously nevertheless it is below the threshold in the case with qq, see fig.5. 4.1.4. effect of quorum quenching on the stress response mechanism. a crucial assumption underlying our model is taking into account the response of qs to antibiotics as stressor. considering qs as a stress response mechanism enables our model to describe resistance of biofilm to a small dosage of antibiotics. it was shown in [31] that this is the result of qs upregulation and production of a more protected biofilm. to study the effect of qq on this mechanism is the objective of this section. for this purpose, four cases are considered: (1) without qq and without the stress response mechanism (2) without qq and with the stress response mechanism (3) with the stress response mechanism and qq activation at t = 0 (4) with the stress response mechanism and qq activation at t = 8, the time of adding antibiotics a low dosage of antibiotics, 10 times the half saturation concentration c∞ = 0.34 [gm −3] is added to the system at t = 8 and the switching threshold is set to τ = 20[nm]. to turn off the stress response mechanism, the corresponding parameter µs[nmd −1] is set to zero. the total values of biomass volume fractions and ahl for these four cases are given in fig.6. without qq, for µs = 0 the bacteria are a model of quorum quenching in biofilms. 133 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 5 10 15 20 a (t )+ b (t ) t w/o stress, w/o qq w stress w/o qq w stress, qq at t=10 w stress, qq at t=8 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 5 10 15 20 a (t ) t w/o stress, w/o qq w stress w/o qq w stress, qq at t=10 w stress, qq at t=8 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 5 10 15 20 b (t ) t w/o stress, w/o qq w stress w/o qq w stress, qq at t=10 w stress, qq at t=8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 5 10 15 20 s (t ) t w/o stress, w/o qq w stress w/o qq w stress, qq at t=10 w stress, qq at t=8 figure 6. biomass volume fractions a(t), b(t), ahl s(t) without and with qq (added at two different times t = 0 and t = 8). c∞ = 0.34 [gm −3] and s = 1 is the qs induction threshold. swiftly and completely eradicated by the antibacterial agent. consequently, the concentration of ahl reduces. in the same system but with accounting for the stress response mechanism upon exposure to antibiotics, the ahl concentration increases and surpasses the threshold almost everywhere resulting in the formation of a more protected biofilm. by adding qq at either t = 0 or t = 8, it counteracts the response of qs to antibiotics and does not allow the ahl concentration to surpass the threshold of qs upregulation. nevertheless, the biofilm is eradicated earlier if qq is added at t = 0 because of the production of down-regulated biomass. 4.2. numerical results: the effect of environmental conditions. new in our current model, vis-a-vis the model in [31] on which it is based is the quorum quenching mechanism. therefore, it is important to investigate the role of the parameters that affect quorum quenching. these are primarily: • νq: signal inactivation rate, the higher it is the more effective is qq • kq: quenching half saturation signal concentration: the higher it is, the smaller is s/(kq +s), i.e. less effective is qq • γq: quencher decay rate, the higher it is, the faster are quenchers degraded, the less effective is qq • q∞: bulk concentration of quenchers, the higher it is the more effective is qq • λ: controls how fast quenchers are added to the system (see also the discussion below in the next subsection) • dq: diffusion coefficient of quenchers, controls how fast quenchers spread in the domain; in the absence of reactions with the signal or the biomass that degrade quenchers a homogeneous distribution is reached quickly, so that this parameter plays only a small role 134 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi more specifically, γq,q∞ determine which value of q will be attained. the rate at which the signal is degraded by quenching depends proportionally on q, and on νq. so, changing q via γq,q∞ has similar effect as changing νq and only one of those factors needs to be investigated. kq controls how quenching depends on the signal concentration but the effect is limited: the smaller kq is, the less important it will be since s/(kq + s) ≈ 1. for large kq (much higher than the signal concentrations that are attained, kq � s =⇒ kq + s ≈ kq), quenching activity ∼ skq depends proportionally on s. q∞,λ are entirely controlled by the experimental setup, νq,kq,γq likely depend on the biological characteristics of a specific system and probably also on the experimental setup. under this light, owing to the high computational cost of a full fledged sensitivity analysis, we only include one of the parameters γq,q∞,νq that directly (and in a very similar manner) affect the quenching rate. more specifically we choose q∞ that is entirely controlled by the experimenter. we also include the parameter λ that is related to the role of external mass transfer. 4.2.1. external mass transfer. the effect of the fluid flow velocity on the supply of nutrients, antibiotics, qq to the system, and on the removal of autoinducers from the system is considered in our model indirectly by the external concentration boundary layer thickness λ[mm]. large values of λ[mm] correspond to slow bulk flow while low values of λ[mm] mimick fast flows. by increasing the external mass transfer, i.e. decreasing the value of λ[mm] more qq is delivered; see also [18, 31] for a discussion of boundary conditions. on the other hand, at smaller λ[mm] more ahl is washed out and more antibiotics are provided which regulates the qs upregulation. thus, it is not straightforward to predict whether increasing the external mass transfer can prevent or delay qs upregulation. to study this question, we varied the value of λ[mm] from 0 to 2[mm] and computed the total value of biomass volume fractions, ahl, and qq. the results are plotted in fig.7 and snapshots of biofilm structure and ahl concentration are given in figs.8-11. the results are also compared to the base case without qq. we assume in these simulations that antibiotics are added at t = 10 and qq is added continuously, from the onset at t = 0, as in our earlier illustrative simulations in section 4.1.1. before antibiotics are added, a smaller value of λ[mm] means more nutrient supply, which makes the biofilm bigger. conversely, after the time at which antibiotics has been first added, an increased nutrient and an increased antibiotics supply compete. the biofilm that was largest at the onset of treatment remains largest after the treatment although active biomass is totally removed for all tested values of λ after t = 15. at the beginning and before adding antibiotics the total values of biomass volume fractions are the same in the system with and without qq, but after adding antibiotics the value of the active biomass in the system with qq reduces more because the biofilm is mostly down-regulated and can be removed faster. in the system with qq, increasing the external mass transfer brings more qq into the system and also increases the ahl washout. thus, the ahl concentration stays below the threshold for all values of parameter λ. nevertheless, for λ = 0 an instantaneous spike in the signal is found upon adding antibiotics which exceeds the qs threshold (shown by dashed line). noteworthy here is having another spike at the beginning and a different trend in the temporal behavior of ahl compared with the case without qq. by increasing λ the total value of ahl increases if qq is deactivated while with qq, ahl has the maximum value for λ = 0. the reason for the instantaneous jump at t ≈ 0.1 is the low concentration of qq that cannot prevent qs and the time course of ahl has the same trend as in the case without qq. as time passes, q(t) increases and the interaction between qs and qq begins and the maximum value of ahl that was correspondent to λ = 2[mm] becomes minimum. the different behavior is caused by the small positive feedback due to a mostly down-regulated biofilm in a model of quorum quenching in biofilms. 135 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 5 10 15 20 a (t )+ b (t ) t λ=0 [mm] λ=0.5 [mm] λ=1 [mm] λ=2 [mm] 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 5 10 15 20 a (t ) t λ=0 [mm] λ=0.5 [mm] λ=1 [mm] λ=2 [mm] 0 0.02 0.04 0.06 0.08 0.1 0.12 0 5 10 15 20 b (t ) t λ=0 [mm] λ=0.5 [mm] λ=1 [mm] λ=2 [mm] 0 2 4 6 8 10 12 14 16 0 5 10 15 20 s (t ) t λ=0 [mm] λ=0.5 [mm] λ=1 [mm] λ=2 [mm] 0 0.1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 20 s (t ) t λ=0 [mm] λ=0.5 [mm] λ=1 [mm] λ=2 [mm] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 q (t ) t λ=0 [mm] λ=0.5 [mm] λ=1 [mm] λ=2 [mm] figure 7. biomass volume fractions a(t), b(t), ahl s(t) and qq (q(t)) in a system with (solid lines) and without qq (dashed lines). qq is added at t = 0 and antibiotics at t = 10, c∞ = 2 [gm −3]. s = 1 is the qs induction threshold. panel for s(t) in the third row shows the total value of ahl in a system with qq. the system with qq. by decreasing λ, the production of ahl by antibiotics and biomass species and decay of ahl by qq and through washout increases. however, under the given parameter conditions, the increase in the production of ahl is more than that in the decay of ahl. thus with qq, total value of ahl is maximum for λ = 0, see figs.7. visualizations of the spatial structure of the biofilm for different values of λ at t = 13 are depicted in fig.8. the biofilm size decreases as λ increases due to the nutrient limitation. for λ = 2[mm] we observe that for both cases with and without qq, an inactivation occurs only in a small inner region of the biofilm due to the limitation in antibiotics supply. nevertheless, the inactive zone is larger if qq is added. for λ = 1[mm], the distribution of active and inactive cells in the biofilm is relatively homogeneous in the system with qq because the biofilm is homogeneously down-regulated. however, without qq the qs upregulation makes the inner parts of the biofilm up-regulated which are more resistant to antibiotics. moreover, due to the penetration limitation, antibiotics do not reach the inner 136 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi regions of biofilm very well. thus, we note gradients of active biomass from the center to the outer regions and formation of niches in the inner core of the biofilm where bacteria are protected. for the other two values of λ, the gradients are smaller and so is the fraction of active biomass in the system without qq. while with qq, as ahl concentration is below the threshold and there is enough nutrient, biofilm grows and becomes disinfected homogeneously, see figs. 8-9. to investigate how the concentration of ahl changes spatially in the systems with and without qq, we plot a visualization of ahl concentration at t = 13 for these two cases in figs 10-11. we observe that by increasing λ in the system with qq, the concentration of ahl decreases and has the maximum value within the biofilm. however, without qq increasing λ decreases the washout of ahl and makes the gradient of the ahl concentration less steep. 4.2.2. limitation of quorum quenching bulk concentration. as discussed above, besides λ there are other parameters in our model that affect the interaction between qs and qq. these are: νq, kq, γq, dq, and q∞. among these parameters only q∞ can be entirely controlled externally by an experimenter, the others depend primarily on the biological characteristics of a specific system and probably also on the experimental setup. our main objectives in this sections are to investigate whether the concentration at which qq is added can change the qs behavior of a biofilm that is already up-regulated at the time when antibiotics are administered, and to find a range of qq concentration under which quenching is efficient for the simulation setup investigated here. for this purpose we vary the qq bulk concentration from q∞ = 0.5[gm−3] to q∞ = 50[gm −3] over a specific time interval. we assume that antibiotics and qq both are added at t = 10. the time course of biomass volume fractions and ahl is given in fig.12. by increasing the qq bulk concentration, the signal concentration s(t) drops down below the threshold earlier, which causes a back transformation from up-regulated to down-regulated biomass. hence, the antibiotics kill bacteria more effectively and less active cells remain at t = 20. furthermore, the results show that quenching is not effective below a specific qq concentration (q∞ = 10[gm −3] in the current study) and changing the qq bulk concentration around very small values does not result in a notable difference in the amount of active bacterial cells a(t) + b(t). nevertheless, increasing the bulk concentration of qq unlimitedly does not necessarily improve the quenching because of the saturation phenomenon in the qs-qq interaction. this suggests that to obtain an optimum result in terms of inhibiting or delaying the qs upregulation, a particular range of qq bulk concentration should be used that varies depending on the simulation setup and other parameter values. in our study 10 < q∞ ≤ 30[gm−3] leaves the minimum active cells at the end of the experiment. noteworthy here is a small rise in a(t) at t ≈ 15 for q∞ = 10[gm−3] and 10 < t < 15 for q∞ = 30, 50[gm −3]. the reason for this behavior is that after adding qq it takes a while for q(t) to diffuse within the liquid and biofilm and qs inhibitions to begin. this initiation time is smaller for higher concentrations of qq, see fig.12. nevertheless, the value of a at the time of stopping the treatment is almost zero in all cases. 5. discussion it is well-known nowadays, that bacteria may use their quorum sensing system to control several mechanisms which may help themselves to defend against antibiotics treatment. this may concern a direct resistance of a part of the bacterial population against antibiotics, or an indirect enhanced protection against treatment, e.g. by mechanical protection of the biofilm ([48, 49]). additionally, more a model of quorum quenching in biofilms. 137 λ = 0 λ = 0.5[mm] figure 8. biofilm at t = 13 with (top) and without (bottom) qq for various λ. shown is the relative fraction of active biomass, r := a+b i+a+b . λ = 1[mm] λ = 2[mm] figure 9. biofilm at t = 13 with (top) and without (bottom) qq for various λ. shown the relative fraction of active biomass, r := a+b i+a+b . and more bacterial species develop resistances against antibiotics in general, which creates a big need for alternative treatment methods against pathogenic bacteria. thus qq, avoiding qs-upregulation and 138 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi λ = 0 λ = 0.5[mm] figure 10. ahl concentration at t = 13 with (top) and without (bottom) qq for various λ. λ = 1[mm] λ = 2[mm] figure 11. ahl concentration at t = 13 with (top) and without (bottom) qq for various λ. by that pathogeneity of the bacteria, can be seen as alternative to the classical antibiotics treatment a model of quorum quenching in biofilms. 139 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 5 10 15 20 a (t )+ b (t ) t q ∞ =0.5 [gm -3 ] q ∞ =2 [gm -3 ] q ∞ =5 [gm -3 ] q ∞ =10 [gm -3 ] q ∞ =30 [gm -3 ] q ∞ =50 [gm -3 ] 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 5 10 15 20 a (t ) t q ∞ =0.5 [gm -3 ] q ∞ =2 [gm -3 ] q ∞ =5 [gm -3 ] q ∞ =10 [gm -3 ] q ∞ =30 [gm -3 ] q ∞ =50 [gm -3 ] 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 5 10 15 20 b (t ) t q ∞ =0.5 [gm -3 ] q ∞ =2 [gm -3 ] q ∞ =5 [gm -3 ] q ∞ =10 [gm -3 ] q ∞ =30 [gm -3 ] q ∞ =50 [gm -3 ] 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 s (t ) t q ∞ =0.5 [gm -3 ] q ∞ =2 [gm -3 ] q ∞ =5 [gm -3 ] q ∞ =10 [gm -3 ] q ∞ =30 [gm -3 ] q ∞ =50 [gm -3 ] figure 12. biomass volume fractions a(t), b(t), and ahl s(t). antibiotics and qq are added at t = 10 with bulk concentration c∞ = 2 [gm −3] and various q∞. s = 1 is the induction threshold. (e.g. [2]). one could argue now for complete replacement of antibiotics by qq, but the combination may be even more successful, as combination therapies are applied in different contexts: combining different antibiotics or combining completely different types of therapies as for tumour treatment. moreover, sometimes combining different therapies even better avoids the development of resistances against treatments. especially due to the important role of the biofilm around the bacterial cells and colonies, spatial effects are very important to be taken into account. this does not only concern the spatial arrangement of the neighbouring bacterial cells or microcolonies, but also the biofilm acting as a physical barrier and reducing the accessibility by all chemical substances, ahl, antibiotics and qq. as already taken into account in [31], the model approach provides some options as e.g. the consideration of qs as a stress response, in this case the presence of antibiotics as stressor, leading to an increased ahl production. this may result in a larger proportion of the more-resistant up-regulated biomass fraction, thus an effect counteracting the application of antibiotics. especially at that point, the additional application of a qq process or substance, applied at a suitable time point or time interval, may help a lot. a typical question in this context is of course how to choose a good, if not the best possible time schedule as treatment strategy, and to check if varying it makes a big difference at all. this could be shown e.g. in section 4.1.1: there is not much difference in the total biomass, if qq was added first or not until the antibiotic treatment was started. in both cases, the biomass was reduced significantly after the antibiotic was added. thus, the combined treatment strategy can be 140 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi very successful. but one can also observe that not all situations which prove to be helpful in the model, can be applied realistically: as it could be seen in section 4.1.2, the addition of qq would work most efficiently if applied at the very beginning. but in practical life, one may only know after a while that bacteria are present and a treatment should be started. the provided model approach allows for first examinations of such situations, even in a quantitative way if ranges of parameter values are known for the species of interest. for our simulation studies, parameter values were chosen mainly in a range suitable for the the soil bacterium p. putida, but other gram-negative bacteria with a lux-type qs system may act in similar ranges and the goal of the present study focuses on the general behavior, not on very concrete quantitative statements. by that, this very general approach could even be adapted and used for different chemicals and gene regulation systems going forward. our pde approach considers both, the time-dynamics as well as the spatial effects and therefore also describes heterogeneous, realistic situations. it includes both, a refined model component for the growth of biomass and biofilm, as well as several prototypic qs and qq phenomena in their time dynamics and a refined model for the multiple interactions between antibiotics and both, biomass growth and qs (as analysed already in greater detail in [31]). some effects even could not be observed without having this spatial structure included: in figure 3(c) a situation can be observed, where the ahl concentration is on a low total level, but nevertheless there is a clear gradient in its concentration level visible which strongly influences the behavior of bacteria in the biofilm via the nonlinearities. a homogeneous model would not show here any effect. as mentioned above, the role of stressors which increase the qs activity, may be important. figure 6 shows such an example. although, this effect might be quantitatively overestimated, it is clearly visible that qq could be useful when it comes to controlling the bacteria, even if it is added late. we also considered the external mass transfer and were able to obtain expectable results, i.e. stronger or weaker effects of qq if the mass transfer, which is responsible for supplying nutrients, antibiotics, qq and removing ahl, was larger or smaller. apart from using potential qq players as treatment, agents acting as quorum quenchers might also appear in natural settings or might be produced by other cells and organisms [12]. examples for this are known from very different species, even plants and animals [2], but not much quantitative information is available yet. in our model, we introduced a qq which acts like an enzyme degrading the qs signal. even for the ahl-type qs signals, there are many different sources of enzymatic degradation known, not only bacteria but also eukaryotes [4, 32]. models of other, maybe more specific quorum quenching systems, might require different formulations. in the parallel study [26] we investigated in simulations a mechanism in which quorum quenchers decay when they inactivate signals. this introduced additional parameters (the qq degradation rate and a corresponding half saturation concentration). depending on the values of these parameters this extended model behaves like the one we presented and discussed here (small qq degradation rate), in other parameter ranges (e.g. high qq degradation rates, small half saturation concentrations), the qq mechanism might become ineffective if qq becomes limited. where to apply such models of qq interactions? obviously, the present model is mainly suited for laboratory conditions, where most players can be controlled well and no external conditions, e.g. limitations in concentrations, need to be accounted for. taking into account e.g. patients’ needs, goes beyond this study, as this would involve too many other interactions and side conditions, such as physiological effects that remain yet to be investigated, making it difficult to study the role of qq in medical treatment with the current model. a model of quorum quenching in biofilms. 141 possible extensions of the interaction structure could be to adapt it to more specific bacterial species, e.g. by adapting the structure of the underlying qs system and the ahl production, or by considering different activity levels of biomass or other types of interactions between qs and qq. this may modify the corresponding interaction terms, but not the general structure of the model system. taken together, this prototype model may yield some first insights about the potential combination of antibiotics and quorum quenching as treatment and better control against pathogenic bacteria, and can serve as a basis for further studies. 6. conclusion we propose a nonlinear model to describe the interaction between quorum sensing and quorum quenching in biofilms as an inhibitory enzyme that can disrupt quorum sensing upregulation, and the role of quorum quenching as a potential adjuvant for antibiotic therapy. as quorum sensing increases the resistance of biofilms to antibiotics through various mechanisms, its prevention can make bacterial biofilms more vulnerable to chemical treatments. in our study, we investigated the effect of environmental conditions and time of adding quorum quenching on its influence on quorum sensing upregulation. the main findings are listed as follows: • quorum quenching counteracts the stress response mechanism and hinders the onset of quorum sensing upregulation. this leaves the biofilms in an unprotected mode of growth for a prolonged period. however upon turning off the treatment for a very long time, upregulation may occur and a biofilm in a protected mode is formed. • adding quorum quenching at the beginning prevents quorum sensing upregulation and bacteria cannot be synchronized for group behavior. this keeps the biofilm better susceptible to antibiotics. • in the case of adding quorum quenching either at the time of adding antibiotics or after that, a high concentration of quorum quenching should be used to interfere with quorum sensing. • increasing the external mass transfer which corresponds to fast fluid velocity increases the nutrient and antibiotic supply as well as quorum quenching. on the one hand, it increases washout of signal molecules. upon quorum sensing disruption, the biofilm is down-regulated, hence, grows faster and is more vulnerable to antibiotics. by decreasing λ, the ahl production dominates the qq-dependent decay of ahl. however, since all bacterial cells are inactivated, the signal molecules diminish as well. • in a system without quorum quenching, a periodic application of antibiotics is not efficient in the sense of removing biofilms. on the other hand, with quorum quenching the quorum sensing induced upregulation is prevented and a periodic administration of the same amount of antibiotics as in the case without qq can remove bacteria. references [1] j.n. anderl, m.j. franklin, p.s. stewart, 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[57] j. zhao, q. wang, three-dimensional numerical simulations of biofilm dynamics with quorum sensing in a flow cell. bull. math. biol. 79(4) (2017), 884-919. a model of quorum quenching in biofilms. 145 appendix a. existence of solutions let   ∂ni = ∂na = ∂nb = ∂nn = ∂nc = ∂ns = ∂nq = 0, at x = 0,l and y = 0 i = a = b = 0, at y = h n + λ∂nn = n∞, c + λ∂nc = c∞ at y = h s + λ∂ns = 0, q + λ∂nq = q∞ at y = h (a.1) be the boundary conditions for the pde system (2.2) and the initial conditions satisfy the following properties   a(0, ·) = a0, b(0, ·) = b0, i(0, ·) = i0, n(0, ·) = n0, c(0, ·) = c0, s(0, ·) = s0, q(0, ·) = q0 i0,a0,b0,n0,c0,s0,q0 ∈ l∞(ω) ‖a0 + b0 + i0‖l∞(ω) = 1 − δ0 < 1 0 ≤ c ≤ c∞, 0 ≤ n0 ≤ n∞, 0 ≤ s0, 0 ≤ q0 ≤ q∞ (a.2) where 0 < δ0 < 1. we have remark a.1. the boundary conditions (a.1) differ from the boundary conditions (2.2) for the components i, a, and b at the top boundary. as discussed in [13, 18, 35], solutions of the degenerate problem (2.2) exist for all t > 0 if robin or dirichlet conditions are applied somewhere along the boundary to keep the biomass density there below unity. moreover to prevent un-physical boundary condition effects that occur when the biofilm interface reaches the top boundary, our numerical solutions stop long before it happens. thus, solutions that satisfy the homogeneous neumann condition also satisfy the homogeneous dirichlet condition, i.e. solutions to problem (2.2) with boundary conditions (2.2) and (a.1) are identical and the existence proof covers our simulation periods as our solutions satisfy both (2.2) and (a.1) simultaneously. theorem a.1. the system (2.2) with boundary conditions (a.1) and initial conditions (a.2) possesses a solution in the sense of distributions in l∞(r+ ×ω)×l∞(r+ ×ω)×l∞(r+ ×ω)×l∞(r+ ×ω)× l∞(r+ × ω) ×l∞(r+ × ω) ×l∞(r+ × ω). this solution satisfies almost everywhere a ≥ 0, b ≥ 0, i ≥ 0 and a + b + i < 1, as well as 0 ≤ c ≤ c∞, 0 ≤ n ≤ n∞, 0 ≤ s, and 0 ≤ q ≤ q∞. proof. in order to show the existence of non-negative solutions, we employ the regularization idea introduced in [13, 30, 35] and define the regularized, non-degenerate quasi-linear diffusion-reaction 146 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi system   ∂i ∂t = ∇(d�(m)∇i) +βa cn1a k n1 c + c n1 + βb cn1b k n1 c + c n1 ∂a ∂t = ∇(d�(m)∇a) +µa na kn + n −βa cn1a k n1 c + c n1 +ψ τn2b τn2 + sn2 −ω sn2a τn2 + sn2 −kaa ∂b ∂t = ∇(d�(m)∇b) +µb nb kn + n −βb cn1b k n1 c + c n1 −ψ τn2b τn2 + sn 2 + ω sn2a τn2 + sn2 −kbb ∂n ∂t = ∇(dn (m)∇n) −νa na kn + n −νb nb kn + n ∂c ∂t = ∇(dc(m)∇c) −δa cn1a k n1 c + c n1 − δb cn1b k n1 c + c n1 −θc ∂s ∂t = ∇(ds(m)∇s) +σ0(a + b) + µs(a + b) c ḱc + c +σs sn2 τn2 + sn2 b −νqq s kq + s −γss ∂q ∂t = ∇(dq(m)∇q) −γqq (a.3) where the regular diffusion coefficient is defined as d�(m) :=   d�a if m < 0, d (m + �)a (1 −m)b if 0 ≤ m ≤ 1 − �, d�−b if m > 1 − �, (a.4) and show that solutions of the regularized system (a.3) converges to solutions of the original degenerate problem (2.2) if � → 0. since the pde system (a.3) is regular, the existence of solutions is concluded using standard arguments, e.g., those in ladyženskaja et al. [36] and following the positivity criterion in [13] these solutions denoted by (i�,a�,b�,n�,c�,s�,q�) are non-negative. in order to show the boundedness of biomass volume fractions by unity (which is required physically and mathematically), we add the equations for the biomass volume fractions to obtain ∂m� ∂t = ∇(d�(m�)∇m�) + (µaa� + µbb�) n� kn + n� −kaa� −kbb� (a.5) where m� := i� + a� + b�. introducing µ = max{µa,µb} and k = min{0,ka,kb} yields ∂m� ∂t ≤∇(d�(m�)∇m�) + µ n�m� kn + n� (a.6) defining the following single-species model with solutions (m̃�,ñ�, c̃�, s̃�,q̃�) a model of quorum quenching in biofilms. 147   ∂m̃� ∂t = ∇(d�(m̃�)∇m̃�) + µ ñ�m̃� kn + ñ� ∂ñ� ∂t = ∇(dn (m̃�)∇ñ�) −ν ñ�m̃� kn + ñ� ∂c̃� ∂t = ∇(dc(m̃�)∇c̃�) −δ c̃n1� m̃� kn1c + ñ n1 � −θc̃� ∂s̃� ∂t = ∇(ds(m̃�)∇s̃�) + σm̃� + µ̄s m̃�c̃� ḱc + c̃� −νq q̃�s̃� kq + s̃� −γss̃� ∂q̃� ∂t = ∇(dq(m̃�)∇q̃�) −γqq̃� (a.7) gives m̃� as the upper solution for m�, i.e., m̃� ≥ m�. using the same arguments in [13, 35, 47], it can be shown that m̃� is bounded by unity indicating the boundedness of m� by a value less than one. hence, due to the non-negativity of i�,a�,b�, we conclude that 0 ≤ i�,a�,b� < 1. to complete the proof, we need to show that solution of the regularized problem (a.3) is convergent to the solutions of the degenerate problem (2.2) as � → 0. this can be done by constructing an upper bound solution and employing comparison theorem. we refer for the technical details of the procedure to [35, 47]. � appendix b. numerical discretisation for spatial discretization, we introduce a uniform grid of size n × m for the rectangular domain [0, 1] × [0,h/l] and integrate each equation of the resulting nondimensionalized system over each grid cell. for instance, integrating the equation for a over grid cell with index (i,j) and using the divergence theorem yields d dt ∫ vi,j a dx dy = ∫ ∂vi,j jnds + ∫ vi,j r1a(n,c,s)a dx dy + ∫ vi,j r2a(s)b dx dy, (b.1) for i = 1, ...,n and j = 1, ...,m. here, vi,j represents the domain of the grid cell, jn = d(m)∂na denotes the outward normal flux across the grid cell boundary, and r1a(n,c,s) = n kn + n −βa cn1 kn1c + c n1 −ω sn2 1 + sn2 −ka and r2a(s) = ψ 1 1 + sn2 stand for the reaction terms. to evaluate the area integrals in (b.1), we evaluate the dependent variables at the center of the grid cells, ai,j(t) := a(t,xi,yj) ≈ a ( t, ( i− 1 2 ) ∆x, ( j − 1 2 ) ∆x ) (b.2) for i = 1, ...,n and j = 1, ...,m with ∆x = 1/n = h/lm; the value of all other dependent variables at the center of the grid cells can be evaluated in a similar way. the integrals in (b.1) are evaluated by the midpoint rule and the line integral is evaluated by considering every edge of the grid cell separately. to 148 v. freingruber, c. kuttler, h.j. eberl, and m. ghasemi this end, the diffusion coefficient d(m) in the midpoint of the cell edge is approximated by arithmetic averaging from the neighbouring grid cell center points, and the derivative of a across the cell edge by a central finite difference. these result in the following ordinary differential equation for a in cell (i,j): d dt ai,j = 1 ∆x ( ji+ 1 2 ,j + ji−1 2 ,j + ji,j+ 1 2 + ji,j−1 2 ) + r1ai,j ai,j + r2ai,j bi,j, (b.3) where r1ai,j = ni,j kn + ni,j −βa cn1i,j kn1c + c n1 i,j −ω sn2i,j 1 + sn2i,j −ka and r2ai,j = ψ 1 1 + sn2i,j and for the fluxes we have, accounting for the boundary conditions, ji,j+ 1 2 = { 1 2∆x ( d(mi,j+1) + d(mi,j) ) (ai,j+1 −ai,j) for j < m, − 2 ∆x d(0)ai,m for j = m, (b.4) ji,j−1 2 = { 0 for j = 1, 1 2∆x ( d(mi,j−1) + d(mi,j) ) (ai,j−1 −ai,j), for j > 1. (b.5) ji+ 1 2 ,j = { 0 for i = n, 1 2∆x ( d(mi+1,j) + d(mi,j) ) (ai+1,j −ai,j) for i < n, (b.6) ji−1 2 ,j = { 0 for i = 1, 1 2∆x ( d(mi−1,j) + d(mi,j) ) (ai−1,j −ai,j) for i > 1, (b.7) the spatial discretization of the equations for the other dependent variables i,b,n,c,s,q follows the same principle. the major difference is for the substrates for which we have a robin boundary condition at the top of the domain instead of homogeneous neumann condition. we refer the reader to [30] for a detailed description of the changes to the discretization that this introduces. introducing the lexicographical grid ordering π : {1, ...,n}×{1, ...,m}→{1, ...,nm} , (i,j) 7→ p = (j − 1)n + i (b.8) and the vector notation i = (i1, ...,inm ), a = (a1, ...,anm ), b = (b1, ...,bnm ), n = (n1, ...,nnm ), c = (c1, ...,cnm ), s = (s1, ...,snm ) and q = (q1, ...,qnm ) with (i,a,b,n,c,s,q)p := (i,a,b,n,c,s,q)π(i,j) = (i,a,b,n,c,s,q)i,j for i = 1, ...,n, j = 1, ...,m, give the coupled system of 7 ·n ·m ordinary differential equations a model of quorum quenching in biofilms. 149   di dt = dii + r1i a + r2i b + bi da dt = daa + r1aa + r2ab + ba db dt = dbb + r1b a + r2b b + bb dn dt = dnn + r1n a + r2n b + bn dc dt = dcc + r1c a + r2c b + r3c c + bc ds dt = dss + r1s a + r2s b + r3s s + bs dq dt = dqq + r1qq + bq (b.9) maxwell institute of mathematical sciences and department of mathematics, heriot-watt university email address: vf10@hw.ac.uk zentrum mathematik, technische universität münchen email address: kuttler@ma.tum.de department of mathematics and statistics, university of guelph email address: heberl@uoguelph.ca corresponding author, department of applied mathematics, university of waterloo email address: m23ghase@uwaterloo.ca 1. introduction 2. mathematical model 2.1. model assumptions 2.2. governing equation 3. numerical methods and simulation setting 4. numerical simulations 4.1. model validation experiments 4.2. numerical results: the effect of environmental conditions 5. discussion 6. conclusion references appendix a. existence of solutions appendix b. numerical discretisation mathematics in applied sciences and engineering https://ojs.lib.uwo.ca/mase online first, pp.1-24 https://doi.org/10.5206/mase/15949 optimal actuator placement for control of vibrations induced by pedestrian-bridge interactions martin deosborns arop, henry kasumba, juma kasozi, and fredrik berntsson abstract. in this paper, an optimal actuator placement problem with a linear wave equation as the constraint is considered. in particular, this work presents the frameworks for finding the best location of actuators depending upon the given initial conditions, and where the dependence on the initial conditions is averaged out. the problem is motivated by the need to control vibrations induced by pedestrian-bridge interactions. an approach based on shape optimization techniques is used to solve the problem. specifically, the shape sensitivities involving a cost functional are determined using the averaged adjoint approach. a numerical algorithm based on these sensitivities is used as a solution strategy. numerical results are consistent with the theoretical results, in the two examples considered. 1. introduction an actuator is a device that introduces or prevents motion in a control system [10]. in this work, an actuator is defined as a device that prevents motion in a control system. optimal actuator placement problems involve the question of finding the optimal location of the subdomain [23]. they arise naturally in many practical applications, for example, in seismic inversion [20], placement of loudspeakers for ideal acoustics [11], and medical applications [2]. there are extensive works on the optimal actuator placement problems governed by linear ordinary differential equations in the literature, see [9, 22] and the references therein. from among the earlier publications in this direction, we quote the work in [9], where the optimal placement of actuators and sensors for gyroelastic bodies is studied based on controllability and observability criteria. another important study is by van de wal and de jager [22], where a linear system is solved using controllability and observability gramians. the optimal placement of actuators in dynamical systems governed by heat, advection, and wave equations has also received a growing amount of attention. in [21], an actuator and sensor placement problem is considered using an advection equation with an application in building systems. the authors proposed a gramian criterion, where the degree of controllability and observability is maximized with respect to the least controllable and observable states. an optimal actuator design and placement problem for a linear heat equation is investigated in [10] using a shape and topology optimization approach. the authors parametrized the actuators by considering controls over some subsets of the domains using indicator functions. in [7] and [8], optimal stabilizations of the one-dimensional wave equation are investigated using a genetic algorithm and frequential analysis approach, respectively. furthermore, the optimal location of controllers for the one-dimensional wave equation is studied in [16] as an exact controllability problem received by the editors 31 january 2023; accepted 7 august 2023; published online 20 august 2023. 2020 mathematics subject classification. primary 54c40, 14e20; secondary 46e25, 20c20. key words and phrases. functionals, wave equation, optimal actuator placement, shape optimization, finite difference. m. d. arop was supported in part by sida bilateral programme (2015–2022) with makerere university; project 316. 1 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15949 2 m. d. arop, h. kasumba, j. kasozi, and f. berntsson using the frequential analysis approach. in addition, the optimal location of the support of the control for the one-dimensional wave equation as an exact controllability problem is studied in [14]. inspired by the work in [14], we study an optimal actuator placement problem for linear wave dynamics by using shape optimization techniques. in particular, we extend the techniques presented in [10] to a dynamic system governed by the linear wave equation. numerical realization of the problem is achieved by using a finite difference method, see e.g., [13]. in this paper, we determine the optimal actuator placement for the stabilization of pedestrian-bridge vibrations. more precisely, we use a shape optimization approach to find the optimal actuator location so that the vibrations induced by pedestrian-bridge interactions are controlled. the remainder of this paper is organized as follows. in section 2, we fix the notations utilized in the sequel and formulate the state and optimization problems. section 3 is devoted to proving wellposedness and deriving the optimality system for our optimization problems. in section 4, we derive the shape derivatives of the optimization problems. numerical tests that illustrate the theoretical results are given in section 5. the paper ends with concluding remarks and future work. 2. formulation of the problem 2.1. notations. let g be either the domain ω or its boundary ∂ω. then, we define l2(g) as a linear space of all measurable functions y : g → r such that ‖y‖l2(g) := (∫ g |y|2 dx )1 2 < ∞. the standard sobolev space of order m ∈ r+ ∪{0}, denoted by hm(g), is defined as hm(g) := {y ∈ l2(g)|dγy ∈ l2(g), for all 0 ≤ |γ| ≤ m}, where dγ is the weak partial derivative and γ is a multi-index. the norm ‖ · ‖hm(g) associated with hm(g) is given by ‖y‖hm(g) := √√√√ ∑ |γ|≤m ∫ g |dγy|2 dx. for a functional space x, we denote by lp(0,t; x) (1 ≤ p < ∞) the space of measurable functions y : [0,t] → x such that ‖y‖lp(0,t;x) := (∫ t 0 ‖y(·, t)‖px dt )1 p < ∞, where t is the final time. the space of essentially bounded functions from [0,t] into x is denoted by l∞(0,t; x) and is equipped with the norm ess supt∈[0,t]‖y(·, t)‖x, where ess sup denotes the essential supremum. the duality pairing between h10 (ω) and h −1(ω) will be denoted by 〈·, ·〉h−1(ω),h10 (ω) while the inner product in r2 will be denoted by (·, ·). we denote the control space by u := l2(0,t; l2(ω)) and the collection of measurable subdomains of ω by e(ω). we shall use l2(l2(ω)),l2(h10 (ω)) and l∞(h10 (ω)) as the short forms for l 2(0,t; l2(ω)),l2(0,t; h10 (ω)) and l ∞(0,t; h10 (ω)), respectively. 2.2. setup of the problem. in this work, we consider the problem of controlling vibrations induced by pedestrian-bridge interactions, see figure 1. the vibrations y(x,t) at position x and time t are governed by the wave equation: optimal actuator placement for control of vibrations 3 ω1 ω2 ω figure 1. control of vibrations on the domain ω using the supports at ω := ω1 ∪ω2. ∂2y ∂t2 −4y = χωu, (x,t) ∈ ω × (0,t], y = 0, (x,t) ∈ ∂ω × (0,t], (2.1) y(x, 0) = f(x), ∂y ∂t (x, 0) = g(x), x ∈ ω, where u = u(x,t) denotes the control variable, χω the characteristic function for the domain ω ⊂ ω, and x ∈ r2. the domain ω represents the location of the actuators. it is not known where these supports should be placed in order to control the vibrations on the bridge. the goal is to determine the optimal location of these supports. the vibrations may depend on the initial conditions f and g, control variable u, and subdomain ω. this leads to the cost functional j : e(ω)×uad×h10 (ω)×l2(ω) → r defined by j(ω,u,f,g) := ∫ t 0 1 2 ‖yu,f,g,ω(·, t)‖2l2(ω) + 1 2 ∥∥∥∥dyu,f,g,ωdt (·, t) ∥∥∥∥2 l2(ω) + α 2 ‖χωu(·, t)‖2l2(ω)dt, (2.2) where α > 0 is a given parameter and uad is the admissible set of controls consisting of a closed and convex subset of u. the first and second terms in (2.2) suggest that we minimize the vibrations and speed, respectively while the third term is the control cost. remark 2.1. the notation χωu(x,t) is used to stress the fact that u(x,t) is zero outside of ω. let ω,f and g be fixed. then by taking the infimum of the cost j over all controls u ∈ uad, we obtain the functional j1 : e(ω) ×h10 (ω) ×l2(ω) → r defined by j1(ω,f,g) := inf u∈uad j(ω,u,f,g). (2.3) note that the shape functional j1 depends on the initial conditions f and g. to overcome such a dependence, we introduce a functional j2 : e(ω) → r defined by j2(ω) := sup f∈k1,g∈k2 j1(ω,f,g), (2.4) where k1 and k2 denote weakly compact subsets of h 1 0 (ω) and l 2(ω) defined by k1 := {f : ‖f‖h10 (ω)) ≤ 1} and k2 := {g : ‖g‖l2(ω)) ≤ 1}, respectively. these conditions are used to average out the dependence of j1 on the initial conditions, and overcome overflow for large values of f and g. after introducing the two functionals in (2.3) and (2.4), we now study the problems of finding a minimum cost functional for a fixed ω ⊂ ω and a lipschitz vector field x. 4 m. d. arop, h. kasumba, j. kasozi, and f. berntsson definition 2.1. the optimal actuator placement problems related to j1 and j2 are defined by the minimization problems: inf x∈r2 j1((id + x)(ω),f,g) (2.5) and inf x∈r2 j2((id + x)(ω)), (2.6) where f ∈ k1, g ∈ k2 and (id + x)(ω) := {x + x : x ∈ ω}, respectively. 3. well-posedness of the functionals to simplify the analysis, we reformulate the wave equation as a system. note that by setting ∂yu,f,g,ω ∂t = vu,f,g,ω, we can rewrite (2.1) as the following first-order system:  ∂yu,f,g,ω ∂t −vu,f,g,ω = 0, (x,t) ∈ ω × (0,t], ∂vu,f,g,ω ∂t −4yu,f,g,ω −χωu = 0, (x,t) ∈ ω × (0,t], yu,f,g,ω(x, 0) = f(x), vu,f,g,ω(x, 0) = g(x), x ∈ ω, yu,f,g,ω = 0, (x,t) ∈ ∂ω × (0,t]. (3.1) this reformulation is useful in the derivation of the optimality system and the discretization of the optimization problems. the well-posedness of (3.1) and hence, (2.1) is guaranteed by the following lemma: lemma 3.1. let f ∈ h10 (ω),g ∈ l2(ω) and χωu ∈ l2(l2(ω)). then the problem  〈 ∂vu,f,g,ω ∂t ,φ 〉 h−1(ω),h10 (ω) + ∫ ω oyu,f,g,ω ·oφ dx = ∫ ω χωuφ dx, ( ∂yu,f,g,ω ∂t ,ψ ) = ( vu,f,g,ω,ψ ) , (3.2) for all φ ∈ l2(h10 (ω)) and ψ ∈ l2(l2(ω)) for a.e. t ∈ (0,t] with yu,f,g,ω(x, 0) = f(x), vu,f,g,ω(x, 0) = g(x), has a unique weak solution yu,f,g,ω ∈ l2(h10 (ω)) and vu,f,g,ω ∈ l2(l2(ω)) with ∂vu,f,g,ω ∂t ∈ l2(h−1(ω)). moreover, yu,f,g,ω ∈ l∞(h2∩h10 (ω)) and vu,f,g,ω ∈ l∞(h10 ∩l2(ω)), and there exists a constant c > 0 that depends on ω and t such that ‖yu,f,g,ω‖l∞(h10 (ω)) + ‖v u,f,g,ω‖l∞(l2(ω)) ≤ c ( ‖χωu‖l2(l2(ω)) + ‖f‖h10 (ω) + ‖g‖l2(ω) ) . (3.3) proof. it is well known that problem (3.2) has a unique and stable weak solution yu,f,g,ω ∈ l∞(h10 (ω))∩ l2(h10 (ω)) and v u,f,g,ω ∈ l∞(l2(ω)) ∩l2(l2(ω)), see e.g., [6, chap. 7] . � now, we establish the convergence of the sequence of solutions to (3.1). optimal actuator placement for control of vibrations 5 lemma 3.2. suppose that {fn} is a sequence in k1 that converges weakly in h10 (ω) to f ∈ k1, {gn} is a sequence in k2 that converges weakly in l 2(ω) to g ∈ k2 and {un} is a sequence in uad that converges weakly to a function u ∈ uad. then: yun,fn,gn,ω → yu,f,g,ω in l2(h10 (ω)) as n →∞, vun,fn,gn,ω → vu,f,g,ω in l2(l2(ω)) as n →∞. proof. note that inequality (3.3) implies that the sequences {yun,fn,gn,ω} and {vun,fn,gn,ω} are bounded in l2(h2(ω) ∩ h10 (ω)) and l2(h10 (ω) ∩ l2(ω)), respectively. by rellich-kondrachov theorem (see e.g.,[1]), we can extract the subsequences again denoted by {yun,fn,gn,ω} and {vun,fn,gn,ω} such that {yun,fn,gn,ω} converges weakly to yu,f,g,ω in l2(h2(ω)∩h10 (ω)) and strongly to yu,f,g,ω in l2(h10 (ω)), and {vun,fn,gn,ω} converges weakly to vu,f,g,ω in l2(h10 (ω)) and strongly to vu,f,g,ω in l2(l2(ω)). thus, replacing (u,f,g,ω) by (un,fn,gn,ω) in problem (3.2), we may pass to the limits and obtain by the uniqueness that y = yu,f,g,ω and v = vu,f,g,ω. � in the following lemma, we check that the optimization problem (2.3) is well-posed. lemma 3.3. problem (2.3) admits a unique optimal solution u. proof. we refer to [19, chap. 1]. � the notation uf,g,ω will be used to indicate that u depends on f,g,ω. lemma 3.4. suppose that {fn} is a sequence in h10 (ω) that converges weakly to f in h10 (ω) and {gn} is a sequence in l2(ω) that converges weakly to g in l2(ω). then we have ufn,gn,ω → uf,g,ω in uad as n →∞, where uf,g,ω solves (2.3). proof. since ufn,gn,ω minimizes j with (ω,f,g) replaced by (ω,fn,gn), for all u ∈ uad and n ≥ 0, it follows from (3.3) that we must have 1 2 ∫ t 0 ‖yu fn,gn,ω,fn,gn,ω(·, t)‖2l2(ω) + ‖v ufn,gn,ω,fn,gn,ω(·, t)‖2l2(ω) + α‖χωu fn,gn,ω(·, t)‖2l2(ω)dt ≤ 1 2 ∫ t 0 ‖yu,fn,gn,ω(·, t)‖2l2(ω) + ‖v u,fn,gn,ω(·, t)‖2l2(ω) + α‖χωu(·, t)‖ 2 l2(ω)dt, (3.4) ≤ c(‖χωu‖2l2(l2(ω)) + ‖fn‖ 2 h10 (ω) + ‖gn‖2l2(ω)). this implies that {un} := {ufn,gn,ω} is bounded in uad. by rellich-kondrachov theorem, we can extract a subsequence {unk} such that unk ⇀ u in uad as k →∞. since u is a unique solution of j(ω, ·,f,g), the whole sequence {un} converges weakly to u in uad as n →∞. thus, using lemma 3.2 and by weak lower semicontinuity of norms, we may pass to the limit infimum in (3.4) to obtain∫ t 0 ‖yu,f,g,ω(·, t)‖2l2(ω) + ‖v u,f,g,ω(·, t)‖2l2(ω) + α‖χωu(·, t)‖ 2 l2(ω)dt ≤ ∫ t 0 ‖yu,f,g,ω(·, t)‖2l2(ω) + ‖v u,f,g,ω(·, t)‖2l2(ω) + α‖χωu(·, t)‖ 2 l2(ω)dt, (3.5) for all u ∈ uad. so, we must have u = uf,g,ω and since uf,g,ω is the minimizer of j(ω, ·,f,g) (see e.g., lemma 3.3), the whole sequence {un} converges weakly to uf,g,ω. therefore, un ⇀ uf,g,ω in uad. as a consequence of weak lower semicontinuity, we must have ‖uf,g,ω‖l2(l2(ω)) ≤ limk→∞ inf ‖unk‖l2(l2(ω)) = ‖u f,g,ω‖l2(l2(ω)). 6 m. d. arop, h. kasumba, j. kasozi, and f. berntsson thus, it follows from (3.5) that the norm ‖ufn,gn,ω‖l2(l2(ω)) converges to ‖uf,g,ω‖l2(l2(ω)). the weak convergence and norm convergence of (un) imply that u fn,gn,ω → uf,g,ω in uad as n →∞. � the following result will be used to characterize the optimal solution u. theorem 3.5. suppose that uad = u. then we have the following optimality system: ∂yu,f,g,ω ∂t −vu,f,g,ω = 0, (x,t) ∈ ω × (0,t], ∂vu,f,g,ω ∂t −4yu,f,g,ω −χωu = 0, (x,t) ∈ ω × (0,t], yu,f,g,ω(x, 0) = f, vu,f,g,ω(x, 0) = g, x ∈ ω, yu,f,g,ω = 0, (x,t) ∈ ∂ω × (0,t], (3.6) ∂pu,f,g,ω ∂t −wu,f,g,ω = −vu,f,g,ω, (x,t) ∈ ω × (0,t], ∂wu,f,g,ω ∂t −4pu,f,g,ω = −yu,f,g,ω, (x,t) ∈ ω × (0,t], pu,f,g,ω(x,t) = 0, wu,f,g,ω(x,t) = 0, x ∈ ω, pu,f,g,ω = 0, (x,t) ∈ ∂ω × (0,t] (3.7) and αχωu−χωpu,f,g,ω = 0, (x,t) ∈ ω × (0,t], (3.8) where pu,f,g,ω ∈ l2(h10 (ω)),wu,f,g,ω ∈ l2(l2(ω)) and (yu,f,g,ω,vu,f,g,ω,u,pu,f,g,ω,wu,f,g,ω) solves (3.6)–(3.8). proof. the optimality system (3.6)–(3.8) can be easily proved using standard techniques, see e.g., [12, theorem 2.1], [19, chap. 3]. � remark 3.1. let uad ( u. then, instead of (3.8), we find the variational inequality∫ ω×[0,t] (αχωu−χωpu,f,g,ω)(u−u) dxdt ≥ 0, for all u ∈ uad. (3.9) the optimal solution u is now characterized using (3.9). in the following lemma, the well-posedness of j2 is checked. lemma 3.6. let k1 and k2 be two weakly compact sets containing the respective origins. then for every ω ∈ e(ω), we can find f ∈ k1 and g ∈ k2 satisfying ‖f‖h10 (ω) ≤ 1, ‖g‖l2(ω) ≤ 1 and j2(ω) = j1(ω,f,g). proof. note that 0 ∈ uad. let f ∈ k1 and g ∈ k2 with fixed ω ∈ e(ω). then in the absence of control, using (3.3) we have j1(ω,f,g) = min u∈uad j(ω,u,f,g) ≤ ∫ t 0 1 2 ‖y0,f,g,ω(·, t)‖2l2(ω) + 1 2 ‖v0,f,g,ω(·, t)‖2l2(ω) dt, ≤ c(‖f‖2h10 (ω) + ‖g‖ 2 l2(ω)) ≤ cr 2, (3.10) where r = √ 2. since f ∈ k1 and g ∈ k2, it follows that fr ∈ k1 and g r ∈ k2 with ‖fr‖h10 (ω) ≤ 1, ‖ g r ‖l2(ω) ≤ 1. next, we show that j2(ω) = j1(ω,f,g). from (2.4), we have j2(ω) = sup f∈k1,g∈k2 ∫ t 0 1 2 ‖yu f,g,ω,f,g,ω(·, t)‖2l2(ω) + 1 2 ‖vu f,g,ω,f,g,ω(·, t)‖2l2(ω) + α 2 ‖χωuf,g,ω(·, t)‖2l2(ω) dt. (3.11) optimal actuator placement for control of vibrations 7 let {fn} ⊂ k1, ‖fn‖h10 (ω) ≤ 1 and {gn} ⊂ k2, ‖gn‖l2(ω) ≤ 1 be maximizing sequences. then, (3.11) can be written as j2(ω) = lim n→∞ ∫ t 0 1 2 ‖yu fn,gn,ω,fn,gn,ω(·, t)‖2l2(ω) + 1 2 ‖vu fn,gn,ω,fn,gn,ω(·, t)‖2l2(ω) + α 2 ‖χωufn,gn,ω(·, t)‖2l2(ω) dt. (3.12) since {fn} and {gn} are bounded in k1 and k2, respectively, a subsequence {fnk} converges weakly to f ∈ k1; {gnk} converges weakly to g ∈ k2. since {fn}⊂ k1 and {gn}⊂ k2, the limit elements satisfy ‖f‖h10 (ω) ≤ limk→∞ inf ‖fnk‖h10 (ω) ≤ 1, ‖g‖l2(ω) ≤ limk→∞ inf ‖gnk‖l2(ω) ≤ 1, by lower semicontinuity of norms. thus, ‖f‖h10 (ω) ≤ 1 and ‖g‖l2(ω) ≤ 1. since {fnk} and {gnk} are bounded in h10 (ω) and l 2(ω), respectively, fnk ⇀ f ∈ h 1 0 (ω) and gnk ⇀ g in l 2(ω). from lemma 3.2, we note that {yun,fn,gn,ω} converges strongly to yu,f,g,ω in l2(h10 (ω)) and {vun,fn,gn,ω} converges strongly to vu,f,g,ω in l2(l2(ω)), and from lemma 3.4, ufn,gn,ω → uf,g,ω in uad as n → ∞. thus, by lower semicontinuity, we have norm convergence. hence, we may pass to the limit in (3.12) and obtain j2(ω) = 1 2 ∫ t 0 ‖yu f,g,ω,f,g,ω(·, t)‖2l2(ω) + ‖v uf,g,ω,f,g,ω(·, t)‖2l2(ω) + α‖χωu f,g,ω(·, t)‖2l2(ω)dt = j1(ω,f,g). since f ∈ k1 and g ∈ k2 satisfy ‖f‖h10 (ω) ≤ 1, ‖g‖l2(ω) ≤ 1, maxf∈k1,g∈k2 j1(ω,f,g) =: j2(ω) = j1(ω,f,g), it follows that the map ω 7→ j2(ω) is well-posed. � 4. sensitivity analysis of the functionals 4.1. shape derivative. in order to compute the shape derivatives of j1 and j2, we introduce a perturbation of the identity. consider the space c̊0,1(ω,r2) of lipschitz vector fields vanishing on ∂ω. we define a perturbation of the identity tτ (x) by tτ (x) := x + τx(x), where x ∈ ω, x ∈ c̊0,1(ω,r2) and τ is the perturbation parameter [5, p.175]. in view of the perturbation of the identity, we give the definition of a shape derivative of j as follows. definition 4.1. the directional derivative of j at ω ∈ e(ω) in the direction x ∈ c̊0,1(ω,r2) is defined by dj(ω)(x) := lim τ↘0 j(tτ (ω)) −j(ω) τ , provided the limit exists. remark 4.1. the cost functional j is shape differentiable at ω if x 7→ dj(ω)(x) is linear and continuous for all x ∈ c̊0,1(ω,r2), see e.g., [5] and [4]. 4.2. sensitivity of the state equation. the space-time cylinder and its boundary will be denoted by ωt := ω × (0,t] and γt := γ × (0,t], respectively. the sensitivity of the solution of (3.1) is given in the following lemma. 8 m. d. arop, h. kasumba, j. kasozi, and f. berntsson lemma 4.1. let tτ = id + τx,τ ≥ 0. suppose that ω is perturbed such that ωτ := tτ (ω),ω ∈ e(ω). then on the perturbed domain ωτ × (0,t] with ωτ := tτ (ω),τ ≥ 0, we have ∂yu,f,g,τ ∂t −vu,f,g,τ = 0 in ωt , (4.1) ∂vu,f,g,τ ∂t − 1 ζ(τ) div(a(τ)∇yu,f,g,τ ) = χωu in ωt , (4.2) yu,f,g,τ (x, 0) = f(x) ◦ tτ, vu,f,g,τ (x, 0) = g(x) ◦ tτ in ω, (4.3) yu,f,g,τ = 0 on γt , (4.4) where a(τ) := ζ(τ)(∂tτ ) −1(∂tτ ) −>, ζ(τ) := |det(∂tτ )|. (4.5) proof. in view of (3.1) with ωτ := tτ (ω),ω ∈ e(ω), we have ∂yu,f,g,ωτ ∂t −vu,f,g,ωτ = 0 in ωt , (4.6) ∂vu,f,g,ωτ ∂t −4yu,f,g,ωτ = χωτu in ωt , (4.7) yu,f,g,ωτ (x, 0) = f(x),vu,f,g,ωτ (x, 0) = g(x) in ω, (4.8) yu,f,g,ωτ = 0 on γt , (4.9) where ωτ ⊂ ω. thus, considering (4.7) on the perturbed domain ωτ × (0,t] with ωτ = tτ (ω),τ ≥ 0, we get the perturbed weak formulation:∫ ωτ×(0,t] ∂vu,f,g,ωτ ∂t ϕ dxτdt + ∫ ωτ×(0,t] ∇yu,f,g,ωτ ·∇ϕ dxτdt = ∫ ωτ×(0,t] χωτuϕ dxτ dt, (4.10) for all ϕ ∈ l2(h10 (ωτ )) with (yu,f,g,ωτ ,vu,f,g,ωτ ) satisfying (4.6)–(4.9). next, employing a change of variables induced by ωτ := tτ (ω) in (4.10) gives∫ ωt ζ(τ) ∂(vu◦t −1 τ ,f,g,ωτ ◦ tτ ) ∂t (ϕ◦ tτ ) dxdt + ∫ ωt ζ(τ)o(yu◦t −1 τ ,f,g,ωτ ◦ tτ ) ·o(ϕ◦ tτ ) dxdt = ∫ ωt ζ(τ)(χωτu◦ tτ )(ϕ◦ tτ ) dxdt, for all ϕ ∈ l 2(h10 (ωτ )). (4.11) applying the chain rule (see e.g., [17, p.63]) in (4.11) together with χωτ = χω◦t−1τ and the perturbed variables (see e.g., [5, p.523]) yu,f,g,τ = yu◦t −1 τ ,f,g,ωτ ◦ tτ, vu,f,g,τ = vu◦t −1 τ ,f,g,ωτ ◦ tτ, (4.12) yield ∫ ωt ζ(τ) ∂vu,f,g,τ ∂t (ϕ◦ tτ ) + ζ(τ)(∂tτ )−>∇yu,f,g,τ · (∂tτ )−>∇(ϕ◦ tτ ) dxdt = ∫ ωt ζ(τ)(χωu)(ϕ◦ tτ ) dxdt, for all ϕ ∈ l2(h10 (ωτ )). (4.13) from (4.5), equality (4.13) simplifies to∫ ωt ζ(τ) ∂vu,f,g,τ ∂t (ϕ◦ tτ ) dxdt + ∫ ωt a(τ)∇yu,f,g,τ ·∇(ϕ◦ tτ ) dxdt = ∫ ωt ζ(τ)(χωu)(ϕ◦ tτ ) dxdt, for all ϕ ∈ l2(h10 (ωτ )). (4.14) optimal actuator placement for control of vibrations 9 since (4.14) is true for all ϕ ∈ l2(h10 (ωτ )), it follows that for all φ ∈ l2(h10 (ω)) the function φ◦ t−1τ belongs to l2(h10 (ωτ )). so, testing (4.14) with ϕ := φ◦t−1τ for an arbitrary φ ∈ l2(h10 (ω)), we obtain∫ ωt ζ(τ) ∂vu,f,g,τ ∂t (φ◦ t−1τ ◦ tτ ) + a(τ)∇y u,f,g,τ ·∇(φ◦ t−1τ ◦ tτ ) dxdt = ∫ ωt ζ(τ)(χωu)(φ◦ t−1τ ◦ tτ ) dxdt, for all φ ∈ l 2(h10 (ω)). (4.15) rewriting (4.15), we have∫ ωt ζ(τ) ∂vu,f,g,τ ∂t φ dxdt + ∫ ωt a(τ)∇yu,f,g,τ ·∇φ dxdt = ∫ ωt ζ(τ)χωuφ dxdt, for all φ ∈ l2(h10 (ω)). (4.16) similarly, considering (4.6) on ωτ × (0,t], it can be shown that:∫ ωt ζ(τ) ∂yu,f,g,τ ∂t ψ − ζ(τ)vu,f,g,τψ dxdt = 0, for all ψ ∈ l2(l2(ω)). (4.17) thus, after mapping back (4.16) and (4.17), and using (4.8)–(4.9) in (4.12), we have (4.1)–(4.4). � in the following essential lemma, the sequence {τn}∞n=1 will be necessary. lemma 4.2. let x ∈ c̊0,1(ω,r2). (a) then as τn → 0+, we have ζ(τn) − 1 τn → div(x) strongly in l∞(ω), (4.18) a(τn) − i τn → div(x)i −∂x −∂x> strongly in l∞(ω,r2×2), (4.19) where i is the 2-dimensional identity matrix. (b) suppose that {ψn} is a sequence in h10 (ω) converging weakly to ψ ∈ h10 (ω). (i) then for all ψ ∈ h10 (ω), we have as τ → 0+, ψn ◦ tτ → ψ strongly in h10 (ω). (4.20) (ii) if {τn} is a null sequence, then as n →∞ we have ψn ◦ tτn − ψn τn ⇀ ∇ψ · x weakly in h10 (ω). (4.21) proof. the results of the convergence (4.18), (4.19), and (4.21) are proved in [17]: lemma 2.31, p.107 and proposition 2.72, respectively while (4.20) is proved in [5, p.527]. � remark 4.2. there are constants c1,c2 > 0 such that for all x ∈ ω and τ ∈ [0,τx],τx ≥ 0, c1 ≤ ζ(τ)(x), c2|ζ|2 ≤ a(τ)(x)ζ · ζ, (4.22) for all ζ ∈ r2, see e.g., [5, p.559]. the following lemma gives the a-priori estimates for yu,f,g,ωτ , yu,f,g,τ , vu,f,g,ωτ and vu,f,g,τ . lemma 4.3. for all (u,f,g,ω) ∈ uad × h10 (ω) × l2(ω) × e(ω), there exists a constant c > 0, such that ‖yu,f,g,ωτ‖l2(h10 (ω)) + ‖v u,f,g,ωτ‖l2(l2(ω)) ≤ c ( ‖χωτu‖l2(l2(ω)) + ‖f‖h10 (ω) + ‖g‖l2(ω) ) , (4.23) 10 m. d. arop, h. kasumba, j. kasozi, and f. berntsson ‖yu,f,g,τ‖l2(h10 (ω)) + ‖v u,f,g,τ‖l2(l2(ω)) ≤ c ( ‖χωu‖l2(l2(ω)) + ‖f‖h10 (ω) + ‖g‖l2(ω) ) . (4.24) proof. note that (4.23) is a consequence of (3.3) and the proof is omitted here. we prove (4.24) as follows. by a change of variables, we have∫ t 0 ‖yu,f,g,τ (·, t)‖2l2(ω) + ‖∇y u,f,g,τ (·, t)‖2l2(ω) dt = ∫ ωt |yu,f,g,τ|2 + |∇yu,f,g,τ|2 dxdt, = ∫ ωt ζ−1(τ)|yu,f,g,τ ◦t−1τ | 2 + ζ−1(τ)∇yu,f,g,τ ◦t−1τ ·∇y u,f,g,τ ◦t−1τ dxdt, (4.25) = ∫ ωt ζ−1(τ)|yu◦t −1 τ ,f,g,ωτ |2 + a−1(τ)∇yu◦t −1 τ ,f,g,ωτ ·∇yu◦t −1 τ ,f,g,ωτ dxdt, (4.22) ≤ c ∫ ωt |yu◦t −1 τ ,f,g,ωτ |2 + ∇yu◦t −1 τ ,f,g,ωτ ·∇yu◦t −1 τ ,f,g,ωτ dxdt, (4.23) ≤ c ( ‖χωτu◦ t −1 τ ‖ 2 l2(l2(ω)) + ‖f‖ 2 h10 (ω) + ‖g‖2l2(ω) ) . using χωτ = χω ◦t−1τ and the natural norm on h1(ω), i.e.,∫ t 0 ‖yu,f,g,τ (·, t)‖2l2(ω) + ‖∇y u,f,g,τ (·, t)‖2l2(ω) dt = ‖y u,f,g,τ‖2l2(h1(ω)), in (4.25) (see e.g., [3, p.39]), we obtain the desired inequality. � for the continuity results of (u,f,g,τ) 7→ yu,f,g,τ and (u,f,g,τ) 7→ vu,f,g,τ , we prove the lemma that follows. lemma 4.4. for every (ω1,u1,f1,g1), (ω2,u2,f2,g2) ∈ e(ω) × uad × h10 (ω) × l2(ω), with (y1,v1) and (y2,v2) being the corresponding solutions to (4.6)–(4.9), there is a constant c > 0, independent of (ω1,u1,f1,g1) and (ω2,u2,f2,g2), such that ‖y1 −y2‖l2(h10 (ω)) + ‖v1 −v2‖l2(l2(ω)) ≤ c ( ‖χω1u1 −χω2u2‖l2(l2(ω)) + ‖f1 −f2‖h10 (ω) + ‖g1 −g2‖l2(ω) ) . (4.26) proof. since (y1,v1) and (y2,v2) solve (4.6)–(4.9), it follows that they satisfy ∂yk ∂t −vk = 0 in ωt , ∂vk ∂t −4yk = χωkuk in ωt , yk(x, 0) = fk(x),vk(x, 0) = gk(x) in ω, yk = 0 on γt , for all k = 1, 2. let y12 := y1 −y2 and v12 := v1 −v2. then (y12,v12) satisfies ∂y12 ∂t −v12 = 0 in ωt , ∂v12 ∂t −4y12 = χω1u1 −χω2u2 in ωt , y12(x, 0) = f1(x) −f2(x),v12(x, 0) = g1(x) −g2(x) in ω, y12 = 0 on γt . optimal actuator placement for control of vibrations 11 hence, (4.26) follows from (3.3). � the following lemma is an immediate consequence of lemma 4.4. lemma 4.5. let ω ∈ e(ω) be given. suppose that for all τn ∈ (0,τx], un,u ∈ uad, fn,f ∈ h10 (ω) and gn,g ∈ l2(ω), un ⇀ u in uad, fn ⇀ f in h 1 0 (ω),gn ⇀ g in l 2(ω), τn → 0, as n →∞. then: yun,fn,gn,τn → yu,f,g,ω in l2(h10 (ω)) as n →∞, vun,fn,gn,τn → vu,f,g,ω in l2(l2(ω)) as n →∞. proof. using inequality (4.24), we see that the sequences {yun,fn,gn,τn} and {vun,fn,gn,τn} are bounded in l2(h2(ω) ∩h10 (ω)) and l2(h10 (ω) ∩l2(ω)), respectively. by rellich-kondrachov theorem, we can extract subsequences {yunk,fnk,gnk,τnk} and {vunk,fnk,gnk,τnk} such that {yunk,fnk,gnk,τnk} converges weakly to yu,f,g,ω in l2(h2(ω) ∩h10 (ω)) and strongly to yu,f,g,ω in l2(h10 (ω)), and {v unk,fnk,gnk,τnk} converges weakly to vu,f,g,ω in l2(h10 (ω)) and strongly to v u,f,g,ω in l2(l2(ω)). from (4.16) and (4.17), it is known that (yk,vk) with yk := y unk,fnk,gnk,τnk and vk := v unk,fnk,gnk,τnk , k ∈{0}∪n satisfies the variational formulations∫ ωt ζ(τnk) ∂vk ∂t ϕ + a(τnk)∇yk ·∇ϕ dxdt = ∫ ωt ζ(τnk)χωunkϕ dxdt,∫ ωt ζ(τnk) ∂yk ∂t ψ dxdt− ∫ ωt ζ(τnk)vkψ dxdt = 0, (4.27) for all ϕ ∈ l2(h10 (ω)) and ψ ∈ l2(l2(ω)) with yk(x, 0) = fnk(x) ◦ tτnk and vk(x, 0) = gnk(x) ◦ tτnk in ω. from lemma 4.2, it follows that fnk(x) ◦ tτnk → f(x) in h 1 0 (ω) and gnk(x) ◦ tτnk → g(x) in l2(ω) as k → ∞. thus, we have y(x, 0) = f(x) and v(x, 0) = g(x). using the weak convergence of {unk}, {yk}, {vk} and the strong convergence in lemma 4.2, i.e., ζ(τnk) → 1 in l ∞(ω), a(τnk) → i in l∞(ω,r2×2) as k →∞, we pass to the limits in (4.27) and obtain∫ ωt ∂v ∂t ϕ + ∇y ·∇ϕ dxdt = ∫ ωt χωuϕ dxdt,∫ ωt ∂y ∂t ψ dxdt− ∫ ωt vψ dxdt = 0, (4.28) for all ϕ ∈ l2(h10 (ω)) and ψ ∈ l2(l2(ω)) with y(x, 0) = f(x),v(x, 0) = g(x). furthermore, since (4.28) with y(x, 0) = f(x),v(x, 0) = g(x) admits a unique solution, we must have y = yu,f,g,ω and v = vu,f,g,ω. thus, the sequences {yn} and {vn} converge to y = yu,f,g,ω in l2(h10 (ω)) and v = vu,f,g,ω in l2(l2(ω)), respectively. this finishes the proof. � the following lemmas will be employed in the proof of the theorem that follows. lemma 4.6. for every null-sequence {τn} in [0,τx], every sequence {fn} in k1 converging weakly in h10 (ω) to f ∈ k1 and for every sequence {gn} in k2 converging weakly in l2(ω) to g ∈ k2, we have ufn,gn,τn → uf,g,ω in uad as n →∞. proof. we proceed as follows. note that ωτn, u fn,gn,ωτn and ufn,gn,τn represent the perturbed domain, optimal control solution, and perturbed optimal control, respectively. since ufn,gn,τn = ufn,gn,ωτn ◦tτn (see e.g., [5, p.523]) and ufn,gn,ωτn → uf,g,ω in uad by lemma 3.4, the desired result follows from lemma 4.2. � 12 m. d. arop, h. kasumba, j. kasozi, and f. berntsson in the sequel, we denote the set of maximizers by x2(ω). lemma 4.7. for every null sequence {τn} in [0,τx] and every sequence {fn,gn} with (fn,gn) ∈ x2(ωτn), we can find a subsequence {fnk,gnk}, such that fnk ⇀ f in h 1 0 (ω) and gnk ⇀ g in l 2(ω) as k →∞, where (f,g) ∈ x2(ω). proof. it is easy to prove this from (2.4) and the proof is left out. � 4.3. averaged adjoint equations. let τ ∈ [0,τx] be fixed. then the mapping t−1τ : uad → uad, u 7→ t−1τ ◦ u is a bijection between uad and uad that preserves the binary operations. as a consequence and using the change of variables tτ , it is easy to show that inf u∈uad j(ωτ,u,f,g) = 1 2 inf u∈uad ∫ ωt ζ(τ) ( |yu,f,g,τ|2 + |vu,f,g,τ|2 + α|u|2 ) dxdt. note that p ∈ l2(h10 (ω)) and w ∈ l2(l2(ω)). by choosing lagrange multipliers φ = p and ψ = w, we can incorporate (4.1)–(4.4) in the formulation of the following lagrangian functional. definition 4.2. define the parametrized lagrangian h̃ : [0,τx] ×uad ×k1 ×k2 ×h10 (ω) ×l2(ω) ×h10 (ω) ×l2(ω) → r by h(τ,u,f,g) := ∫ ωt 1 2 ζ(τ) ( (yu,f,g,τ )2 + (vu,f,g,τ )2 + α(u)2 ) dxdt + ∫ ωt ζ(τ) ∂vu,f,g,τ ∂t pu,f,g,τ + a(τ)∇yu,f,g,τ ·∇pu,f,g,τ − ζ(τ)χωupu,f,g,τ + ζ(τ) ∂yu,f,g,τ ∂t wu,f,g,τ − ζ(τ)vu,f,g,τwu,f,g,τ dxdt (4.29) + ∫ ω ζ(τ)(yu,f,g,τ (x, 0) −f ◦ tτ )wu,f,g,τ (x, 0) + ζ(τ)(vu,f,g,τ (x, 0) −g ◦ tτ )pu,f,g,τ (x, 0)dx, where h(τ,u,f,g) := h̃(τ,u,f,g,yu,f,g,τ,vu,f,g,τ,pu,f,g,τ,wu,f,g,τ ). in the sequel, the following definition is used. definition 4.3. given τ ∈ [0,τx], 0 ≤ s ≤ 1 and (u,f,g) ∈ uad ×k1 ×k2. we define the averaged adjoint equations associated with yu,f,g,τ and yu,f,g,ω; vu,f,g,τ and vu,f,g,ω as: find pu,f,g,τ ∈ l2(h10 (ω)) and wu,f,g,τ ∈ l2(l2(ω)) such that∫ 1 0 ∂yh̃(τ,u,f,g,sy u,f,g,τ + (1 −s)yu,f,g,ω,vu,f,g,τ,pu,f,g,τ,wu,f,g,τ )(φ)ds = 0, (4.30) for all φ ∈ l2(h10 (ω)), and∫ 1 0 ∂vh̃(τ,u,f,g,y u,f,g,τ,svu,f,g,y,τ + (1 −s)vu,f,g,ω,pu,f,g,τ,wu,f,g,τ )(ψ)ds = 0, (4.31) for all ψ ∈ l2(l2(ω)), where ∂yh̃ and ∂vh̃ denote the partial derivatives of h̃ with respect to y and v, respectively. the following lemmas will be important in the proof of the theorem that follows. optimal actuator placement for control of vibrations 13 lemma 4.8. the averaged adjoint equations (4.30) and (4.31), associated with yu,f,g,τ and yu,f,g,ω; vu,f,g,τ and vu,f,g,ω are given by∫ ωt −ζ(τ)φ ∂wu,f,g,τ ∂t dxdt + ∫ ωt a(τ)∇φ ·∇pu,f,g,τ dxdt = − ∫ ωt 1 2 ζ(τ)(yu,f,g,τ + yu,f,g,ω)φ dxdt, for all φ ∈ l2(h10 (ω)) (4.32) and ∫ ωt −ζ(τ)ψ ( ∂pu,f,g,τ ∂t + wu,f,g,τ ) dxdt = − ∫ ωt 1 2 ζ(τ)(vu,f,g,τ + vu,f,g,ω)ψ dxdt, (4.33) for all ψ ∈ l2(l2(ω)), respectively. proof. since yu,f,g,τ 7→ h̃(τ,u,f,g,yu,f,g,τ,vu,f,g,τ,pu,f,g,τ,wu,f,g,τ ) is affine, h̃ is gâteaux differentiable with respect to y, see e.g., [19, p.200]. thus, it is easy to see that (4.32) and (4.33) hold. � the lemma that follows is a direct consequence of lemmas 4.5 and 4.8. lemma 4.9. for all τn ∈ (0,τx], un ∈ uad, fn ∈ k1 and gn ∈ k2, such that un ⇀ u in uad, fn ⇀ f in h 1 0 (ω),gn ⇀ g in l 2(ω), τn → 0, as n →∞, where u ∈ uad, f ∈ k1 and g ∈ k2, we have pun,fn,gn,τn → pu,f,g,ω in l2(h10 (ω)) as n →∞, wun,fn,gn,τn → wu,f,g,ω in l2(l2(ω)) as n →∞, with pu,f,g,ω ∈ l2(h10 (ω)) and wu,f,g,ω ∈ l2(l2(ω)) satisfying the adjoint equations∫ ωt −φ ∂wu,f,g,ω ∂t dxdt + ∫ ωt ∇φ ·∇pu,f,g,ω dxdt = − ∫ ωt yu,f,g,ωφ dxdt,∫ ωt −ψ ∂pu,f,g,ω ∂t −ψwu,f,g,ω dxdt = − ∫ ωt vu,f,g,ωψ dxdt, for all φ ∈ l2(h10 (ω)) and ψ ∈ l2(l2(ω)) with pu,f,g,ω(x,t) = 0 and wu,f,g,ω(x,t) = 0 a.e. in ω. proof. using (3.6)–(3.7) and the estimate in [6, p.391-393, theorem 6] , we have the a-priori bound for the adjoint given by ‖pu,f,g,ω‖l2(h10 (ω)) + ∥∥∥∥∂pu,f,g,ω∂t ∥∥∥∥ l2(l2(ω)) ≤ c‖vu,f,g,τ + vu,f,g,ω‖l2(l2(ω)). (4.34) using similar arguments as in lemma 4.5 and replacing (u,f,g,τ) by (un,fn,gn,τn) in (4.32) and (4.33), and passing to the limits as n →∞, we have the desired result. � 4.4. directional derivative of max-min functions. let h : [0,τx] ×uad ×k1 ×k2 → r be a function. then, we define the max-min function h : [0,τx] → r by h(τ) := sup f∈k1,g∈k2 inf u∈uad h(τ,u,f,g). in the following lemma, we seek to find out sufficient conditions for the existence of the limit d d` h(0+) := lim τ↘0+ h(τ) −h(0) `(0) , for any function ` : [0,τx] → r such that `(τ) > 0 for τ ∈ (0,τx], and `(0) = 0. lemma 4.10. assume that the following hypotheses hold. 14 m. d. arop, h. kasumba, j. kasozi, and f. berntsson (h0) the problem inf u∈uad h(τ,u,f,g) admits a unique optimal solution u. (h1) the set of maximizers x2(ω) := {(f,g) : sup f∈k1,g∈k2 inf u∈uad h(τ,u,f,g) = inf u∈uad h(τ,uτ,f,g,f,g)} is nonempty for all τ ∈ [0,τx]. (h2) for all f ∈ k1,g ∈ k2 and τ ∈ [0,τx], the partial derivatives lim τ↘0 h(τ,uτ,f,g,f,g) −h(0,uτ,f,g,f,g) `(τ) and lim τ↘0 h(τ,u0,f,g,f,g) −h(0,u0,f,g,f,g) `(τ) exist and are equal. (h3) for all τn ∈ [0,τx] and (fn,gn) ∈ x2(ωn), there exist subsequences {τnk} and {fnk,gnk} with fnk ⇀ f in h 1 0 (ω) and gnk ⇀ g in l 2(ω) as k →∞ and (f,g) ∈ x2(ω), such that lim k→∞ h(τnk,unk,fnk,gnk) −h(0,unk,fnk,gnk) `(τnk) = ∂`h(0 +,u0,f,g,f,g) and lim k→∞ h(τnk,u fnk,gnk,0,fnk,gnk) −h(0,u fnk,gnk,0,fnk,gnk) `(τnk) = ∂`h(0 +,uf,g,0,f,g). then, we have d d` h(τ)|τ=0+ = max (f,g)∈x2(ω) ∂`h(0 +,u0,f,g,f,g). proof. we refer to [5, p.524] and [18]. � in the following theorem, we derive the directional derivative of j2 for `(τ) = τ. theorem 4.11. the directional derivative of j2(ω) at ω in the direction x ∈ c̊0,1(ω,r2) is given by dj2(ω)(x) = max (f,g)∈x2(ω) ∫ ωt s1 ( yf,g,ω,vf,g,ω,pf,g,ω,wf,g,ω,uf,g,ω ) : ∂x + s0(f,g) · x dxdt, (4.35) where s1 ( yf,g,ω,vf,g,ω,pf,g,ω,wf,g,ω,uf,g,ω ) := ( 1 2 |yf,g,ω|2 + 1 2 |vf,g,ω|2 + α 2 |uf,g,ω|2 −vf,g,ω ∂pf,g,ω ∂t −yf,g,ω ∂wf,g,ω ∂t + ∇yf,g,ω ·∇pf,g,ω −χωuf,g,ωpf,g,ω −vf,g,ωwf,g,ω − 1 t gpf,g,ω(x, 0) − 1 t fwf,g,ω(x, 0) ) i −∇yf,g,ω ⊗∇pf,g,ω −∇pf,g,ω ⊗∇yf,g,ω, s0(f,g) := − 1 t ( ∇fwf,g,ω(x, 0) + ∇gpf,g,ω(x, 0) ) , (4.36) and the adjoint (pf,g,ω,wf,g,ω) satisfies (3.6)–(3.7). optimal actuator placement for control of vibrations 15 proof. since j1 and j2 are well-posed, it follows that (h0) and (h1) are satisfied. next, we check that (h2) and (h3) hold. using the fundamental theorem of calculus on averaged adjoint equations (4.30)–(4.31), it is easy to see that h̃(τ,u,f,g,yu,f,g,τ,vu,f,g,τ,pu,f,g,τ,wu,f,g,τ ) = h̃(τ,u,f,g,yu,f,g,ω,vu,f,g,ω,pu,f,g,τ,wu,f,g,τ ). (4.37) since j(ωτ,u◦ t−1τ ,f,g) = h̃(τ,u,f,g,yu,f,g,τ,vu,f,g,τ,pu,f,g,τ,wu,f,g,τ ), it follows from (4.37) that j(ωτ,u◦ t−1τ ,f,g) = h̃(τ,u,f,g,y u,f,g,ω,vu,f,g,ω,pu,f,g,τ,wu,f,g,τ ). hence, j1(ωτ,f,g) = inf u∈uad h̃(τ,u,f,g,yu,f,g,ω,vu,f,g,ω,pu,f,g,τ,wu,f,g,τ ). (4.38) choosing u := uf,g,τ in (4.38) with (τ,u,f,g,yu,f,g,ω,vu,f,g,ω,pu,f,g,τ,wu,f,g,τ ) replaced by (τn,un,fn,gn,y un,fn,gn,ω,vun,fn,gn,ω,pun,fn,gn,τn,wun,fn,gn,τn) and substituting in (4.29), we have h(τn,un,fn,gn) = ∫ ωt 1 2 ζ(τn)(|yun,fn,gn,ω|2 + |vun,fn,gn,ω|2 + α|un|2) dxdt + ∫ ωt ζ(τn) ∂vun,fn,gn,ω ∂t pun,fn,gn,τn + a(τn)∇yun,fn,gn,ω ·∇pun,fn,gn,τn dxdt + ∫ ωt ( − ζ(τn)χωunpun,fn,gn,τn + ζ(τn) ∂yun,fn,gn,ω ∂t wun,fn,gn,τn − ζ(τn)vun,fn,gn,ωwun,fn,gn,τn ) dxdt + ∫ ω [ ζ(τn) ( yun,fn,gn,ω(x, 0) −fn ◦ tτn ) wun,fn,gn,τn(x, 0) + ζ(τn) ( vun,fn,gn,ω(x, 0) −gn ◦ tτn ) pun,fn,gn,τn(x, 0) ] dx. (4.39) from (4.5) as τn → 0+, we have ζ(τn) → 1,a(τn) → i. utilizing this result in (4.39), and re-arranging the terms, we obtain h(τn,un,fn,gn) −h(0,un,fn,gn) τn = ∫ ωt ζ(τn) − 1 τn · 1 2 ( |yun,fn,gn,ω|2 + |vun,fn,gn,ω|2 + α|un|2 ) dxdt + ∫ ωt ζ(τn) − 1 τn ∂vun,fn,gn,ω ∂t pun,fn,gn,τn + a(τn) − i τn ∇yun,fn,gn,ω ·∇pun,fn,gn,τn − ζ(τn) − 1 τn χωunp un,fn,gn,τn dxdt + ∫ ωt ζ(τn) − 1 τn ( ∂yun,fn,gn,ω ∂t wun,fn,gn,τn −vun,fn,gn,ωwun,fn,gn,τn ) dxdt + ∫ ω ζ(τn) − 1 τn (( yun,fn,gn,ω(x, 0) −fn ◦ tτn ) wun,fn,gn,τn(x, 0) + ( vun,fn,gn,ω(x, 0) −gn ◦ tτn ) pun,fn,gn,τn(x, 0) ) dx − ∫ ω ( fn ◦ tτn −fn τn wun,fn,gn,τn(x, 0) + gn ◦ tτn −gn τn pun,fn,gn,τn(x, 0) ) dx. (4.40) 16 m. d. arop, h. kasumba, j. kasozi, and f. berntsson note that gn ◦ tτn = ∂ ∂t ( yu◦t −1 τn ,fn,gn,ω(x, 0) ◦ tτn ) and gn ◦ tτn −gn τn = ∂ ∂t ( yu◦t −1 τn ,fn,gn,ω(x, 0) ◦ tτn −y u◦t−1τn ,fn,gn,ω(x, 0) τn ) since tτ is independent of t. using these results, lemmas 4.2 and 4.9, the right-hand side of (4.40) converges to ∫ ωt div(x) ( 1 2 |yf,g,ω|2 + 1 2 |vf,g,ω|2 + α 2 |uf,g,ω|2 + ∂vf,g,ω ∂t pf,g,ω + ∂yf,g,ω ∂t wf,g,ω −vf,g,ωwf,g,ω + ∇yf,g,ω ·∇pf,g,ω −χωuf,g,ωpf,g,ω ) dxdt − ∫ ωt ( ∂x∇yf,g,ω ·∇pf,g,ω + ∂xt∇yf,g,ω ·∇pf,g,ω + 1 t ∇f · xwf,g,ω(x, 0) + 1 t ∇g · xpf,g,ω(x, 0) ) dxdt. (4.41) integrating the fourth and fifth terms of (4.41) by partial integration in time t, and using the facts that pf,g,ω(x,t) = 0,wf,g,ω(x,t) = 0, a: b = ∑2 i,l=1 ailbil and a⊗ b: a = a ·ab, a,b ∈ r 2, a,b ∈ r2×2, we have ∫ ωt (( 1 2 |yf,g,ω|2 + 1 2 |vf,g,ω|2 + α 2 |uf,g,ω|2 −vf,g,ω ∂p f,g,ω ∂t −yf,g,ω ∂w f,g,ω ∂t + ∇yf,g,ω ·∇pf,g,ω −χωuf,g,ωpf,g,ω −vf,g,ωwf,g,ω − 1t gp f,g,ω(x, 0) − 1 t fwf,g,ω(x, 0) ) i −∇yf,g,ω ⊗∇pf,g,ω −∇pf,g,ω ⊗∇yf,g,ω ) : ∂x − 1 t ( ∇fwf,g,ω(x, 0) + ∇gpf,g,ω(x, 0) ) · x dxdt. thus, we have the tensor representation (4.35)–(4.36). hence, lim n→∞ h(τn,un,fn,gn) −h(0,un,fn,gn) τn = ∫ ωt s1 ( yf,g,ω,vf,g,ω,pf,g,ω,wf,g,ω,uf,g,ω ) : ∂x + s0(f,g) · x dxdt. (4.42) suppose that un,0 := u fn,gn,0. then similarly, modifying un as un,0, we obtain lim n→∞ h(τn,un,0,fn,gn) −h(0,un,0,fn,gn) τn = ∫ ωt s1 ( yf,g,ω,vf,g,ω,pf,g,ω,wf,g,ω,uf,g,ω ) : ∂x + s0(f,g) · x dxdt. (4.43) let {fn} and {gn} be constant sequences. then, it is clearly seen that h(τn,un,fn,gn)−h(0,un,fn,gn) in (4.42) and h(τn,un,0,fn,gn) − h(0,un,0,fn,gn) in (4.43) are equal. hence, (h2) is satisfied. utilizing lemma 4.7, we obtain lhs of (4.42) and (4.43) as ∂τh(0 +,u0,f,g,f,g) and ∂τh(0 +,uf,g,0,f,g), respectively. hence, (h3) is satisfied. � as a consequence of theorem 4.11, we obtain the directional derivative of j1. optimal actuator placement for control of vibrations 17 corollary 4.12. let the hypotheses of theorem 4.11 hold. let (f,g) ∈ h10 (ω) ×l2(ω) := v be given. then the directional derivative of j1(ω,f,g) at ω in the direction x ∈ c̊0,1(ω,r2) is given by dj1(ω,f,g)(x) = ∫ ωt s1 ( yf,g,ω,vf,g,ω,pf,g,ω,wf,g,ω,uf,g,ω ) : ∂x + s0(f,g) · xdxdt, (4.44) where s1 ( yf,g,ω,vf,g,ω,pf,g,ω,wf,g,ω,uf,g,ω ) and s0(f,g) are defined by (4.36). proof. for a constant r > 0, we note that max f∈k1,g∈k2 ‖f‖ h1 0 (ω) ≤r,‖g‖ l2(ω) ≤r j1(ω,f,g) = r 2 max f∈ 1 r k1,g∈ 1rk2 ‖f‖ h1 0 (ω) ≤1,‖g‖ l2(ω) ≤1 j1(ω,f,g). (4.45) from (4.45) and by the hypotheses of theorem 4.11, we deduce that f r ∈ k1 and gr ∈ k2 with ‖f r ‖h10 (ω) ≤ 1 and ‖ g r ‖l2(ω) ≤ 1. thus, we have the singleton {k1,k2} := {(f,g)}. so, for all ω ∈ e(ω), we have j2(ω) = max f∈k1,g∈k2 j1(ω,f,g) = j1(ω,f,g). hence, we deduce that x2(ω) = {(f,g)}. since x2(ω) is a singleton, (4.44) follows by theorem 4.11. � as a further consequence of theorem 4.11, we write (4.35) as an integral over ∂ω. to this end, we require that ω and ω are c2 domains. additionally, for any two sets ω and ω, the notation ω b ω will be used to mean that ω is compactly contained in ω. in other words, ω b ω if ω ⊂ ω and ω is compact. corollary 4.13. let f ∈ k1,g ∈ k2 and x ∈ c̊0,1(ω,r2) be given. assume that ω b ω and ω are c2 domains. (a) given (f,g) ∈ x2(ω), define ŝ1(f,g) and ŝ0(f,g) by ŝ1(f,g) := ∫t 0 s1(f,g)(s) ds and ŝ0(f,g) := ∫t 0 s0(f,g)(s) ds, respectively. then we have ŝ1(f,g)|ω∈ w 1,1(ω,r2×2), ŝ1(f,g)|ω\ω∈ w 1,1(ω \ω,r2×2), ŝ0(f,g)|ω∈ l2(ω,r2), (4.46) −div(ŝ1(f,g)) + ŝ0(f,g) = 0 a.e. in ω ∪ (ω \ω). (4.47) moreover, (4.35) can be written as dj2(ω)(x) = max (f,g)∈x2(ω) − ∫ ∂ω ∫ t 0 uf,g,ω(t)pf,g,ω(t)(x ·ν) dtds, (4.48) for x ∈ c̊0,1(ω,r2), with ν the outer normal to ω. we denote the jump of ŝ1(f,g)ν across ∂ω by [ŝ1(f,g)ν] := ŝ1(f,g)|ων − ŝ1(f,g)|ω\ων. (b) we have that (4.44) can be written as dj1(ω,f,g)(x) = − ∫ ∂ω ∫ t 0 uf,g,ω(t)pf,g,ω(t)(x ·ν) dtds, (4.49) for x ∈ c̊0,1(ω,r2). we begin by stating an important lemma, the so-called nagumo’s lemma (see e.g., [15]) before proving corollary 4.13. the outer normal to ∂r2 will be denoted by ν. 18 m. d. arop, h. kasumba, j. kasozi, and f. berntsson lemma 4.14. let ω ⊂ r2 be a bounded domain of class ck, k ≥ 1. suppose that x ∈ c̊0,1(r2,r2) is a vector field satisfying x(x) ·ν(x) = 0, for all x ∈ ∂r2. then the flow φτ of x satisfies φτ (ω) = ω and φτ (∂ω) = ∂ω, for all τ. proof of corollary 4.13. we prove (4.46)–(4.48) as follows. by nagumo’s lemma, we have dj2(ω)(x) = 0, for all x ∈ c1c (ω,r2). using this condition and definitions of ŝ1(f,g) and ŝ0(f,g) in (4.35), we see that ∫ ω ŝ1(f,g) : ∂x + ŝ0(f,g) · x dx = 0, (4.50) for all x ∈ c1c (ω,r2). integrating the first term in (4.50) by partial integration and using x|∂ω= 0, we have ∫ ω (−div(ŝ1(f,g)) + ŝ0(f,g)) · x dx = 0, (4.51) for all x ∈ c1c (ω,r2). since x ∈ c1c (ω,r2), applying the fundamental lemma of calculus of variations on (4.51) gives (4.47). further, since y,p ∈ h2(ω) ∩h10 (ω) follows from elliptic regularity theory (see e.g., [6, p.317] ), we have that (4.46) holds. thus, noting that ω = ω∪(ω\ω) and by partial integration, we have for all x ∈ c1c (ω,r2), dj2(ω)(x) = max (f,g)∈x2(ω) ∫ ω ŝ1(f,g) : ∂x + ŝ0(f,g) · x dx, = max (f,g)∈x2(ω) (∫ ω ( − div(ŝ1(f,g)) + ŝ0(f,g) ) · x dx + ∫ ω\ω ( − div(ŝ1(f,g)) + ŝ0(f,g) ) · x dx + ∫ ∂ω [ŝ1(f,g)ν] · x ds ) , (4.47) = max (f,g)∈x2(ω) ∫ ∂ω [ŝ1(f,g)ν] · x ds. (4.52) since (4.46) holds, it follows that tτ (f,g) := ŝ1(f,g) + ∫ t 0 χωu f,g,ω(t)pf,g,ω(t) dt ∈ w 1,1(ω,r2×2). (4.53) so, tτ (f,g)ν = 0 on ∂ω. hence, it is easy to see from (4.53) that [ŝ1(f,g)ν] = − (∫ t 0 χωu f,g,ω(t)pf,g,ω(t) dt ) ν. (4.54) since x and ν are independent of time t, substituting (4.54) in (4.52) gives dj2(ω)(x) = max (f,g)∈x2(ω) − ∫ ∂ω ∫ t 0 uf,g,ω(t)pf,g,ω(t)(x ·ν) dtds, as was to be proved. the proof of (4.49) is similar to the proof of corollary 4.12. � 4.5. gradient algorithm for optimal actuator placement. here, we present the steps of a gradient-based algorithm for optimal actuator placement. the version of the algorithm is summarized in algorithm 1. it is important to note that we can also use j2 in this algorithm to investigate the optimal actuator placement by replacing j1 with j2. optimal actuator placement for control of vibrations 19 algorithm 1 shape derivative-based gradient algorithm for optimal actuator placement require: ω0 ∈ e(ω),f,g, tolerance ε > 0,k = 0,λ,d0 := −∇j1(ω0,f,g). while |dk| ≥ ε do if j1((id + λdk)(ωk),f,g) < j1(ωk,f,g) then dk = −∇j1(ωk,f,g) ωk+1 = (id + λdk)(ωk) k := k + 1 end if end while return optimal actuator placement ωk+1 5. numerical examples 5.1. discretization. let step sizes be h in space and 4t in time, i.e., 4x1 = 4x2 = h and tk = k4t. then, discretizing (3.6) and (3.7) using finite differences, we have for k = 1, 2, . . . ,m − 1 yk+1h = y k h + 4tv k h, vk+1h = v k h + ary k h + 4tχωu k h, (5.1) y1h = fh, v1h = gh, and for k = m,m − 1, . . . , 2 pk−1h = p k h + 4t(w k h − v k h), wk−1h = w k h + arp k h −4ty k h, (5.2) pmh = 0, wmh = 0, respectively, where yh = (y11,y12, . . . ,y(n−1)2(n−1)2 ) >, vh = (v11,v12, . . . ,v(n−1)2(n−1)2 ) >, uh = (u11,u12, . . . ,u(n−1)2(n−1)2 ) >, fh = (f11,f12, . . . ,f(n−1)2(n−1)2 ) >, gh = (g11,g12, . . . ,g(n−1)2(n−1)2 ) >, ph = (p11,p12, . . . ,p(n−1)2(n−1)2 ) >, wh = (w11,w12, . . . ,w(n−1)2(n−1)2 ) >,r = 4t h2 and ar =   b i 0 . . . 0 i b i . . . ... 0 i . . . . . . 0 ... . . . . . . b i 0 . . . 0 i b   with b =   −4 1 0 . . . 0 1 −4 1 . . . ... 0 . . . . . . . . . 0 ... . . . 1 −4 1 0 . . . 0 1 −4   and i is the identity matrix. the matrix ar is of size (n − 1)2 × (n − 1)2 while matrices b, i and 0 are of size (n − 1) × (n − 1). the discrete functionals of j1 and j2 are j1,h(ω, fh, gh) = 1 2 min uh∈uad ∫ t 0 yh(t) >yh(t) + vh(t) >vh(t) + αχωuh(t) >χωuh(t)dt (5.3) 20 m. d. arop, h. kasumba, j. kasozi, and f. berntsson and j2,h(ω) = max fh,gh j1,h(ω, fh, gh), (5.4) respectively. the discrete derivatives j1,h and j2,h are given by dj1,h(ω, fh, gh)(x) = − ∫ ∂ω ∫ t 0 uh(s,t) >ph(s,t)(x ·ν) dtds (5.5) and dj2,h(ω)(x) = max fh,gh j1,h(ω, fh, gh), for x ∈ c̊0,1(ω,r2), respectively. the vector b ∈ r2 has components bj := − ∫ ∂ω ∫ t 0 uh(s,t) >ph(s,t)(ej ·ν) dtds,j = 1, 2, where ej is the jth element of the standard basis of r2. 5.2. examples. we illustrate the actuator placement optimizations for two cases of initial conditions f and g. in all the experiments, the actuators ω1, and ω2 each of fixed size 0.2 × 0.2 are placed on the domain and moved along the descent direction x1 = x2. we consider two actuators without overlap such that they move into their optimal locations. we set the tolerance ε to 10−4 and n to 8. example 5.21. we consider the case y(x1,x2, 0) = sin πx1 sin πx2, 0 ≤ x1,x2 ≤ 1, v(x1,x2, 0) = πc 20 sin πx1 sin πx2, 0 ≤ x1,x2 ≤ 1, so that the initial speed v(x1,x2, 0) varies with the speed of wave 1 ≤ c ≤ 20π . first, we start by investigating the optimal actuator placement using j1,h. for initial actuators ω1,ω2 centered at (0.4, 0.4) and (0.825, 0.825), respectively, a shape optimization algorithm 1 is utilized. the results are presented in figure 2. it is observed from figure 2 that as the actuators move toward the optimal locations in the subsequent iterations (see figure 2(a)), the functional j1,h decays until a stationary point is reached, see figure 2(b). the optimization algorithm converges after 120 iterations. the optimal actuators are centered at (0.325, 0.325) and (0.75, 0.75), respectively. next, we perform numerical experiments using j2,h but with initial actuators ω1,ω2 centered at (0.2, 0.2) and (0.825, 0.825), respectively. algorithm 1 is run until the set criterion is achieved. the results are depicted in figure 3. from this figure, we see that as the actuators move toward the optimal locations, see figure 3(a), the functional j2,h decays until a stationary point is reached, see figure 3(b). the convergence of the optimization algorithm occurs after 73 iterations. the final actuator locations are found at (0.325, 0.325) and (0.75, 0.75), respectively. this is consistent with the result obtained by using j1,h. lastly, the results of the experiments to investigate the influence of the wave speed are shown in figure 4 and table 1. from table 1, we see that when ω1 is placed at (0.4, 0.4), and ω2 at (0.825, 0.825) (see figure 4(a)), the least values of both j1,h and j2,h are obtained. furthermore, it is observed that j1,h increases with an increase in the wave speed c, see figure 4 and table 1. we also note from table 1 that the least values of j2,h(ω) after 120 iterations are the same. this confirms the fact that the dependence of j1,h on the initial conditions is averaged out. optimal actuator placement for control of vibrations 21 (a) (b) figure 2. (a) the initial actuator center locations: (0.4, 0.4), (0.825, 0.825) (red) and final actuator center locations: (0.325, 0.325), (0.75, 0.75) (blue). (b) the history of cost functional j1,h, as the actuators move from the initial to the optimal actuator locations. the speed of the wave is set to c = 1. (a) (b) figure 3. (a) the initial actuator center locations: (0.2, 0.2), (0.825, 0.825) (red) and final actuator center locations: (0.325, 0.325), (0.75, 0.75) (blue). (b) the history of cost functional j2,h, as the actuators move from the initial to the optimal actuator locations. the speed of the wave is set to c = 1. example 5.22. in this example, we set y(x1,x2, 0) = x1x2(1 −x1)(1 −x2), 0 ≤ x1,x2 ≤ 1, v(x1,x2, 0) = 1 2 sin(x1(1 −x1)x2(1 −x2)), 0 ≤ x1,x2 ≤ 1, so that the initial conditions of the dynamics satisfy dirichlet boundary conditions. therefore, the optimal actuator center locations are expected at points different from the boundary of the domain. first, we start by investigating the optimal actuator placement using j1,h. with initial actuator center locations ω1,ω2 at (0.2, 0.2) and (0.825, 0.825), respectively, the results are shown in figure 5. the minimum value of j1,h occurs when the actuators are placed at (0.3, 0.3) and (0.7, 0.7). 22 m. d. arop, h. kasumba, j. kasozi, and f. berntsson (a) (b) figure 4. (a) the initial actuator center locations: (0.4, 0.4), (0.825, 0.825) (red) and final actuator center locations: (0.325, 0.325), (0.75, 0.75) (blue). (b) demonstration of the influence of wave speed on the history of cost functional j1,h, as the actuators move from the initial to the final actuator locations. table 1. the minimum values of functionals j1,h and j2,h after 120 iterations for the given speed of wave c. the measure of the cost of control is set to α = 10−4. c j1,h(ω,f,g) j2,h(ω) 1 78.8529 0.4307 3 80.2176 0.4307 5 83.4128 0.4307 next, we perform a numerical experiment using j2,h. with initial actuator center locations at (0.2, 0.2) and (0.825, 0.825), we run algorithm 1 until the set criterion is achieved. the results are given in figure 6. from this figure, we see that the functional j2,h decays until a stationary point is reached. the convergence of the optimization algorithm occurs after 50 iterations. the final actuator locations are found at (0.3, 0.3) and (0.7, 0.7), see figure 6(a). this is in agreement with the result obtained by using j1,h. 6. conclusion in this paper, we proved important results for the differentiability of functionals j1 and j2. the shape derivative is derived using the averaged adjoint approach. we also developed a shape derivative-basedgradient algorithm for determining the optimal actuator placement for the control of vibrations induced by pedestrian-bridge interactions. the algorithm is constructed by embedding the shape sensitivities in a gradient-based method. the numerical results presented illustrate the potential of the shape sensitivities for solving the optimal actuator placement problem whenever the actuator’s width is fixed in advance. the optimal actuator design for the wave equation is under our next research study plan. the term ”design” here means picking the best domain ω by parametrizing the set of admissible domains. optimal actuator placement for control of vibrations 23 (a) (b) figure 5. (a) the initial actuator center locations: (0.2, 0.2), (0.825, 0.825) (red) and final actuator center locations: (0.3, 0.3), (0.7, 0.7) (blue). (b) demonstration of the history of cost functional j1,h, as the actuators move from the initial to the optimal actuator locations. (a) (b) figure 6. (a) the initial actuator center locations: (0.2, 0.2), (0.825, 0.825) (red) and final actuator center locations: (0.3, 0.3), (0.7, 0.7) (blue). 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[23] s. yang and k. morris, comparison of linear-quadratic and controllability criteria for actuator placement on a beam, 2014 american control conference, ieee, portland, 2014, pp. 4069–4074. m. d. arop, corresponding author, department of mathematics, makerere university, kampala, uganda current address: department of mathematics, muni university, arua, uganda email address: d.arop@muni.ac.ug h. kasumba, department of mathematics, makerere university, kampala, uganda email address: henry.kasumba@mak.ac.ug j. kasozi, department of mathematics, makerere university, kampala, uganda email address: juma.kasozi@mak.ac.ug f. berntsson, department of mathematics, linköping university, linköping, sweden email address: fredrik.berntsson@liu.se 1. introduction 2. formulation of the problem 2.1. notations 2.2. setup of the problem 3. well-posedness of the functionals 4. sensitivity analysis of the functionals 4.1. shape derivative 4.2. sensitivity of the state equation 4.3. averaged adjoint equations 4.4. directional derivative of max-min functions 4.5. gradient algorithm for optimal actuator placement 5. numerical examples 5.1. discretization 5.2. examples 6. conclusion acknowledgements references