www.biomathforum.org/biomath/index.php/biomath ORIGINAL ARTICLE Permanence and periodic solution for a modified Leslie-Gower type predator-prey model with diffusion and non constant coefficients A. Moussaoui1, M. A. Aziz Alaoui2, R. Yafia3 1Department of Mathematics, University of Tlemcen, Tlemcen, Algeria moussaoui.ali@gmail.com 2Normandie University, Le Havre, France ULH, LMAH, FR CNRS 3335, Le Havre, France aziz.alaoui@univ-lehavre.fr 3Université Ibn Zohr, Faculté Polydisciplinaire, Ouarzazate, Morocco yafia@yahoo.fr Received: 5 March 2017, accepted: 10 July 2017, published: 19 July 2017 Abstract—In this paper we study a predator-prey system, modeling the interaction of two species with diffusion and T -periodic environmental parameters. It is a Leslie-Gower type predator-prey model with Holling-type-II functional response. We establish some sufficient conditions for the ultimate bound- edness of solutions and permanence of this system. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Numerical simulations are presented to illustrate the results. Keywords-Reaction-diffusion equations, Predator- prey model, Functional response, Permanence. I. INTRODUCTION AND MATHEMATICAL MODEL The dynamical properties of the predator-prey models can be used to analyze the relations be- tween the prey and predator and to predicate whether they can coexist. As we known, one of the earliest and also the best known predator- prey models is the Leslie-Gower model [16], [17], which is a modificiation of the Lotka-Volterra model [22]. The Leslie-Gower type model can be described by the following autonomous bi- dimensional system [16], [17]  du dt = u(a− bu) −αuv, dv dt = v ( c− βv u ) , (1) where u is the population of the prey and v is the population of the predator. In (1) we assume the prey grows logistically with carrying capacity K = a b and intrinsic growth rate a in the absence of predation. The predation is assumed to be proportional to the population size of the prey. The predator grows logistically with intrinsic growth rate c and carrying capacity c β u(t) proportional to the population size of prey (or prey abundance). Citation: A. Moussaoui, M. A. Aziz Alaoui, R. Yafia, Permanence and periodic solution for a modified Leslie-Gower type predator-prey model with diffusion and non constant coefficients, Biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 Page 1 of 8 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2017.07.107 A. Moussaoui et al., Permanence and periodic solution for a modified Leslie-Gower type ... The parameter β is a measure of the food quality that the prey provides for conversion into predator birth. The stability of the interior equilibrium is studied in [25] by numerical methods. Lindstrom [18] investigated the nonexistence, existence and limit cycles. Hsu and Huang [13] prove that all the solutions are bounded and positive if their initial values are in the first quadrant, and study the globally asymptotical stability of the interior equilibrium using Liapunov function and LaSalle’s invariance principle. Aziz-Alaoui and Daher Okiye [3] argued that a suitable predator-prey model should incorpo- rate some kind of functional response, while the predator species could have other food resource. Basing on those assumption, they proposed a predator-prey model with modified Leslie-Gower and Holling-type II schemes [12] as follow:  du dt = u(a− bu) − αuv u + k1 , dv dt = v ( c− βv u + k2 ) , (2) where k1 is the half-saturation constant in the Holling-type II functional response and k2 is a measure of alternative prey densities in the en- vironment, allowing the predator to persist when the prey population disappears. The authors inves- tigated the boundedness and global stability of the system (2). Nindjin et al. [23] further incorporated the time delay to the system considered in [3], and they showed that time delay plays important role on the dynamic behaviors of the system. Yafia et al. [27] studied the limit cycle bifurcated from time delay. For more works on Leslie-Gower predator-prey model, one could refer to [1], [4], [7], [8], [14], [26], [28], [29] and the references cited therein. To achieve further understanding it is now essential to consider more general and hence more ”dif- ficult” models. We will focus here on the case in which the biological or environmental param- eters are time-periodic, and will assume that the species are free to move at random throughout some bounded habitat. Under these assumptions, we model the species interaction via a system of reaction-diffusion equations of the form  ∂u ∂t −d1∆u=u ( a(t,x)−b(t,x)u ) − α(t,x)uv u+k1(t,x) , ∂v ∂t −d2∆v =v ( c(t,x)− β(t,x)v u+k2(t,x) ) , (3) where the function u(t,x) and v(t,x) determine the densities of prey and predator, respectively, at a point x and time t. Here the equations are assumed to be satisfied in a cylinder x ∈ Ω̄, 0 < t < ∞, where Ω is an open, bounded, smooth domain in Rn. These equations are supplemented with homogeneous Neumann boundary conditions ∂u ∂n = ∂v ∂n = 0 on ∂Ω × (0,∞). where n is the outward unit vector of the boundary ∂Ω which we assume is smooth, and the following nonnegative initial values u(0,x) = u0(x) ≥ 0, v(0,x) = v0(x) ≥ 0 in Ω. The various coefficients on the right-hand side depend on both t and x modelling the fact that effects vary in both time and space. The periodicity of coefficients models seasonal fluctuations. d1 and d2, are positive diffusion coefficients reflecting the non-homogeneous dispersion of populations. Many authors studied the qualitative properties of this system, but for the case for which the pa- rameters are constant, using Neumann or Dirichlet boundary conditions, see [8], [24], [27], [29]. Motivated by the papers mentioned above, we deal here with the permanence and existence of periodic solutions of the diffusive system (3). The content of this paper is as follows. In section 2, we give conditions for the ultimate boundedness of solutions and permanence of the system. In Section 3, we establish conditions for the existence of a unique periodic solution of the system. Nu- merical simulations are presented in Section 4 to illustrate the feasibility of our results. Biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 Page 2 of 8 http://dx.doi.org/10.11145/j.biomath.2017.07.107 A. Moussaoui et al., Permanence and periodic solution for a modified Leslie-Gower type ... II. BOUNDEDNESS AND PERMANENCE We analyze the permanence (dissipation and persistence) of system (3) with non-negative initial functions, this ensures the long-term survival (i.e., will not vanish in time) of all components of system (3), under some conditions. We first recall a well known result on the logistic equation. Lemma 2.1: [30]. Assume that u(t,x) is de- fined by  ∂u ∂t =d1∆u+ru ( 1− u K ) , x∈Ω, t>0, ∂u ∂n = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u0(x) > 0, x ∈ Ω, (4) then, limt→∞u(t,x) = K. Throughout the paper we always assume that: (H): Functions a(t,x), b(t,x), c(t,x), α(t,x), β(t,x) and ki(t,x), (i = 1, 2) are bounded positive-valued functions on R+ × Ω̄ con- tinuously differentiable in t and x, and are periodic in t with a period T > 0. For a bounded function φ(t,x), we denote φm = inf (t,x)∈R+×Ω φ(t,x), φM = sup (t,x)∈R+×Ω φ(t,x). A. Dissipation Proposition 2.2: All the solutions of (3) initiated in the positive octant are nonnegative and satisfy lim sup t→+∞ max x∈Ω̄ u(t,x) ≤ aM bm , lim sup t→+∞ max x∈Ω̄ v(t,x) ≤ cM βm (aM bm + kM2 ) . Proof The nonnegativity of the solutions of (3) is ob- vious since the initial value is nonnegative. We consider now the second part of the theorem. For convenience, we denote a = a(t,x), and similar meaning to b,c,α,β,k1 and k2. From the first equation of system (3), we have ∂u ∂t = d1∆u + u ( a− bu ) − αuv u + k1 , ≤ d1∆u + u ( aM − bmu ) . (5) From the comparison principle of the parabolic equations [9], [11], it is easy to verify that u(t,x) ≤ u(t), where u(t) is the spatially homo- geneous solutions of  ∂u ∂t =d1∆u+u ( aM−bmu ) , x ∈ Ω, t > 0 ∂u ∂n = 0, x ∈ ∂Ω, t > 0 u(0,x) = u∗ (6) where u∗ = max x∈Ω u(0,x). This implies, by using Lemma 2.1, that lim sup t→+∞ max x∈Ω̄ u(t,x) ≤ aM bm . Then, for ε > 0 there exists T1 > 0 such that u(t,x) ≤ η1 for t > T1, (7) where η1 = a M bm + ε. Therefore, from the second equation of system (3) and (7) and using the same reasoning, we have ∂v ∂t = d2∆v + v ( c− βv u + k2 ) ≤ d2∆v + v ( cM − βmv η1 + k M 2 ) for t > T1. Hence there exists T2 > T1 such that for any t > T2 v(t,x) ≤ η2 (8) where η2 = c M βm (a M bm + ε + kM2 ) + ε, which implies lim sup t→+∞ max x∈Ω̄ v(t,x) ≤ cM βm ( aM bm + kM2 ) . Therefore, any positive solution of system (3) is ultimately bounded, which completes the proof. Biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 Page 3 of 8 http://dx.doi.org/10.11145/j.biomath.2017.07.107 A. Moussaoui et al., Permanence and periodic solution for a modified Leslie-Gower type ... B. Persistence Definition 2.3: [5], [6] System (3) is said to be persistent if for any positive initial data (u0(x),v0(x)), there exist positive constants ξ1 = ξ1(u0,v0),ξ2 = ξ2(u0,v0), such that the solution (u(t,x),v(t,x)) of (3) satisfies lim inf t→+∞ min x∈Ω̄ u(t,x) ≥ ξ1, lim inf t→+∞ min x∈Ω̄ v(t,x) ≥ ξ2 Proposition 2.4: Assume that amkm1 β m > αMcM (aM bm + kM2 ) (9) then system (3) is persistent. Proof From (3), (7) and (8), it follows that for t ≥ T2, ∂u ∂t = d1∆u + u(a− bu) − αuv u + k1 ≥ d1∆u + u(am − bMu) − αMη2u km1 = d1∆u + u ( am − αMη2 km1 − bMu ) then from the comparison principle of the parabolic equations, it is easy to verify that u(t,x) ≥ u(t), where u(t) is the spatially homo- geneous solutions of  ∂u ∂t =d1∆u + u ( am− αMη2 km1 −bMu ) , x ∈ Ω, t > 0, ∂u ∂n = 0,x ∈ ∂Ω, t > 0, u(x, 0) = u∗ (10) where u∗ = min x∈Ω u(0,x). Thanks to lemma 2.1, we obtain, lim inf t→+∞ min x∈Ω̄ u(t,x) ≥ 1 bM ( am − αMη2 km1 ) . Hence, there exists T3 > T2 such that for any t > T3, u(t,x) ≥ ξ1 (11) where, ξ1 = 1 bM ( am − αMη2 km1 −ε ) . From the predator equation, it follows that ∂v ∂t = d2∆v + v ( c− βv u + k2 ) ≥ d2∆v + v ( cm − βMv km2 ) . Hence, there exists T4 > 0 such that for any t > T4 v(t,x) > ξ2, (12) where ξ2 = cmkm2 βM −ε. Therefore, from (11) and (12), we obtain, lim inf t→+∞ min x∈Ω̄ u(t,x) ≥ 1 bM ( am − αMcM βm (a M bm + kM2 ) km1 ) , lim inf t→+∞ min x∈Ω̄ v(t,x) ≥ cmkm2 βM . (13) Thus, system (3) is persistent, which completes the proof of Proposition 2.4. A direct application of Proposition 2.2 and Proposition 2.4 gives the following result. Proposition 2.5: (Permanence) If condition (9) holds, there exist positive constants 0 < ζ < η, such that, ζ ≤ lim inf t→∞ min x∈Ω̄ u(t,x) ≤ lim sup t→∞ max x∈Ω̄ u(t,x) ≤ η ζ ≤ lim inf t→∞ min x∈Ω̄ v(t,x) ≤ lim sup t→∞ max x∈Ω̄ v(t,x) ≤ η That is, model (3) is permanent. Biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 Page 4 of 8 http://dx.doi.org/10.11145/j.biomath.2017.07.107 A. Moussaoui et al., Permanence and periodic solution for a modified Leslie-Gower type ... III. PERIODIC SOLUTIONS A very basic and important problem in the study of a population growth models with periodic environment is the global existence and stabil- ity of positive periodic solutions, which plays a similar role as a globally stable equilibrium for autonomous models [5], [11], [15], [19], [21]. In this section, we derive sufficient conditions that guarantee existence, uniqueness and global stability of a T -periodic positive solution of system (3). For this aim, we consider the matrix M, which reads as: M =   2 ( aM−2bmζ− α mkm1 ζ (η+kM1 ) 2 ) βMη2 (ζ+km2 ) 2 βMη2 (ζ+km2 ) 2 2 ( cM−2βmζ ζ+k m 2 (η+kM2 ) 2 )   (14) where ζ and η are the bounds of any non-zero arbitrary solution of system (3), initialing with non-negative function, given by Proposition 2.5. Proposition 3.1: Assume that condition (9) holds, that is system (3) is permanent, if µ(M) < 0, (15) where µ(M) is the maximal eigenvalue of the matrix M. Then, system (3) has a unique glob- ally asymptotic stable strictly positive T -periodic solution. Proof Let (u1(t,x),v1(t,x)) and (u2(t,x),v2(t,x)) be two solutions of system (3), by Proposition 2.5, these solutions are bounded by constants ζ and η, where ζ = min{ξ1,ξ2} and η = max{η1,η2}, defined in section 2. Consider the function U(t)= ∫ Ω ( (u1(t,x)−u2(t,x))2+(v1(t,x)−v2(t,x))2 ) dx (16) One has, dU(t) dt = 2 ∫ Ω (u1 −u2) ( ∂u1 ∂t − ∂u2 ∂t ) dx +2 ∫ Ω (v1 −v2) ( ∂v1 ∂t − ∂v2 ∂t ) dx = 2d1 ∫ Ω (u1 −u2)∆(u1 −u2)dx +2d2 ∫ Ω (v1 −v2)∆(v1 −v2)dx +2 ∫ Ω (u1 −u2)[( u1(a−bu1)−αu1v1u1+k1 ) − ( u2(a−bu2)−αu2v2u2+k1 )] dx +2 ∫ Ω (v1−v2) [( v1(c− βv1u1+k2 ) − ( v2(c− βv2u2+k2 )] dx := I1 + I2 + I3 + I4 (17) It follows from the boundary condition in (3) that I1 + I2 = 2d1 ∫ ∂Ω (u1 −u2)∇(u1 −u2)dη −2d1 ∫ Ω (∇(u1 −u2))2dx +2d2 ∫ ∂Ω (v1 −v2)∇(v1 −v2)dη −2d2 ∫ Ω (∇(v1 −v2))2dx =−2d1 ∫ Ω (∇(u1−u2))2dx−2d2 ∫ Ω (∇(v1−v2))2dx. ≤ 0. For the third and fourth term in (17), we have I3 +I4 = 2 ∫ Ω (u1−u2) [( (u1−u2) ( a−b(u1 +u2) ) −α u1u2(v1 −v2) + k1(u1v1 −u2v2 (u1 + k1)(u2 + k1) ] dx +2 ∫ Ω (v1 −v2) [ c(v1 −v2) −β (v21u2 −v 2 2u1) + k2(v1 −v2)(v1 + v2) (u1 + k2)(u2 + k2) dx ] . Note that v21u2−v 2 2u1 = (v1−v2)(v1u2+v2u1)−v1v2(u1−u2) and u1v1 −u2v2 = u1(v1 −v2) + v2(u1 −u2). Therefore I3 + I4 = 2 ∫ Ω (u1 −u2)2 [ a− b(u1 + u2) − αk1v2(u1+k1)(u2+k1) ] dx +2 ∫ Ω (v1−v2)2 [ c−β (v1u2 + v2u1) + k2(v1 + v2) (u1 + k2)(u2 + k2) dx ] Biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 Page 5 of 8 http://dx.doi.org/10.11145/j.biomath.2017.07.107 A. Moussaoui et al., Permanence and periodic solution for a modified Leslie-Gower type ... +2 ∫ Ω (u1 −u2)(v1 −v2)[ βv1v2 (u1+k2)(u2+k2) − αu1 (u1+k1) ] dx ≤ 2 ∫ Ω (u1−u2)2 [ aM−2bmζ− αmkm1 ζ (η + kM1 ) 2 ] dx +2 ∫ Ω (v1−v2)2 [ cM−2βmζ ζ + km2 (η + kM2 ) 2 ] dx +2 ∫ Ω ∣∣∣(u1−u2)(v1−v2)∣∣∣ βMη2 (ζ + km2 ) 2 dx ≤ µ(M) ∫ Ω [ (u1−u2)2+(v1−v2)2 ] dx. Using (15) yields, U(t) ≤ U(0)eµ(M)t → 0 as t →∞ (18) Thus, we have proved that ‖u1(t,x)−u2(t,x)‖→ 0 and ‖v1(t,x) −v2(t,x)‖→ 0 as t →∞, where ‖.‖ denotes the norm of the space L2(Ω). Let p > n a positive integer and w(t,w0) = (u(t,x,u0,v0),v(t,x,u0,v0)). By applying ex- actly the same reasoning as in [2], we prove that for some γ ∈ ( 1 2 + n 2p , 1), the solution {w(t,w0)} is relatively compact in the space C1+θ(Ω̄,R2), for 0 < θ < 2γ − 1 −n/p. Therefore, lim t→∞ sup x∈Ω |u1(t,x) −u2(t,x)| = 0, lim t→∞ sup x∈Ω |v1(t,x) −v2(t,x)| = 0. (19) Now we consider the sequence (u(kT,x,u0,v0),v(kT,x,u0,v0)) = w(kT,w0). Then, {w(kT,w0),k ∈ N} is compact in the space C(Ω̄) × C(Ω̄). Let ω̄ be a limit point of this sequence, then w(T,w̄) = w̄. Indeed, it follows, from w(T,w(knT,w0)) = w(knT,w(T,w0)) and ω(knT,w(T,ω0))−w(knT,ω0) → 0 as kn →∞, that ‖w(T,w̄)−w̄‖C≤‖w(T,w̄)−w(T,w(knT,w0))‖C +‖w(T,w(knT,w0)) −w(knT,w0)‖C +‖w(knT,w0) − w̄‖C → 0 as n →∞. The sequence {w(kT,w0),k ∈ N} has a unique limit point, otherwise, there are two limit points w̄ = lim t→∞ w(knT,w0) and ŵ = lim t→∞ w(knT,w0). But, thanks to (19) and ŵ = w(knT,ŵ), we get ‖w̄−ŵ‖C ≤ ‖w̄−w(knT,w0)‖C +‖w(knT,w0)−ŵ‖C → 0 as n →∞. (20) Thus, w̄ = ŵ. Hence, the solution (u(t,x, ū, v̄),v(t,x, ū, v̄)) is the unique periodic solution of system (3). Finally, due to (19), we conclude that this periodic solution is globally asymptotically stable. IV. NUMERICAL SIMULATIONS In this section, numerical simulations for a given parameters range of system (3) are done to support our analytical results obtained in Sections 3 and 4. We consider system (3) with d1 = 0.5,d2 = 0.8, a = 2 + 0.5 sin(2πt), b = 4 + 0.5 sin(2πt), α = 0.03 + 0.02sin(2πt), k1 = 1 + 0.2 sin(2πt), k2 = 1 + 0.5 sin(2πt), c = 1 + 0.6 sin(2πt) and β = 1 + 0.8 sin(2πt) . Obviously, all the parameters have a common period T = 1 in t, By a direct computation, we can prove that all conditions in proposition 4 are satisfied. Then, system (3) has a unique positive 1-periodic solution u(t,x),v(t,x) which is glob- ally asymptotically stable. By applying Matlab to simulate, we can obtain Figures 1-4. From these figures, we see that system (3) is permanent and has positive periodic solution. V. CONCLUSION The interacting species play important roles in real ecosystem. In this paper, we have studied time-periodic Leslie-Gower type predator- prey model with diffusion and Holling-type-II functional response whose growth rates and interaction rates are periodic functions of time. We have obtained sufficient conditions for the persistence of (3) in Proposition 2.4. The conditions are given in term of parameters of the model. Biologically speaking, we may expect the coexistence when the predator growth rate is sufficiently small, or if the predation rate α is small enough. Biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 Page 6 of 8 http://dx.doi.org/10.11145/j.biomath.2017.07.107 A. Moussaoui et al., Permanence and periodic solution for a modified Leslie-Gower type ... 0 0.5 1 1.5 2 0 5 10 15 20 0 0.1 0.2 0.3 0.4 xt u (t ,x ) Fig. 1. Periodic prey solution with respect to the time and space variables. 0 2 4 6 8 10 12 14 16 18 20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 t u (t ,x ) Fig. 2. The section of Fig. 1 with x = 0. Next, we have investigated sufficient conditions which ensure the existence of positive T -periodic solutions of (3) in Proposition 3.1. The conditions are given in term of the largest eigenvalue of certain matrix. Our study demonstrates how parameters of the model which are not constant but vary in response to environmental fluctuations, influence a species prosperity, and gives some valuable suggestions for saving the two species and regulating populations when the ecological and environmental parameters are affected by periodic factors such as the season switching. Numerical simulations are carried out to support our theoretical results. 0 0.5 1 1.5 2 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 xt v( t, x) Fig. 3. Periodic prey solution with respect to the time and space variables. 0 2 4 6 8 10 12 14 16 18 20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 t v( t, x) Fig. 4. The section of Fig. 3 with x = 0. Acknowledgments The authors would like to thank the editor and anonymous referees for their careful reading of the manuscript and valuable suggestions to improve the quality of this work. REFERENCES [1] P. Aguirre, E. Gonzalez-Olivares, E. 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Yang, Global asymptotical stability and persistent property for a diffusive predator-prey system with modi- fied Leslie-Gower functional response, Nonlinear Analy- sis: Real World Applications, 14, 1323–1330, 2013. [29] W. S. Yang,Y. Q. Li, Dynamics of a diffusive predator- prey model with modified Leslie-Gower and Holling-type III schemes, Comput. Math. Appl., 65, 1727-1737, 2013. [30] Q. Ye, Introduction to Reaction-Diffusion Equations, Science Press, Beijing, 2013. [31] Q. Yue, Dynamics of a modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge, SpringerPlus 5: 461, (2016). DOI 10.1186/s40064-016-2087-7. Biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 Page 8 of 8 http://dx.doi.org/10.11145/j.biomath.2017.07.107 Introduction and mathematical model Boundedness and Permanence Dissipation Persistence Periodic solutions Numerical simulations Conclusion References