Original article Biomath 1 (2012), 1210231, 1–7 B f Volume ░, Number ░, 20░░ BIOMATH ISSN 1314-684X Editor–in–Chief: Roumen Anguelov B f BIOMATH h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / Biomath Forum Predator-Prey Model with Prey Harvesting, Holling Response Function of Type III and SIS Disease Jean Jules Tewa∗, Ramses Djidjou Demasse† and Samuel Bowong‡ ∗ National Advanced School of Engineering, University of Yaounde I , UMI 209 UMMISCO, GRIMCAPE, Yaounde, Cameroon Email: tewajules@gmail.com † Faculty of Science, University of Yaounde I UMI 209 UMMISCO, GRIMCAPE, Yaounde, Cameroon Email: dramsess@yahoo.fr ‡ Faculty of Science, University of Douala UMI 209 UMMISCO, GRIMCAPE, Yaounde, Cameroon Email: sbowong@gmail.fr Received: 28 June 2012, accepted: 23 October 2012, published: 28 December 2012 Abstract—The populations of prey and predator interact with prey harvesting. When there is no predator, the logistic equation models the behavior of the preys. For interactions between preys and predators, we use the generalized Holling response function of type III. This function which models the consumption of preys by predators is such that the predation rate of predators increases when the preys are few and decreases when they reach their satiety. Our main goal is to analyze the influence of a SIS infectious disease in the community. The epidemiological SIS model with simple mass incidence is chosen, where only susceptibles and infectious are counted. We assume firstly that the disease spreads only among the prey population and secondly that it spreads only among the predator population. There are many bifurcations as: Hopf bifurcation, transcritical bifurcation and saddle-node bifurcation. The results indicate that either the disease dies out or persists and then, at least one population can disappear because of infection. For some particular choices of the parameters however, there exists endemic equilibria in which both populations survive. Numerical simulations on MATLAB and SCILAB are used to illustrate our results. Keywords-Predator; Prey; Infectious disease; Response function; Bifurcation; Global Stability I. INTRODUCTION There are many epidemiological or ecological models [6], [7], [8], [9], [10], [11], [5] in the literature and also many models which encompass the two fields [3], [4], [8], [9], [10], [11], [12]. Dynamic models for infectious diseases are mostly based on compartment structures that were initially proposed by Kermack and McKendrick (1927,1932) and developed later by many other researchers. The main questions regarding population dynamics concern the effects of infectious diseases in regulating natural populations, decreasing their population sizes, reducing their natural fluctuations, or causing destabi- lizations of equilibria into oscillations of the population states. With the Holling function response of type III, it is well known that the predators increase their searching activity when the prey density increases. Generally, if x denotes the density of prey population, the Holling function of type I is φ1(x) = r x where r is the intrinsic growth rate of preys. The Holling function of type II is φ2(x) = B ω0 x 1 + B ω1 x , where ω0 and ω1 denote respectively the time taking by a predator to search and Citation: J. Tewa, R. D. Demasse, S. Bowong, Predator-Prey Model with Prey Harvesting, Holling Response Function of Type III and SIS Disease, Biomath 1 (2012), 1210231, http://dx.doi.org/10.11145/j.biomath.2012.10.231 Page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.10.231 J. Tewa et al., Predator-Prey Model with Prey Harvesting, Holling Response Function of Type III... capture preys, B is the predation rate per unit of time. In the models considered in this work, the Holling function of type III is used for interactions between predators and preys : φ3(x) = m x2 a x2 + b x + 1 [2], where m and a are positive constants, b is an arbitrary constant. This function models the consumption of preys by predators. It is well known that with this function, the predation rate of predators increases when the preys are few and decreases when they reach their satiety (a predator increases his searching activity when the prey density increases). The functions φ1, φ2 and φ3 are respectively also referred to as Lotka-Volterra, Michaelis-Menten and sigmoidal response functions. Generally, there are more macroparasitic infections which can affect only preys, only predators or both preys and predators. Our goal in this paper is to analyze the influence of a SIS infectious disease which spreads only in one of the two populations. The models considered and analyzed here are different from all the models in the literature. Moreover, we use numerical simulations on MATLAB and SCILAB to illustrate our results. II. THE MODEL FORMULATION The model (s1) is obtained from the classic Lotka- Volterra model with simple mass action when the disease spreads only inside the prey population. In this model, the infected preys do not reproduce and there is no dis- ease related mortality. The model (s2) is obtained when the disease spreads only inside the predator population. These models are respectively  ẋ = r̃ ( 1 − x k̃ ) x − m̃x2y ãx2 + b̃x + 1 − λ̃x z +γ̃z − h̃1, ż = λ̃x z − γ̃z − m̃1z 2y ãz2 + b̃z + 1 , (s1) ẏ = c̃m̃x2y ãx2 + b̃x + 1 − m̃2z 2y ãz2 + b̃z + 1 − d̃y, x ≥ 0, z ≥ 0, y ≥ 0,  ẋ = r̃ ( 1 − x k̃ ) x − m̃x2y ãx2 + b̃x + 1 − η̃1x 2ω ãx2 + b̃x + 1 −h̃1, ẏ = c̃m̃x2y ãx2 + b̃x + 1 − d̃ y − δ̃ y ω + µ̃ ω, (s2) ω̇ = ẽm̃x2ω ãx2 + b̃x + 1 + δ̃yω − (µ̃ + d̃)ω, x ≥ 0, y ≥ 0, ω ≥ 0. where the variables z and ω denotes respectively the infected preys and infected predators, r̃ denotes the intrinsic growth rate of preys, d̃ is the natural death rate of predators, k̃ is the capacity of environment to support the growth of preys, h̃1 is the rate of preys’s harvesting, γ̃ and µ̃ are the recover rates of infected preys and infected predators respectively, λ̃ is the adequate contact rate between susceptible preys and infected preys while δ̃ is the adequate contact rate between susceptible predators and infected predators. We also assume that infected predators still can catch preys at a different rate η̃1 than sound ones. The parameter η̃1 can be thought to be less than m̃, if the disease affects the ability in hunting of the predators or larger than m̃, if we want to emphasize that the interactions with infected predators cause the preys to die for the disease even if they are not caught. ã and b̃ are positive constants. m̃ > 0 and m̃1 > 0 denote the adequate predation rate between predators and preys. c̃ and ẽ denote the conversion coefficients. m̃2 can be negative (conversion of prey’s biomass into predator’s biomass) or positive (bad effect of the infected preys for the predator population due to disease). Trough the linear transformation and time scaling (X, Z, Y, W, T ) = ( x k̃ , z k̃ , y c̃k̃ , ω ẽk̃ , c̃m̃k̃2t ) , the follow- ing simplified systems are obtained from (s1) and (s2),   ẋ = ρx(1 − x) − p(x) y − λ x z + γ z − h1, ż = λ x z − γ z − m1 p(z) y, ẏ = p(x) y − m2 p(z) y − d y, x ≥ 0; y ≥ 0; z ≥ 0, (1)   ẋ = ρx(1 − x) − p(x) y − η1 p(x) ω − h1, ẏ = p(x)y − dy − δ y ω + µ ω, ω̇ = e p(x)ω + δ1 y ω − µ1 ω, x ≥ 0; y ≥ 0; ω ≥ 0, (2) where the parameters are defined as follow ρ = r̃ c̃m̃k̃2 , η1 = η̃1ẽ c̃m̃ , η2 = η̃2ẽ c̃m̃ , h1 = h̃1 c̃m̃k̃3 , λ = λ̃ c̃m̃k̃ , γ = γ̃ c̃m̃k̃2 , m1 = m̃1 m̃ , m2 = m̃2 c̃m̃ , m3 = m̃3 c̃m̃ , d = d̃ c̃m̃k̃2 , δ = δ̃ẽ c̃m̃k̃ , µ = µ̃ẽ c̃2m̃k̃2 , e = ẽ c̃ , δ1 = δ̃ m̃k̃ , µ1 = µ̃ + d̃ c̃m̃k̃2 , a = ãk̃2, b = b̃k̃, p(x) = x2 a x2 + b x + 1 . (3) Systems (1) and System (2) are new and different from all the models in the literature. These models without disease give us the same system which has been analyzed without disease in [1]. Biomath 1 (2012), 1210231, http://dx.doi.org/10.11145/j.biomath.2012.10.231 Page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2012.10.231 J. Tewa et al., Predator-Prey Model with Prey Harvesting, Holling Response Function of Type III... III. RESULTS A. Results for the Model (1) with Disease only in Prey Population Let us set u1(x) = ρx(1 − x) − h1 (1 + m)p(x) − md , R0 = m2(p(η) − d)u21(η) a(p(η) − d)2u21(η) + b(λη − γ)(p(η) − d)u1(η) + (λη − γ)2 the basic reproduction number, and x1 = 1 − √ 1 − 4 h1 ρ 2 , x2 = 1 + √ 1 − 4 h1 ρ 2 , xz = γ λ , x0 = 1 2 , z0 ∈ R∗+, (4) the expressions of the positive real values x0, x1, x2, xz . Theorem 1: The equilibrium points of System (1), according to the values of the parameters, are given as follow : • When h1 > ρ 4 , then there is no equilibrium point. • When h1 = ρ 4 , then the unique equilibrium is B0(x0, 0, 0) which is a double point if d 6= 1 a + 2b + 4 and triple point if d = 1 a + 2b + 4 . • When h1 < ρ 4 and a d ≥ 1, then the equilibria are B1(x1, 0, 0) and B2(x2; 0; 0). • When h1 < ρ 4 ; a d < 1 and x3 = x1, then B1(x1, 0, 0) is a double point and B2(x2, 0, 0) ex- ists. • When h1 < ρ 4 ; a d < 1 and x3 = x2, then B1(x1, 0, 0) is simple and B2(x2, 0, 0) is a double point. • When h1 < ρ 4 ; a d < 1 and x3 ∈]x1; x2[, then B1(x1, 0, 0); B2(x2, 0, 0) and B3(x3, 0, y3) exist, where y3 = ρx3(1 − x3) − h1 d > 0. • When h1 < ρ 4 ; a d < 1 and x3 ∈ [0; x1[∪]x2; +∞[, then B1(x1, 0, 0) and B2(x2, 0, 0) exist. • When h1 < ρ 4 ; ad < 1; x4 ∈]η; x2[, x2 > max ( x3; γ λ ) and R0 > 1, then B1(x1, 0, 0); B2(x2, 0, 0) and B4(x4, z4, y4) exist, where x4 > 0, z4 > 0 and y4 > 0. Proof : These equilibria are obtained by setting the right hand side of (1) equals to zero. For y = 0 one has equation ρx2 − ρx + h1 = 0. Then we have B0, B1 and B2. For z = 0, one has p(x) = d ⇐⇒ (1 − a d)x2 − b dx−d = 0. We deduce x3 and then B3. The condition for existence of B4 is p(z) = 1 m2 (p(x) − d) > 0 ie p(x) − d > 0 ⇐⇒ a d < 1 and x ∈]x3, +∞[. Concerning the stability of these equilibria, the fol- lowing theorem hold. Theorem 2: Let’s consider System (1). • The equilibria B0 and B1 are always unstable. • The equilibrium B2 is stable if one of the following conditions is satisfied : h1 < ρ 4 , γ λ ≥ x2 and p(x2) ≤ d, or h1 < ρ 4 , γ λ < x2, p(x2) = d and p′′(x2) ≤ 0. • The equilibrium B3 is stable if one of the following conditions is satisfied. h1 < ρ 4 , ad < 1, x3 ∈]x1; x2[ and x3 = γ λ , or h1 < ρ 4 , ad < 1, x3 ∈]x1; x2[, x3 < γ λ and d > 1 a + 2b + 4 , or h1 < ρ 4 , ad < 1, x3 ∈ ]x1; x2[, x3 < γ λ , d < 1 a + 2b + 4 and χ0(x3) < 0, where χ0(x3) is the eigenvalue of x3. • The equilibrium point B4(x4, z4, y4) is asymptoti- cally stable if and only if the following conditions hold : a2 < 0; a2a1 + a0 > 0 and a1a0 > 0, where  a2 = ρ(1 − 2x4) − p′(x4)y4 − λz4 +λx4 − γ − m1p′(z4)y4, a1 = − [ρ(1 − 2x4) − p′(x4)y4 − λz4] × [λx4 − γ − m1p′(z4)y4] −λm1p(z4)y4 − p(x4)p′(x4)y4, a0 = − [ρ(1 − 2x4) − p′(x4)y4 − λz4] × [λx4 − γ − m1p′(z4)y4] + λm2p(x4)p′(z4)y4z4 +p′(x4)y4m1p(z4)(λx4 − γ) +p(x4)p′(x4)y4(λx4 − γ − m1p′(z4)y4). (5) Proof : The eigenvalues of the jacobian matrix J(B0) are χ1 = 0; χ2 = λx0 − γ and χ3 = p(x0) − d. a) If γ λ < 1 2 or d < 1 a + 2b + 4 = p(x0), then χ2 > 0 or χ3 > 0 and B0 is unstable. b) If γ λ > 1 2 and d = 1 a + 2b + 4 = p(x0), then χ2 < 0 and χ3 = 0. Hence, the stability of B0 is given by the center manifold theorem. The translation (u1, u2, u3) = (x − x0, z, y) brings the singular point B0 to the origin. In the neighborhood of the origin and, since h1 = ρ 4 , System (1) has a new form. The Jacobian matrix J(B0) is not diagonalizable and the passage matrix to the Jordan’s basis is Biomath 1 (2012), 1210231, http://dx.doi.org/10.11145/j.biomath.2012.10.231 Page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2012.10.231 J. Tewa et al., Predator-Prey Model with Prey Harvesting, Holling Response Function of Type III... P =   −1 0 10 0 −1 0 1 0  . By the transformation (v1, v2, v3)T = P −1(u1, u2, u3)T , the system becomes:  v̇1 = v2 + p′(x0)(v1v3 − v23) + p′′(x0) 2 (v21 + v 2 3 − 2v1v3)v2 + m1p ′′(0) 2 v2v 2 3 + O(|(v1, v2, v3)| 4), v̇2 = v3 + p′(x0)(v1v3 − v23) + p′′(x0) 2 (v21 + v 2 3 − 2v1v3)v2 − m2p ′′(0) 2 v2v 2 3 + O(|(v1, v2, v3)| 4), v̇3 = χ2v3 − λ(v1v3 − v23) + m1p ′′(0) 2 v2v 2 3 +O(|(v1, v2, v3)|4). (6) We can now find that the center manifold is given by W c = {v3 = 0}. Therefore, the system (6) is topologically equivalent, around the origin, to the following system:  v̇1 = v2 + p′′(x0) 2 v21v2 + O(|(v1, v2)| 4), v̇2 = O(|(v1, v2)|4), v̇3 = O(|(v1, v2)|4). Then, the singular point B0 is unstable. c) If γ λ = 1 2 and d = 1 a + 2b + 4 = p(x0), then χ2 = 0 and χ3 = 0. Applying the center manifold theory as previously, B0 is unstable. d) If γ λ = 1 2 and d > 1 a + 2b + 4 = p(x0), we have χ2 = 0 and χ3 < 0. Applying the center manifold theory as previously, B0 is unstable. The stability of B1 is obtained with jacobian matrix. The stability of B2 is obtained using the center manifold theorem. Taking into account the fact that p(x3) = d, one find that the characteristic polynomial of the linearized system around the singular point B3 is Q(χ) = (χ − λx3 + γ) [ −χ2 + (ρ(1 − 2x3) − p′(x3)y3)χ ] −d(χ − λx3 + γ)p′(x3)y3. The discriminant of Q(χ) is ∆3(h1) = ( ρ(1 − 2x3) − p′(x3)y3 )2 − 4dp′(x3)y3. (7) a) If x3 > γ λ , then the eigenvalue χ1 = λx3 −γ > 0. Hence, B3 is unstable. b) If x3 < γ λ , then χ1 < 0. b1) When ∆3(h1) = 0 the Jacobian matrix at B3 has a double eigenvalue χ0(x3) := ρ(1 − 2x3) − p′(x3)y3 2 . (8) • If d ≥ 1 a + 2b + 4 , then x3 > 1 2 . From where χ0(h1) < 0. Therefore, the singular point B3 is stable. • If d < 1 a + 2b + 4 , then: When χ0(h1) < 0 (resp. χ0(h1) > 0) the singular point B3 is stable (resp. unstable). b2) When ∆3 > 0 the eigenvalues of the Jacobian matrix at B3 are χ1 < 0, χ2 = χ0(h1) − √ ∆3 2 and χ3 = χ0(h1) + √ ∆3 2 . We have, χ2χ3 = dp′(h1)y3 > 0 and χ2 + χ3 = χ0(h1), where χ0(h1) is defined by (8). • If d ≥ 1 a + 2b + 4 , then the singular point B3 is stable. • If d < 1 a + 2b + 4 , then: When χ0(h1) < 0 (resp. χ0(h1) > 0) the singular point B3 is stable (resp. unstable). b3) If ∆3 < 0, then the eigenvalues of the Jacobian matrix at B3 are χ1 < 0, χ2 = χ0(h1) − i √ −∆3 2 and χ3 = χ0(h1) + i √ −∆3 2 , where χ0(h1) is defined by (8). If d ≥ 1 a + 2b + 4 , then the singular point B3 is stable. If d < 1 a + 2b + 4 and χ0(h1) < 0 then, the singular point B3 is stable. If d < 1 a + 2b + 4 and χ0(h1) > 0 then, the singular point is unstable. If d < 1 a + 2b + 4 and χ0(h1) = 0 then, the real central and stable spaces are respec- tively defined by Ec = 〈(1, 0, 0); (0, 0, 1)〉 and Es = 〈 (1, −1 − dp′(x3)y3 χ21 , p′(x3)y3 χ1 ) 〉 . Then applying the center manifold theorem it comes that the singular point B3 is unstable. The stability of B4 is obtained using the Routh-Hurwitz conditions. Biomath 1 (2012), 1210231, http://dx.doi.org/10.11145/j.biomath.2012.10.231 Page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2012.10.231 J. Tewa et al., Predator-Prey Model with Prey Harvesting, Holling Response Function of Type III... B. Results for the Model (2) with Disease only inside Predator Population Let us set u2(x) = e δ [ µ1 e − p(x) ] and v2(x) = (p(x) − d)u2(x) e [ d e − p(x) ] . Let x5 the eventual positive root of equation p(x5) = d e and the function g2(x) = ρx(1 − x) − h1 − p(x)u2(x) − η1p(x)v2(x). Hypothesis 1 : The attack of non-infected predators is more important than the one of the infected predators i.e. e = ẽ c̃ ≤ 1. Theorem 3: The equilibria of System (2), where x0; x1 and x2 are given by (4), according to the values of the parameters, are given as follow. • When h1 > ρ 4 , then there is no equilibrium point. • When h1 = ρ 4 , then C0(x0; 0; 0) is a double point if d 6= 1 a + 2b + 4 and triple point if d = 1 a + 2b + 4 . • When h1 < ρ 4 and a d ≥ 1, then C1(x1; 0; 0) and C2(x2; 0; 0) exist. • When h1 < ρ 4 ; a d < 1 and x3 = x1, then C1(x1; 0; 0) is a double point and C2(x2; 0; 0) ex- ists. • When h1 < ρ 4 ; a d < 1 and x3 = x2, then C1(x1; 0; 0) exists and C2(x2; 0; 0) is a double point. • When h1 < ρ 4 ; a d < 1 and x3 ∈]x1; x2[, then the equilibria are C1(x1; 0; 0); C2(x2; 0; 0) and C3(x3; y3; 0), where y3 = ρx3(1 − x3) − h1 d > 0. • When h1 < ρ 4 ; a d < 1 and x3 ∈ [0; x1[∪]x2; +∞[, then the equilibria are C1(x1; 0; 0) and C2(x2; 0; 0). • When h1 < ρ 4 ; a d < 1; a d e ≥ 1, x6 ∈ ]x1; x2[∩]x3; +∞[; x2 > x3 or h1 < ρ 4 ; a d e < 1, x6 ∈]x1; x2[∩]x3; x5[; x2 > x3; x1 < x5, then the equilibria are C1(x1; 0; 0); C2(x2; 0; 0) and C4(x6; y6; ω6), y6 = u2(x6) and ω6 = v2(x6). Proof : The equilibria C0, C1, C2 and C3 are obtained in the same way as in theorem 1, setting the right hand side of the system equals to zero. Equilibrium C4 exists when the previous conditions are satisfied. Concerning the Stability analysis of these equilibria, the following theorem holds. Theorem 4: Let’s consider the System (2) and sup- pose that Hypothesis 1 holds. • The equilibria C0 and C1 are always unstable. • The equilibrium C2 is stable if h1 < ρ 4 and p(x2) < d. • The equilibrium C3 is stable if and only if one of these conditions is satisfied : h1 < ρ 4 , ad < 1, x3 ∈ ]x1; x2[ and y3 = e δ1 ( µ1 e − d ) or h1 < ρ 4 , ad < 1, x3 ∈]x1; x2[, y3 < e δ1 ( µ1 e − d ) , d > p(x0), or h1 < ρ 4 , ad < 1, x3 ∈]x1; x2[, y3 < e δ1 ( µ1 e − d ) , d < p(x0), ξ0(x3) < 0. • The singular point C4(x6, y6, ω6) is asymptotically stable if and only if the following conditions are satisfied : b2 < 0; b2b1 + b0 > 0 and b1b0 > 0, where  b2 = ρ(1 − 2x6) − p′(x6)(y6 + η1ω6) +p(x6) − d − δω6; b1 = − (ρ(1 − 2x6) − p′(x6)(y6 + η1ω6)) × (p(x6) − d − δω6) + δ1ω6(µ − δy6) −p(x6)p′(x6)y6 − eη1p(x6)p′(x6)ω6; b0 = ep(x6)p′(x6)ω6 [δy6 − µ + η1(p(x6) − d − δω6)] −δ1η1p(x6)p′(x6)y6ω6 −δ1ω6(µ − δy6) (ρ(1 − 2x6) − p′(x6)(y6 + η1ω6)) . (9) Proof : The stability of C0 is deduce as for B0 in theorem 2. The jacobian matrix always has a positive eigenvalue. Then, C1 is unstable. We obtain the stability of C2 and C3 applying the same arguments as for B2 and B3 in theorem 2. The stability of C4 is obtained using the Routh-Hurwitz conditions. IV. HOPF BIFURCATION Let us introduce the following parameters h10 = ρx3 bx3 + 2 [ 2ax33 + (b − a)x 2 3 + 1 ] , (10) and Π = 1 16 [ p(2)(x3) + p (3)(x3) ] − (p′(x3))2 4 √ −∆3(h10) . (11) Recalling (4), the flow of System (1) and System (2) respectively undergo a supercritical Hopf bifurcation around h10 given by the following result Theorem 5: (Hopf bifurcation) Let h1 < ρ 4 ; ad < 1; x3 ∈]x1, min ( 1 2 , γ λ ) [. Thanks to Hypothesis 1. Then, a unique stable curve of periodic solution bifurcates from the singular points B3 and C3 into the regions h1 > h10 if Π < 0 or h1 < h10 if Π > 0. The singular points B3 and C3 are stable for h1 < h10 and unstable for Biomath 1 (2012), 1210231, http://dx.doi.org/10.11145/j.biomath.2012.10.231 Page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2012.10.231 J. Tewa et al., Predator-Prey Model with Prey Harvesting, Holling Response Function of Type III... h1 ≥ h10. This correspond to supercritical stable Hopf bifurcation. Proof : The proof can be obtained as in [13]. V. NUMERICAL SIMULATIONS 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 Time (t) x( t), z (t) , y (t) Non infected preys Infected preys Non infected predators (a) 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 0 1 2 Infected preys (z) Non infected preys (x) N on in fe ct ed p re da to rs (y ) Initial condition Trajectory Initial condition Trajectory Initial condition Trajectory Equlibrium B1 Equilibrium B2 Invariant axis ( 1) (b) Fig. 1. Phase portraits of System (1) for h1 < ρ 4 ; γ λ = x2 and p(x2) < d. B1 and B2 are unstable. The axis x = γ λ is stable. 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 0 0.5 1 1.5 2 Infected preys (z) Non infected preys (x) N on in fe ct ed p re da to rs (y ) Initial condition Trajectory Equilibrium B1 Equilibrium B2 (a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.50 0.5 1 1.5 2 2.5 3 Time (t) x( t), z (t) , y (t) Non infected preys Infected preys Non infected predators (b) Fig. 2. Phase portraits of System (1). The case (a) corresponds to h1 < ρ 4 ; γ λ < x2 and p(x2) < d. The case (b) corresponds to h1 = ρ 4 ; γ λ > 1/2 and d = 1 a + 2b + 4 . Unstability of B1 and B2. 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 Time (t) x( t), y (t) , (t ) Non infected preys Non infected predators Infected predators (a) 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Non infected predators (y)Non infected preys (x) In fe ct ed p re da to rs ( ) Initial condition Trajectory Equilibrium C1 Equilibrium C2 (b) Fig. 3. Phase portraits of System (2) for h1 < ρ 4 and d > p(x2). Stability of C2. 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 Time (t) x( t), y (t) , (t ) Non infected preys Non infected predators Infected predators (a) 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 0 10 20 30 40 Non infected predators (y)Non infected preys (x) In fe ct ed p re da to rs ( ) Initial condition Trajectory Equilibrium C1 Equilibrium C2 Equilibrium C3 (b) Fig. 4. Phase portraits of System (2) for h1 < ρ 4 and d < p(x2). Unstability of C1, C2 and C3. (a) (b) Fig. 5. Phase portraits of System (1). The case (a) corresponds to h1 < ρ γ λ ( 1 − γ λ ) . The case (b) corresponds to h1 > ρ γ λ ( 1 − γ λ ) and γ λ < 1 2 . Illustration of saddle-node bifurcation phenomenon. VI. CONCLUSION Our goal was to analyze the modifications on a preda- tor prey model (generalized Gause model) with prey har- vesting and Holling response type III : m x2 a x2 + b x + 1 , to account for a disease spreading among one of the two species. The simple epidemiological model SIS has been chosen, where only susceptibles and infectives are counted. The results indicate that either the disease dies out, leaving only neutral cycles of generalized Gause model, or one species disappears and all individuals in the other one eventually become infected. For some particular choices of the parameters however, endemic equilibria in which both populations survive seem to arise. REFERENCES [1] R.M. Etoua and C. Rousseau, Bifurcation analysis of a general- ized Gause model with prey harvesting and a generalized Holling response function of type III , J. Differ. Equations 249, No. 9, 2316–2356 (2010), ISSN 0022–0396. [2] A.D. Bazykin, A. Iosifovich Khibnik and B. Krauskopf, Nonlin- ear Dynamics of Interacting Populations, World Scientific, 1998, 193 pages. [3] K.P. Hadeler and H.I. Freedman, Predator-prey populations with parasitic infection, J. Math. Biol. 27, (1989) 609–631. http://dx.doi.org/10.1007/BF00276947 [4] E. 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Biomath 1 (2012), 1210231, http://dx.doi.org/10.11145/j.biomath.2012.10.231 Page 7 of 7 http://dx.doi.org/10.1016/j.amc.2011.10.085 http://dx.doi.org/10.1093/imammb/dqp007 http://dx.doi.org/10.1016/j.ecocom.2010.04.001 http://dx.doi.org/10.11145/j.biomath.2012.10.231 Introduction The Model Formulation Results Results for the Model (??) with Disease only in Prey Population Results for the Model (??) with Disease only inside Predator Population Hopf Bifurcation Numerical Simulations Conclusion References