Predator-Prey Model with Prey Harvesting , Holling Response Function of Type III and SIS Disease

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. I NTRODUCTION 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 destabilizations 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, ifx denotes the density of prey population, the Holling function of type I isφ1(x) = r x wherer is the intrinsic growth rate of preys. The Holling function of type II is φ2(x) = B ω0 x 1 + B ω1 x , whereω0 andω1 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 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 andφ3 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. T HE MODEL FORMULATION The model (s1) is obtained from the classic LotkaVolterra 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 disease 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− m̃x 2y 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− m̃x 2y 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 variablesz and ω denotes respectively the infected preys and infected predators, ̃ denotes the intrinsic growth rate of preys,̃ d is the natural death rate of predators,̃k is the capacity of environment to support the growth of preys,̃ h1 is the rate of preys’s harvesting, γ̃ andμ̃ are the recover rates of infected preys and infected predators respectively,̃ λ is the adequate contact rate between susceptible preys and infected preys while δ̃ s 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 thañ m, 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 andb̃ are positive constants.̃ m > 0 andm̃1 > 0 denote the adequate predation rate between predators and preys. c̃ and ẽ denote the conversion coefficients.̃ m2 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

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 destabilizations 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 φ 1 (x) = r x where r is the intrinsic growth rate of preys.The Holling function of , where ω 0 and ω 1 denote 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 : φ 3 (x) = m x 2 a x 2 + 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 φ 1 , φ 2 and φ 3 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 disease related mortality.The model (s2) is obtained when the disease spreads only inside the predator population.These models are respectively where the variables z and ω denotes respectively the infected preys and infected predators, r denotes the intrinsic growth rate of preys, d is the natural death rate of predators, k is the capacity of environment to support the growth of preys, h1 is the rate of preys's harvesting, γ and μ are the recover rates of infected preys and infected predators respectively, λ is the adequate contact rate between susceptible preys and infected preys while δ 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 η1 than sound ones.The parameter η1 can be thought to be less than m, if the disease affects the ability in hunting of the predators or larger than m, 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 b are positive constants.m > 0 and m1 > 0 denote the adequate predation rate between predators and preys.c and ẽ denote the conversion coefficients.m2 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 , c mk 2 t , the following simplified systems are obtained from (s1) and (s2), where the parameters are defined as follow (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].

A. Results for the Model (1) with Disease only in Prey Population
the basic reproduction number, and the expressions of the positive real values x 0 , x 1 , x 2 , x z .
Theorem 1: The equilibrium points of System (1), according to the values of the parameters, are given as follow : , then there is no equilibrium point.
• The equilibria B 0 and B 1 are always unstable.
• The equilibrium B 2 is stable if one of the following conditions is satisfied : and χ 0 (x 3 ) < 0, where χ 0 (x 3 ) is the eigenvalue of x 3 .• The equilibrium point B 4 (x 4 , z 4 , y 4 ) is asymptotically stable if and only if the following conditions hold : a 2 < 0; a 2 a 1 + a 0 > 0 and a 1 a 0 > 0, where Proof : The eigenvalues of the jacobian matrix J(B 0 ) = p(x 0 ), then χ 2 < 0 and χ 3 = 0. Hence, the stability of B 0 is given by the center manifold theorem.The translation (u 1 , u 2 , u 3 ) = (x − x 0 , z, y) brings the singular point B 0 to the origin.In the neighborhood of the origin and, since By the transformation (v 1 , v 2 , v 3 ) T = P −1 (u 1 , u 2 , u 3 ) T , the system becomes: We can now find that the center manifold is given by W c = {v 3 = 0}.Therefore, the system ( 6) is topologically equivalent, around the origin, to the following system: Then, the singular point B 0 is unstable.= p(x 0 ), we have χ 2 = 0 and χ 3 < 0. Applying the center manifold theory as previously, B 0 is unstable.
The stability of B 1 is obtained with jacobian matrix.The stability of B 2 is obtained using the center manifold theorem.Taking into account the fact that p(x 3 ) = d, one find that the characteristic polynomial of the linearized system around the singular point B 3 is b1) When ∆ 3 (h 1 ) = 0 the Jacobian matrix at B 3 has a double eigenvalue , then x 3 > 1 2 .From where χ 0 (h 1 ) < 0. Therefore, the singular point B 3 is stable.
, then the singular point B 3 is stable.
and χ 0 (h 1 ) = 0 then, the real central and stable spaces are respectively defined by E c = (1, 0, 0); (0, 0, 1) and ) .Then applying the center manifold theorem it comes that the singular point B 3 is unstable.
The stability of B 4 is obtained using the Routh-Hurwitz conditions.
Hypothesis 1 : The attack of non-infected predators is more important than the one of the infected predators i.e. e = ẽ c ≤ 1. Theorem 3: The equilibria of System (2), where x 0 ; x 1 and x 2 are given by ( 4), according to the values of the parameters, are given as follow.
, then there is no equilibrium point.
then the equilibria are C 1 (x 1 ; 0; 0); C 2 (x 2 ; 0; 0) and C 4 (x 6 ; y 6 ; ω 6 ), y 6 = u 2 (x 6 ) and ω 6 = v 2 (x 6 ).Proof : The equilibria C 0 , C 1 , C 2 and C 3 are obtained in the same way as in theorem 1, setting the right hand side of the system equals to zero.Equilibrium C 4 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 suppose that Hypothesis 1 holds.
• The equilibria C 0 and C 1 are always unstable.
• The equilibrium C 2 is stable if h 1 < ρ 4 and p(x 2 ) < d.
• The equilibrium C 3 is stable if and only if one of these conditions is satisfied : ) Proof : The stability of C 0 is deduce as for B 0 in theorem 2. The jacobian matrix always has a positive eigenvalue.Then, C 1 is unstable.We obtain the stability of C 2 and C 3 applying the same arguments as for B 2 and B 3 in theorem 2. The stability of C 4 is obtained using the Routh-Hurwitz conditions.

IV. HOPF BIFURCATION
Let us introduce the following parameters and Recalling (4), the flow of System (1) and System (2) respectively undergo a supercritical Hopf bifurcation around h 10 given by the following result Theorem 5: (Hopf bifurcation) Let h 1 < ρ 4 ; ad < 1; Proof : The proof can be obtained as in [13].
V. NUMERICAL SIMULATIONS   , 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.