Dynamic behaviors of the periodic predator–prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect

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Abstract

In this paper, a predator–prey system which based on a modified version of the Leslie–Gower scheme and Holling-type II scheme with impulsive effect are investigated, where all the parameters of the system are time-dependent periodic functions. By using Floquet theory of linear periodic impulsive equation, some conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are obtained. It is proved that the system can be permanent if all the trivial and semi-trivial periodic solutions are linearly unstable. We use standard bifurcation theory to show the existence of nontrivial periodic solutions which arise near the semi-trivial periodic solution. As an application, we also examine some special case of the system to confirm our main results.

Introduction

A natural predator–prey system is described as follows:x˙(t)=ax(t)-φ(x(t))y(t),y˙(t)=-by(t)+γφ(x(t))y(t),where a,b,γ are positive constants, φ(x) is the functional response of the predator y(t). Recently the dynamics of a predator–prey system is studied in many papers, for example, Amine et al. [1], Tang [22], López-Gómez et al. [15] and Rinaldi et al. [19]. In this paper, we consider a predator–prey model which incorporates a modified version of the Leslie–Gower functional response as well as that of the Holling-type II. About Holling-type II we can see [14], and about Leslie–Gower scheme we can see [2], [3], [7], [12], [13], [24].

The predator–prey model describes a prey population x which serves as food for a predator y, however, due to seasonal effects of weather, temperature, food supply, mating habits, contact with predators and other resource or physical environmental quantities, we can assume temporal variation to be cyclic or periodic. The rate equations for the two components of system with periodic coefficients can be written as follows:x˙(t)=r1(t)-b1(t)x(t)-a1(t)y(t)x(t)+k1(t)x(t),y˙(t)=r2(t)-a2(t)y(t)x(t)+k2(t)y(t),where x and y represent the population densities at time t; b1(t),ri(t),ai(t),ki(t)(i=1,2) are model parameters assuming only positive values, r1(t) is the growth rate of prey x, b1(t) measures the strength of competition among individuals of species x, a1(t) is the maximum value which per capita reduction rate of x can attain, k1(t) (respectively, k2(t)) measures the extent to which environment provides protection to prey x (respectively, to predator y), r2(t) describes the growth rates of y, and a2(t) has a similar meaning to a1(t).

The Leslie–Gower formulation is based on the assumption that reduction in a predator population has a reciprocal relationship with per capita availability of its preferred food. Indeed, Leslie [10] introduced a predator–prey model where the carrying capacity of the predator's environment is proportional to the number of prey. This interesting formulation for the predator dynamics has been discussed by Leslie and Gower in [11] and by Pielou in [17]. It is dydt=r2y(1-y/αx), in which the growth of the predator population is of logistic form (i.e., dydt=r2y(1-y/C)), but the conventional ‘C’, which measures the carrying capacity set by the environmental resources is C=αx, proportional to prey abundance (α is the conversion factor of prey into predators). The term y/αx of this equation is called Leslie–Gower term. It measures the loss in the predator population due to rarity (per capita y/x) of its favorite food. In the case of severe scarcity, y can switch over to other populations but its growth will be limited by the fact that its most favorite food (x) is not available in abundance. This situation can be taken care of by adding a positive constant d to the denominator. Hence, the equation above becomes dydt=r2y(1-y/(αx+d)), and thus, dydt=y(r2-(r2/α)(y/(x+d/α))); that is the second equation of system (1.2), dydt=(r2-a2y/(x+k2))y; see [2].

The above system can be considered as a representation of an insect pest-spider food chain, nature abounds in systems which exemplify this model; see [24].

However, the ecological system is often affected by environmental changes and other human activities. In many practical situations, it is often the case that predator or parasites are released at some transitory time slots and harvest or stock of the species are seasonal or occur in regular pulses. These short-time perturbations are often assumed to be in the form of impulses in the modelling process. Consequently, impulsive differential equations (hybrid dynamical systems) provide a natural description of such systems, see [9]. Equations of this kind are found in almost every domain of applied sciences. Numerous examples are given in Bainov's and his collaborators’ books [4]. They generally describe phenomena which are subject to steep and/or instantaneous changes. Some impulsive equations have been recently introduced in population dynamics, such as vaccination [5], [21], chemotherapeutic treatment of disease [8], [16], chemostat [6], birth pulse [20], [23].

In this paper, we consider the following T-periodic predator–prey system with impulsive effectsx˙(t)=r1(t)-b1(t)x(t)-a1(t)y(t)x(t)+k1(t)x(t),y˙(t)=r2(t)-a2(t)y(t)x(t)+k2(t)y(t),tτk,kZ+,x(τk+)=(1+hk)x(τk),y(τk+)=(1+gk)y(τk),t=τk,kZ+,where b1(t),ai(t),ri(t),ki(t)(i=1,2) are continuous T-periodic functions such that b1(t)>0,ai(t)>0,ri(t)>0,ki(t)>0(i=1,2) and Z+={1,2,}. Assume that hk,gk(kZ+) are constants and there exists an integer q>0 such that hk+q=hk,gk+q=gk,τk+q=τk+T. With model (1.3), we can take into account the possible exterior effects under which the population densities change very rapidly. For instance, impulsive reduction of the population density of a given species is possible after its partial destruction by catching or by poisoning with chemicals used in agriculture (hk<0orgk<0). A natural constraint in this case is 1+hk>0,1+gk>0 for all kZ+. An impulsive increase of the density is possible by artificial breeding of the species or release some species (hk>0,gk>0).

The purpose of this paper is to investigate the dynamic behaviors of predator–prey system in a periodic environment with impulsive perturbations. Model (1.3) exhibits three types of T-periodic component-wise nonnegative solutions, the solution (0,0), usually known as the trivial one; those with one component vanishing, often known as the semi-trivial positive solutions; and the coexistence states, which are the solutions with both components positive. In Section 2, we give some notations and preliminaries. The linear stability conditions of trivial periodic solution and semi-trivial periodic solutions are given in Section 3, we also get some sufficient conditions for the permanence of the system. In Section 4, by using standard techniques of bifurcation theory, we show the existence of nontrivial periodic solutions which arise near the semi-trivial periodic solution. In final section, we conclude our paper and give an specific example to confirm our main results.

Section snippets

Notations and preliminaries

Let JR,PC(J,R) denotes the set of functions ψ:JR which are continuous for tJ,tτk, and have discontinuities of the first kind at the points τkJ where they are continuous from the left. PC(J,R) denotes the set of functions ψ:JR with a derivative dψdtPC(J,R). Throughout this paper we consider the Banach spaces of T-periodic functions PCT={ψPC([0,R],R)ψ(0)=ψ(T)}. Under the supremum norm: ψPC=sup{|ψ(t)|:t[0,T]} and PCT={ψPC([0,T],R)|ψ(0)=ψ(T)} under the supremum norm: ψPCT=max{ψPC

Extinction and permanence

In this section, we give the following characterization of the linear stability of the trivial and semi-trivial states. Firstly, we present the Floquet theory for the linear T-periodic impulsive equation.dxdt=A(t)x,tτk,tR,Δx=Bkx,t=τk,kZ.Then we introduce the following conditions:

  • H1:

    A(·)PC(R,Cn×n) and A(t+T)=A(t)(tR),

  • H2:

    BkCn×n,det(E+Bk)0,τk<τk+1(kZ),

  • H3:

    There exists a qN such that Bk+q=Bk,τk+q=τk+T(kZ).

Let Φ(t) be a fundamental matrix of (3.1), then there exists a unique nonsingular matrix MCn×n

Existence of the positive T-periodic solution of (1.3) and bifurcation

In this section, we deal with the problem of the existence of strictly positive (componentwise) periodic solution of (1.3) by means of bifurcation theory of Rabinowitz [18], near the semi-trivial periodic solution (θ[r1,b1],0). In Section 3, we see that, when parameter [r1] passes through some critical value, the stability of semi-trivial solution (θ[r1,b1],0) changes.

Fixing [r1]>1Tlnk=1q11+hk and regarding r2 as bifurcation parameter, we shall show that μ*=1Tlnk=1q11+gk is a bifurcation

Conclusions and example

In this paper, we have investigated the dynamic behaviors of a periodic predator–prey model with modified Leslie–Gower and Holling-type II schemes and impulsive effect. We have shown that there exist three types of T-periodic component-wise nonnegative solutions: trivial solution (0,0), semi-trivial periodic solutions (θ[r1,b1],0) and (0,θ[r2,a2/k2]), nontrivial T-periodic solutions which bifurcate from the semi-trivial branch corresponding to the coexistence of the two species. The stability

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This work is supported by the National Natural Science Foundation of China (No. 10471117), the Henan Innovation Project for University Prominent Research Talents (No. 2005KYCX017) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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