Permanence and extinction of a periodic predator–prey delay system with functional response and stage structure for prey

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Abstract

In this paper, we consider a periodic coefficients predator–prey system with functional response and infinite delay, in which the prey has a history that takes them through two stages, immature and mature. Sufficient conditions which guarantee the permanence and extinction of the system are obtained. Finally, we give an example.

Introduction

It is well known that past history as well as current conditions can influence population dynamics and such interactions has motivated the introduction of delays in population growth. There are several books [1], [2], [3], [4]devoted to investigations of the dynamic behavior of delay differential equations. Recently, Stage structure models have received much attraction [6], [7], [8], [9], [10], [11], [19]. This is not only because they are much more simple than the models governed by partial differential equations but also they can exhibit phenomena similar to those of partial differential models [6], and many important physiological parameters can be incorporated. The single species model with stage structure was studied by Aiello and Freedman [7]. Two species models with stage structure were investigated by Wang and Chen [8], Xiao and Chen [9] and Magnusson [10]. Zhang, Chen and Neumann [12] proposed the following autonomous stage structure predator–prey system:x1=αx2-r1x1-βx1-ηx12-β1x1x3,x2=βx1-r2x2,x3=x3(-r+kβ1x1-η1x3),where α, β, β1, η, η1, r, r1, r2 and k are all positive constants, k is a digesting constant. Sufficient conditions which ensure the permanence of two species and extinction of one or two species are obtained.

On the other hand, since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a periodically varying environment are considered as important selective forces on systems in a fluctuating environment. More realistic and interesting models should take into account both the seasonality of the changing environment [13], [14]. This motivated Cui and Song [15] to consider the following periodic nonautonomous predator–prey model with stage structure for prey:x1˙=a(t)x2-b(t)x1-d(t)x12-p(t)x1y,x2˙=c(t)x1-f(t)x22,y˙=y[-g(t)+h(t)x1-q(t)y],where a(t), b(t), c(t), d(t), f(t), g(t), h(t), p(t) and q(t) are all continuous positive ω-periodic functions. x1 and x2 denote the density of immature and mature population (prey) respectively, and y is the density of the predator that only prey on x1 (immature prey). They obtained a set of sufficient and necessary condition which guarantee the permanence of the above system. Recently, maybe stimulated by the works of Teng and Chen [16], Cui and Sun [18] further incorporated infinite delay to system (2) and investigated the following model:x1˙=a(t)x2-b(t)x1-d(t)x12-p(t)x1-0k12(s)y(t+s)ds,x2˙=c(t)x1-f(t)x22,y˙=y-g(t)+h(t)-0k21(s)x1(t+s)ds-q(t)-0k22(s)y(t+s)ds.Under the assumption that the coefficients in (3) are all ω-periodic and continuous for t  0, a(t), b(t), c(t), d(t) and f(t) are all positive, p(t), h(t) and q(t) are nonnegative, and 0ωq(t)dt>0, 0ωg(t)dt0. The functions kij(s) (i, j = 1, 2) defined on R-=(-,0] are nonnegative and integrable, -0kij(s)=1. By using analysis technique, they obtained a set of sufficient and necessary conditions which guarantee the permanence of the system.

Noticing that (1), (2), (3) are modified from the classical Lotka–Volterra predator–prey system:x=x(a-bx-cy),y=y(-d+ex-fy).A predators functional response is its per capita feeding rate on prey. Holling [20], [21] suggested that the predator should not be able to consume an unlimited number of prey as the prey population increases. That is, in the Lotka–Volterra equations, the number of prey consumed per predator is unlimited as the prey population increases. The number of prey removed is cxy, so that the number of prey eaten per predator is unlimited as x increases to infinity. Holling proposed three models of the rate of prey capture per predator as a function of prey population density: Types I, II, and III. In 2001, Skalski and Gilliam reviewed the literature on functional response curves and presented statistical evidence from 19 predator–prey systems that three predatordependent functional responses [22], [23], [24], [25], i.e., models that are functions of both prey and predator abundance because of predator interference, can provide better descriptions of predator feeding over a range of predator–prey abundances. No single functional response best described all of the data sets.

To our knowledge, seldom did scholars consider the stage structure predator–prey system with functional response and infinite delay. This paper is largely motivated by the above-mentioned fact. We consider the following system:x1˙=a(t)x2-b(t)x1-d(t)x12-p(t)x12(1-e-γx1)P+x12-0k12(s)y(t+s)ds,x2˙=c(t)x1-f(t)x22,y˙=y-g(t)+h(t)-0k21(s)x12(t+s)(1-e-γx1(t+s))P+x12(t+s)ds-y-q(t)-0k22(s)y(t+s)ds2,where x1 and x2 denote the density of immature and mature population (prey) respectively. y is the density of the predator that only prey on x1. The coefficients in (5) are all continuous positive ω-periodic for t  0. P, γ are positive constants. The functions kij(s) (i, j = 1, 2) defined on R-=(-,0] are nonnegative and integrable, -0kij(s)=1. x12P+x12(Hollingtype) and 1-e-γx1(Ivlevtype) represent the functional response of predator to prey. The biological background for (5) can be found in [3], [15], [16].

Let C+={ϕ=(ϕ1,ϕ2,ϕ3):ϕi(t)iscontinuousandnonnegativeonR-andϕi(0)>0,i=1,2,3}. In this paper, we always assume that solutions of (5) satisfy the initial conditions:xi(s)=φi(s),y(s)=ψ(s),(i=1,2),(φ1,φ2,ψ)C+,s(-,0].The main purpose of this paper is to find a set of easily verifiable sufficient conditions for the permanence and extinction of the system (5). The present paper is organized as follows. In Section 2, we introduce some notations and definitions, give some preliminary results needed in later sections, and then state the main results of this paper. We then prove, in Section 3, the main results of (5) by using analysis technique. Finally, in Section 4, we work out an example.

Section snippets

Main results

Let f(t) be a continuous ω-periodic function defined on [0, +∞), we setAω(f)=ω-10ωf(t)dt,fU=maxt[0,ω]f(t),fL=mint[0,ω]f(t).

Definition 2.1

The system x˙=F(t,x), xRn is said to be permanent if there are constants M  m > 0 such that every positive solution x(t)=(x1(t),,xn(t))R+n={(x1,,xn):xi>0,i=1,,n} of this system, satisfiesmliminftxi(t)limsuptxi(t)M.

Lemma 2.2

[19] The systemx1˙=a(t)x2-b(t)x1-d(t)x12,x2˙=c(t)x1-f(t)x22,has a positive ω-periodic solution (x1(t),x2(t)) which is globally asymptotically stable

Proof of main results

Lemma 3.1

There exist positive constants Mx and My such thatlimsuptxi(t)Mx,limsupty(t)My.

Proof

Obviously, R+3 is a positively invariant set of system (5). Given any positive solution (x1(t), x2(t), y(t)) of (5) with initial conditions (6), we havex1˙a(t)x2-b(t)x1-d(t)x12,x2˙=c(t)x1-f(t)x22.Next, we consider the following auxiliary equationsu1˙=a(t)u2-b(t)u1-d(t)u12,u2˙=c(t)u1-f(t)u22,by Lemma 2.2, it follows that (12) has a globally asymptotically stable positive ω-periodic solution (x1(t),x2(t)). Let (u

Example

Example 4.1

x1˙=22+sintx2-x1-x12-12x12(1-e-2x1)2+x12-0k12(s)y(t+s)ds,x2˙=2x1-2-cost(2+sint)2x22,y˙=y-(1-1e)+3(2+sint)-0k21(s)x12(t+s)(1-e-2x1(t+s))2+x12(t+s)ds-y--0k22(s)y(t+s)ds2.

Example 4.2

x1˙=22+sintx2-x1-x12-12x12(1-e-2x1)2+x12-0k12(s)y(t+s)ds,x2˙=2x1-2-cost(2+sint)2x22,y˙=y-(3-cost2)+(e2-1)(2+sint)e2-0k21(s)x12(t+s)(1-e-2x1(t+s))2+x12(t+s)ds-y-5+sint3-0k22(s)y(t+s)ds2.We consider the subsystem of (53), (54):x1˙=22+sintx2-x1-x12,x2˙=2x1-2-cost(2+sint)2x22.Obviously, (55) admits an unique positive 2π

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