# The Effect of Impulsive Diffusion on Dynamics of a Stage-Structured Predator-Prey System

Authors: Jianjun Jiao
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

We investigate a predator-prey model with impulsive diffusion on predator and stage structure on prey. The globally attractive condition of prey-extinction periodic solution of the system is obtained by the stroboscopic map of
the discrete dynamical system. The permanent condition of the system is also obtained by the theory of impulsive delay differential equation. The results indicate that the discrete time delay has influence on the dynamical behaviors of the system. Finally, some numerical simulations are carried out to support the analytic results.


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## Body

## 1. Introduction

The dispersal is a ubiquitous phenomenon in the natural world. It is well recognized that the spatial distribution of populations and population dynamics are much affected by spatial heterogeneity and population mobility [1]. The fragmented landscapes are common because the populations of most species occupy mosaic habitats and because of rapid destruction of natural habitats. Briggs and Hoopes [2] identify three mechanisms whereby limited dispersal of hosts and parasitoids combined with other features, such as spatial and temporal heterogeneity, can promote persistence and stability of populations. It is important for us to understand the ecological and evolutionary dynamics of populations mirrored by the large number of mathematical models devoting to it in the scientific literatures [3–6]. In recent years, the analysis of these models focuses on the coexistence of population and local (or global) stability of equilibria [7–13]. Spatial factors play a fundamental role on the persistence and stability of the population, although the complete results have not yet been obtained even in the simplest one-species case. Most previous papers focused on the population dynamical system modeled by the ordinary differential equations; if the population dynamics with the effects of spatial heterogeneity are modeled by a diffusion process, it will be very interesting. While in practice, it is often the case that diffusion occurs in regular pulse. For example, when winter comes, birds will migrate between patches in search for a better environment, whereas they do not diffuse in other seasons, and the excursion of foliage seeds occurs at fixed period of time every year. Thus, impulsive diffusion provides a more natural description. Lately theories of impulsive differential equations [14] have been introduced into population dynamics. Impulsive differential equations are found in almost a domain of applied science [15–24]. Newly, persistence and stability of population dynamical system involving time delay have been discussed by some authors; see, for example, [8, 25, 26] and references cited therein. They obtained some sufficient conditions that guarantee permanence of population or stability of positive equilibria or positive periodic solutions.The organization of this paper is as follows. In the next section, we introduce the model and background concepts. In Section3, some important lemmas are presented. In Section 4, we give the conditions of global attractivity and permanence for system (2.3). In Section 5, A brief discussion is given in the last section to conclude this work.


## 2. The Model

Stage-structured models were analyzed in many literatures [7, 27–33]. The following stage-structured Holling mass defence predator-prey model with impulsive perturbations on predators was introduced by Jiao et al. [31]:(2.1)x1′(t)=rx2(t)-re-wτ1x2(t-τ1)-wx1(t),x2′(t)=re-wτ1x2(t-τ1)-βx2(t)1+ax2+bx22x3(t)-d3x2(t)-d4x22(t),x3′(t)=kβx2(t)1+ax2+bx22x3(t)-dx3(t),t≠nτ,Δx1(t)=0,Δx2(t)=0,Δx3(t)=μ,t=nτ,n=1,2,…,(φ1(ζ),φ2(ζ),φ3(ζ))∈C+=C([-τ1,0],R+3),φi(0)>0,i=1,2,3,
where x1(t), x2(t) represent the immature and mature pest densities, respectively, x3(t) denotes the density of nature enemy, τ1 represents a constant time to maturity, and r,w,d3,d4,d,k,a,b and β are positive constants. This model is derived as follows. We assume that at any time t>0, birth into the immature population is proportional to the existing mature population with proportionality constant r. We then assume that the death rate of immature population is proportional to the existing immature population with proportionality constant w. w(w>d), d3, and d(d>d3) are called the death coefficient of x1(t), x2(t), and x3(t), respectively. We assume that the death rate of mature populations are of a logistic nature, that is, proportional to the square of the population with proportionality constant d4. k>0 is the rate of conversing prey into predator. Δx3(t)=x3(t+)-x3(t), μ≥0 is the releasing amount of natural enemies at t=nτ, n∈Z+ and Z+={1,2,…}, τ and is the period of the impulsive immigration of the predator.Diffusive predator-prey models were analyzed in literatures [16, 34, 35]. In [16], Hui and Chen considered the single species model with impulsive diffusion as follows:(2.2)dx1(t)dt=x1(t)(a1-b1x1(t)),dx2(t)dt=x2(t)(a2-b2x2(t)),t≠nτ,Δx1(t)=d1(x2(t)-x1(t)),Δx2(t)=d2(x1(t)-x2(t)),t=nτ,n=1,2,…,
where we suppose that the system is composed of two patches connected by diffusion and xi(i=1,2) is the density of species in the ith patch. Intrinsic rate of natural increase and density dependence rate of prey population in the first habitat are denoted by ai(i=1,2), and di(i=1,2) is the dispersal rate in the ith patch. It is assumed here that the net exchange from the jth patch to ith patch is proportional to the difference xj-xi of population densities. The pulse diffusion occurs every τ period (τ>0). The system evolves from its initial state without being further affected by diffusion until the next pulse appears; Δxi=xi(nτ+)-xi(nτ), and xi(nτ+) represents the density of population in the ith patch immediately after the nth diffusion pulse at time t=nτ, while xi(nτ) represents the density of population in the ith patch before the nth diffusion pulse at time t=nτ, n=0,1,2,…;  ri,ki, and di(i=1,2) are positive constants.Motivated by these ideals from all of the above, we assume that the predator population diffuses between the two patches, and the prey population exists only in one patch. Then, we consider a delayed predator-prey model with impulsive diffusion on predator and stage structure on prey as follows:(2.3)dx1(t)dt=rx2(t)-re-wτ1x2(t-τ1)-wx1(t),dx2(t)dt=re-wτ1x2(t-τ1)-dx2(t)-βx2(t)y1(t),dy1(t)dt=-a1y1(t)+kβx2(t)y1(t),dy2(t)dt=y2(t)(a2-b2y2(t)),t≠nτ,Δx1(t)=0,Δx2(t)=0,Δy1(t)=D(y2(t)-y1(t)),Δy2(t)=D(y1(t)-y2(t)),t=nτ,n=1,2,…,
with initial condition (2.4)(φ1(ζ),φ2(ζ),φ3(ζ),φ4(ζ))∈C+=C([-τ1,0],R+4),φi(0)>0,i=1,2,3,4,
where x1(t), x2(t) represent the immature and mature prey population densities, respectively. τ1 represents a constant time to maturity, that is, immature individuals and mature individuals are divided by age τ1. It is assumed that the system is composed of two patches connected by diffusion and occupied by a single species and yi(t)(i=1,2) is the density of predator species in the ith patch. Death rate of the predator population in the first patch is denoted by a1>0. Intrinsic rate of natural increase and density dependence rate of predator population in the second patch are denoted by a2>0 and b2>0; a2/b2 denotes the carrying capacity in the second patch. 0<D<1 is dispersal rate of the predator population between two patches. It is assumed here that the net exchange from the jth patch to ith patch is proportional to the difference yj-yi of predator population densities. β is the prey capture rate by mature predator. k is the rate of conversion of nutrients into the reproduction rate of the mature predator. w is the death rate of immature prey population in the first patch. d is the death rate of mature prey population in first patch. The pulse diffusion occurs every τ>0 period. The system evolves from its initial state without being further affected by diffusion until the next pulse appears. Δyi(nτ)=yi(nτ+)-yi(nτ), where yi(nτ+) represents the density of population in the ith patch immediately after the nth diffusion pulse at time t=nτ, while yi(nτ) represents the density of population in the ith patch before the nth diffusion pulse at time t=nτ(n=0,1,2,…).Obviously, (2.3) can be simplified as follows:(2.5)dx2(t)dt=re-wτ1x2(t-τ1)-dx2(t)-βx2(t)y1(t),dy1(t)dt=-a1y1(t)+kβx2(t)y1(t),dy2(t)dt=y2(t)(a2-b2y2(t)),t≠nτ,Δx2(t)=0,Δy1(t)=D(y2(t)-y1(t)),Δy2(t)=D(y1(t)-y2(t)),t=nτ,n=1,2,…,
with initial condition
(2.6)(φ2(ζ),φ3(ζ),φ4(ζ))∈C+′=C([-τ1,0],R+3),φi(0)>0,i=2,3,4.


## 3. The Lemmas

The solution of (2.3), denoted by x(t)=(x1(t),x2(t),y1(t),y2(t))T, is a piecewise continuous function x:R+→R+4. x(t) is continuous on (nτ,(n+1)τ], n∈Z+, and x(nτ+)=limt→nτ+x(t) exists. Obviously, the global existence and uniqueness of solutions of (2.3) is guaranteed by the smoothness of properties of f, which denotes the mapping defined by right-side of system (2.3) (see Lakshmikantham et al. [36]). Before having the main results, we need some lemmas which will be used next.According to the biological meanings, it is assumed thatx1(t)≥0, x2(t)≥0, y1(t)≥0, and y2(t)≥0. Now, we will show that all solutions of (2.3) are uniformly ultimately bounded.Lemma 3.1.
There exists a constantM>0 such that x1(t)≤M/k, x2(t)≤M/k, y1(t)≤M, and y2(t)≤M, for each solution (x1(t),x2(t),y1(t),y2(t)) of (2.3) with all large enough t.Proof.
DefineV(t)=kx1(t)+kx2(t)+y1(t)+y2(t). Denote d¯=min{a1,r-d,w}. When t≠nτ, we have
(3.1)D+V(t)+d¯V(t)=-k(w-d¯)x1(t)-k[(r-d)-d¯]x2(t)-(a1-d¯)y1(t)+y2(t)[(a2+d¯)-b2y2(t)]≤y2(t)((a2+d¯)-b2y2(t))≤M0,
where M0=(a2+d¯)2/4b2. When t=nτ, we also have V(nτ+)=V(nτ). By Lemma 2.3 in [14], for t∈(nτ,(n+1)τ], we obtain
(3.2)V(t)≤V(0)exp(-d¯t)+∫0tM0exp(-d¯(t-s))ds=V(0)exp(-d¯t)+M0d¯(1-exp(-d¯t))<V(0)exp(-d¯t)+M0d¯(1-exp(-d¯t))→M0d¯,ast→∞.
So,V(t) is uniformly ultimately bounded. Hence, by the definition of V(t), there exists a constant M>0 such that x1(t)≤M/k, x2(t)≤M/k, y1(t)≤M, y2(t)≤M for all t large enough. The proof is complete.Ifx2(t)=0, the subsystem of (2.3) is obtained as follows:(3.3)dy1(t)dt=-a1y1(t),dy2(t)dt=y2(t)(a2-b2y2(t)),t≠nτ,Δy1(t)=D(y2(t)-y1(t)),Δy2(t)=D(y1(t)-y2(t)),t=nτ,n=1,2,….It is easy to solve the first two equations of system (3.3) between pulses(3.4)y1(t)=y1(nτ+)e-a1(t-nτ),nτ<t≤(n+1)τ,y2(t)=a2ea2(t-nτ)y1(nτ+)a2+b2(ea2(t-nτ)-1)y1(nτ+),nτ<t≤(n+1)τ.By considering the last two equations of System (3.3), we obtain the following stroboscopic map of system (3.3):(3.5)y1((n+1)τ+)=(1-D)e-a1τy1(nτ+)+Da2ea2τy2(nτ+)a2+b2(ea2τ-1)y2(nτ+),y2((n+1)τ+)=(1-D)a2ea2τy2(nτ+)a2+b2(ea2τ-1)y2(nτ+)+De-a1τy1(nτ+).Then, two lemmas are obtained as follows.Lemma 3.2 (see [37, Theorem 3.1]).
Ifa2τ<ln((1-(1-D)e-a1τ)/(1-(1-2D)e-a1τ)), the trivial equilibrium (0,0) of (3.5) is globally asymptotically stable. If a2τ>ln((1-(1-D)e-a1τ)/(1-(1-2D)e-a1τ)), there exists a unique positive equilibrium (y1*,y2*), which is globally asymptotically stable, where y1*=Dea2τy2*/((1-D)ea2τ-(1-2D)e(a2-a1)τ) and y2*=a2[(1-D)(ea2τ+e-a1τ)-(1-2D)e(a2-a1)τ-1]/b2(ea2τ-1)(1-(1-D)e(a2-a1)τ).Lemma 3.3 (see [37, Theorem 3.2]).
Ifa2τ>ln((1-(1-D)e-a1τ)/(1-(1-2D)e-a1τ)), then (3.3) has a τ-periodic positive solution (y1(t)̃,y2(t)̃), which is globally asymptotically stable. Here (y1(t)̃,y2(t)̃) can be expressed as
(3.6)y1(t)̃=y1*e-a1(t-nτ),nτ<t≤(n+1)τ,y2(t)̃=a2ea2(t-nτ)y2*a1+b2(ea2(t-nτ)-1)y2*,nτ<t≤(n+1)τ,
where y1* and y2* are defined as in Lemma 3.2.Lemma 3.4 (see [38]).
Consider the following delay equation:(3.7)dx(t)dt=c1x(t-τ)-c2x(t).
one assumes that c1,c2,τ>0, x(t)>0 for -τ≤t≤0. Assuming that c1<c2, then
(3.8)limt→∞x(t)=0.


## 4. The Dynamics

From the above discussion, we can easily know that there exists a prey-extinction boundary periodic solution(0,0,y1(t)̃,y2(t)̃) of system (2.3). In this section, we will prove that the prey-extinction boundary periodic solution (0,0,y1(t)̃,y2(t)̃) of system (2.3) is globally attractive.Theorem 4.1.
If(4.1)a2τ>ln1-(1-D)e-a1τ1-(1-2D)e-a1τ,re-wτ1<d+Dβea2τy2*(1-D)ea2τ-(1-2D)e(a2-a1)τ,
hold, the prey-extinction boundary periodic solution (0,0,y1(t)̃,y2(t)̃) of (2.3) is globally attractive. Here, y2*=a2[(1-D)(ea2τ+e-a1τ)-(1-2D)e(a2-a1)τ-1]/b2(ea2τ-1)[1-(1-D)e(a2-a1)τ].Proof.
It is clear that the global attractivity of the predator-extinction boundary periodic solution(0,0,y1(t)̃,y2(t)̃) of system (2.3) is equivalent to the global attractivity of predator-extinction boundary periodic solution (0,y1(t)̃,y2(t)̃) of system (2.5). So, we devote ourselves to investigate system (2.5). Since Dkβy2*ea2τ-wτ1<d[(1-D)ea2τ-(1-2D)e(a2-a1)τ], we can choose ɛ0 sufficiently small such that
(4.2)kβe-wτ1[Dea2τy2*(1-D)ea2τ-(1-2D)e(a2-a1)τ+ɛ0]<d.
Since we know from the second equation of system (2.5) that dy1(t)/dt≤-a1y1(t), we consider the following comparison an impulsive differential system:
(4.3)dy1′(t)dt=-a1y1′(t),dy2′(t)dt=y2′(t)(a2-b2y2′(t)),t≠nτ,Δy1′(t)=D(y2′(t)-y1′(t)),Δy2′(t)=D(y1′(t)-y2′(t)),t=nτ,n=1,2,….
In view of Lemma 3.3 and (3.6), we obtain the τ-periodic solution of system (4.3)
(4.4)y1′(t)̃=y1′*e-a1(t-nτ),nτ<t≤(n+1)τ,y2′(t)̃=a2ea2(t-nτ)y2′*a2+b2(ea2(t-nτ)-1)y2′*,nτ<t≤(n+1)τ,
which is globally asymptotically stable, where y1′*=Dea2τy2′*/((1-D)ea2τ-(1-2D)e(a2-a1)τ) and y2′*=a2((1-D)(ea2τ+e-a1τ)-(1-2D)e(a2-a1)τ-1)/b2(ea2τ-1)(1-(1-D)e(a2-a1)τ).
From Lemma3.3 and comparison theorem of impulsive equation [36], we have y1(t)≤y1′(t) and y1′(t)→y1(t)̃ as t→∞. Then there exists an integer k2>k1, t>k2 such that
(4.5)y1(t)≤y1′(t)≤y1(t)̃+ɛ0,nτ<t≤(n+1)τ,n>k2,
that is,
(4.6)y1(t)<y1(t)̃+ɛ0≤y1*+ɛ0=Δϱ,nτ<t≤(n+1)τ,n>k2.
From (2.5), we get
(4.7)dx2(t)dt≤re-wτ1x2(t-τ1)-(d+βϱ)x2(t),t>nτ+τ1,n>k2.
Consider the following comparison differential system:
(4.8)dx2′(t)dt=re-wτ1x2′(t-τ1)-(d+βϱ)x2′(t),t>nτ+τ1,n>k2.
So we have re-wτ1<d+βϱ. According to Lemma 3.4, we have limt→∞x2′(t)=0.
Let(x2(t),y1(t),y2(t)) be the solution of system (2.5) with its initial conditions and x2(ζ)=φ2(ζ)(ζ∈[-τ1,0]), y2′(t) is the solution of system (4.8) with initial condition x2′(ζ)=φ2(ζ)(ζ∈[-τ1,0]). By the comparison theorem, we have
(4.9)limt→∞x2(t)<limt→∞x2′(t)=0.
Incorporating into the positivity of x2(t), we know that limt→∞x2(t)=0. Therefore, for any ɛ1>0 (sufficiently small), there exists an integer k3(k3τ>k2τ+τ1) such that x2(t)<ɛ1 for all t>k3τ.
For system (2.5), we have
(4.10)-a2y1(t)≤dy1(t)dt≤-(a2-kβɛ1)y1(t).
Then, z2(t)≤y1(t)≤z2′(t) and z2(t)→y1(t)̃, z2′(t)→y1(t)̃ as t→∞, where (z1(t),z2(t)) and (z1′(t),z2′(t)) are the solutions of
(4.11)dz1(t)dt=-a1z1(t),dz2(t)dt=z2(t)[a2-b2z2(t)],t≠nτ,Δz1(t)=D(z2(t)-z1(t)),Δz2(t)=D(z1(t)-z2(t)),t=nτ,n=1,2,…,(4.12)dz1′(t)dt=-(a1-βɛ1)z1′(t),dz2′(t)dt=z2′(t)(a2-b2z2′(t)),t≠nτ,Δz1′(t)=D(z2′(t)-z1′(t)),Δz2′(t)=D(z1′(t)-z2′(t)),t=nτ,n=1,2,…,
respectively. Here
(4.13)z1′(t)̃=z1′*e-(a1-kβɛ1)(t-nτ),nτ<t≤(n+1)τ,z2′(t)̃=a2ea2(t-nτ)z2′*a2+b2(ea2(t-nτ)-1)z2′*,nτ<t≤(n+1)τ,
where
(4.14)z1′*=Dea2τz2′*(1-D)ea2τ-(1-2D)e(a2-(a1-kβɛ1))τ,z2′*=a2((1-D)(ea2τ+e-(a1-kβɛ1)τ)-(1-2D)e(a2-(a1-kβɛ1))τ-1)b2(ea2τ-1)(1-(1-D)e(a2-(a1-kβɛ1))τ).
Therefore, for anyɛ2>0, there exists an integer k4, n>k4 such that z2(t)̃-ɛ2<y1(t)<z2′(t)̃+ɛ2,z1(t)̃-ɛ2<y2(t)<z1′(t)̃+ɛ2. Let ɛ1→0. So we have y1(t)̃-ɛ2<y1(t)<y1(t)̃+ɛ2,y2(t)̃-ɛ2<y2(t)<y2(t)̃+ɛ2, for t large enough, which implies y1(t)→y1(t)̃ and y2(t)→y2(t)̃ as t→∞. This completes the proof.The next work is to investigate the permanence of the system (2.3). Before starting the following theorem, we give the following definition.Definition 4.2.
System (2.3) is said to be permanent if there are constants m,M>0 (independent of initial value) and a finite time T0, such that for all solutions (x1(t),x2(t),y1(t),y2(t)) with all initial values x1(0+)>0, x2(0+)>0, y1(0+)>0, y2(0+)>0, m≤x1(t)≤M/k, m≤x2(t)≤M/k, m≤y1(t)≤M, m≤y2(t)≤M, holds for all t≥T0. Here T0 may depend on the initial values (x1(0+),x2(0+),y1(0+),y2(0+)).Theorem 4.3.
If(4.15)a2τ>ln1-(1-D)e-(a1-βx2*)τ1-(1-2D)e-(a1-βx2*)τ,re-wτ1>d+Dβv1*e(a1τ-wτ1)(1-D)ea1τ-(1-2D)e(a1-(a2+βx2*))τ,
hold, there is a positive constant g such that each positive solution (x2(t),y1(t),y2(t)) of (2.5) satisfies x2(t)≥g for large enough t. x2* can be confirmed by the equation
(4.16)(re-wτ1-d)[(1-D)ea2τ-(1-2D)e(a2-(a1+βx2*))τ]=Dβv1*ea2τ-wτ1.
Here
(4.17)v1*=a2((1-D)(ea2τ+e-(a1+βx2*)τ)-(1-2D)e(a2-(a1+βx2*))τ-1)b2(ea2τ-1)(1-(1-D)e(a2-(a1+βx2*))τ).Proof.
The fourth equation of (2.5) can be rewritten as
(4.18)dx2(t)dt=[re-wτ1-(d+βy1(t))]x2(t)-re-wτ1ddt∫t-τ1tx2(u)du.
Let us consider any positive solution(x2(t),y1(t),y2(t)) of system (2.5). According to (4.18), Q(t) is defined as
(4.19)Q(t)=x2(t)+re-wτ1∫t-τ1tx2(u)du.
We calculate the derivative of Q(t) along the solution of (2.5)
(4.20)dQ(t)dt=[re-wτ1-(d+βy1(t))]x2(t).
Since Dkβv1*e(a1τ-wτ1)/((1-D)ea1τ-(1-2D)e(a1-(a2+βy2*))τ)>d, we can easily know that there exists sufficiently small ɛ>0 such that
(4.21)kβe-wτ1(Dv1*ea1τ(1-D)ea1τ-(1-2D)e(a1-(a2+βy2*))τ-ɛ)>d.
We claim that for any t0>0, it is impossible that x2(t)<x2* for all t>t0. Suppose that the claim is not valid. Then there is a t0>0 such that x2(t)<x2* for all t>t0. It follows from the first equation of (2.5) that for all t>t0,
(4.22)dy1(t)dt<-(a2-kβx2*)y1(t).
Consider the following comparison an impulsive system for all t>t0(4.23)dv1(t)dt=-(a1-kβx2*)v1(t),dv2(t)dt=v2(t)(a2-b2v2(t)),t≠nτ,Δv1(t)=D(v2(t)-v1(t)),Δv2(t)=D(v1(t)-v2(t)),t=nτ,n=1,2,….
By Lemma 3.2 and the condition a2τ>ln((1-(1-D)e-(a1+βx2*)τ)/(1-(1-2D)e-(a1+βx2*)τ)), we obtain
(4.24)v1(t)̃=v1*e-(a1+βx2*)(t-nτ),nτ<t≤(n+1)τ,v2(t)̃=a2ea2(t-nτ)v2*a2+b2(ea2(t-nτ)-1)v2*,nτ<t≤(n+1)τ,
where
(4.25)v1*=Dea2τv2*(1-D)ea2τ-(1-2D)e(a2-(a1-βx2*))τ,v2*=a2((1-D)(ea2τ+e-(a1-βx2*)τ)-(1-2D)e(a2-(a1-βx2*))τ-1)b2(ea2τ-1)(1-(1-D)e(a2-(a1-βx2*))τ),
is the unique positive periodic solution of (4.23), which is globally asymptotically stable. By the comparison theorem for impulsive differential equation [36], we know that there exists t1(>t0+τ1) such that the inequalities y1(t)≤v1(t)̃+ɛ and y2(t)≤v2(t)̃+ɛ hold for t≥t1. Thus y1(t)≤v1*+ɛ for all t≥t1. We make notation as σ=Δv1*+ɛ for convenience. Also
(4.26)re-wτ1>d+βσ.
Then
(4.27)dQ(t)dt>x2(t)[re-wτ1-(d+βσ)]
for all t>t1. Setting x2m=mint∈[t1,t1+τ1]x2(t), we will show that x2(t)≥x2m for all t≥t1. Supposing the contrary, then there is a T0>0 such that x2(t)≥x2m for t1≤t≤t1+τ1+T0, x2(t1+τ1+T0)=x2m and x2′(t1+τ1+T0)<0. Hence, the first equation of system (2.5) implies that
(4.28)x2′(t1+τ1+T0)=re-wτ1x2(t1+T0)-dx2(t1+τ1+T0)-βx2(t1+τ1+T0)y1(t1+τ1+T0)≥[re-wτ1-(d+βσ)]x2m>0.
This is a contradiction. Thus, x2(t)≥x2m for all t>t1. As a consequence, Q′(t)>x2m[re-wτ1-(d+βσ)]>0 for all t>t1. This implies that as t→∞, Q(t)→∞. It is a contradiction to Q(t)≤M(1+τ1re-wτ1). Hence, the claim is complete.
By the claim, we are left to consider two case. First,x2(t)≥x2* for all t large enough. Second, x2(t) oscillates about x2* for large enough t.
Define(4.29)g=min{x2*2,g1},
where g1=x2*e-(d+dM)τ1. We hope to show that x2(t)≥g for all t large enough. The conclusion is evident in the first case. For the second case, let t*>0 and ξ>0 satisfy x2(t*)=x2(t*+ξ)=x2* and x2(t)<x2* for all t*<t<t*+ξ, where t* is sufficiently large such that x2(t)>σ for t*<t<t*+ξ, x2(t) is uniformly continuous. The positive solutions of (2.5) are ultimately bounded, and x2(t) is not affected by impulses. Hence, there is a T(0<t<τ1 and T is dependent on the choice of t*) such that x2(t*)>x2*/3 for t*<t<t*+T. If ξ<T, there is nothing to prove. Let us consider the case T<ξ<τ1. Since x2′(t)>-(d+βM)x2(t) and x2(t*)=x2*, it is clear that x2(t)≥g1 for t∈[t*,t*+τ1]. Then, proceeding exactly as the proof for the above claim, we see that x2(t)≥g1 for t∈[t*+τ1,t*+ξ]. Because the kind of interval, t∈[t*,t*+ξ] is chosen in an arbitrary way (we only need t* to be large). We concluded that x2(t)≥g for all large t. In the second case, in view of our above discussion, the choice of g is independent of the positive solution, and we proved that any positive solution of (2.5) satisfies x2(t)≥g for all sufficiently large t. This completes the proof of this theorem.Theorem 4.4.
If(4.30)a2τ>ln1-(1-D)e-(a1-βx2*)τ1-(1-2D)e-(a1-βx2*)τ,re-wτ1>d+Dβv1*e(a1τ-wτ1)(1-D)ea1τ-(1-2D)e(a1-(a2+βx2*))τ,
hold, system (2.3) is permanent. Here
(4.31)v1*=a2((1-D)(ea2τ+e-(a1+βx2*)τ)-(1-2D)e(a2-(a1+βx2*))τ-1)b2(ea2τ-1)(1-(1-D)e(a2-(a1+βx2*))τ).Proof.
Let(x1(t),x2(t),y1(t),y2(t)) be any solution of system (2.3). From system (2.3) and Theorem 4.3, we have
(4.32)dx1(t)dt=x1(t)(a1-b1x1(t)),dx2(t)dt≥-a2x2(t),t≠nτ,Δx1(t)=D(x2(t)-x1(t)),Δx2(t)=D(x1(t)-x2(t)),t=nτ,n=1,2,….
By the same argument as those in the proof of Theorem4.1, we have that x1(t)≥x1′′*-ɛ=p1, and x2(t)≥x1′′*e-a2τ-ɛ=p2, and ɛ>0 is small enough. Here
(4.33)x1′′*=a2((1-D)(ea2τ+e-a1τ)-(1-2D)e(a2-a1)τ-1)b2(ea2τ-1)(1-(1-D)e(a2-a1)τ),(4.34)x2′′*=Dea2τx1′′*(1-D)ea2τ-(1-2D)e(a2-a1)τ.
In view of Theorem4.1, the third equation of system (2.3) becomes
(4.35)dy1(t)dt≥r(p2g-Me-wτ1)-wy1(t).
It is easy to obtain y1(t)≥δ-ɛ, and ɛ>0 is small enough, where δ=r(p2g-Me-wτ1)/w-ɛ. By Theorem 4.3 and the above discussion, system (2.3) is permanent. The proof of Theorem 4.4 is complete.


## 5. Discussion

In this paper, we investigate a delayed predator-prey model with impulsive diffusion on predator and stage structure on prey. We analyze that the prey-extinction periodic solution of system (2.3) is globally attractive, and we also obtain the permanent condition of system (2.3). It is assumed that x2(0)=5.2,y1(0)=5.0, y2(0)=5.6, r=1, τ1=3, d=3, a1=1, a2=1, b2=1, β=4, k=0.9, D=0.5, w=1, and τ=1. Then the prey-extinction periodic solution is a global attractivity (see Figure 1). We also assume that x2(0)=5.2, y1(0)=5.0, y2(0)=5.6, r=1, τ1=1, d=3, a1=1, b1=1, a2=1, b2=1, β=4, k=0.9, D=0.5, w=1, τ=1, then, system (2.5) is permanent (see Figure 2). From Theorems 4.1 and 4.4, we can easily guess that there must exist a threshold τ1*. If τ1>τ1*, the prey-extinction periodic solution (x1(t)̃,x2(t)̃,0,0) of (2.3) is globally attractive. If τ1<τ1*, system (2.3) is permanent. The results show that the diffusion and the mature time of the prey play important roles for the permanence of system (2.3) and provide tactical basis for the biological resource protection.Dynamical behavior of system (2.5) on prey-extinction periodic solution with x2(0)=5.2, y1(0)=5.0, y2(0)=5.6, r=1, τ1=3, d=3, a1=1, a2=1, b2=1, β=4, k=0.9, D=0.5, w=1, and τ=1. (a) Time-series of x2(t), (b) time-series of y1(t), (c) time-series of y2(t).
(a)(b)(c)Dynamical behavior of the permanence of system (2.5) with x2(0)=5.2, y1(0)=5.0, y2(0)=5.6, τ1=1, d=3, a1=1, a2=1, b2=1β=4, k=0.9, D=0.5, w=1, and τ=1. (a) Time-series of x2(t). (b)Time-series of y1(t). (c) Time-series of y2(t).
(a)(b)(c)



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*Source: 716932-2010-10-26.xml*