We investigate a predator-prey model with stage structure for the predator and periodic constant impulsive perturbations. Conditions for extinction of prey and immature predator are given. By using the Floquet theory and small amplitude perturbation skills, we consider the local stability of prey, immature predator eradication periodic solution. Furthermore, by using the method of numerical simulation, the influence of the impulsive control strategy on the inherent oscillation is investigated, which shows rich complex dynamic (such as periodic doubling bifurcation, periodic halving bifurcation, nonunique attractors, chaos, and periodic windows).

1. Introduction

Stage structure model has received much attention in the recent years [1–3]. In this paper, based on the model of [1], we study the dynamic complexities of a predator-prey system with stage structure for the predator and periodic constant impulsive perturbations. The model takes the form

ẋ(t)=x(t)(r-ax(t)-by2(t)1+mx(t)),ẏ1(t)=kbx(t)y2(t)1+mx(t)-(D+v1)y1(t),t≠(n+l-1)T,t≠nT,ẏ2(t)=Dy1(t)-v2y2(t),Δx(t)=-δx(t),Δy1(t)=-δ1y1(t),t=(n+l-1)T,Δy2(t)=-δ2y2(t),Δx(t)=0,Δy1(t)=0,t=nT,Δy2(t)=q,
where x(t) is the density of prey at time t,y1(t),y2(t) are the densities of immature and mature predators at time t, respectively. All parameters are positive constants. bx/(1+mx) is the response function of the mature predator. r is the intrinsic growth rate of the prey, v1(v2) is the death rate of the immature (mature) predator, constant k denotes the coefficient in converting prey into a new immature predator, and the constant D denotes the rate of immature predator becoming mature predator. 0<l≤1, Δx(t)=x(t+)-x(t),Δy1(t)=y1(t+)-y1(t),Δy2(t)=y2(t+)-y2(t),and0<δ≤1(0<δ1≤1,0<δ2≤1) represents the fraction of prey (immature predator, mature predator) which dies at t=(n+l-1)T, T is the period of the impulsive effect, and q>0 is the release amount of mature predator at t=nT.

Recently, it is of great interests to investigate chaotic impulsive differential equations about biological control [4–6]. Gao and Chen [7] and Tang and Chen [8] investigated dynamic complexities in a single-species model with stage structure and birth pulses.

2. Preliminaries

In this section, we give some definitions and lemmas which will be useful to our results.

Let R+=[0,∞),R+3={x∈R3:x≥0}. Denote by f=(f1,f2,f3) the map defined by the right hand of the first, second, and third equations of system (1.1). Let V:R+×R+3→R+, and let V be said to belong to class V0 if

V is continuous at ((n-1)T,(n+l-1)T]×R+3⋃((n+l-1)T,nT]×R+3 for each x∈R+3,n∈N,lim(t,y)→((n+l-1)T+,x)V(t,y)=V((n+l-1)T+,x), and lim(t,y)→(nT+,x)V(t,y)=V(nT+,x) exist;

V is locally Lipschitzian in x.

Definition 2.1.

Let V∈V0, then for (t,x)∈((n-1)T,(n+l-1)T]×R+3⋃((n+l-1)T,nT]×R+3, the upper right derivative of V(t,x) with respect to the impulsive differential system (1.1) is defined as
D+V(t,x)=limh→0+sup1h[V(t+h,x+hf(t,x))-V(t,x)].

Lemma 2.2.

Let X(t) be a solution of system (1.1) with X(0+)≥0, then X(t)≥0 for all t≥0. And further X(t)>0,t>0 if X(0+)>0.

Lemma 2.3.

Suppose V∈V0. Assume that
D+V(t,X)≤g(t,V(t,x)),t≠nT,t≠(n+l-1)T,V(t,X(t+)≤ψn(V(t,x))),t=nTort=(n+l-1)T,
where g:R+×R+→R is continuous in ((n-1)T,(n+l-1)T]×R+⋃((n+l-1)T,nT]×R+ and for u∈R+,n∈N, then lim(t,v)→(nT+,u)g(t,v)=g(nT+,u) and lim(t,v)→((n+l-1)T+,u)g(t,v)=g((n+l-1)T+,u) exist, ψn:R+→R+ is nondecreasing. Let r(t)=r(t,t0,u0) be maximal solution of the scalar impulsive differential equation
u̇(t)=g(t,u(t)),t≠nT,t≠(n+l-1)T,u(t+)=ψn(u(t)),t=nTort=(n+l-1)T,u(0+)=u0,
existing on [0,∞), then V(0+,x0)≤u0, implies that V(t,x(t))≤r(t),t≥t0, where x(t)=x(t,t0,u0) is any solution of system (1.1).

In the following, we give some basic properties about the following subsystem (2.4) and (2.7) of system (1.1):

(y2*(0+)=y2*(nT+)=q/(1-(1-δ2)exp(-v2T)),y2*(lT+)=(q(1-δ2)exp(-v2lT))/(1-(1-δ2)exp(-v2T))) is a positive periodic solution of system (2.4). Since

y2(t)={(1-δ2)(n-1)(y2(0+)-q1-(1-δ2)exp(-v2T))exp(-v2t)+y2*(t)(n-1)T<t≤(n+l-1)T,(1-δ2)n(y2(0+)-q1-(1-δ2)exp(-v2T))exp(-v2t)+y2*(t)(n+l-1)T<t≤nT,
is the solution of system (2.4) with initial value y20≥0, where t∈((n-1)T,(n+l-1)T)⋃((n+l-1)T,nT),n∈N, we get the following.

Lemma 2.4.

Let y2*(t) be a positive periodic of system (2.4), and for every solution y2(t) of system (2.4) with y20, one has |y2(t)-y2*(t)|→0, when t→∞,
ẏ1(t)=-(D+v1)y1(t),t≠(n+l-1)T,Δy1(t)=-δ1y1(t),t=(n+l-1)T,y1(0+)=y10.

Lemma 2.5.

In subsystem (2.7) exists a equilibrium y1*(t)=0.

Therefore, we obtain a prey, immature predator eradication periodic solution (0,0,y2*(t)) of system (1.1).

3. Extinction and Boundary

In this section, we study the stability of prey, immature predator eradication periodic solution of system (1.1), and we investigate the boundary of system (1.1).

Theorem 3.1.

Let (x(t),y1(t),y2(t)) be any solution of system (1.1), then (0,0,y2*(t)) is locally asymptotically stable if
rT-bq(1-δ2exp(-v2lT)-(1-δ2)exp(-v2T))v2(1-(1-δ2)exp(-v2T))<ln11-δ.

Proof.

The local stability of periodic solution (0,0,y2*(t)) may be determined by considering the behavior of small amplitude perturbations of the solution. Define x(t)=u(t),y1(t)=u(t),y2(t)=w(t)+y2*(t), then
(u(t)v(t)w(t))=Φ(t)(u(0)v(0)w(0))0≤t<T,
where Φ(t) satisfies
Φ̇(t)=(r-by2*00kby2*-(D+v1)00D-v2)Φ(t),
and Φ(0)=0, the identity matrix. The impulsive perturbations of system (1.1) become
(u((n+l-1)T+)v((n+l-1)T+)w((n+l-1)T+))=(1-δ0001-δ10001-δ2)(u((n+l-1)T)v((n+l-1)T)w((n+l-1)T)),(u(nT+)v(nT+)w(nT+))=(100010001)(u(nT)v(nT)w(nT)).
The stability of the periodic solution (0,0,y2*(t)) is determined by the eigenvalues of
M=(1-δ0001-δ10001-δ2)(100010001)Φ(T),
where μ1=(1-δ1)exp(-(D+v1)T)<1,μ2=(1-δ2)exp(-v2T)<1,μ3=(1-δ)exp(∫0T(r-by2*(t))dt) according to Floquet theory, the solution (0,0,y2*(t)) is locally stable if |μ3|<1, this is to say, rT-(bq(1-δ2exp(-v2lT)-(1-δ2)exp(-v2T)))/v2(1-(1-δ2)exp(-v2T))<ln1/(1-δ).

Theorem 3.2.

There exists a positive constant M such that x(t)≤M,y1(t)≤M,y2(t)≤M for each solution (x(t),y1(t),y2(t)) of system (1.1) with all t large enough.

Proof.

Let X(t)=(x(t),y1(t),y2(t)) be any solution of system (1.1). Define function V(t)=V(t,X(t))=kx(t)+y1(t)+y2(t), when t≠(n+l-1)T,t≠nT, we select 0<L<min{v1,v2}, then
D+V(t)+LV(t)=k(r+L)x-kax2+(L-v1)y1+(L-v2)y2≤k(r+L)x-kax2≤M0,
where M0=k(r+L)2/4a. When t=(n+l-1)T,V((n+l-1)T+)≤V((n+l-1)T),t=nT,V(nT+)≤V(nT)+q.

According to Lemma 2.3, for t≥0 we have
V(t)≤V(0)e-Lt+M0l(1-e-Lt)+qe-L(t-T)1-eLT+qeLTeLT-1→M0l+qeLTeLT-1(t→∞).

Therefore V(t) is ultimately bounded, and we obtain that each positive solution of system (1.1) is uniformly ultimately bounded.

4. Numerical Analysis

In this section, we study the influence of impulsive period T and impulsive perturbation q on complexities of the system (1.1).

Let r=1,a=1,b=4,k=2,m=0.8,D=0.4,v1=v2=0.2,l=0.5. Since the corresponding continuous system (1.1) cannot be solved explicitly and system (1.1) cannot be rewritten as equivalent difference equations, it is difficult to study them analytically. So we have to study system (1.1) numerically integrated by stroboscopically sampling one of the variables over a range of T,q values. The bifurcation diagram provides a summary of essential dynamical behavior of system.

From Theorem 3.1, we know that the prey, immature predator eradication periodic solution (0,0,y2*(t)) is locally stable if q>qmax=0.7397, which is shown in Figure 1. when q=0.74, we find that the variable y2(t) oscillates in a stable cycle. In contrast, the variables x(t),y1(t) rapidly decrease to zero.

The dynamics of system (1.1) with r=1,a=1,b=4,k=2,m=0.8,D=0.4,v1=v2=0.2,l=0.5,δ=0.1,δ1=0.5,δ2=0.1,q=0.74,T=14.5 and initial values x=8.5,y1=4.8,y2=2.8. (a) Time series of prey; (b) time series of immature predator; (c) time series of mature predator; (d) phase portrait of system (1.1).

The bifurcation diagram (Figure 2) shows that with T increasing from 12.5 to 16, system (1.1) has rich dynamics including periodic doubling bifurcation, chaos, periodic windows, and nonunique attractors. When T<T1≈12.97, system (1.1) has a stable T-periodic solution. When T>T1, it is unstable and there is a cascade of periodic doubling bifurcations leading to chaos (Figure 3) (chaotic area with periodic windows). Figure 4 shows the phenomena of non-unique attractors [9]: different attractors can coexist, which one of the attractors is reached depends on the initial values.

Bifurcation diagrams of system (1.1) showing the effect of T with r=1,a=1,b=4,k=2,m=0.8,D=0.4,v1=v2=0.2,l=0.5,δ=0.1,δ1=0.5,δ2=0.1,q=0.5,12.5≤T≤16 and initial values x=8.5,y1=4.8,y2=2.8.

Period doubling cascade: (a) phase portrait of T-periodic solution for T=12.8; (b) phase portrait of 2T-periodic solution for T=13.5; (c) phase portrait of 4T for T=13.8.

Nonunique attractor at T=13.7: (a) Phase portrait of 3T-periodic solution with initial values x=8.5,y1=4.8,y2=2.8. (b) Phase portrait of 2T-periodic solution with initial values x=18.5,y1=14.8,y2=12.8.

Non-unique attractor at T=15.22: (a) Phase portrait of 3T-periodic solution with initial values x=8.5,y1=4.8,y2=2.8. (b) Phase portrait of chaotic periodic solution with initial values x=18.5,y1=14.8,y2=12.8.

The bifurcation diagram (Figure 6) shows that with q increasing from 0.01 to 0.8, system (1.1) has rich dynamics including periodic doubling bifurcation, chaos, periodic windows, and periodic halving bifurcation. When q<q1≈0.08, system (1.1) has a stable 2T-periodic solution. When q>q1, it is unstable and there is a cascade of periodic doubling bifurcations leading to chaos with periodic windows (Figure 8) which is followed by a cascade of periodic halving bifurcation from chaos to periodic solution.

Bifurcation diagrams of system (1.1) showing the effect of q with r=1,a=1,b=4,k=2,m=0.8,D=0.4,v1=v2=0.2,l=0.5,δ=0.1,δ1=0.5,δ2=0.1,T=13.7,0.01≤q≤0.8 and initial values x=8.5,y1=4.8,y2=2.8.

Period doubling cascade: (a) phase portrait of 2T-periodic solution for q=0.05; (b) phase portrait of 4T-periodic solution for q=0.08; (c) phase portrait of 4T-periodic solution for q=0.085.

Period halving cascade: (a) phase portrait of T-periodic solution for q=0.49; (b) phase portrait of 2T-periodic solution for q=0.52; (c) phase portrait of T-periodic solution for q=0.6.

5. Conclusion

In this paper, we investigated the dynamic complexities of a predator-prey system with impulsive perturbations and stage structure for the predator. Conditions for the local asymptotical stability of prey, immature predator eradication periodic solution (0,0,y2*(t)) were given by using the Floquet theory and small amplitude perturbation skills. Using the method of numerical simulation, the influence of impulsive control strategy on inherent oscillation showed that there exists complexity for system (1.1) including periodic doubling bifurcation, periodic halving bifurcation, nonunique attractors, chaos, and periodic windows. All these results showed that dynamical behavior of system (1.1) becomes more complex under periodically impulsive control strategy.

WangW.ChenL.A predator-prey system with stage-structure for predatorXiaoY. N.ChenL.Global stability of a predator-prey system with stage structure for the predatorMagnussonK. G.Destabilizing effect of cannibalism on a structured predator-prey systemZhangS.ChenL.A Holling II functional response food chain model with impulsive perturbationsZhangS.TanD.ChenL.Dynamic complexities of a food chain model with impulsive perturbations and Beddington-DeAngelis functional responseLiuX.ChenL.Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predatorGaoS.ChenL.Dynamic complexities in a single-species discrete population model with stage structure and birth pulsesTangS.ChenL.Multiple attractor stage-structure population model with birth pulsesTangS.ChenL.Chaos in functional response host-parasitoid ecosystem models