# Seasonal Effects on a Beddington-DeAngelis Type Predator-Prey System with Impulsive Perturbations

Authors: Hunki Baek; Younghae Do
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

We study a Beddington-DeAngelis type predator-prey system with impulsive perturbation and seasonal effects. First, we numerically observe the influence of seasonal effects on the system without impulsive perturbations. Next, we find the conditions for the local and global stabilities of prey-free periodic solutions by using Floquet theory for the impulsive equation and small amplitude perturbation skills, and for the permanence of the system via comparison theorem. Finally, we show that seasonal effects and impulsive perturbation can give birth to various kinds of dynamical behavior of the system including chaotic phenomena by numerical simulations.


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## Body

## 1. Introduction and Model Formulation

In ecology, one of main goals is to understand the dynamical relationship between predator and prey. Such relationship can be represented by the functional response which refers to the change in the density of prey attached per unit time per predator as the prey density changes. One of well-known functional responses is Beddington-DeAngelis functional response introduced by Beddington [1] and DeAngelis et al. [2], independently. It is similar to Holling type II functional response but contains an extra term describing mutual interference by predators. In fact, there are much significant evidences to suggest that functional responses with predator interference occur quite frequently in laboratory and natural systems [3]. Thus, we can establish a predator-prey model with Beddington-DeAngelis functional response as the following form [1, 4, 5]:
(1.1)x′(t)=rx(t)(1-x(t)K)-ax(t)y(t)by(t)+x(t)+c,y′(t)=-dy(t)+eax(t)y(t)by(t)+x(t)+c,
where x(t), y(t) represent the population densities of prey and predator at time t, respectively. In this system, the prey grows according to a logistic growth with intrinsic growth rate r and K is called the carrying capacity of the prey. For parameters setting, a is the per-capita rates of predation of the predator, the constants e,d are the conversion rate and the death rate of the predator, respectively, and the term by measures the mutual interference between predators.It is necessary and important to consider models with periodic ecological parameters which might be quite naturally exposed such as those due to seasonal effects of weather or food supply [6]. Thus when the environmental factors that affect various parameters of the ecological model fluctuate periodically, then the corresponding parameters should be taken as periodic functions of time [7]. There are a number of ways to apply periodic perturbation in ecological models. Especially, one of the most popular ways to describe periodic phenomena is to use the sine (or cosine) wave or sinusoid function which describes a wave-like function of time with peak deviation from center and angular frequency [8–15]. Thus we consider the intrinsic growth rate r in system (1.1) as periodically varying function of time due to seasonal variation and adopt the sine wave as mentioned above to investigate the seasonality on the system. In fact the seasonality is superimposed as follows:
(1.2)r0=r(1+ϵsin(ωt)),where the parameterϵ represents the degree of seasonality; λ=rϵ is the magnitude of the perturbation in r0, ω is the angular frequency of the fluctuation caused by seasonality. Since r0 is assumed to be positive, we have 0≤ϵ≤1. With this idea of periodic forcing, we consider the following predator-prey system with periodic variation in the intrinsic growth rate of the prey:
(1.3)x′(t)=rx(t)(1-x(t)K)-ax(t)y(t)by(t)+x(t)+c+λx(t)sin(ωt),y′(t)=-dy(t)+eax(t)y(t)by(t)+x(t)+c,
where λ and ω represent the magnitude and frequency of the forcing term, respectively. Of course, a number of researchers [14, 16, 17] have studied that dynamical systems with simple dynamic behavior in the constant parameter case display very complex behavior including chaos when they are periodically perturbed. In this context, in Section 2 we illustrate numerical simulations for system (1.3) to show the existence of limit circles and various kinds of dynamical behaviors including chaos. For this reason, system (1.3) reflects more realistic situation than system (1.1).There are still some other periodic perturbations such as fire, flood, and mating habits or harvesting seasons which are not suitable to be considered continually. Suppose that with the pest outbreak, for example, there are many ways to beat agricultural pests. One of important ways is biological control leading reduction in pest population from the actions of other living organisms, often called natural enemies or beneficial species. As we know, anther important method for pest control is chemical control. Pesticides can reduce farmer's financial losses by preventing crop losses to insects and other pests. Such control tactics should be used not continuously but impulsively. There are many literatures on systems dealing with impulsive controls [10, 11, 13–15, 18–21]. Thus, we consider the following predator-prey system with adding periodic constant impulsive immigration of the predator regarded as natural enemy of the prey (pest) to system (1.3) and spraying pesticides (harvesting) on all species at the same times
(1.4)x′(t)=rx(t)(1-x(t)K)-ax(t)y(t)by(t)+x(t)+c+λx(t)sin(ωt),y′(t)=-dy(t)+eax(t)y(t)by(t)+x(t)+c,t≠nτ,x(t+)=(1-p1)x(t),t=nτ,y(t+)=(1-p2)y(t)+q,(x(0+),y(0+))=(x0,y0),
where τ is the period of the impulsive immigration or stock of the predator, 0≤p1, p2<1 present the fraction of the prey and the predator which die due to the harvesting or pesticides, and so forth, and q>0 is the size of immigration or stock of the predator.If we takeb=0, system (1.4) can be expressed as the Holling-type II predator-prey system with impulsive perturbations and seasonal effects as follows:
(1.5)x′(t)=rx(t)(1-x(t)K)-ax(t)y(t)x(t)+c+λx(t)sin(ωt),t≠nτ,y′(t)=-dy(t)+eax(t)y(t)x(t)+c,x(t+)=(1-p1)x(t),t=nτ,y(t+)=(1-p2)y(t)+q,(x(0+),y(0+))=(x0,y0).
While if c=0, then system (1.4) can be expressed as the ratio-dependent predator-prey system with impulsive perturbations and seasonal effects as follows:
(1.6)x′(t)=rx(t)(1-x(t)K)-ax(t)y(t)by(t)+x(t)+λx(t)sin(ωt),t≠nτ,y′(t)=-dy(t)+eax(t)y(t)by(t)+x(t),x(t+)=(1-p1)x(t),t=nτ,y(t+)=(1-p2)y(t)+q,(x(0+),y(0+))=(x0,y0).
We will investigate system (1.4) together with systems (1.5) and (1.6). Impulsive differential equations such as (1.4) are found in almost every domain of applied science and have been studied in many investigations [10, 11, 14, 16, 17]. Especially, Zhang and Chen [15] considered system (1.4) when p2, λ=0. They investigated abundance of complex dynamics for system (1.4) when p2, λ=0 and suggested a more executable way for observing chaos and coexistence of attractors. They also gave a threshold that classifies between the permanence and the stability of prey-free solutions for system (1.4). However, in case p2, λ≠0, system (1.4) has not been studied yet. Thus, the purpose of this paper is to find conditions for the stability of prey-free periodic solutions. Also, we show that the system is permanent under some conditions. In addition, using numerical simulations various kinds of dynamical phenomena are discussed in Section 4.


## 2. Numerical Analysis of System (1.3)

In this section we will numerically study the influence of the seasonality parameterλ on system (1.3). For this, we fix parameters ω=2π, a=3.2118, b=0.0246, c=3.3667, d=3.7798, e=3.3189, r=2.3553, K=9.7016 and we choose (x0,y0)=(1,1) as an initial point. It follows from [16] that system (1.3) with these parameters has a unique stable limit cycle when λ=0. Since the corresponding continuous system (1.3) cannot be solved explicitly and system (1.3) cannot be rewritten as equivalent difference equations, it is difficult to study them analytically. However, the influence of λ may be documented by stroboscopically sampling one of the variables over a range of λ values. Thus we numerically integrate system (1.3) and seek the behavior of the solutions. The bifurcation diagram provides a summary of essential dynamical behavior of system. Indeed the points that are plotted will represent either fixed or periodic sinks or other attracting sets including chaos. It shows the birth, evolution, and death of the attracting sets. In Figure 1, we illustrate bifurcation diagrams of system (1.3) to examine significant changes in the set of fixed or periodic points or other sets of interest. As is evident from Figures 1 and 2, the solutions are still periodic for values of λ in the range 0≤λ<λ1(≈1.5) and quasiperiodic motions appear when λ>λ1 (see Figure 2(b)). Periodic windows are intermittently scattered. Also Figures 3(a) and 3(b) show the route to chaos through the cascade of period doubling. Moreover, although the magnitude λ of seasonality increases, the solutions are stable and even they become periodic cycles like case λ=0 after λ>λ2(≈5.6) (see Figure 3(c)). We can also catch sight of the existence of occurrences of sudden changes in Figure 1 when λ≈0.84,1.04,1.25,3.26,3.9, and so forth. They can lead to nonunique attractors. For example, there exist at least three different attractors according to initial values when λ=3.9 (see Figure 4). This result shows that the seasonality in just one parameter can give rise to multiple attractors. Thus, these numerical examples show that the dynamical behavior of system (1.3) is more abundant than that of system (1.1).Bifurcation diagrams of system (1.3). (a) x is plotted for λ, (b) y is plotted for λ.
(a)(b)Phase portraits of solutions for system (1.3). (a) λ=1.4 (a 3τ-periodic solution), (b) λ=1.7.
(a)(b)Phase portraits of solutions for system (1.3). (a) λ=2.6 (a 2τ-periodic solution), (b) λ=4.8 (chaotic motion), (c) λ=6.0 (a τ-periodic solution).
(a)(b)(c)Coexistence of solutions whenλ=3.9, and (a) (x0,y0)=(1,1), (b) (1.54,1.65), and (c) (4.79,1.43), respectively.
(a)(b)(c)


## 3. Mathematical Analysis

In this section we give some notations, definitions, and lemmas which will be useful for our main results.Denoteℕ the set of all of nonnegative integers, ℝ+=[0,∞), ℝ+*=(0,∞), ℝ+2={x=(x,y)∈ℝ2:x,y≥0}, and f=(f1,f2)T the right-hand side of system (1.4). Let V:ℝ+×ℝ+2→ℝ+, then V is said to be in a class V0 if(1)
V is continuous on (nτ,(n+1)τ]×ℝ+2, and lim(t,y)→(nτ,x),t>nτV(t,y)=V(nτ+,x) exists;(2)
V is locally Lipschitzian in x.Definition 3.1.
LetV∈V0, (t,x)∈(nτ,(n+1)τ]×ℝ+2. The upper right derivatives of V(t,x) with respect to the impulsive differential system (1.4) are defined as
(3.1)D+V(t,x)=lim suph→0+1h[V(t+h,x+hf(t,x))-V(t,x)].Remark 3.2.
(1) The solution of system (1.4) is a piecewise continuous function x:ℝ+→ℝ+2; that is, x(t) is continuous on (nτ,(n+1)τ],n∈ℕ, and x(nτ+)=limt→nτ+x(t) exists. (2) The smoothness properties of f guarantee the global existence and uniqueness of the solutions of system (1.4) (see [22] for the details).Definition 3.3.
System (1.3) is said to be permanent if there exist positive constants m, M, and T0 such that every positive solution (x(t),y(t)) of system (1.4) with x0,y0>0 satisfies m≤x(t)≤M and m≤y(t)≤M for t>T0.We will use the following important comparison theorem on an impulsive differential equation [22].Lemma 3.4 (see [22]).
SupposeV∈V0 and
(3.2)D+V(t,x)≤g(t,V(t,x)),t≠nτ,V(t,x(t+))≤ψn(V(t,x)),t=nτ,
then g:ℝ+×ℝ+→ℝ is continuous on (nτ,(n+1)τ]×ℝ+ and for u(t)∈ℝ+, n∈ℕ, lim(t,y)→(nτ+,u)g(t,y)=g(nτ+,u) exists, ψn:ℝ+→ℝ+ is nondecreasing. Let r(t) be the maximal solution of the scalar impulsive differential equation
(3.3)u′(t)=g(t,u(t)),t≠nτ,u(t+)=ψn(u(t)),t=nτ,u(0+)=u0,
existing on [0,∞). Then V(0+,x0)≤u0 implies that V(t,x(t))≤r(t), t≥0, where x(t) is any solution of (3.2).We now indicate a special case of Lemma3.4 which provides estimations for the solution of impulsive differential inequalities. For this, we let PC(ℝ+,ℝ)(PC1(ℝ+,ℝ)) denote the class of real piecewise continuous (real piecewise continuously differentiable) functions defined on ℝ+.Lemma 3.5 (see [22]).
Let the functionu(t)∈PC1(ℝ+,ℝ) satisfy the inequalities
(3.4)u′(t)≤f(t)u(t)+h(t),t≠τk,t>0,u(τk+)≤αku(τk)+θk,k≥0,u(0+)≤u0,
where f,h∈PC(ℝ+,ℝ) and αk≥0, θk, and u0 are constants and (τk)k≥0 is a strictly increasing sequence of positive real numbers. Then, for t>0,
(3.5)u(t)≤u0(∏0<τk<tαk)exp(∫0tf(s)ds)+∫0t(∏s≤τk<tαk)exp(∫stf(γ)dγ)h(s)ds+∑0<τk<t(∏τk<τj<tαj)exp(∫τktf(γ)dγ)θk.Similar result can be obtained when all conditions of the inequalities in the Lemmas3.4 and 3.5 are reversed.Using Lemma3.5, it is easy to prove that the solutions of system (1.4) with strictly positive initial value remain strictly positive as follows.Lemma 3.6.
The positive quadrant(ℝ+*)2 is an invariant region for system (1.4).Proof.
Let(x(t),y(t)):[0,t0)→ℝ2 be a solution of system (1.4) with a strictly positive initial value (x0,y0). By Lemma 3.5, we can obtain that, for 0≤t<t0,
(3.6)x(t)≥x0(1-p1)[t/τ]exp(∫0tg1(s)ds),y(t)≥y0(1-p2)[t/τ]exp(∫0tg2(s)ds),
where g1(s)=r(1-x(s)/K)-(a/c)y(s)-λ and g2(s)=-d. Thus, x(t) and y(t) remain strictly positive on [0,t0).Now, we give the basic properties of the following impulsive differential equation considered the absence of the prey:(3.7)y′(t)=-dy(t),t≠nτ,y(t+)=(1-p2)y(t)+q,t=nτ,y(0+)=y0.
Solving the first equation of (3.7) between pulses implies
(3.8)y(t)=y(nτ+)exp(-d(t-nτ)),t∈(nτ,(n+1)τ].
Substituting it in the second equation of (3.7), the following difference equation is obtained:
(3.9)y((n+1)τ)=((1-p2)y(nτ)+q)exp(-dτ).
Then a periodic solution y*(t) of (3.7) is given by
(3.10)y*(t)=qexp(-d(t-nτ))1-(1-p2)exp(-dτ),t∈(nτ,(n+1)τ],n∈ℕ.Thus we can easy obtain the following results.Lemma 3.7.
( 1)y*(t)=qexp(-d(t-nτ))/(1-(1-p2)exp(-dτ)), t∈(nτ,(n+1)τ], n∈ℕ, and y*(0+)=q/(1-(1-p2)exp(-dτ)) is a positive periodic solution of (3.7).
(2)y(t)=(1-p2)n+1(y0-qexp(-dτ)/(1-(1-p2)exp(-dτ)))exp(-dτ)+y*(t) is a general solution of (3.7) with y0≥0, t∈(nτ,(n+1)τ] and n∈ℕ.
(3) For every solutiony(t) and every positive periodic solution y*(t) of system (3.7), it follows that y(t) tends to y*(t) as t→∞. Thus, the complete expression for the prey-free periodic solution of system (1.4) is obtained (0,y*(t))=(0,qexp(-d(t-nτ))/(1-(1-p2)exp(-dτ))) for t∈(nτ,(n+1)τ].Now, we discuss the stability of the prey-free periodic solution(0,y*(t)).Theorem 3.8.
(1) The prey-free periodic solution(0,y*(t)) of system (1.4) is locally asymptotically stable if
(3.11)(r+λ)τ+abdlnbqexp(-dτ)+c(1-(1-p2)exp(-dτ))bq+c(1-(1-p2)exp(-dτ))<ln11-p1.
(2) Moreover,(0,y*(t)) is globally asymptotically stable if
(3.12)(r+λ)τ+abdlnrbqexp(-dτ)+(K(r+λ)+rc)(1-(1-p2)exp(-dτ))rbq+(K(r+λ)+rc)(1-(1-p2)exp(-dτ))<ln11-p1.Proof.
To show the local stability of the prey-free periodic solution(0,y*(t)) of system (1.4), consider the following impulsive differential system:
(3.13)x1′(t)=rx1(t)(1-x1(t)K)-ax1(t)y1(t)by1(t)+x1(t)+c+λx1(t),t≠nτ,y1′(t)=-dy1(t)+eax1(t)y1(t)by1(t)+x1(t)+c,x1(t+)=(1-p1)x1(t),t=nτ,y1(t+)=(1-p2)y1(t)+q,(x1(0+),y1(0+))=(x0,y0).
By Lemma 3.4, x(t)≤x1(t) and y(t)≤y1(t), where (x(t),y(t)) is a solution of system (1.4). Note that if (x1(t),y1(t)) is locally stable, then so is (x(t),y(t)). It is easy to see that the periodic solution (0,y1*(t)) of (3.13) is the same as that of (1.4). That is, y1*(t)=y*(t). The local stability of the periodic solution (0,y1*(t)) may be determined by considering the behavior of small amplitude perturbations of the solution. Define x1(t)=u(t), y1(t)=y1*(t)+v(t). Then they may be written as
(3.14)(u(t)v(t))=Φ(t)(u(0)v(0)),0≤t≤τ,
where Φ(t) satisfies
(3.15)dΦdt=(r+λ-ay1*(t)by1*(t)+c0eay1*(t)by1*(t)+c-d)Φ(t),
and Φ(0)=I, where I is the identity matrix. The linearization of the third and fourth equations of system (1.4) becomes
(3.16)(u(nτ+)v(nτ+))=(1-p1001-p2)(u(nτ)v(nτ)).
Note that all eigenvalues of
(3.17)(1-p1001-p2)Φ(τ)
are μ1=(1-p1)exp(∫0τr+λ-ay1*(t)/(by1*(t)+c)dt) and μ2=(1-p2)exp(-dτ)<1.
Since∫0τay1*(t)/(by1*(t)+c)dt=-(a/bd)ln((bqexp(-dτ)+c(1-(1-p2)exp(-dτ)))/(bq+c(1-(1-p2)exp(-dτ)))), the condition |μ2|<1 is equivalent to (3.11). According to Floquet theory [22], (0,y*(t))=(0,y1*(t)) is locally asymptotically stable.
It is easy to see that the solution(0,y*(t)) is locally stable if condition (3.12) holds. Now, to prove the global stability of the pest-free periodic solution, let (x(t),y(t)) be a solution of system (1.4). From (3.12), we can select a sufficiently small number ϵ1>0 satisfying
(3.18)ρ=(1-p1)exp(∫0τr+λ-ra(y*(t)-ϵ1)rb(y*(t)-ϵ1)+K(r+λ)+rϵ1+rcdt)<1.
It follows from the first equation in (1.4) that x′(t)≤x(t)(r+λ-(r/K)x(t)) for t≠nτ. From Lemma 3.4, we have x(t)≤u(t), where u(t) is a solution of the following impulsive differential equation:
(3.19)u′(t)=u(t)(r+λ-rKu(t)),t≠nτ,u(t+)=(1-p1)u(t),t=nτ,u(0+)=x0.
Since u(t)→K(r+λ)/r as t→∞ if x0>0, x(t)≤K(r+λ)/r+ϵ for any ϵ>0 with t large enough. For simplicity, we may assume that x(t)≤K(r+λ)/r+ϵ1 for all t>0. Since y′(t)≥-dy(t), it follow from Lemma 3.4 that y(t)≥v(t)>y*(t)-ϵ1 for t sufficiently large, where v(t) is a solution of the following impulsive differential equation:
(3.20)v′(t)=-dv(t),t≠nτ,v(t+)=(1-p2)v(t)+q,t=nτ,v(0+)=y0.
For simplicity, we may suppose that y(t)≥v(t)>y*(t)-ϵ1 for all t≥0. From system (1.4), we obtain
(3.21)x′(t)≤x(t)(r+λ-a(y*(t)-ϵ1)b(y*(t)-ϵ1)+K(r+λ)/r+ϵ1+c),t≠nτ,x(t+)=(1-p1)x(t),t=nτ.
Integrating (3.21) on (nτ,(n+1)τ], we get
(3.22)x((n+1)τ)≤(1-p1)x(nτ)exp(∫nτ(n+1)τr+λ-ra(y*(t)-ϵ1)rb(y*(t)-ϵ1)+K(r+λ)+rϵ1+rcdt)=x(nτ)ρ
and hence x((n+1)τ)≤x(nτ)ρn which implies that x(nτ)→0 as n→∞. Further, we obtain, for t∈(nτ,(n+1)τ],
(3.23)x(t)≤(1-p1)x(nτ)exp(∫nτtr+λ-ra(y*(t)-ϵ1)rb(y*(t)-ϵ1)+K(r+λ)+rϵ1+rcdt)≤x(nτ)exp((r+λ+acϵ1)τ)
which implies that x(t)→0 as t→∞. Now, take a sufficiently small number ϵ2>0 satisfying ϵ2<d. Since limt→∞x(t)=0, we may assume that x(t)≤ϵ2 for all t≥0. It follows from the second equation in (1.4) that, for t≠nτ,
(3.24)y′(t)≤y(t)(-d+eacϵ2).
Thus, by Lemma 3.4, we induce that y(t)≤ỹ*(t), where ỹ*(t) is the periodic solution of (3.7) with d changed into d-(ea/c)ϵ2. By taking sufficiently small ϵ1 and ϵ2, we obtain that y(t) tends to y*(t) as t→∞.Using the similar method to the proof of Theorem3.8, we obtain the following theorems.Theorem 3.9.
For system (1.5), the periodic solution (0,y*(t)) is locally asymptotically stable if (r+λ)τ-aq(1-exp(-dτ))/cd(1-(1-p2)exp(-dτ))<ln(1/(1-p1)), and moreover, it is globally asymptotically stable if (r+λ)τ-raq(1-exp(-dτ))/d(K(r+λ)+rc)(1-(1-p2)exp(-dτ))<ln(1/(1-p1)).Theorem 3.10.
For system (1.6), the periodic solution (0,y*(t)) is locally asymptotically stable if (r+λ-a/b)τ<ln(1/(1-p1)), and moreover, it is globally asymptotically stable if (r+λ)τ+(a/bd)ln((rbqexp(-dτ)+K(r+λ)(1-(1-p2)exp(-dτ)))/(rb+K(r+λ)(1-(1-p2)exp(-dτ))))<ln(1/(1-p1)).Now, we prove the boundedness of system (1.4).Theorem 3.11.
There is anM>0 such that x(t),y(t)≤M for all t large enough, where (x(t),y(t)) is a solution of system (1.4).Proof.
Letx(t)=(x(t),y(t)) be a solution of system (1.4) and let V(t,x)=ex(t)+y(t). Then V∈V0. If t≠nτ, then we obtain
(3.25)D+V+βV=-erKx(t)2+e(r+λsin(ωt)+β)x(t)+(β-d)y(t),V(nτ+)≤V(nτ)+q and V(0+)=ex0+y0. Clearly, the right-hand side of (3.25) is bounded by a constant M0>0 if 0<β<d. So we can choose β0 such that
(3.26)D+V≤-β0V+M0,t≠nτ,V(nτ+)≤V(nτ)+q.
From Lemma 3.4, we obtain that
(3.27)V(t)≤(V(0+)-M0β0)exp(-β0t)+q(1-exp(-(n+1)β0τ))1-exp(-β0τ)exp(-β0(t-nτ))+M0β0
for t∈(nτ,(n+1)τ]. Therefore, V(t) is bounded by a constant M>0 for sufficiently large t. Hence x(t)≤M, y(t)≤M for a solution (x(t),y(t)) with all t large enough.The boundedness of systems (1.5) and (1.6) can be obtained from Theorem 3.11.Next, we investigate the permanence of system (1.4).Theorem 3.12.
System (1.4) is permanent if
(3.28)(r-λ)τ+abdlnbqexp(-dτ)+c(1-(1-p2)exp(-dτ))bq+c(1-(1-p2)exp(-dτ))>ln11-p1.Proof.
Letx0,y0>0. Consider the following system:
(3.29)x2′(t)=rx2(t)(1-x2(t)K)-ax2(t)y2(t)by2(t)+x2(t)+c-λx2(t),t≠nτ,y2′(t)=-dy2(t)+eax2(t)y2(t)by2(t)+x2(t)+c,x2(t+)=(1-p1)x2(t),t=nτ,y2(t+)=(1-p2)y2(t)+q,(x2(0+),y2(0+))=(x0,y0).
It follows from Lemma 3.4 that x(t)≥x2(t) and y(t)≥y2(t). From Theorem 3.11, we may assume that x2(t)<M, y2(t)<M, for all t≥0 large enough and M>(r-λ)/a. Let m2=qexp(-dτ)/(1-(1-p2)exp(-dτ))-ϵ2, ϵ2>0. From Lemmas 3.4 and 3.7, we obtain y2(t)≥m2 for all t large enough. Thus we will show that x2(t) has a lower bound m1>0 for all t large enough. We will do this in the following two steps.
Step 1.
From (3.28), we can choose m3>0, ϵ1>0 small enough such that δ<d and η=(1-p1)exp((r-λ-(r/K)m3)τ+(a/bd)ln(A/B)-(aϵ1/c)τ)>1, where δ=eam3/c, A=bqexp(-dτ)+c(1-(1-p2)exp(-dτ)) and B=bq+c(1-(1-p2)exp(-dτ)). Suppose that x2(t)<m3 for all t. Then, from the second equation of system (3.29), we obtain y2′(t)≤y2(t)(-d+(ea/c)x2(t))≤y2(t)(-d+δ). By Lemmas 3.4 and 3.7, we get y2(t)≤u(t) and u(t)→u*(t) as t→∞, where u(t) is the solution of
(3.30)u′(t)=(-d+δ)u(t),t≠nτ,u(t+)=(1-p2)u(t)+q,t=nτ,u(0+)=u0>0,
and u*(t)=qexp((-d+δ)(t-nτ))/(1-(1-p2)exp((-d+δ)τ)), t∈(nτ,(n+1)τ]. Then there exists T1>0 such that y2(t)≤u(t)≤u*(t)+ϵ1 for t≥T1. So, if t≠nτ, t≥T1, then
(3.31)x2′(t)≥x2(t)(r-λ-rKm3-au*(t)+aϵ1bu*(t)+bϵ1+c)≥x2(t)(r-λ-rKm3-au*(t)bu*(t)+c-aϵ1c),
and if t=nτ, t≥T1, then x(t+)=(1-p1)x(t). Let N1∈ℕ and N1τ≥T1. Integrating (3.31) on (nτ,(n+1)τ], n≥N1, we have x2((n+1)τ)≥(1-p1)x2(nτ)exp(∫nτ(n+1)τr-λ-(r/K)m3-au*(t)/(bu*(t)+c)-(aϵ1/c)dt)≥x2(nτ)exp(η). Then x2((N1+k)τ)≥x2(N1τ)exp(kη)→∞ as k→∞ which is a contradiction to the boundedness of x2(t). Hence there exists a t1>0 such that x2(t1)≥m3.Step 2.
Ifx2(t)≥m3 for all t≥t1, then we are done. If not, we may let t*=inft>t1{x2(t)<m3}. Then x2(t)≥m3 for t∈[t1,t*) and, by the continuity of x2(t), we have x2(t*)=m3. Suppose that t*∈[n1τ,(n1+1)τ) for some n1∈ℕ. Select n2,n3∈ℕ such that n2τ(δ-d)<ln(ϵ1/(M+p)) and (1-p1)n2exp((n2+1)η1τ)exp(n3η)>1, where η1=r-λ-(r/K)m3-aM<0. Let T=n2τ+n3τ. Then we have only to consider two possible cases for t∈(t*,(n1+1)τ].Case 1 (x2(t)<m3 for t∈(t*,(n1+1)τ]).
In this case we will show that there existst2∈[(n1+1)τ,(n1+1)τ+T] such that x2(t2)≥m3. Suppose not, that is, x2(t)<m3, t∈[(n1+1)τ,(n1+1+n2+n3)τ]. Then x2(t)<m3 for all t∈(t*,(n1+1+n2+n3)τ]. By (3.30) with u((n1+1)τ+)=y2((n1+1)τ+), we obtain
(3.32)u(t)=(1-p1)n1+1(u((n1+1)τ+)-p1-(1-p2)exp(-d+δ))×exp((-d+δ)(t-(n1+1)τ))+u*(t)
for t∈(nτ,(n+1)τ], n1+1≤n≤n1+n2+n3. So we get |u(t)-u*(t)|≤(M+p)exp((-d+δ)n2τ)<ϵ1 and y2(t)≤u(t)≤u*(t)+ϵ1 for t∈[(n1+1+n2)τ,(n1+1+n2+n3)τ]. Also we obtain that x2′(t)≥x2(t)(r-λ-(r/K)m3-(au*(t)/(bu*(t)+c)-aϵ1/c) if t≠nτ and x2(t+)=(1-p1)x2(t) if t=nτ, for t∈[(n1+1+n2)τ,(n1+1+n2+n3)τ]. Similarly to Step 1, we have
(3.33)x2((n1+1+n2+n3)τ)≥x2((n1+1+n2)τ)exp(n3η).
Since (3.29) and y2(t)≤M we have, for all t∈(t*,(n1+1+n2)τ], x2′(t)≥x2(t)(r-λ-(r/K)m3-aM)=η1x2(t) if t≠nτ and x2(t+)=(1-p1)x2(t) if t=nτ. Integrating it on [t*,(n1+1+n2)τ] we obtain that
(3.34)x2((n1+1+n2)τ)≥m3(1-p1)n2exp(η1((n1+1+n2)τ-t*))≥m3(1-p1)n2exp(η1(n2+1)τ).
Thus x2((n1+1+n2+n3)τ)≥m3(1-p1)n2exp(η1(n2+1)τ)exp(n3η)>m3 which is a contradiction.
Now, lett̅ = inft>t*{x2(t)≥m3}. Then x2(t)≤m3 for t*≤t<t̅ and x2(t̅)=m3. So, we have, for t∈[t*,t̅), x2′(t)≥x2(t)(r-λ-(r/K)m3-aM) if t≠nτ and x2(t+)=(1-p1)x2(t) if t=nτ. By the integration of it on [t*,t) for (t*≤t≤t̅), we can get that x2(t)≥x2(t*)(1-p1)1+n1+n2exp(η1(t-t*))≥m3(1-p1)1+n1+n2exp(η1(1+n2+n3)τ)≡m1>0.Case 2 (there is at′∈(t*,(n1+1)τ] such that x2(t′)≥m3).
Lett̲=inft>t*{x2(t)≥m3}. Then x2(t)≤m3 for t∈[t*,t̲) and x2(t̲)=m3. For t∈[t*,t̲), x2′(t)≥x2(t)(r-λ-(r/K)m3-aM) if n≠nτ. Integrating the equation on [t*,t̲)(t*≤t̲≤t), we can get that x2(t)≥x2(t*)exp(η1(t-t*))≥m3exp(η1τ)≥m1.
Thus, in both cases the similar argument can be continued sincex2(t)≥m3 for some t>t1. This completes the proof.Applying the method used in the proof of Theorem3.12 to systems (1.5) and (1.6), we obtain the following results.Theorem 3.13.
System (1.5) is permanent if (r-λ)τ-aq(1-exp(-dτ))/cd(1-(1-p2)exp(-dτ))>ln(1/(1-p1)).Theorem 3.14.
System (1.6) is permanent if (r-λ-a/b)τ>ln(1/(1-p1)).


## 4. Numerical Analysis of Seasonal Effect and Impulsive Perturbation

In this section we will study the influence of impulsive perturbation and seasonal effects on system (1.4), and the relationship between seasonal effects and impulsive perturbation. For this, we take the same parameters as those in Section 2, p1=p2=0 and τ=10.First, we display bifurcation diagrams for system (1.4) as p increases from 0 to 20 about λ=0 and λ=1 in Figure 5. From Figures 5(a) and 5(b), we see that system (1.4) experiences quasiperiodic oscillation (see Figure 6(a)) when p<p1(≈0.5). However, when p>p1, we see that there is a cascade of periodic bifurcation (see Figure 6(b)) leading to chaos (see Figure 6(c)), which is followed by a cascade of periodic halving bifurcation from chaos to periodic solutions (see Figure 7). Figures 5(c) and 5(d) clearly show that with p increasing from 0 to 20, system (1.4) experiences process of periodic oscillating→periodic doubling→chaos→periodic halving. Figure 8 displays two different strange attractors. Next, Figure 9 illustrates bifurcation diagrams for different values of the pulse p and λ as a bifurcation parameter. It follows from Figures 9(a) and 9(b) that system (1.4) experiences process of periodic oscillating with different periods→periodic doubling→chaos→periodic windows with periodic halving cascade →τ-periodic solutions. Figure 10 exhibits two different strange attractors. It follows from Figures 9(c) and 9(d) that system (1.4) undergoes chaotic motions when λ<λ1(≈0.51). When λ>λ1, chaotic motions suddenly disappear and appear as τ-periodic solutions. There are also periodic doubling and halving phenomena. Finally, we investigate the relationship between p, λ, and τ in a view of controlling the population density of the prey and predator. As seen in Figure 11, we figure out that the longer the period τ is, the larger the permanence region is and the smaller the stability region is. That means that we should release the predator within a short period, or the impulsive perturbations of the predator should be occurred at short intervals, to eradicate the prey. On the contrary, the impulsive perturbations of the predator should be occurred at long-time intervals for coexistence of the prey and the predator. If we choose (p,λ,τ)=(5,1,1.5), we can see coexistence of the prey and predator as shown in Figure 12.Bifurcation diagrams of system (1.4). (a)-(b) x,y are plotted for p when λ=0, (c)-(d) x,y are plotted for p when λ=1.
(a)(b)(c)(d)Phase portraits of solutions for system (1.4) when λ=0. (a) p=0.4, (b) p=3.1 (a 4τ-periodic solution), (c) p=4.
(a)(b)(c)Phase portraits of solutions for system (1.4) when λ=0. (a) p=10.4 (a 8τ-periodic solution), (b) p=11 (a 4τ-periodic solution).
(a)(b)Phase portraits of solutions for system (1.4) when λ=1. (a) p=0.35, (b) p=6.9.
(a)(b)Bifurcation diagrams of system (1.4). (a)-(b) x,y are plotted for λ when p=3, (c)-(d) x,y are plotted for λ when p=10.
(a)(b)(c)(d)Phase portraits of solutions for system (1.4) when p=3. (a) λ=2.3, (b) λ=6.2.
(a)(b)Regions induced from Theorems3.8 and 3.12. (a) τ=0.1, (b) τ=0.5, (c) τ=1.5, (d) τ=2.
(a)(b)(c)(d)(a) Phase portrait of system (1.4) with p=5, λ=1, and τ=1.5. (b)-(c) Time series of x and y.
(a)(b)(c)


## 5. Conclusion

In this paper, we have investigated the effects of periodic forcing in the intrinsic growth rate of the prey and impulsive perturbations on a predator-prey system with the Beddington-DeAngelis functional response. We have shown that there exists an asymptotically stable prey-free periodic solution if the magnitudeλ of seasonality is less than some critical value and have found parameter regions which system (1.4) is permanent. Numerical results have shown that system (1.4) can give birth to various kinds of dynamical behaviors. Especially, the prey and the predator can coexist even if there are seasonal effects on the prey. In addition, conditions for the stability of prey-free solution and for the permanence of Holling-type II or ration-dependent predator-prey systems have been obtained. Thus we have improved the results of [15].



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*Source: 695121-2009-10-29.xml*