# Complex Dynamical Behaviors in a Predator-Prey System with Generalized Group Defense and Impulsive Control Strategy

Authors: Shunyi Li
Journal: Discrete Dynamics in Nature and Society
 (2013)

---

## Abstract

A predator-prey system with generalized group defense and impulsive control strategy is investigated. By using Floquet theorem and small amplitude perturbation skills, a local asymptotically stable prey-eradication periodic solution is obtained when the impulsive period is less than some critical value. Otherwise, the system is permanent if the impulsive period is larger than the critical value. By using bifurcation theory, we show the existence and stability of positive periodic solution when the pest eradication lost its stability. Numerical examples show that the system considered has more complicated dynamics, including (1) high-order quasiperiodic and periodic oscillation, (2) period-doubling and halving bifurcation, (3) nonunique dynamics (meaning that several attractors coexist), and (4) chaos and attractor crisis. Further, the importance of the impulsive period, the released amount of mature predators and the degree of group defense effect are discussed. Finally, the biological implications of the results and the impulsive control strategy are discussed.


---

## Body

## 1. Introduction

In population dynamics, a functional response of the predator to the prey density refers to the change in the density of prey attached per unit time per predator as the prey density changes and it is assumed to be monotonically increasing in most predator-prey systems. For example, Holling type I, II, and III functional response [1]
(1)f1(x,y)=rx,f2(x,y)=rxa+bx,f3(x,y)=rx2a+bx2,
and the sigmoidal type response function [2]
(2)f4(x,y)=rx2(a+x)(b+x),
and Ivlev type response function [3]
(3)f5(x,y)=r(1-e-ax).
The previous functional responses are prey dependent. But, both predator and prey densities have an effect on the response, such as Beddington-DeAngelis functional response [4, 5]
(4)f6(x,y)=rxa+bx+cy
and modified Holling type II and type III response functions [6]
(5)f7(x,y)=rx(a+bx)(b+y),f8(x,y)=rx2(a+bx2)(b+y).However, some experimental and observational evidence shown that the functional response is not always monotonically increasing, such as Holling type IV [7]
(6)f9(x,y)=rxa+bx+cx2
and f10(x,y)=αxe-βx [8]. Group defense is a term used to describe the phenomenon whereby predation is decreased, or even prevented altogether, due to the increased ability of the prey to better defend or disguise itself when it exists in enough large numbers [9–11]. The buffalo group defense was modeled using a generalized group defense in [12],
(7)f11(x,y)=αx1+hxβ,
where β is a positive integer whose value determines the degree of antipredator behavior and group defense.Recently, it is of great interest to investigate complex dynamics for impulsive perturbations in populations dynamics. In particular, the impulsive prey-predator population models have been investigated by many researchers. The results of studies of the dynamics of a predator-prey model with nonmonotonic functional response, such as Holling type IV functional response with respect to an impulsive control strategy, were presented in [13–24]. To the best of our knowledge, there are few papers studying the group defense predator-prey with impulsive effect, where the antipredator behavior and group defense effect described by nonmonotonic functional response. Zhang et al. [25] considered a predator-prey system with defensive ability of prey by Holling type IV functional response and impulsive perturbations on the predator:
(8)x′(t)=rx(t)(1-x(t)k)-x(t)y(t)a1+x2(t),y′(t)=y(t)(-d+μx(t)a1+x2(t)),t≠nT,x(nT+)=x(nT),y(nT+)=y(nT)+τ,X(0+)=x0=(x0,y0)T,t=nT.
The conditions for the local asymptotically stable prey-eradication periodic solution and permanence of the system are obtained; a series of complex phenomena are displayed by numerical simulation. Furthermore, based on this work, Pei et al. [26] investigated a one-prey multi-predator model with defensive ability of the prey by introducing impulsive biological control strategy:
(9)x′(t)=rx(t)(1-x(t)k)-∑i=1m‍x(t)y(t)ai+x2(t),yi′(t)=yi(t)(∑i=1m‍μix(t)ai+x2(t)-di),t≠nT,x(nT+)=x(nT),yi(nT+)=yi(nT)+pi,X0=(x(0+),y1(0+),…,ym(0+))T=(x0,y01,…,y0m)T,t=nT.
And it shown that the multi-predator impulsive control strategy is more effective than the classical one and makes the dynamical behaviors of the system more complex. Recently, a predator-prey system with impulsive effect and group defense with the nonmonotone function f10(x,y)=αxe-βx was studied by Li et al. [27],
(10)x′(t)=x(t)(a-bx(t))-αx(t)y(t)e-βx(t),y′(t)=kαx(t)y(t)e-βx(t)-y(t)d,t≠nT,Δx(t)=-px(t),Δy(t)=q,t=nT.
They proved that there exists a locally stable pest-eradication periodic solution when the impulsive period is less than certain critical values; otherwise, the system is permanent. Some complicated dynamics, such as quasiperiodic oscillation, bifurcation, and attractor crisis, were shown by numerical simulations.In this paper, we study a predator-prey system with impulsive effect and generalized group defense with the nonmonotone functionf11(x,y)=αx/(1+hxβ):
(11)x′(t)=x(t)(a-bx(t))-αx(t)1+hxβ(t)y(t),y′(t)=kαx(t)1+hxβ(t)y(t)-dy(t),t≠nT,Δx(t)=-p1x(t),Δy(t)=-p2y(t)+q,t=nT,
where x(t) and y(t) represent the prey and the predator populations at time t, respectively; a,b,α,h,β,andk are positive. a is the intrinsic rate of increase of the prey and d is the death rate of the predator, a/b is the carrying capacity of the prey, β>1 is the degree of anti-predator behavior and group defense, and k(0<k<1) is the rate of conversing prey into predator. Δx(t)=x(t+)-x(t),Δy(t)=y(t+)-y(t), T is the periodic of the impulse for predator in order to eradicate target pests, protect nontarget pest (or harmless insect) from extinction and drive target pest to extinction, or control target pest at acceptably low level to prevent an increasing pest population from causing an economic loss. n∈ℕ+,ℕ+={1,2,…},pi>0(i=1,2) is the proportionality constant which represents the rate of mortality due to the applied pesticide; for example, impulsive reduction of the population is possible by harvesting or by poisoning with chemicals used in agriculture. q>0 is the number of predators released each time, for example, by artificial breeding of the species or release of some species.The paper is  arranged as follows. In Section2, some notations and Lemmas are given. In Section 3, using the Floquet theory of impulsive equation and small amplitude perturbation skills, we will prove the local stability of prey-eradication periodic solution when the impulsive period is less than some critical value and give the condition of permanence. In Section 4, by using bifurcation theory, the existence and stability of positive periodic solution are  studied when T is close to the critical value T0. In Section 5, the results of numerical examples are shown, and some rich dynamic behaviors are obtained; the effects of the impulsive period, the released amount of mature predators and the coefficient of group defense effect are discussed. Finally, the conclusions are discussed briefly  in Section 6.


## 2. Preliminaries

In this section, we will give some definitions, notations,  and lemmas which will be useful for our main results.Letℝ+=[0,∞),ℝ+2={x∈ℝ2∣x≥0}. Denote by f=(f1,f2) the map defined by the right hand of the first two equations of system (11), and denote by  ℕ the set of all nonnegative integers. Let V:ℝ+×ℝ+2→ℝ+, then V is said to belong to class V0 if(1)
Vis continuous in (t,x)∈(nT,(n+1)T]×ℝ+2 and for each x∈ℝ+2,n∈ℕ,  lim(t,y)→(nT+,x)V(t,y)=V(nT+,x) exists,(2)
V is locally Lipschitzian in x.Definition 1.
LetV∈V0; then for (t,x)∈(nT,(n+1)T]×ℝ+2, the upper right derivative of V(t,x) with respect to the impulsive differential system (11) is defined as
(12)D+V(t,x)=limh→0+sup1h[V(t+h,x+hf(t,x))-V(t,x)].Definition 2.
System (11) is said to be permanent if there exist two positive constants m,M and T0 such that each positive solution (x(t),y(t)) of the system (11) satisfies m≤x(t)≤M,m≤y(t)≤M, for all t>T0.
The solution of system (11) is a piecewise continuous function x:ℝ+↦ℝ+2, x(t) is continuous on (nT,(n+1)T],n∈ℕ, and x(nT+)=limt→nT+x(t) exists; the smoothness properties of f guarantee the global existence and uniqueness of solutions of system (11); for details see [28, 29]. The following lemma is obvious.Lemma 3.
LetX(t) be a solution of system (11) with X(0+)≥0; then X(t)≥0 for all t≥0 and further X(t)>0 for all t≥0 if X(0+)>0.And we will use the following important comparison theorem on impulsive differential equation [29].Lemma 4.
SupposeV∈V0. Assume that
(13)D+V(t,x)≤g(t,V(t,x)),t≠nT,V(t,x(t+))≤ψn(V(t,x)),t=nT,
where g:ℝ+×ℝ+↦ℝ is continuous in (nT,(n+1)T]×ℝ+, and for u∈ℝ+,n∈ℕ, lim(t,y)→(nT+,u)=g(nT+,u) exists; ψn:ℝ+→ℝ+ is nondecreasing. Let r(t) be the maximal solution of the scalar impulsive differential equation
(14)u′(t)=g(t,u(t)),t≠nT,u(t+)=ψn(u(t)),t=nT,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 system (11).Finally, we give some basic properties about the following subsystem of system (11):
(15)y′(t)=-dy(t),t≠nT,Δy(t)=-p2y(t)+q,t=nT,y(0+)=y0≥0.
Clearly, when t∈(nT,(n+1)T],n∈ℕ,
(16)y*(t)=qexp[-d(t-nT)]1-(1-p2)exp(-dT),y*(0)=q1-(1-p2)exp(-dT),
is a positive periodic solution of system (15). Since
(17)y(t)=(y(0+)-q1-(1-p2)exp(-dT))×exp(-dt)+y*(t)
is the solution of system (15) with initial value y0≥0, where t∈(nT,(n+1)T],n∈ℕ; then one can get the following.Lemma 5.
Lety*(t) be a positive periodic solution of system; (15) and every solution y(t) of system (15) with y0≥0, one has  |y(t)-y*(t)|→0, when t→∞.
Therefore, one obtains the pest-eradication periodic solution(18)(0,y*(t))=(0,qexp[-d(t-nT)]1-(1-p2)exp(-dT))
for t∈(nT,(n+1)T].


## 3. Extinction and Permanence

Firstly, we study the stability of prey-eradication periodic solution.Theorem 6.
LetX(t)=(x(t),y(t)) be any solution of system (11); then X(t)=(0,y*(t)) is locally asymptotically stable provided that
(19)aT-αq[1-exp(-dT)]d[1-(1-p2)exp(-dT)]<ln(11-p1).Proof.
The local stability of periodic solutionX(t)=(0,y*(t)) may be determined by considering the behavior of small amplitude perturbations of the solution. Consider
(20)x(t)=u(t),y(t)=y*(t)+v(t).
There may be written
(21)(u(t)v(t))=Φ(t)(u(0)v(0)),0≤t<T,
where Φ(t) satisfies
(22)dΦdt=(a-αy*(t)0kαy*(t)-d)Φ(t),
and Φ(0)=I, the identity matrix. The linearization of the third and fourth equations of system (11) becomes
(23)(u(nT+)v(nT+))=(1-p1001-p2)(u(nT)v(nT)).
Hence, if both eigenvalues of
(24)M=(1-p1001-p2)Φ(t)
have absolute values less than one, then the periodic solution X(t)=(0,y*(t)) is locally stable. Since all eigenvalues of M are
(25)μ1=(1-p2)exp(-dT)<1,μ2=(1-p1)exp(∫0T‍(a-αy*(t))dt),μ2<1 if and only if
(26)aT-αq[1-exp(-dT)]d[1-(1-p2)exp(-dT)]<ln(11-p1).
According to Floquet theory [28] of impulsive differential equation, the prey-eradication solution X(t)=(0,y*(t)) is locally stable. This completes the proof.Theorem 7.
There exists a constantM>0, such that x(t)≤M,y(t)≤M for each solution X(t)=(x(t),y(t)) of system (11) with all t being large enough.Proof.
LetV(t)=kx(t)+y(t). It is clear that V∈V0. We calculate the upper right derivative of V(t,x) along a solution of system (11) and get the following impulsive differential equation:
(27)D+V(t)∣(11)+LV(t)=kx(a+L-bx)-(d-L)y,t≠nT,V(t+)≤V(t)+q,t=nT.
Let 0<L<d;  then kx(a+L-bx)-(d-L)y is bounded. Select L0 and M0 such that
(28)D+V(t)≤-L0V(t)+M0,t≠nT,V(t+)≤V(t)+q,t=nT,
where L0 and M0 are two positive constants. According to Lemma 4, we have
(29)V(t)≤M0L0+(V(0+)-M0L0)×exp(-L0t)-q1-exp(-L0T)exp(-L0t)+q1-exp(-L0T)exp(-L0(t-nT)),
where t∈(nT,(n+1)T]. Hence
(30)limt→∞V(t)≤M0L0+q1-exp(-L0T).
Therefore, V(t,x) is ultimately bounded. We obtain that each positive solution of system (11) is uniformly ultimately bounded. This completes the proof.In the following, we investigate the permanence of system (11).Theorem 8.
System (11) is permanent if
(31)aT-αq[1-exp(-dT)]d[1-(1-p2)exp(-dT)]>ln(11-p1).Proof.
SupposeX(t)=(x(t),y(t)) is a solution of system (11) with X0>0. From Theorem 7 we may assume that  x(t)≤M,y(t)≤M,  and  M>a/b,t≥0. Let
(32)m2=qexp(-dT)1-(1-p2)exp(-dT)-ε2,ε2>0.
According to Lemmas 4 and 5, we have y(t)>m2 for all t large enough. In the following, we want to find m1 such that x1(t)≥m1 for all t large enough. We will do it in the following two steps for convenience.
Step1. Since(33)aT-αq[1-exp(-dT)]d[1-(1-p2)exp(-dT)]>ln(11-p1),
we can select  m3>0,ε1>0 small enough such that m3<a/b, δ=kαm3<d, and
(34)σ=(a-bm3-αε1)T-αq[1-exp((-d+δ)T)](d-δ)[1-(1-p2)exp((-d+δ)T)]-ln(11-p1)>0.
We will prove there existst1∈(0,∞) such that x(t1)≥m3. Otherwise, according to the above assumption, we get y′(t)≤y(t)(-d+δ), and by Lemmas 4 and 5, we have y(t)≤z(t) and z(t)≤z*(t), where
(35)z*(t)=qexp[(-d+δ)(t-nT)]1-(1-p2)exp((-d+δ)T),t∈(nT,(n+1)T],
and z(t) is the solution of the following equation:
(36)z′(t)=z(t)(-d+δ),t≠nT,Δz(t)=-p2z(t)+q,t=nT,z(0+)=y0≥0.
Therefore, there exists a T1>0 such that
(37)y(t)≤z(t)≤z*(t)+ε1,x′(t)≥x(t)(a-bm3-(z*(t)+ε1)).
Let N1∈ℕ and  let N1T≥T1. We can get
(38)x′(t)≥x(t)(a-bm3-(z*(t)+ε1)),t≠nT,Δx(t)=-p1x(t),t=nT.
Integrating (38) on (nT,(n+1)T](n≥N1), we have
(39)x((n+1)T)≥x(nT+)exp(∫nT(n+1)T‍(a-bm3-α(z*(t)+ε1))dt)=(1-p1)x(nT)×exp[αq[1-exp((-d+δ)T)](d-δ)[1-(1-p2)exp((-d+δ)T)](a-bm3-αε1)T-×expa-αq[1-exp((-d+δ)T)](d-δ)[1-(1-p2)exp((-d+δ)T)]]=x(nT)exp(σ).
Then x((N1+h)T)≥x(N1T)exp(hσ)→∞ as h→∞, which is a contradiction to the boundedness of x(t). Hence there exists a t1>0 such that x(t1)≥m3.
Step2. If x(t1)≥m3 for all t≥t1, then our aim is obtained.
Hence we only need to consider those solutions which leave the regionR={X(t)∈ℝ2+:x(t)<m3} and reenter again. Let t*=inft≥t1{x(t)<m3}. Then t* is impulsive point or nonimpulsive point.
Case   1. If t* is impulsive point, there exist a n1∈ℕ such that t*=n1T. Then x(t)≥m3 for t∈[t1,t*] and
(40)(1-p1)m3≤x(t*+)=(1-p1)x(t*)<m3.
Choose n2,n3∈ℕ such that
(41)n2T>T2=ln(ε1/(M+q))(-d+δ),(1-p)n2+1exp((n2+1)σ1T)exp(n3σ)>1,
where σ1=a-bm3-αM<0. Let T¯=(n2+n3)T. Then, there exists a t2∈[(n1+1)T,(n1+1)T+T¯] such that x(t2)≥m3. Otherwise x(t)<m3,  t∈[(n1+1)T,(n1+1)T+T¯]. Consider (36) with z((n1+1)T+)=y((n1+1)T+); we have
(42)z(t)=(z(n1+1)T+-q1-(1-p2)exp((-d+δ)T))×exp[(-d+δ)(t-(n1+1)T)]+z*(t),
where t∈(nT,(n+1)T],n1+1≤n≤(n1+1)+n2+n3. Then
(43)|z(t)-z*(t)|<(M+q)exp(-(d-δ)n2T)<ε1,y(t)≤z(t)≤z*(t)+ε1
for (n1+1+n2)T≤t≤(n1+1)T+T¯, which implies (39) holds for (n1+1+n2)T≤t≤(n1+1)T+T¯. As in Step 1, we have
(44)x((n1+1+n2+n3)T)≥x((n1+1+n2)T)exp(n3σ).
The first and third equations of system (11) given
(45)x′(t)>x(t)(a-bm3-αM)=σ1x(t),t≠nT,x(t+)=(1-p1)x(t),t=nT.
Integrating the above equation on [t*,(n1+1+n2)T], we can get
(46)x((n1+1+n2)T)≥(1-p)n2+1m3exp(σ1(n2+1)T),
and thus
(47)x((n1+1+n2+n3)T)≥m3(1-p)n2+1exp((n2+1)σ1T)exp(n3σ)>m3,
a contradiction.
Lett-=inft≥t*{x(t)≥m3}; then x(t-)≥m3. For t∈[t*,t-), we have
(48)x(t)≥m3exp(σ1(1+n2+n3)T)(1-p)1+n2+n3≜m1.
For t>t-, the same arguments can be continued since x(t-)≥m3.
Case   2. If t* is nonimpulsive point, then x(t)≥m3 for t∈[t1,t*) and x(t*)=m3; suppose t*∈(n1′T,(n1′+1)T),n1′∈ℕ. There are two possible cases for t∈(t*,(n1′+1)T).
Case   2.1.   x(t)≤m3 for all t∈(t*,(n1′+1)T). As in step 1, we can prove that there must be a t2′∈[(n1′+1)T,(n1′+1)T+T¯] such that x(t2′)>m3; Let t-=inft≥t*{x(t)≥m3}, then x(t-)≤m3 and x(t-)=m3. For t∈[t*,t-), we have
(49)x(t)≥m3exp(σ1(1+n2+n3)T)(1-p)1+n2+n3≜m1.
For t>t-, the same arguments can be continued since x(t-)≥m3.
Case   2.2. There exists a t∈(t*,(n1′+1)T¯) such that x(t)>m3. Let tˇ=inft>t*{x(t)>m3}; then x(t)≤m3 for t∈(t*,tˇ) and x(tˇ)=m3. For t∈(t*,tˇ), (45) holds true; integrating (45) on t∈(t*,tˇ), we have
(50)x(t)≥x(t*)exp[σ1(t-t*)]≥m3exp(σ1T)>m1.
Since x(tˇ)≥m3 for t>tˇ, the same arguments can be continued.
Hencex(t)≥m1 for all t>t1. The proof is completed.Remark 9.
Let(51)f(T)=aT-αq[1-exp(-dT)]d[1-(1-p2)exp(-dT)]-ln(11-p1).
Since f(0)=-ln(1/(1-p1))<0, f(T)→+∞ as T→+∞, and
(52)f′′(T)=(αqp2d[[1-(1-p2)exp(-dT)]exp(-dT)α×p2d+2(1-p2)exp(-2dT)])×([1-(1-p2)exp(-dT)]3)-1>0,
so f(T)=0 has a unique positive root, denoted by  Tmax. From Theorems 6 and 8 we know that Tmax is a threshold. If T<Tmax, then pest-eradication periodic solution (0,y*(t)) is asymptotically stable; if T>Tmax, then system (11) is permanent.Remark 10.
Ifp1=p2=0,q=0; that is, there are without taking any pest-management strategy, large numbers of preys (pest) would coexisting with predators (natural enemy). If q=0,0<p1,p2<1, that is, there is periodic spraying pesticide (or harvesting) only. Thus, we can easily obtain that Tmax′=ln(1/(1-p1))/a<Tmax is the threshold. If p1=p2=0,q>0; that is, there is periodic releasing of predator (natural enemy) only, without periodic spraying pesticide (or harvesting). We can easily get that Tmax′′=αq/(ad)<Tmax is the threshold. Comparing with the classic methods (such as biological control or chemical control), the integrated pest management (IPM) is a better one, since Tmax>Tmax′ and Tmax>Tmax′′. Some numerical examples will be given in Section 5.


## 4. Bifurcation and Existence of Positive Periodic Solution

In this section, we deal with the existence of a nontrivial periodic solution to system (11) near the prey-eradication periodic solution (0,y*(t)) via bifurcation.Letx1(t)=y(t),x2(t)=x(t); system (11) becomes as follows:
(53)x1′(t)=x1(t)(kαx2(t)1+hx2β(t)-d)≜F1(x1(t),x2(t)),x2′(t)=x2(t)(a-bx2(t)-αx1(t)1+hx2β(t))≜F2(x1(t),x2(t)),t≠nT,x1(nT+)=(1-p2)x1(nT)+q≜Θ1(x1(nT),x2(nT)),x2(nT+)=(1-p1)x2(nT)≜Θ2(x1(nT),x2(nT)),t=nT.
All notations used in this section are the same as those in [30]. Let Φ be the flow associated to (53); we have x(t)=Φ(t,x0),0<t≤T, where x(t)=(x1(t),x2(t)),x0=x(0+),
(54)d0′=1-(∂Θ2∂x2∂Φ2∂x2)(T0,x0),
where T0 is the root of d0′=0,
(55)a0′=1-(∂Θ1∂x1∂Φ1∂x1)(T0,x0),b0′=-(∂Θ1∂x1∂Φ1∂x2+∂Θ1∂x2∂Φ2∂x2)(T0,x0),∂Φ1(t,x0)∂x1=exp(∫0t‍∂F1(ξ(r))∂x1dr),∂Φ2(t,x0)∂x2=exp(∫0t‍∂F2(ξ(r))∂x2dr),∂Φ1(t,x0)∂x2=∫0t‍exp(∫ut‍∂F1(ξ(r))∂x1dr)(∂F1(ξ(u))∂x2)××exp(∫0u‍∂F2(ξ(r))∂x2dr)du,∂2Φ2(t,x0)∂x1∂x2=∫0t‍exp(∫ut‍∂F2(ξ(r))∂x2dr)(∂2F1(ξ(u))∂x1∂x2)××exp(∫0u‍∂F2(ξ(r))∂x2dr)du,∂2Φ2(t,x0)∂x22=∫0t‍exp(∫ut‍∂F2(ξ(r))∂x2dr)(∂2F2(ξ(u))∂x22)××exp(∫0u‍∂F2(ξ(r))∂x2dr)du+∫0t‍exp(∫ut‍∂F2(ξ(r))∂x2dr)(∂2F2(ξ(u))∂x1∂x2)××∫0u‍exp(∫pu‍∂F1(ξ(r))∂x1dr)(∂F1(ξ(u))∂x2)×××××exp(∫0p‍∂F2(ξ(r))∂x2dr)dpdu,∂2Φ2(t,x0)∂T-∂x2=∂F2(ξ(t))∂x2exp(∫0t‍∂F2(ξ(r))∂x2dr),∂2Φ1(T0,x0)∂T=y′(T0),B=-∂2Θ2∂x1∂x2(∂Φ1(T0,X0)∂T-∂Φ1(T0,X0)∂x1+∂Φ1(T0,X0)∂x11a0′∂Θ1∂x1∂Φ1(T0,X0)∂T-)×∂Φ2(T0,X0)∂x2-∂Θ2∂x2(∂2Φ2(t,X0)∂T-∂x2+∂2Φ2(t,X0)∂x1∂x2×∂Φ2(T0,X0)∂x2-∂Θ2∂x2+××1a0′∂Θ1∂x1∂Φ1(T0,X0)∂T-),C=-2∂2Θ2∂x1∂x2(-b0′a0′∂Φ1(T0,X0)∂x1+∂Φ1(T0,X0)∂x2)×∂Φ2(T0,X0)∂x2-∂2Θ2∂x22(∂Φ2(T0,X0)∂x2)2+2∂Θ2∂x2b0′a0′∂2Φ2(T0,X0)∂x1∂x2-∂Θ2∂x2∂2Φ2(T0,X0)∂x22,
where ξ(t)=(y*(t),0).Lemma 11 (see [30]).
If|1-a0′|<1 and  d0′=0, then one has the following.(a)
IfBC≠0, then one has a bifurcation. Moreover, one has a bifurcation of a nontrivial periodic solution of  (53) if BC<0 and a subcritical case if BC>0.(b)
IfBC=0, then one has an undetermined case.In order to apply Lemma11, we compute the following:
(56)d0′=1-(1-p1)exp(∫0T‍(a-αy*(r))dr)=1-(1-p1)exp[aT0-αq(1-exp(-dT))d[1-(1-p2)exp(-dT)]].
If  d0′=0, this corresponds to T0 satisfying
(57)aT0=αq(1-exp(-dT0))d[1-(1-p2)exp(-dT0)]+ln(11-p1).
Further, we can get
(58)a0′=1-exp(-dT0)>0,b0′=-kα(1-p2)exp(∫0T0‍y*(r)dr)<0,∂Φ1(T0,x0)∂x1=exp(-dT0)>0,∂Φ2(T0,x0)∂x2=exp[aT0-αq(1-exp(-dT))d[1-(1-p2)exp(-dT)]]=11-p1>0,∂Φ1(T0,x0)∂x2=kα∫0T0‍exp(-d(T0-u))y*(u)kα××exp(∫0u‍(a-αy*(r))dr)du>0,∂2Φ2(T0,x0)∂T-∂x2=(a-αq(1-exp(-dT0))d[1-(1-p2)exp(-dT0)])11-p1,∂2Φ2(T0,x0)∂x1∂x2=-kαT01-p1<0,∂Φ1(T0,x0)∂T=y′(T0)=-dqexp(-dT0)1-(1-p2)exp(-dT0)<0.
Note that
(59)∂2F2(ξ(u))∂x22=-(2b+αy*(u))<0,∂2F2(ξ(u))∂x1∂x2=-α<0,∂F1(ξ(u))∂x2=kαy*(u)>0;
then
(60)∂2Φ2(T0,x0)∂x22<0.
Since
(61)∂Θ1∂x2=∂Θ2∂x1=0,∂Θ1∂x1=1-p2,∂Θ2∂x2=1-p1,
it is easy to verify that C>0 and
(62)B=-[a-αqexp(-dT0)d[1-(1-p2)exp(-dT0)]---a+kαdqT0(1-p2)exp(-dT0)[1-(1-p2)exp(-dT0)]2].
In order to determine the sign of B, let
(63)f(t)=a-αqexp(-dt)1-(1-p2)exp(-dT0).
We have
(64)f′(t)=αqexp(-dt)d[1-(1-p2)exp(-dT0)]>0.
Thus, we can conclude that f(T0)>0, since
(65)∫0T0‍f(t)dt=aT0-αqexp(-dt)d[1-(1-p2)exp(-dT0)]=ln(11-p1)≥0,
and f(t) is strictly increasing. Therefore, we have B<0.  In view of  T0=Tmax and according to Lemma 11, we obtain the following result.Theorem 12.
System (11) has a positive periodic solution if   T>T0 and  T is close to T0, where T0 satisfies
(66)aT0=αq(1-exp(-dT0))d[1-(1-p2)exp(-dT0)]+ln(11-p1),
and the nontrivial periodic solution is supercritical case via bifurcation, which means that the positive periodic solution is stable.


## 5. Numerical Analysis

In this section, we will study the impulsive effect on system (11) and show that the impulsive perturbations cause complicated dynamical behavior for system (11). The influence of T,q, and β may be documented by stroboscopically sampling one of the variables over a range of their values. Stroboscopic map is a special case of the Poincaré map for periodically forced system or periodically pulsed system. Fixing points of the stroboscopic map correspond to periodic solutions of system (11) having the same period as the pulsing term; periodic points of period k about stroboscopic map correspond to entrained periodic solutions of system (11) having exactly k times the period of the pulsing; invariant circles correspond to quasi-periodic solutions of system (11); system (11) possibly appear chaotic (or strange) attractors.Example 13.
Leta=3.1,  b=1.5,  α=1.05,k=0.85,n=2.15,h=0.97,d=0.3,p1=0.85,p2=0.55,and q=0.2 with initial value X(0)=(0.5,0.5).From Remark10, large numbers of preys (pest) could coexist with predators (natural enemy) with periodic oscillations, if we  are not taking pest-management strategy (p1=0,p2=0,q=0) (Figure 1(a)). q=0,p1=0.85,andp2=0.55; that is, there are periodic spraying pesticide (or harvesting) only, without releasing of predator (natural enemy), large numbers of preys (pest) coexist with periodic oscillation, but predators (natural enemy) rapidly decrease to zero when T=0.66<Tmax (Figure 1(b)).  p1=0,p2=0,q=0.2; that is, there is periodic releasing of predator (natural enemy) only; without spraying pesticide (or harvesting), a few of preys (pest) coexist with predators (natural enemy) when T=0.66<Tmax (Figure 1(c)). We cannot make the prey population x(t) eradicate when T=0.66. From Theorem 6, we know that the prey-eradication periodic solution is asymptotically stable provided that T<Tmax≈0.6776. A typical prey-eradication periodic solution of the system (11) is shown in Figure 1(d), where we observe how the variable y(t) oscillates in a stable cycle. In contrast, the prey population x(t) rapidly decreases to zero when T=0.66<Tmax≈0.6776. Hence, the integrated pest management (IPM) is better than the classic methods (such as biological control or chemical control).Time series of system (11) when T=0.66<Tmax≈0.6776. (a) p1=0,p2=0,andq=0, without taking any pest-management strategy; (b) p1=0.85,p2=0.55,andq=0, with spraying pesticide (or harvesting) only; (c) p1=0,p2=0,andq=0.2, with releasing of predator (natural enemy) only; (d) p1=0.85,p2=0.55,andq=0.2, with taking integrated pest-management strategy.
(a)
(b)
(c)
(d)According to Theorem12, if the impulsive periodic T>Tmax and is close to Tmax, the prey eradication solution becomes unstable, there is a supercritical bifurcation, then the prey and predator can coexist on a stable positive periodic solution when T=0.68>Tmax≈0.6776 (Figure 2). Therefore, in order to control the pest populations, we would choose an appropriate impulsive periodic T. T>Tmax and close to Tmax would be a better one.Time series of system (11) when T=0.68>Tmax≈0.6776; prey population x and predator population y coexist with periodic oscillations.
(a)
(b)Letq=0.55 and fix other parameter sets of values; we have displayed bifurcation diagrams for the pest population x and the predator population y for impulsive period T over [1,11] and [6,11]. We find that by increasing the impulsive period T, system (11) undergoes a process of period-doubling cascade → chaos → crisis and high-order periodic oscillations (Figure 3). When T increases from 6 to 7, there is a cascade of period-doubling bifurcations leading to chaos (Figure 4). When T=8.62,  the chaos suddenly disappears and a T-periodic solution  appears, then the T-periodic solution abruptly disappears and the chaos abruptly appears again when T=9.08, these constituting several types of crises (Figure 5). However, when T=8.62 and T=9.08, it appears that attractors are nonunique, coexistence of stranger attractor with T-periodic solution (Figure 6). Obviously, which one of the attractors is reached depends on the initial values.Bifurcation diagrams of system (11): (a) prey population x and predator population y with impulsive period T over [1,11]; (b) prey population x and predator population y with impulsive period T over [6,11].
(a)
(b)Period doubling bifurcation leads to chaos:T,2T,and4T periodic solutions and chaos when T=6.4,6.6,6.8,and7.0, respectively.
(a)
(b)
(c)
(d)Crises are shown. There is a crisis that the chaos suddenly disappears whenT=8.61,8.62, and there is a crisis that the chaos suddenly appears when T=9.07,9.08.
(a)
(b)
(c)
(d)Different attractors’ coexistence: solution with initial valueX(0)=(0.5,2.5) will finally tend to strange attractor, solution with X(0)=(0.6,2.5) will finally tend to T periodic solution when  T=8.62,  solution with initial value X(0)=(1.1,2.5) will finally tend to strange attractor, and solution with X(0)=(1.2,2.5) will finally tend to T periodic solution when  T=9.08, respectively.
(a)
(b)
(c)
(d)Example 14.
Leta=3.1,b=1.5,α=1.05,k=0.85,n=2.15,h=0.97,d=0.3,p1=0.85,p2=0.55, andT=8 with initial value X(0)=(0.5,0.5). We investigate the effect of q on the system (11). Figure 7 showed bifurcation diagrams obtained by stroboscopically sampling the pest population x and the predator population y for q over [0.1,3.1]. The resulting bifurcation diagrams clearly showed that system (11) has rich dynamics, including period-doubling bifurcation, period-halving bifurcation, and chaos. When q increases from 0.9 to 2.2, there is a period-halving bifurcation leading to a T-periodic solution (Figure 8).Bifurcation diagrams of system (11): prey population x and predator population y with q over  [0.1,3.1].
(a)
(b)Period-halving bifurcation leads to aT-periodic solution of system (11): chaos and 16T,8T,4T,2T,andT periodic solutions when  q=0.9,0.95,0.98,1.0,1.1,and2.2, respectively.
(a)
(b)
(c)
(d)
(e)
(f)Example 15.
Leta=3.1,b=1.5,α=1.05,k=0.85,n=2.15,h=0.97,d=0.3,p1=0.85,p2=0.55,q=1.2,andT=8 with initial value X(0)=(0.5,0.5). We consider the effect of β on the system (11). The resulting bifurcation diagrams (Figure 9), the pest population x, and the predator population y for β over [2.0,6.0] clearly showed that system (11) has complex dynamics, such as period-doubling bifurcation, high-order periodic oscillation, and chaos. In Figure 10, the typical high-order oscillation of system (11) is shown: 7T,12T,17T,and3T periodic solutions when β=2.8,3.05,3.4,and5.5, respectively. Further, Figure 11 shown the maximin and mean amount of prey population x and predator population y of system (11) with β over [2.0,5.5].Bifurcation diagrams of system (11): prey population x and predator population y with β over [2.0,6.0].
(a)
(b)High-order oscillations of system (11): 7T,12T,17T,and3T periodic solutions when β=2.8,3.05,3.4,and5.5, respectively.
(a)
(b)
(c)
(d)The maximin and mean amount of prey populationx and predator population y of system (11) with β over [2.0,5.5], respectively.
(a)
(b)From bifurcation diagrams in  Figures3, 7, and 9, we can easily see that the dynamical behavior of these three cases is very complicated, which includes (1) high-order quasi-periodic and periodic oscillations, (2) period-doubling bifurcation, (3) period-halving bifurcations, (4) nonunique dynamics (meaning that several attractors coexist), and (5) cries (the phenomenon of “crisis” in chaotic attractors can suddenly appear or disappear, or change size discontinuously as a parameter smoothly varies).


## 6. Conclusion

In this paper, we have investigated a predator-prey system with generalized group defense and concerning impulsive control strategy for pest control in detail. We have shown that there exists an asymptotically stable pest-eradication periodic solution if the impulsive period is less than the critical valueTmax. If we choose our impulsive control strategy, in order to drive the pest to extinction, we can determine the impulsive period T according to the effect of the chemical pesticides on the populations and the cost of releasing natural enemies such that T<Tmax.But, in a real world, complete eradication of pest populations is generally not possible, nor is it biologically or economically desirable. A good-pest control program should reduce pest population to levels acceptable to the public. WhenT>Tmax, the stability of the pest-eradication periodic solution is lost, system (11) is permanent, and there exists a nontrivial periodic solution when T is  close to Tmax. The smaller the period, the fewer the pests. Therefore, we can control the pest population below some economic threshold (ET is defined as the pest population level that produces damage equal to the costs of preventing damage) by choosing appropriate impulsive period T and the number of mature predator released q, according to the degree of antipredator behavior and group defense β, making an integrated pest-management strategy every period T. Then, the periodic releasing of natural enemies and spraying pesticides change the properties of the system without impulses and our results suggest an effective approach in the pest control.Numerical results show that system (11) can take on various kinds of periodic fluctuations and several types of attractor coexistence and is dominated by high-order periodic oscillations,  quasi-periodic oscillations, and chaotic oscillations. These results imply that the presence of pulses destroys equilibria, initiates multiple attractors, quasi-periodic oscillations, and chaos, and makes the dynamical behaviors more complex.



---

*Source: 358930-2013-06-19.xml*