# Dynamics of a Beddington-DeAngelis Type Predator-Prey Model with Impulsive Effect

Authors: Sun Shulin; Guo Cuihua
Journal: Journal of Mathematics
 (2013)

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## Abstract

In view of the logical consistence, the model of a two-prey one-predator system with Beddington-DeAngelis functional response and impulsive control strategies is formulated and studied systematically. By using the Floquet theory of impulsive equation, small amplitude perturbation method, and comparison technique, we obtain the conditions which guarantee the global asymptotic stability of the two-prey eradication periodic solution. We also proved that the system is permanent under some conditions. Numerical simulations find that the system appears the phenomenon of competition exclusion.


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## Body

## 1. Introduction

In an ecological system, understanding the dynamical relationships between predator and prey is one of the central goals. One important component of the predator-prey relationships is the predator's functional response. Functional response refers to the change in the density of prey attached per unit time per predator as the prey density changes. In 1965, Holling [1] gave three different kinds of functional response for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more realistic. The Beddington-DeAngelis functional response was introduced by Beddington [2] and DeAngelis et al. [3] in 1975. It is similar to the Holling type functional response but contains an extra term describing mutual interference by predators. Thus, a predator-prey model with Beddington-DeAngelis functional response is as the following form [2, 3]:
(1)dx(t)dt=ax(t)(1-x(t)K)-ex(t)y(t)by(t)+x(t)+c,dy(t)dt=-Dy(t)+mex(t)y(t)by(t)+x(t)+c,
where x(t) and y(t) represent the population density of the prey and the predator at time t, respectively. Usually, K is called the carrying capacity of the prey. The constant a is called intrinsic growth rate of the prey. m is the conversion rate and D is the death rate of the predator. The term by(t) measures the mutual interference between predators. In 2004, Fan and Kuang [4] explored the dynamics of the nonautonomous, spatially homogeneous, and continuous time predator-prey system with the Beddington-DeAngelis functional response. The explorations involved the permanence, extinction, and global asymptotic stability (general nonautonomous case); the existence, uniqueness, and stability of a positive (almost) periodic solution; and a boundary (almost) periodic solution for the periodic (almost periodic) case.It is straightforward to generalize the above two-species model to the situation when two prey species competing for the same resource but the two prey species are predated by one predator species. This results in the following three-species model:(2)dx(t)dt=a1x(t)(1-x(t)+y(t)K)-e1x(t)z(t)b1z(t)+x(t)+c1,dy(t)dt=a2y(t)(1-x(t)+y(t)K)-e2y(t)z(t)b2z(t)+y(t)+c2,dz(t)dt=-Dz(t)+m1e1x(t)z(t)b1z(t)+x(t)+c1+m2e2y(t)z(t)b2z(t)+y(t)+c2.Ifx and y are identical species, then a1=a2=a, e1=e2=e, b1=b2=b, c1=c2=c, and m1=m2=m. Let N=x+y; then, the model that is consistent with (2) should take the form
(3)dN(t)dt=aN(t)(1-N(t)K)-eN(t)z(t)bz(t)+N(t)+c,dz(t)dt=-Dz(t)+meN(t)z(t)bz(t)+N(t)+c.If model (2) is logically sound one, it should reduce itself to (3) in this situation. However, the result is not the case; instead, it becomes
(4)dN(t)dt=aN(t)(1-N(t)K)-ex(t)z(t)bz(t)+x(t)+c-ey(t)z(t)bz(t)+y(t)+c,dz(t)dt=-Dz(t)+mex(t)z(t)bz(t)+x(t)+c+mey(t)z(t)bz(t)+y(t)+c.According to the logical consistence proposed by Arditi and Michalski in 1996 [5], a logically consistent model of the two-prey one-predator interaction should take the form
(5)dx(t)dt=a1x(t)(1-x(t)+y(t)K)-e1x(t)z(t)b1z(t)+x(t)+y(t)+c1,dy(t)dt=a2y(t)(1-x(t)+y(t)K)-e2y(t)z(t)b2z(t)+x(t)+y(t)+c2,dz(t)dt=-Dz(t)+m1e1x(t)z(t)b1z(t)+x(t)+y(t)+c1+m2e2y(t)z(t)b2z(t)+x(t)+y(t)+c2.There are a number of factors in the environment to be considered in predator-prey models. One of the important factors is an impulsive perturbation such as fire and flood; these are not suitable to be considered continuously. The impulsive perturbations bring sudden changes to the system. Let us think of prey as a pest and predator as a natural enemy of prey. There are many ways to beat agricultural pests, for example, biological control such as harvesting on prey and releasing natural enemies [6–11] and chemical control such as spreading pesticides. However, integrated pest management (IPM) is a more effective approach to control pests in farm, which has been proved by experiments (see [12–14]). Such tactics are discontinuous and periodical. With the idea of periodic forcing and impulsive perturbations, Baek [15] has investigated the predator-prey model with periodic constant impulsive immigration of the predator and periodic variation in the intrinsic growth rate of the prey. And an ecological model consisting of two preys and one predator with impulsive release of the predator is studied in [16]. In this paper, we will consider an ecological model consisting of two preys and one predator with impulsive control strategy. We give some assumptions as follows.(H1)
Two prey species compete the same resource.(H2)
Functional response is effected for mutual interference by predators.Thus, the model can be described by the following impulsive differential equations:(6)dx(t)dt=a1x(t)(1-x(t)+y(t)K)-e1x(t)z(t)b1z(t)+x(t)+y(t)+c1,dy(t)dt=a2y(t)(1-x(t)+y(t)K)-e2y(t)z(t)b2z(t)+x(t)+y(t)+c2,dz(t)dt=-Dz(t)+m1e1x(t)z(t)b1z(t)+x(t)+y(t)+c1+m2e2y(t)z(t)b2z(t)+x(t)+y(t)+c2,}t≠nT,x(t+)=(1-p1)x(t),y(t+)=(1-p2)y(t),z(t+)=(1-p3)z(t)+q,}t=nT,(x(0+),y(0+),z(0+))=(x0,y0,z0)=X0,wherex(t), y(t), and z(t) are the densities of the two preys and a predator at time t, respectively. K is the total carrying capacity of two prey species; ai(i=1,2) are the intrinsic growth rate; ei(i=1,2) are the catching rate; bi(i=1,2) scale the impact of predators mutual interference; ci(i=1,2) are saturation constants; mi(i=1,2) denote the efficiency with which resources are converted to new consumers; D is the mortality rate of the predator. T is the period of the impulsive effect; n∈N;N is the set of all nonnegative integers; pi(i=1,2,3) represent the fractions of prey and predator which die due to pesticides; q is the release amount of the predator at t=nT.The paper is arranged as follows. In Section2, we give some notations and lemmas. In Section 3, we consider the local stability and global asymptotic stability of the two-prey eradication periodic solution by using Floquet theory of the impulsive equation, small amplitude perturbation, and techniques of comparison, and in Section 4 we show that the system is permanent. The paper ends with some interesting numerical simulations and conclusions.


## 2. Preliminaries

Firstly, we give some notations, definitions, and lemmas which will be useful for our main results.LetR+=[0,+∞), R+3={X=(x,y,z)∈R3∣x⩾0,y⩾0,z⩾0}. Denoted by f=(f1,f2,f3)T the map defined by the right hand of the former three equations in system (6) and denoted by N the set of all nonnegative integers. Let V:R+×R+3→R+; then, V is said to belong to class V0 if(1)
V is continuous in (nT,(n+1)T]×R+3 and, for each X∈R+3,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+3, the upper right derivative of V(t,X) with respect to the impulsive differential system (6) is defined as
(7)D+V(t,X)=limh→0+sup1h×[V(t+h,X+hf(t,X))-V(t,X)].The solution of system (6) is a piecewise continuous function X:R+→R+3, 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 solution of system (6) (see [17] for the details).Definition 2.
System (6) is said to be permanent if there exists a compact Ω⊂IntR+3 such that every solution (x(t),y(t),z(t)) of system (6) will eventually enter and remain in the region Ω.
The following lemma is obvious.Lemma 3.
LetX(t) be a solution of system (6) with X(0+)⩾0; then, X(t)⩾0 for all t⩾0. And, furthermore, X(t)>0,t>0 if X(0+)>0.We will use an important comparison theorem on impulsive differential equation.Lemma 4 (comparison theorem [17]).
SupposeV∈V0. Assume that
(8)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,v)→(nT+,u)g(t,v)=g(nT+,u) exists and ψn:R+→R+ is nondecreasing. Let r(t) be the maximal solution of the scalar impulsive differential equation
(9)du(t)dt=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) for all t⩾0, where X(t) is any solution of system (6).To research the stability of the prey-free periodic solution of (6), we present the Floquet theory for the linear T-periodic impulsive equations:
(10)dX(t)dt=A(t)X(t),t≠τk,t∈R,X(t+)=(E+Bk)X(t),t=τk,k∈ℤ.
Then, we introduce the following conditions.(H2.1)
A(·)∈PC(R,Cn×n) and A(t+T)=A(t), (t∈R), where PC(R,Cn×n) is the set of all piecewise continuous matrix functions which is left continuous at t=τk and Cn×n is the set of all n×n matrices.(H2.2)
Bk∈Cn×n, det(E+Bk)≠0, and τk<τk+1.(H2.3)
There exists aq∈ℕ such that Bk+q=Bk,τk+q=τk+T, and k∈ℤ.LetΦ(t) be a fundamental matrix of (10); then, there exists a unique nonsingular matrix M∈Cn×n such that
(11)Φ(t+T)=Φ(t)M,(t∈R).For this equality, we call the constant matrixM is the monodromy matrix of (10) (corresponding to the fundamental matrix Φ(t))All monodromy matrices of (10) are similar and have the same eigenvalues. The eigenvalues μ1,…,μn of the monodromy matrices are called the Floquet multipliers of (10).Lemma 5 (Floquet theory [18]).
Let conditions (H2.1)–(H2.3) hold. Then, the linearT-periodic impulsive equation (10) is(1)
stable if and only if all multipliersμj(j=1,…,n) of (10) satisfy the inequality |μj|≤1, and, moreover, to those μj, for which |μj|=1, correspond simple elementary divisors;(2)
asymptotically stable if and only if all multipliersμj(j=1,…,n) of (10) satisfy the inequality |μj|<1;(3)
unstable if|μj|>1 for some j=1,…,n.
Now, one gives some basic properties about the following subsystem of system (6):
(12)dz(t)dt=-Dz(t),t≠nT,z(t+)=(1-p3)z(t)+q,t=nT,z(0+)=z0.Then we can easily obtain the following results.Lemma 6.
(1)z*(t)=qexp(-D(t-nT))/(1-(1-p3)exp(-DT)),t∈(nT,(n+1)T], n∈N, and z*(0+)=q/(1-(1-p3)exp(-DT)) is a positive periodic solution of (12).
(2)z(t)=(1-p3)n(z0-(q/(1-(1-p3)exp(-DT))))exp(-Dt)+z*(t) is the solution of (12) with z0⩾0, t∈(nT,(n+1)T], and n∈N.
(3) All solutionsz(t) of (12) with z0⩾0 tend to z*(t). That is, |z(t)-z*(t)|→0 as t→∞.Therefore, we obtain the complete expression for the two-prey eradication periodic solution(0,0,z*(t)) of system (6). We will study the stability of the two-prey eradication periodic solution of system (6) in the next section.


## 3. Prey Eradication Periodic Solution

Firstly, we show that all solutions of (6) are uniformly ultimately bounded.Theorem 7.
There exists a constantM>0, such that x(t)⩽M, y(t)⩽M, z(t)⩽M for all t large enough, where (x(t),y(t),z(t)) is a solution of system (6).Proof.
DefineV(t,X(t)) such that
(13)V(t,X(t))=m1x(t)+m2y(t)+z(t);
then, V∈V0. We calculate the upper right derivative of V(t,X) along a solution of system (6) and get the following impulsive differential inequality:
(14)D+V(t)+LV(t)⩽(a1+L)m1x(t)-a1m1Kx2(t)+(a2+L)m2y(t)-a2m2Ky2(t)+(L-D)z(t),t≠nT,V(t+)⩽V(t)+q,t=nT.
Let0<L<D; then, D+V(t)+LV(t) is bounded. Select L1 and L2 such that
(15)D+V(t)≤-L1V(t)+L2,t≠nT,V(t+)≤V(t)+q,t=nT,
where L1 and L2 are two positive constants.
According to Lemma4, we have
(16)V(t)⩽(V(0+)-L2L1)exp(-L1t)+q(1-exp(-nL1T))1-exp(-L1T)exp(-L1(t-nT))+L2L1,
where t∈(nT,(n+1)T]. Therefore, V(t,X(t)) is ultimately bounded by a constant and there exists a constant M>0, such that x(t)⩽M, y(t)⩽M, z(t)⩽M for all t large enough. The proof is completed.Now, we study the stability of the two-prey eradication periodic solution of system (6).Theorem 8.
Let(x(t),y(t),z(t)) be any solution of system (6); then, (0,0,z*(t)) is said to be locally asymptotically stable if
(17)a1T+e1Db1ln[1-b1q(1-exp(-DT))c1(1-(1-p3)exp(-DT))+b1q]<ln11-p1,a2T+e2Db2ln[1-b2q(1-exp(-DT))c2(1-(1-p3)exp(-DT))+b2q]<ln11-p2.Proof.
The local stability of periodic solution(0,0,z*(t)) may be determined by considering the behavior of small amplitude perturbation of the solution. Define
(18)x(t)=u(t),y(t)=v(t),z(t)=w(t)+z*(t).
Putting (18) into (6), the linearization of the system becomes
(19)du(t)dt=(a1-e1z*(t)b1z*(t)+c1)u(t),dv(t)dt=(a2-e2z*(t)b2z*(t)+c2)v(t),dw(t)dt=m1e1z*(t)b1z*(t)+c1u(t)+m2e2z*(t)b2z*(t)+c2v(t)-Dw(t),}t≠nT,u(t+)=(1-p1)u(t),v(t+)=(1-p2)v(t),w(t+)=(1-p3)w(t),}t=nT.
Therefore, we have(20)(u(t)v(t)w(t))=Φ(t)(u(0)v(0)w(0)),0⩽t<T,
where Φ(t) satisfies
(21)dΦdt=(a1-e1z*(t)b1z*(t)+c1000a2-e2z*(t)b2z*(t)+c20m1e1z*(t)b1z*(t)+c1m2e2z*(t)b2z*(t)+c2-D)Φ(t)
Φ
(
0
)
=
I, the identity matrix, and
(22)(u(nT+)v(nT+)w(nT+))=(1-p10001-p20001-p3)(u(nT)v(nT)w(nT)).
The stability of the periodic solution(0,0,z*(t)) is determined by the eigenvalues of
(23)Θ=(1-p10001-p20001-p3)Φ(T).
If absolute values of all eigenvalues are less than one, then the periodic solution(0,0,z*(t)) is locally asymptotically stable. All eigenvalues of Θ are as follows:
(24)μ1=(1-p1)exp(∫0T‍(a1-e1z*(t)b1z*(t)+c1)dt),μ2=(1-p2)exp(∫0T‍(a2-e2z*(t)b2z*(t)+c2)dt),μ3=(1-p3)exp(-DT)<1.
According to the Floquet theory of impulsive differential equation,(0,0,z*(t)) is locally asymptotically stable if |μ1|<1 and |μ2|<1; that is to say,(25)a1T+e1Db1ln[1-b1q(1-exp(-DT))c1(1-(1-p3)exp(-DT))+b1q]<ln11-p1,a2T+e2Db2ln[1-b2q(1-exp(-DT))c2(1-(1-p3)exp(-DT))+b2q]<ln11-p2.
This completes the proof.Theorem 9.
Let(x(t),y(t),z(t)) be any solution of system (6); then, (0,0,z*(t)) is said to be globally asymptotically stable if the conditions of Theorem 8 hold and
(26)q⩾max{a1e1(exp(DT)-(1-p3))(b1M+2M+c1),hhhhhhhha2e2(exp(DT)-(1-p3))(b2M+2M+c2)a1e1}.Proof.
By Theorem8, we know that (0,0,z*(t)) is locally asymptotically stable. In the following, we will prove its global attraction. Let V(t)=x(t)+y(t); then, we get
(27)V′∣(1.6)=a1x(t)-a1Kx(t)(x(t)+y(t))-e1x(t)z(t)b1z(t)+x(t)+y(t)+c1+a2y(t)-a2Ky(t)(x(t)+y(t))-e2y(t)z(t)b2z(t)+x(t)+y(t)+c2.
By Theorem 7, there exists a constant M>0, such that x(t)⩽M, y(t)⩽M, and z(t)⩽M for each solution X(t)=(x(t),y(t),z(t)) of system (6) with all t large enough.
Thus,(28)V′∣(1.6)⩽a1x(t)-e1x(t)z(t)b1M+2M+c1+a2y(t)-e2y(t)z(t)b2M+2M+c2.
On the other hand, we have(29)dz(t)dt=m1e1x(t)z(t)b1z(t)+x(t)+y(t)+c1+m2e2y(t)z(t)b2z(t)+x(t)+y(t)+c2-Dz(t)⩾-Dz(t),t≠nT,z(t+)=(1-p3)z(t)+q,t=nT.
By Lemmas 4 and 5, we know that there exists a t1>0, and ϵ>0 small enough, such that z(t)⩾z*(t)-ϵ; for all t⩾t1, we have
(30)z(t)⩾qexp(-DT)1-(1-p3)exp(-DT)-ϵ≜γ.
Then,
(31)V′∣(6)⩽(a1-e1γb1M+2M+c1)x(t)+(a2-e2γb2M+2M+c2)y(t),
if Q1≜a1-(e1γ/(b1M+2M+c1))<0 and Q2≜a2-(e2γ/(b2M+2M+c2))<0; that is to say,
(32)q⩾max{a1e1(exp(DT)-(1-p3))(b1M+2M+c1),hhhhhha2e2(exp(DT)-(1-p3))(b2M+2M+c2)a1e1}.
Then, for t⩾t1, we have
(33)V′∣(6)⩽-QV(t),
where Q=min{|Q1|,|Q2|}, so V(t)→0 and x(t)→0,y(t)→0 as t→∞. Notice that the limit system of the system (6) is exactly as system (10); together with Lemma 5, we know that the two-prey eradication periodic solution (0,0,z*(t)) is a global attractor. The proof is completed.


## 4. Permanence

In this section, we investigate the permanence of the system (6).Theorem 10.
The system (6) is permanent if
(34)a1T+e1Db1ln[1-b1q(1-exp(-DT))c1(1-(1-p3)exp(-DT))+b1q]>ln11-p1,a2T+e2Db2ln[1-b2q(1-exp(-DT))c2(1-(1-p3)exp(-DT))+b2q]>ln11-p2,q≤max{a1e1(exp(DT)-(1-p3))(b1M+2M+c1),hhhhha2e2(exp(DT)-(1-p3))(b2M+2M+c2)a1e1}.Proof.
LetX(t)=(x(t),y(t),z(t)) be any solution of the system (6) with X(0)>0. From Theorem 7, we assume that x(t)⩽M, y(t)⩽M, z(t)⩽M, and M>max{a1c1/e1,a2c2/e2}. Let ζ3=(qexp(-DT))/(1-(1-p3)exp(-DT))-ϵ,ϵ>0. From Lemma 5, clearly, we have z(t)⩾ζ3 for all t large enough. Now, we will find ζ1¯ and ζ2¯ such that x(t)⩾ζ1¯ and y(t)⩾ζ2¯ for t large enough. We will do this in the following two steps.
Step  1. We will prove that there exists a t1∈(0,+∞) such that x(t1)⩾ζ1. Otherwise, x(t)<ζ1 for all t>0.
We can chooseζ1>0,ϵ1>0 to be small enough such that
(35)δ=e1a1ζ1b1+e2a2Mb2<D,η=(1-p1)exp(∫nT(n+1)T‍(a1(1-2MK)hhhhhhhhhhhhhhhh-e1(v*(t)+ϵ1)c1)dt)>1.
From system (6), we can obtain that
(36)dz(t)dt⩽(-D+δ)z(t),t≠nT,z(t+)=(1-p3)z(t)+q,t=nT,z(0+)=z0.
By Lemma 4, we have z(t)⩽v(t) and v(t)→v*(t) as t→+∞, where v(t) is the solution of
(37)dv(t)dt=(-D+δ)v(t),t≠nT,v(t+)=(1-p3)v(t)+q,t=nT,v(0+)=z0,
and v*(t)=qexp((-D+δ)(t-nT))/(1-(1-p3)exp((-D+δ)T)), t∈(nT,(n+1)T], and n∈N. Therefore, there exists a T1>0; when t>T1, we have z(t)⩽v(t)<v*(t)+ϵ1 and
(38)dx(t)dt⩾x(t)(a1(1-2MK)-e1(v*(t)+ϵ1)c1),t≠nT,x(t+)=(1-p1)x(t),t=nT.
Let N1∈N and N1T⩾T1; integrating (38) on (nT,(n+1)T], n⩾N1, we can obtain
(39)x((n+1)T)⩾x(nT+)×exp(∫nT(n+1)T‍(e1(v*(t)+ϵ1)c1a1(1-2MK)hhhhhhhhhhhhhhhhhh-e1(v*(t)+ϵ1)c1)dt)=x(nT)(1-p1)×exp(∫nT(n+1)T‍(e1(v*(t)+ϵ1)c1a1(1-2MK)hhhhhhhhhhhhhhhhhh-e1(v*(t)+ϵ1)c1)dt)=x(nT)η.
Then,x((N1+k)T)⩾x(N1T)ηk→∞ as k→∞, which is a contradiction to the boundedness of x(t). Hence, there exists a t1>0 such that x(t1)⩾ζ1.
Step  2. If x(t)⩾ζ1, for all t⩾t1, then our aim is obtained. Hence, we only need to consider those solutions which leave the region Ω0={x(t)∣x(t)<ζ1} and reenter it again. Letting t*=inft⩾t1{x(t)<ζ1}, we have x(t)⩾ζ1 for t∈[t1,t*]; by the continuity of x(t), we have x(t*)=ζ1. In this step, we have only to consider two possible cases.
Case  1.
t
*
=
n
1
T for some n1∈N; then, (1-p1)ζ1⩽x(t*+)=(1-p1)x(t*)<ζ1. Select n2,n3∈N such that
(40)n2T>ln(ϵ1/(M+q))-D+δ,(1-p1)n2exp(n2σT)ηn3>(1-p1)n2+1exp((n2+1)σT)ηn3>1,
where σ=a1(1-(2M/K))-(e1M/c1)<0. Let T′=n2T+n3T. In this case, we will show that there exists t2∈(t*,t*+T′] such that x(t2)⩾ζ1; otherwise, x(t)<ζ1 for t∈(t*,t*+T′]. Considering (37) with v(t*+)=z(t*+), we have
(41)v(t)=(1-p3)n-(n1+1)×(v(t*+)-q1-(1-p3)exp((-D+δ)T))×exp((-D+δ)(t-t*))+v*(t)
for (n-1)T<t⩽nT and n1+1⩽n⩽n1+1+n2+n3; then,
(42)|v(t)-v*(t)|<(M+q)exp((-D+δ)(t-t*))<(M+q)exp((-D+δ)n2T)<ϵ1,z(t)⩽v(t)⩽v*(t)+ϵ1
for t*+n2T⩽t⩽t*+T′, which implies that (38) holds for t∈[t*+n2T,t*+T′]. As in Step 1, we have
(43)x(t*+T′)⩾x(t*+n2T)ηn3.
Since z(t)⩽M, we have
(44)dx(t)dt⩾x(t)(a1(1-2MK)-e1Mc1)=σx(t),t≠nT,x(t+)=(1-p1)x(t),t=nT,
for t∈[t*,t*+n2T]. Integrating (44) on [t*,t*+n2T], we have
(45)x(t*+n2T)⩾ζ1(1-p2)n2exp(σn2T).
Thus,x(t*+T′)⩾ζ1(1-p2)n2exp(σn2T)ηn3>ζ1, which bring on a contradiction. Now, let t¯=inft⩾t*{x(t)⩾ζ1}. Then, x(t)⩽ζ1 for t∈[t*,t¯) and x(t¯)=ζ1. For t∈[t*,t¯), suppose that t∈(t*+(k-1)T,t*+kT], k∈N, and k⩽n2+n3; from (44), we obtain
(46)x(t)⩾x(t*+)(1-p1)k-1exp((k-1)σT)×exp(σ(t-(t*+(k-1)T)))⩾ζ1(1-p1)kexp(kσT)⩾ζ1(1-p1)n2+n3exp(σ(n2+n3)T)≡ζ1′.
Case  2. t*≠nT, n∈N. Then, x(t)⩾ζ1 for t∈[t1,t*) and x(t*)=ζ1. Suppose that t*∈(n1′T,(n1′+1)T), n1′∈N. There are two possible cases for t∈(t*,(n1′+1)T].
(a)x(t)<ζ1 for all t∈(t*,(n1′+1)T]. In this case, we will show that there exists t2∈[(n1′+1)T,(n1′+1)T+T′] such that x(t2)⩾ζ1. Otherwise, x(t)<ζ1 for all t∈[(n1′+1)T,(n1′+1)T+T′]. Then, x(t)<ζ1 for all (t*,(n1′+1)T+T′]; considering (37) with v((n1′+1)T+)=z((n1′+1)T+), we have
(47)v(t)=(1-p3)n-(n1′+1)(q1-(1-p3)exp((-D+δ)T)v((n1′+1)T+)hhhhhhhhhhhhhhh-q1-(1-p3)exp((-D+δ)T))×exp((-D+δ)(t-(n1′+1)T))+v*(t)
for nT<t⩽(n+1)T and n1′+1⩽n⩽n1′+1+n2+n3. So we get
(48)|v(t)-v*(t)|<(M+q)×exp((-D+δ)(t-(n1′+1)T))<(M+q)exp((-D+δ)n2T)<ϵ1,z(t)⩽v(t)⩽v*(t)+ϵ1
for (n1′+1+n2)T⩽t⩽(n1′+1+n2+n3)T, which implies that (38) holds for t∈[(n1′+1+n2)T,(n1′+1+n2+n3)T]. As in Step 1, we have
(49)x((n1′+1+n2+n3)T)⩾x((n1′+1+n2)T)ηn3.
Integrate (44) on (t*,(n1′+1+n2)T), which yields
(50)x((n1′+1+n2)T)⩾ζ1(1-p1)n2+1exp(σ(n2+1)T).
Thus,x((n1′+1+n2+n3)T)⩾ζ1(1-p1)n2+1exp(σ(n2+1)T)ηn3>ζ1, which is a contradiction. Let t¯=inft⩾t*{x(t)⩾ζ1}. Then, x(t)⩽ζ1 for t∈[t*,t¯) and x(t¯)=ζ1. For t∈[t*,t¯), suppose that t∈(n1′T+(k′-1)T,n1′T+k′T], k′∈N, and k′⩽1+n2+n3; we have x(t)⩾ζ1(1-p1)n2+n3+1exp(σ(n2+n3+1)T)≡ζ¯. Since ζ′>ζ¯, so x(t)⩾ζ¯ for all t∈(t*,t¯).
(b) There existst3∈(t*,(n1′+1)T] such that x(t3)⩾ζ1. Let t^=inft⩾t*{x(t)⩾ζ1}; then, x(t)⩽ζ1 for all t∈[t*,t^) and x(t^)=ζ1. Also, (44) holds for t∈[t*,t^). Integrating the equation on [t*,t)(t*⩽t⩽t^), we can get that x(t)⩾x(t*)exp(σ(t-t*))⩾ζ1exp(σT)⩾ζ¯. Thus, in both cases, we conclude x(t)⩾ζ¯ for all t⩾t1.
Similarly, we can provey(t)⩾ζ2¯ for all t⩾t2.
SetΩ={(x(t),y(t),z(t))∣x(t)⩾ζ1¯,y(t)⩾ζ2¯,z(t)⩾ζ3}. Obviously, we know that the set Ω∈IntR+3 is a global attractor; every solution of system (6) will eventually enter and remain in region Ω. Therefore, system (6) is permanent. The proof is completed.


## 5. Numerical Simulations and Conclusions

To study the dynamics of an ecological model consisting of two preys and one predator with impulsive control strategy, the solution of the system (6) with initial date (x0,y0,z0)=(0.8,0.6,1) in the positive cone is obtained numerically for biologically feasible range of parametric value. In order to verify our theoretical results, let a1=0.8, a2=0.6, K=20, e1=0.844, e2=0.647, m1=0.25, m2=0.2, b1=1.1, b2=1.13, c1=0.06, c2=0.05, D=0.027, p1=0.3, p2=0.25, p3=0.01 and T=10. On the other hand, it follows from Theorem 7 that we can select a positive number M. Thus, by a straightforward computation, we can obtain the two critical values q*=65.83 and q*=3.98. When q=66>q*, it satisfies the condition of Theorem 9, so the prey eradication periodic solution (0,0,z*(t)) is globally asymptotically stable, A type prey eradication periodic solution of the system (6) is shown in Figure 1, where we may observe how the variable z(t) oscillates in a stable cycle. In contrast, the prey x(t) and the prey y(t) rapidly decrease to zero. When q=3.9<q*, it satisfies the condition of Theorem 10, so the system (6) is permanent. That is to say, the preys and the predator can coexist (see Figure 2). Furthermore, we can give the other two critical values q**=5.31 and q**=65.03. When q**<q<q**, it does not satisfy the conditions of Theorems 9 and 10, but numerical simulations illustrate that the system (6) appears to be the principle of competition exclusion. That is to say, the competition between two prey populations give rise to the result that one prey becomes extinct and the other prey population is persisted. For example, letting q=6, the result is as in Figure 3; letting q=60, the result is in as Figure 4.Time series of a solution of model (6) with q=3.9: (a) time series of the prey population x(t); (b) time series of the prey population y(t); (c) time series of the predator population z(t).
(a)
(b)
(c)Time series of a solution of model (6) with q=66: (a) time series of the prey population x(t); (b) time series of the prey population y(t); (c) time series of the predator population z(t).
(a)
(b)
(c)Time series of a solution of model (6) with q=6: (a) time series of the prey population x(t); (b) time series of the prey population y(t); (c) time series of the predator population z(t).
(a)
(b)
(c)Time series of a solution of model (6) with q=60: (a) time series of the prey population x(t); (b) time series of the prey population y(t); (c) time series of the predator population z(t).
(a)
(b)
(c)In this paper, we have investigated effects of impulsive perturbations on a predator-prey model consisting of two preys and one predator with Beddington-DeAngelis functional response. By using the Floquet theorem and small amplitude perturbation skills, we have proved that the periodic solution(0,0,z*(t)) is globally asymptotically stable when the values of impulsive perturbation q are greater than the critical value q*. In addition, it has been shown that model (6) is permanent when the values of impulsive perturbation q are less than the critical value q*. By numerical simulations, we have found that the system (6) appears to be the principle of competition exclusion, when q**<q<q**. However, our model with the logical consistence is different from the model in [16]; that is, dynamic complexities in [16] does not appear in our model. Thus, our results suggest a new control approach which is more suitable than the classical one in the pest control.



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*Source: 826857-2013-12-24.xml*