# Impulsive Perturbations of a Three-Species Food Chain System with the Beddington-DeAngelis Functional Response

Authors: Younghae Do; Hunki Baek; Dongseok Kim
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

The dynamics of an impulsively controlled three-species food chain system with
the Beddington-DeAngelis functional response are investigated using the Floquet
theory and a comparison method. In the system, three species are prey, mid-predator,
and top-predator. Under an integrated control strategy in sense of biological and
chemical controls, the condition for extinction of the prey and the mid-predator
is investigated. In addition, the condition for extinction of only the mid-predator is
examined. We provide numerical simulations to substantiate the theoretical results.


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## Body

## 1. Introduction

Classical two-species continuous time systems such as a Lotka-Volterra system have been used to investigate the interaction between ecological populations. However, in order to understand a complex ecological system, it is necessary to study multispecies systems. For this reason, in this paper, we study three-species food chain system which appears when a top-predator feeds on a mid-predator, which in turn feeds a prey, specially assuming Beddington-DeAngelis functional responses between species [1].In recent decades, the effects of impulsive perturbations on population systems have been widely studied and discussed by a number of researchers [2–19]. Thus, in order to control an ecological environment, a discrete impulsive strategy has been suggested. Especially for the three-species food chain system, two impulsive control methods, biological and chemical controls, have been taken into account. Here, a biological control means impulsive and periodic releasing of top-predator to control lower-level populations and a chemical control means that as a result of spreading pesticide the population of all three-species are impulsively lessened.In this context, the impulsively controlled three-species food chain system with Beddington-DeAngelis functional responses was proposed and studied by Wang et al. [11] and their system can be described as the following impulsively perturbed system:(1.1)dx(t)dt=x(t)(a-bx(t))-c1x(t)y(t)α1+x(t)+β1y(t),dy(t)dt=k1c1x(t)y(t)α1+x(t)+β1y(t)-c2y(t)z(t)α2+y(t)+β2z(t)-d1y(t)dz(t)dt=k2c2y(t)z(t)α2+y(t)+β2z(t)-d2z(t),,t≠(n+l-1)T,t≠nT,Δx(t)=-δ1x(t),Δy(t)=-δ2y(t)Δz(t)=-δ3z(t),,t=(n+l-1)T,Δx(t)=0,Δy(t)=0,Δz(t)=p,t=nT,(x(0+),y(0+),z(0+))=(x0,y0,z0),
where x(t),y(t), and z(t) are the densities of the lowest-level prey, mid-level predator, and top-predator at time t, respectively. In this system, the prey grows according to a logistic growth with an intrinsic growth rate a and a carrying capacity a/b incorporating the Beddington-DeAngelis functional response. For parameters settings, ki(i=1,2) are the conversion efficiencies, di(i=1,2) are the mortality rates of the mid-level predator and the top-predator, ci(i=1,2) are the maximum numbers of preys that can be eaten by a predator per unit of time, αi(i=1,2) are the saturation constants, and βi(i=1,2) scale the impact of the predator interference. For an impulsive control strategy, top-predators are impulsively released in the periodic fashion of the period T by artificial breeding of species, in a fixed number (p>0) at each time, and by introducing an impulsive catching or poisoning of the prey populations, fixed proportions δ1,δ2,δ3 of the prey, mid-level predator, and top predator are degraded in an impulsive and periodic fashion, with the same period, but at different moments. Here, all parameters except l and δi(i=1,2,3) are positive, 0≤l<1, Δw(t)=w(t+)-w(t) for w∈{x,y,z}, and 0≤δ1,δ2,δ3<1.Although the authors in [11] had introduced the important system (1.1) in a sense of impulsive controlling the food chain system, we find that there are many problems in their theoretical results, where they had showed rich dynamical behaviors in the numerical simulations including a quasiperiodic oscillation, narrow periodic widow, wide periodic window, chaotic band, and period doubling bifurcation, symmetry-breaking pitchfork bifurcation, period-halving bifurcation and crises [20–22].The authors in [11] had argued that (1) the prey and mid-predator free periodic solution (0,0,z*(t)) is always unstable without having any condition and (2) the mid-predator free periodic solution for the system is (a/b,0,z*(t)). But, based on our theoretical computation, the periodic solution (a/b,0,z*(t)) can be found only when δ1=0. It means that under an impulsive control of the population system, the solution (a/b,0,z*(t)) is useless and nonmeaningful. In Section 2, we will give a general form (x*(t),0,z*(t)) of the mid-predator free solution. In the case that the prey is impulsively strong poisoned or caught, there is a possibility that the prey will be eradicated. To examine this possibility, we reinvestigate their system (1.1). Finally, we find out that their theoretical results shown in [11] are wrong. In this paper we thus may correct and rebuild their theoretical results, in particular, the conditions for stabilities of the periodic solutions (0,0,z*(t)) and the new mid-predator free periodic solution (x*(t),0,z*(t)).The main purpose of this paper is to reestablish the local and global stability for two periodic solutions(0,0,z*(t)) and (x*(t),0,z*(t)). In addition, we exhibit some numerical examples. To do it, this paper is organized as follows. In Section 2, we first review notations and theorems. Main theorems for two impulsive periodic solutions are given in Section 3. The mathematical proofs for our main results will be provided in Section 4. Conclusions are presented in Section 5.


## 2. Basic Strategy

In this section we will consider definitions, notations, and auxiliary results for impulsively perturbed dynamical systems.


### 2.1. Preliminaries

Let us denoteℕ by the set of all nonnegative integers, ℝ+=[0,∞),ℝ+*=(0,∞),ℝ+3={x=(x(t),y(t),z(t))∈ℝ3:x(t),y(t),z(t)≥0}, and f=(f1,f2,f3)T the mapping defined by the right-hand sides of the first three equations in (1.1).LetV:ℝ+×ℝ+3→ℝ+, then V is said to belong to class V0 if(1)
V is continuous on ((n-1)T,(n+l-1)T]×ℝ+3∪((n+l-1)T,nT]×ℝ+3 for each x∈ℝ+3,n∈ℕ, and two limits 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;(2)
V is locally Lipschitzian in x.Definition 2.1.
ForV∈V0, one defines the upper right Dini derivative of V with respect to the impulsive differential system (1.1) at (t,x)∈((n-1)T,(n+l-1)T]×ℝ+3∪((n+l-1)T,nT]×ℝ+3 by
(2.1)D+V(t,x)=limsup⁡h→0+1h[V(t+h,x+hf(t,x))-V(t,x)].
We suppose thatg:ℝ+×ℝ+→ℝ satisfies the following hypotheses: (H)g is continuous on ((n-1)T,(n+l-1)T]×ℝ+3∪((n+l-1)T,nT]×ℝ+3 and the limits lim(t,y)→((n+l-1)T+,x)g(t,y)=g((n+l-1)T+,x) and lim(t,y)→(nT+,x)g(t,y)=g(nT+,x) exist and are finite for x∈ℝ+ and n∈ℕ.Lemma 2.2 (see [23]).
SupposeV∈V0 and
(2.2)D+V(t,x)≤g(t,V(t,x)),t≠(n+l-1)T,t≠nT,V(t,x(t+))≤ψn1(V(t,x)),t=(n+l-1)T,V(t,x(t+))≤ψn2(V(t,x)),t=nT,
where g:ℝ+×ℝ+→ℝ satisfies (H) and ψn1,ψn2:ℝ+→ℝ+ are nondecreasing for all n∈ℕ. Let r(t) be the maximal solution for the impulsive Cauchy problem
(2.3)u′(t)=g(t,u(t)),t≠(n+l-1)T,t≠nT,u(t+)=ψn1(u(t)),t=(n+l-1)T,u(t+)=ψn2(u(t)),t=nT,u(0+)=u0,
defined on [0,∞). Then V(0+,x0)≤u0 implies that V(t,x(t))≤r(t),t≥0, where x(t) is any solution of (2.2).A similar result can be obtained when all conditions of the inequalities in the Lemma2.2 are reversed. Using Lemma 2.2, it is easy to prove that the positive octant (ℝ+*)3 is an invariant region for the system (1.1) (see Lemma  2.1 in [11]).


### 2.2. Periodic Solutions

In the case in whichy=0, that is, mid-predator is eradicated, the system (1.1) is decoupled and led to two impulsive differential equations (2.4) and (2.6). Let us consider the properties of these impulsive differential equations. The following equation or a subsystem of system (1.1) is a periodically forced system:(2.4)x′(t)=x(t)(a-bx(t)),t≠(n+l-1)T,t≠nT,x(t+)=(1-δ1)x(t),t=(n+l-1)T,x(t+)=x(t),t=nT,x(0+)=x0.
Straightforward computation for getting a positive periodic solution x*(t) of (2.4) yields the analytic form of x*(t):(2.5)x*(t)=aη1exp⁡(at-λ)b[1-η1+η1exp⁡(at-λ)],(n+l-1)T<t≤(n+l)T,
where λ=a(n+l-1)T and η1=((1-δ1)exp(aT)-1)/(exp(aT)-1).In case thatδ1=0, the system (2.4) is the general logistic equation. From the analytic solution form (2.5) we get that η1 should be 1. It implies that x*(t)=a/b.Lemma 2.3 (see [4]).
The following statements hold.(1)
IfaT+ln(1-δ1)>0, then limt→∞|x(t)-x*(t)|=0 for all solutions x(t) of (2.4) starting with x0>0.(2)
IfaT+ln(1-δ1)≤0, then x(t)→0 as t→∞ for all solutions x(t) of (2.4).Next, we consider the impulsive differential equation as follows:(2.6)z′(t)=-d2z(t),t≠nT,t≠(n+l-1)T,z(t+)=(1-δ3)z(t),t=(n+l-1)T,z(t+)=z(t)+p,t=nT,z(0+)=z0.
The system (2.6) is a periodically forced linear system and its positive periodic solution z*(t) will be obtained:(2.7)z*(t)={pexp⁡[-d2(t-(n-1)T)]1-(1-δ3)exp⁡(-d2T),(n-1)T<t≤(n+l-1)T,p(1-δ3)exp⁡[-d2(t-(n-1)T)]1-(1-δ3)exp⁡(-d2T),(n+l-1)T<t≤nT,(2.8)z*(0+)=z*(nT+)=p1-(1-δ3)exp⁡(-d2T),z*((n+l-1)T+)=p(1-δ3)exp⁡(-d2lT)1-(1-δ3)exp⁡(-d2T).
Moreover, we may get that(2.9)z(t)={(1-δ3)n-1zz+z*(t),(n-1)T<t≤(n+l-1)T,(1-δ3)nzz+z*(t),(n+l-1)T<t≤nT,
is a solution of (2.6), where zz=(z(0+)-(p(1-δ3)e-T/(1-(1-δ3)exp(-d2T))))exp(-d2t).From (2.7) and (2.9), we thus get the following result.Lemma 2.4.
For every solutionz(t) and every positive periodic solution z*(t) of (2.6), it follows that z(t) tends to z*(t) as t→∞.


## 2.1. Preliminaries

Let us denoteℕ by the set of all nonnegative integers, ℝ+=[0,∞),ℝ+*=(0,∞),ℝ+3={x=(x(t),y(t),z(t))∈ℝ3:x(t),y(t),z(t)≥0}, and f=(f1,f2,f3)T the mapping defined by the right-hand sides of the first three equations in (1.1).LetV:ℝ+×ℝ+3→ℝ+, then V is said to belong to class V0 if(1)
V is continuous on ((n-1)T,(n+l-1)T]×ℝ+3∪((n+l-1)T,nT]×ℝ+3 for each x∈ℝ+3,n∈ℕ, and two limits 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;(2)
V is locally Lipschitzian in x.Definition 2.1.
ForV∈V0, one defines the upper right Dini derivative of V with respect to the impulsive differential system (1.1) at (t,x)∈((n-1)T,(n+l-1)T]×ℝ+3∪((n+l-1)T,nT]×ℝ+3 by
(2.1)D+V(t,x)=limsup⁡h→0+1h[V(t+h,x+hf(t,x))-V(t,x)].
We suppose thatg:ℝ+×ℝ+→ℝ satisfies the following hypotheses: (H)g is continuous on ((n-1)T,(n+l-1)T]×ℝ+3∪((n+l-1)T,nT]×ℝ+3 and the limits lim(t,y)→((n+l-1)T+,x)g(t,y)=g((n+l-1)T+,x) and lim(t,y)→(nT+,x)g(t,y)=g(nT+,x) exist and are finite for x∈ℝ+ and n∈ℕ.Lemma 2.2 (see [23]).
SupposeV∈V0 and
(2.2)D+V(t,x)≤g(t,V(t,x)),t≠(n+l-1)T,t≠nT,V(t,x(t+))≤ψn1(V(t,x)),t=(n+l-1)T,V(t,x(t+))≤ψn2(V(t,x)),t=nT,
where g:ℝ+×ℝ+→ℝ satisfies (H) and ψn1,ψn2:ℝ+→ℝ+ are nondecreasing for all n∈ℕ. Let r(t) be the maximal solution for the impulsive Cauchy problem
(2.3)u′(t)=g(t,u(t)),t≠(n+l-1)T,t≠nT,u(t+)=ψn1(u(t)),t=(n+l-1)T,u(t+)=ψn2(u(t)),t=nT,u(0+)=u0,
defined on [0,∞). Then V(0+,x0)≤u0 implies that V(t,x(t))≤r(t),t≥0, where x(t) is any solution of (2.2).A similar result can be obtained when all conditions of the inequalities in the Lemma2.2 are reversed. Using Lemma 2.2, it is easy to prove that the positive octant (ℝ+*)3 is an invariant region for the system (1.1) (see Lemma  2.1 in [11]).


## 2.2. Periodic Solutions

In the case in whichy=0, that is, mid-predator is eradicated, the system (1.1) is decoupled and led to two impulsive differential equations (2.4) and (2.6). Let us consider the properties of these impulsive differential equations. The following equation or a subsystem of system (1.1) is a periodically forced system:(2.4)x′(t)=x(t)(a-bx(t)),t≠(n+l-1)T,t≠nT,x(t+)=(1-δ1)x(t),t=(n+l-1)T,x(t+)=x(t),t=nT,x(0+)=x0.
Straightforward computation for getting a positive periodic solution x*(t) of (2.4) yields the analytic form of x*(t):(2.5)x*(t)=aη1exp⁡(at-λ)b[1-η1+η1exp⁡(at-λ)],(n+l-1)T<t≤(n+l)T,
where λ=a(n+l-1)T and η1=((1-δ1)exp(aT)-1)/(exp(aT)-1).In case thatδ1=0, the system (2.4) is the general logistic equation. From the analytic solution form (2.5) we get that η1 should be 1. It implies that x*(t)=a/b.Lemma 2.3 (see [4]).
The following statements hold.(1)
IfaT+ln(1-δ1)>0, then limt→∞|x(t)-x*(t)|=0 for all solutions x(t) of (2.4) starting with x0>0.(2)
IfaT+ln(1-δ1)≤0, then x(t)→0 as t→∞ for all solutions x(t) of (2.4).Next, we consider the impulsive differential equation as follows:(2.6)z′(t)=-d2z(t),t≠nT,t≠(n+l-1)T,z(t+)=(1-δ3)z(t),t=(n+l-1)T,z(t+)=z(t)+p,t=nT,z(0+)=z0.
The system (2.6) is a periodically forced linear system and its positive periodic solution z*(t) will be obtained:(2.7)z*(t)={pexp⁡[-d2(t-(n-1)T)]1-(1-δ3)exp⁡(-d2T),(n-1)T<t≤(n+l-1)T,p(1-δ3)exp⁡[-d2(t-(n-1)T)]1-(1-δ3)exp⁡(-d2T),(n+l-1)T<t≤nT,(2.8)z*(0+)=z*(nT+)=p1-(1-δ3)exp⁡(-d2T),z*((n+l-1)T+)=p(1-δ3)exp⁡(-d2lT)1-(1-δ3)exp⁡(-d2T).
Moreover, we may get that(2.9)z(t)={(1-δ3)n-1zz+z*(t),(n-1)T<t≤(n+l-1)T,(1-δ3)nzz+z*(t),(n+l-1)T<t≤nT,
is a solution of (2.6), where zz=(z(0+)-(p(1-δ3)e-T/(1-(1-δ3)exp(-d2T))))exp(-d2t).From (2.7) and (2.9), we thus get the following result.Lemma 2.4.
For every solutionz(t) and every positive periodic solution z*(t) of (2.6), it follows that z(t) tends to z*(t) as t→∞.


## 3. Main Results

In this section we study the local and global stability of the lowest-level prey and mid-level predator free periodic solution(0,0,z*(t)) and of the mid-level predator free periodic solution (x*(t),0,z*(t)). The authors in [11] had claimed that the solution (0,0,z*(t)) of the impulsive controlled system (1.1) is always unstable. In the biological point of view, this result is suspected in the sense that the prey and mid-predator will be eradicated under enough impulsive control term, especially, δ1. In Section 3.1 we will reinvestigate the stability of the periodic solution (0,0,z*(t)) and correct the misleading results shown in [11], and then the stability of the mid-level predator free periodic solution (x*(t),0,z*(t)) will be studied in Section 3.2.


### 3.1. A Stability of a Periodic Solution with Prey and Mid-Predator Eradication

We theoretically and numerically consider the stability of the periodic solution(0,0,z*(t)) with prey and mid-predator eradications.Theorem 3.1.
The periodic solution(0,0,z*(t)) is unstable if aT+ln(1-δ1)>0, and globally asymptotically stable if aT+ln(1-δ1)<0.The above Theorem3.1 says that in case that δ1 is sufficiently close to 1 to make aT+ln(1-δ1) negative, the pesticide has a negative effect on the growth of prey in a certain period T. To set up a control strategy for impulsive systems, we have to consider the relationship between two important factors, that is, the natural growth rate and (chemical) controlled rate in a controlling period.The proof of this theorem will be provided in Section4, and we may numerically consider the dynamical feature related to Theorem 3.1. To do it, we first fix the parameters: b=0.5,c1=2,c2=2,α1=0.1,α2=0.5,β1=1,β2=0.1,d1=0.1,d2=0.9,k1=0.7,k2=0.5,δ2=0.0001,δ3=0.03,l=0.6,p=2.9. If we choose the parameter a=1,T=1,δ1=0.7, then the value aT+ln(1-δ1)≈-0.3 is negative. It implies that trajectories asymptotically approach the periodic orbit (0,0,z*(t)) as shown in Figure 1. But, for different parameters setting having a positive value aT+ln(1-δ1)>0, we may expect that a typical trajectory with an initial condition near (0,0,z*(t)) is repelled from the periodic solution (0,0,z*(t)). For instance, we choose the parameters a=5,T=8,δ1=0.2 and then aT+ln(1-δ1)≈39.77. A repelling behavior of a trajectory with a starting point near (0,0,z*(t)) is shown in Figure 2. It shows the instability of the prey and mid-predator free solution.The dynamical behavior of the system (1.1) with parameters a=1,T=1,δ1=0.7. (a)–(c) show that a trajectory with a starting point (x0,y0,z0)=(10,1,0.4) approaches to the periodic orbit (0,0,z*(t)). In (d)–(f) the behavior of trajectory with a different starting point (x0,y0,z0)=(100,100,100) is shown.
(a)(b)(c)(d)(e)(f)The dynamical behavior of the system (1.1) with parameters a=5,T=8,δ1=0.2. (a)–(c) show that a trajectory with a starting point (x0,y0,z0)=(0.01,0.01,0.01) near the solution (0,0,z*(t)) is repelled from the periodic solution (0,0,z*(t)) and then approaches to another periodic solution.
(a)(b)(c)As shown in Figures1 and 2, the dynamical behavior of the periodic solution (0,0,z*(t)) is depending on the stability condition, the positiveness or negativeness of the value aT+ln(1-δ1). This stability condition is related to the total growth aT in a period T and the term ln(1-δ1) representing the loss of the prey due to the impulsive control δ1 on the prey. It means that species will be eradicated or grow depending on the sum of a natural growth and an artificial loss (impulsive control).


### 3.2. A Stability of a Periodic Solution with Mid-Predator Eradication

In this section the stability of the periodic solution(x*(t),0,z*(t)) with mid-predator eradication will be considered. In Theorem 3.2, we will mention the conditions for local and global stability of the periodic orbit (x*(t),0,z*(t)). Compared to Theorem 3.1, the positiveness of the value aT+ln(1-δ1) should be added in the condition for being the stable periodic orbit (x*(t),0,z*(t)).Theorem 3.2.
Suppose thataT+ln(1-δ1)>0. Then the periodic solution (x*(t),0,z*(t)) is locally asymptotically stable if the condition
(3.1)(a+bα1)(c2(A1-A2+A3-A4)-d1d2β2T)+k1c1β2d2A5β2d2(a+bα1)<ln⁡(11-δ2)
holds. Moreover, the periodic solution (x*(t),0,z*(t)) is globally asymptotically stable if the condition
(3.2)k1c1A5a+bα1-d1T<ln⁡(11-δ2)
holds. Here, the values Ai are listed:
(3.3)A1=ln⁡[α2(exp⁡(d2T)-1+δ3)+β2pexp⁡((1-l)d2T)],A2=ln⁡[α2(exp⁡(d2T)-1+δ3)+β2pexp⁡(d2T)],A3=ln⁡[α2(exp⁡(d2T)-1+δ3)+(1-δ3)β2p],A4=ln⁡[α2(exp⁡(d2T)-1+δ3)+(1-δ3)β2pexp⁡((1-l)d2T)],A5=ln⁡[bα1δ1exp⁡(aT)+((1-δ1)exp⁡(aT)-1)(a+bα1)exp⁡(aT)(a+bα1)(exp⁡(aT)-1)-aδ1exp⁡(aT)].The proof of this theorem is provided in Section4. To illustrate an numerical example related to Theorem 3.2, let b=0.5,c1=2,c2=2,α1=0.1,α2=0.5,β1=1,β2=0.1,d1=0.1,d2=0.9,k1=0.7,k2=0.5,δ2=0.0001,δ3=0.03,l=0.6,p=3,a=5,T=8, and δ1=0.2. These parameters satisfy the condition (3.1). It thus implies that trajectories asymptotically approach the periodic orbit (x*(t),0,z*(t)). In Figure 3, we numerically show that the periodic orbit (x*(t),0,z*(t)) is a sink.The dynamical behavior of the system (1.1). (a)–(c) show that a trajectory with a starting point (x0,y0,z0)=(10,1,0.4) approaches the periodic orbit (x*(t),0,z*(t)). In (d)–(f), the behavior of trajectory with a different starting point (x0,y0,z0)=(100,100,100) is shown.
(a)(b)(c)(d)(e)(f)


## 3.1. A Stability of a Periodic Solution with Prey and Mid-Predator Eradication

We theoretically and numerically consider the stability of the periodic solution(0,0,z*(t)) with prey and mid-predator eradications.Theorem 3.1.
The periodic solution(0,0,z*(t)) is unstable if aT+ln(1-δ1)>0, and globally asymptotically stable if aT+ln(1-δ1)<0.The above Theorem3.1 says that in case that δ1 is sufficiently close to 1 to make aT+ln(1-δ1) negative, the pesticide has a negative effect on the growth of prey in a certain period T. To set up a control strategy for impulsive systems, we have to consider the relationship between two important factors, that is, the natural growth rate and (chemical) controlled rate in a controlling period.The proof of this theorem will be provided in Section4, and we may numerically consider the dynamical feature related to Theorem 3.1. To do it, we first fix the parameters: b=0.5,c1=2,c2=2,α1=0.1,α2=0.5,β1=1,β2=0.1,d1=0.1,d2=0.9,k1=0.7,k2=0.5,δ2=0.0001,δ3=0.03,l=0.6,p=2.9. If we choose the parameter a=1,T=1,δ1=0.7, then the value aT+ln(1-δ1)≈-0.3 is negative. It implies that trajectories asymptotically approach the periodic orbit (0,0,z*(t)) as shown in Figure 1. But, for different parameters setting having a positive value aT+ln(1-δ1)>0, we may expect that a typical trajectory with an initial condition near (0,0,z*(t)) is repelled from the periodic solution (0,0,z*(t)). For instance, we choose the parameters a=5,T=8,δ1=0.2 and then aT+ln(1-δ1)≈39.77. A repelling behavior of a trajectory with a starting point near (0,0,z*(t)) is shown in Figure 2. It shows the instability of the prey and mid-predator free solution.The dynamical behavior of the system (1.1) with parameters a=1,T=1,δ1=0.7. (a)–(c) show that a trajectory with a starting point (x0,y0,z0)=(10,1,0.4) approaches to the periodic orbit (0,0,z*(t)). In (d)–(f) the behavior of trajectory with a different starting point (x0,y0,z0)=(100,100,100) is shown.
(a)(b)(c)(d)(e)(f)The dynamical behavior of the system (1.1) with parameters a=5,T=8,δ1=0.2. (a)–(c) show that a trajectory with a starting point (x0,y0,z0)=(0.01,0.01,0.01) near the solution (0,0,z*(t)) is repelled from the periodic solution (0,0,z*(t)) and then approaches to another periodic solution.
(a)(b)(c)As shown in Figures1 and 2, the dynamical behavior of the periodic solution (0,0,z*(t)) is depending on the stability condition, the positiveness or negativeness of the value aT+ln(1-δ1). This stability condition is related to the total growth aT in a period T and the term ln(1-δ1) representing the loss of the prey due to the impulsive control δ1 on the prey. It means that species will be eradicated or grow depending on the sum of a natural growth and an artificial loss (impulsive control).


## 3.2. A Stability of a Periodic Solution with Mid-Predator Eradication

In this section the stability of the periodic solution(x*(t),0,z*(t)) with mid-predator eradication will be considered. In Theorem 3.2, we will mention the conditions for local and global stability of the periodic orbit (x*(t),0,z*(t)). Compared to Theorem 3.1, the positiveness of the value aT+ln(1-δ1) should be added in the condition for being the stable periodic orbit (x*(t),0,z*(t)).Theorem 3.2.
Suppose thataT+ln(1-δ1)>0. Then the periodic solution (x*(t),0,z*(t)) is locally asymptotically stable if the condition
(3.1)(a+bα1)(c2(A1-A2+A3-A4)-d1d2β2T)+k1c1β2d2A5β2d2(a+bα1)<ln⁡(11-δ2)
holds. Moreover, the periodic solution (x*(t),0,z*(t)) is globally asymptotically stable if the condition
(3.2)k1c1A5a+bα1-d1T<ln⁡(11-δ2)
holds. Here, the values Ai are listed:
(3.3)A1=ln⁡[α2(exp⁡(d2T)-1+δ3)+β2pexp⁡((1-l)d2T)],A2=ln⁡[α2(exp⁡(d2T)-1+δ3)+β2pexp⁡(d2T)],A3=ln⁡[α2(exp⁡(d2T)-1+δ3)+(1-δ3)β2p],A4=ln⁡[α2(exp⁡(d2T)-1+δ3)+(1-δ3)β2pexp⁡((1-l)d2T)],A5=ln⁡[bα1δ1exp⁡(aT)+((1-δ1)exp⁡(aT)-1)(a+bα1)exp⁡(aT)(a+bα1)(exp⁡(aT)-1)-aδ1exp⁡(aT)].The proof of this theorem is provided in Section4. To illustrate an numerical example related to Theorem 3.2, let b=0.5,c1=2,c2=2,α1=0.1,α2=0.5,β1=1,β2=0.1,d1=0.1,d2=0.9,k1=0.7,k2=0.5,δ2=0.0001,δ3=0.03,l=0.6,p=3,a=5,T=8, and δ1=0.2. These parameters satisfy the condition (3.1). It thus implies that trajectories asymptotically approach the periodic orbit (x*(t),0,z*(t)). In Figure 3, we numerically show that the periodic orbit (x*(t),0,z*(t)) is a sink.The dynamical behavior of the system (1.1). (a)–(c) show that a trajectory with a starting point (x0,y0,z0)=(10,1,0.4) approaches the periodic orbit (x*(t),0,z*(t)). In (d)–(f), the behavior of trajectory with a different starting point (x0,y0,z0)=(100,100,100) is shown.
(a)(b)(c)(d)(e)(f)


## 4. Proofs of Theorems3.1 and 3.2

### 4.1. Proof of Theorem3.1

Proof.
A local stability of the periodic solution(0,0,z*(t)) of the system (1.1) may be determined by considering the behavior of small amplitude perturbations of the solution. Let (x(t),y(t),z(t)) be any solution of the system (1.1). Define x(t)=u(t),y(t)=v(t) and z(t)=w(t)+z*(t). Then they may be written as
(4.1)(u(t)v(t)w(t))=Φ(t)(u(0)v(0)w(0)),
where Φ(t) satisfies
(4.2)dΦdt=(a000-d1-c2z*(t)β2z*(t)+α200k2c2z*(t)β2z*(t)+α2-d2)Φ(t),
and Φ(0)=I is the identity matrix. Therefore, the fundamental solution matrix is
(4.3)Φ(t)=(exp⁡(at)000exp⁡(∫0t[-d1-c2z*(s)β2z*(s)+α2]ds)00exp⁡(∫0tk2c2z*(s)ds)exp⁡(-d2t)).
The resetting impulsive conditions of the system (1.1) become
(4.4)(u((n+l-1)T+)v((n+l-1)T+)u((n+l-1)T+))=(1-δ10001-δ20001-δ3)(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)).
Note that the eigenvalues of
(4.5)S=(1-δ10001-δ20001-δ3)(100010001)Φ(T)
are μ1=(1-δ1)exp(aT), μ2=(1-δ2)exp(-∫0Td1+(c2z*(s)/(β2x*(s)+α2))ds) and μ3=(1-δ3)exp(-d2T). Clearly, μ2<1 and μ3<1. If aT+ln(1-δ1)>0, then μ1>1. Therefore, by Floquet theory [24], the periodic solution (0,0,z*(t)) is unstable.
To prove a global stability of the periodic solution(0,0,z*(t)), first, we assume that aT+ln(1-δ1)<0. Then μ1=(1-δ1)exp(aT)<1. It means that the periodic solution (0,0,z*(t)) is locally asymptotically stable.
Let(x(t),y(t),z(t)) be any solution of (1.1). Take a number ϵ1 with 0<ϵ1<d1α1/k1c1 and let ξ=(1-δ1)exp((-d1+(k1c1/α1)ϵ1)T). Note that 0<ξ<1. From the first equation in (1.1), we get
(4.6)x′(t)=x(t)(a-bx(t))-c1x(t)y(t)α1+x(t)+β1y(t)≤x(t)(a-bx(t)),
for t≠(n+l-1)T and t≠nT. By Lemma 2.2, we obtain x(t)≤x̃(t) for t≥0, where x̃(t) is the solution of (2.4). Using Lemma 2.3, we also get that x̃(t)→0 as t→∞. It implies that there exists a number T1>0 satisfying x(t)≤ϵ1 for t≥T1. Without loss of generality, we may assume that x(t)≤ϵ1 for all t>0. From the second equation in (1.1), we obtain that for t≠(n+l-1)T and t≠nT,
(4.7)y′(t)=-d1y(t)+k1c1x(t)y(t)α1+x(t)+β1y(t)-c2y(t)z(t)α2+y(t)+β2z(t)≤-d1y(t)+k1c1α1x(t)y(t)≤y(t)(-d1+k1c1α1ϵ1).
Integrating both sides of the inequality (4.7) on ((n+l-1)T,(n+l)T], we get
(4.8)y((n+l)T)≤y((n+l-1)T+)exp⁡((-d1+k1c1α1ϵ1)T)=y((n+l-1)T)ξ.
It implies that y((n+l)T)≤y(lT)ξn. Therefore y((n+l)T)→0 as n→∞. We also obtain that for t∈((n+l-1)T,(n+l)T],
(4.9)y(t)≤y((n+l-1)T+)exp⁡((-d1+k1c1α1ϵ1)(t-(n+l-1)T))≤y((n+l-1)T).
It thus implies that y(t)→0 as t→∞.
Now, take0<ϵ2<(d2α2/k2c2) in order to prove that z(t)→z*(t) as t→∞. Since limt→∞y(t)=0, there is a T2>0 such that y(t)≤ϵ2 for t≥T2. For the sake of simplicity, we assume that y(t)≤ϵ2 for all t≥0. From the third equation in (1.1), we get that, for t≠(n+l-1)T and t≠nT,
(4.10)-d2z(t)≤z′(t)=-d2z(t)+k2c2y(t)z(t)α2+y(t)+β2z(t)≤(-d2+k2c2α2ϵ2)z(t).
Thus, by Lemma 2.2, we induce that z̃1(t)≤z(t)≤z̃2(t), where z̃1(t) is the solution of (2.6) and z̃2(t) is also the solution of (2.6) with d2 changed into d2-(k2c2/α2)ϵ2. Using Lemma 2.4 and letting ϵ2→0, z̃1(t) and z̃2(t) tend to z*(t) as t→∞. We thus prove that |z(t)-z*(t)|→0 as t→∞.


### 4.2. Proof of Theorem3.2

To determine the stability of the periodic solution(x*(t),0,z*(t)), we will use the Floquet theory. First, we construct the monodromy matrix and calculate its eigenvalues:
(4.11)S=(1-δ10001-δ20001-δ3)(100010001)Φ(T),
where Φ(t) satisfies
(4.12)dΦdt=(a-2bx*(t)-c1x*(t)α1+x*(t)00-d1+k1c1x*(t)α1+x*(t)-c2z*(t)α2+β2z*(t)00k2c2z*(t)α2+β2z*(t)-d2)Φ(t),
and Φ(0)=I is the identity matrix. Then all eigenvalues of the matrix S are
(4.13)μ1=(1-δ1)exp⁡(∫0Ta-2bx*(t)dt)μ2=(1-δ2)exp⁡(∫0T-d1+k1c1x*(t)α1+x*(t)-c2z*(t)α2+β2z*(t)dt)μ3=(1-δ3)exp⁡(-d2T)<1.
Note that
(4.14)∫0Tx*(t)dt=1b(ln⁡(1-δ1)+aT),∫0Tx*(t)α1+x*(t)dt=1a+bα1ln⁡(bα1(1-η1)+η1(a+bα1)exp⁡(aT)aη1+bα1),∫0Tz*(t)α2+β2z*(t)dt=1β2d2ln⁡((η2+β2p)(η2+β2p(1-δ3)exp⁡(-d2lT))(η2+β2pexp⁡(-d2lT))(η2+β2p(1-δ3)exp⁡(-d2T))),
where η1=((1-δ1)exp(aT)-1)/(exp(aT)-1) and η2=α2(1-(1-δ3)exp(-d2T)).From (4.14) and aT+ln(1-δ1)>0, we get that μ1<1 and μ2 is equivalent to (3.1) in the statement of Theorem 3.2. By the hypothesis of Theorem 3.2, we obtain μ2<1. Finally, based on the Floquet theory [24], we get that (x*(t),0,z*(t)) is locally asymptotically stable.Suppose thataT+ln(1-δ1)>0 and (3.2) hold.Let(x(t),y(t),z(t)) be any solution of (1.1). The condition (3.2) implies
(4.15)μ2≤(1-δ2)exp⁡(∫0T-d1+k1c1x*(t)α1+x*(t)dt)<1.
Thus the periodic solution (x*(t),0,z*(t)) is locally asymptotically stable. Further, we can choose ϵ3>0 such that
(4.16)0<ζ≡(1-δ2)exp⁡(∫0T-d1+k1c1(x*(t)+ϵ3)α1+x*(t)+ϵ3dt)<1.
As the proof of Theorem 3.1, by Lemma 2.2, we obtain x(t)≤x̃2(t) for t≥0, where x̃2(t) is the solution of (2.4). It follows from Lemma 2.3 that there exists a T3>0 such that x(t)≤x*(t)+ϵ3 for t≥T3. Without loss of generality, we may assume that x(t)≤x*(t)+ϵ3 for t≥0. From the second equation in (1.1), we get that for t≠(n+l-1)T and t≠nT,
(4.17)y′(t)=-d1y(t)+k1c1x(t)y(t)α1+x(t)+β1y(t)-c2y(t)z(t)α2+y(t)+β2z(t)≤y(t)(-d1+k1c1(x*(t)+ϵ3)α1+x*(t)+ϵ3).
By integrating both sides of (4.17) on ((n+l-1)T,(n+l)T], we obtain that
(4.18)y((n+l)T)≤y((n+l-1)T+)exp⁡(∫(n+l-1)T(n+l)T-d1+k1c1(x*(t)+ϵ3)α1+x*(t)+ϵ3dt)=y((n+l-1)T)ζ.
It implies that y((n+l)T)≤y(lT)ζn. Finally, we get that y((n+l)T)→0 as n→∞.From the inequality(4.19)y′(t)≤k1c1α1(x*(t)+ϵ3)y(t)≤k1c1α1(abexp⁡(aT)+ϵ3)y(t),
we get
(4.20)y(t)≤y((n+l-1)T)(1-δ2)exp⁡(k1c1α1[abexp⁡(aT)+ϵ3]T),
for t∈((n+l-1)T,(n+l)T]. Consequently y(t)→0 as t→∞.In order to show that|x(t)-x*(t)|→0 as t→∞, we take ϵ4 such that 0<ϵ4<aα1/c1. Since limt→∞y(t)=0, there exists a T5>0 such that y(t)<ϵ4 for t>T5. For the sake of simplicity, we may suppose that y(t)<ϵ4 for all t≥0. Therefore, for t≠(n+l-1)T and t≠nT, we obtain
(4.21)x′(t)=x(t)(a-bx(t))-c1x(t)y(t)α1+x(t)+β1y(t)≥x(t)([a-c1y(t)α1]-bx(t))≥x(t)([a-c1ϵ4α1]-bx(t)).
Thus, from Lemma 2.2, we obtain that x̃1(t)≤x(t), where x̃1(t) is the solution of (2.4) with a changed into a-c1ϵ4/α1. From Lemma 2.3 and taking sufficiently small ϵ4, it is seen that x̃1(t) and x̃2(t) tend to x*(t) as t→∞. Thus, we get |x(t)-x*(t)|→0 as t→∞.Note that-d2z(t)≤z′(t)=-d2z(t)+k2c2y(t)z(t)/(α2+y(t)+β2z(t))≤-d2z(t)+(k2c2/α2)ϵ4 for t≠(n+l-1)T and t≠nT. By using the same process as the proof of Theorem 3.1, we can show that |z(t)-z*(t)|→0 as t→∞.


## 4.1. Proof of Theorem3.1

Proof.
A local stability of the periodic solution(0,0,z*(t)) of the system (1.1) may be determined by considering the behavior of small amplitude perturbations of the solution. Let (x(t),y(t),z(t)) be any solution of the system (1.1). Define x(t)=u(t),y(t)=v(t) and z(t)=w(t)+z*(t). Then they may be written as
(4.1)(u(t)v(t)w(t))=Φ(t)(u(0)v(0)w(0)),
where Φ(t) satisfies
(4.2)dΦdt=(a000-d1-c2z*(t)β2z*(t)+α200k2c2z*(t)β2z*(t)+α2-d2)Φ(t),
and Φ(0)=I is the identity matrix. Therefore, the fundamental solution matrix is
(4.3)Φ(t)=(exp⁡(at)000exp⁡(∫0t[-d1-c2z*(s)β2z*(s)+α2]ds)00exp⁡(∫0tk2c2z*(s)ds)exp⁡(-d2t)).
The resetting impulsive conditions of the system (1.1) become
(4.4)(u((n+l-1)T+)v((n+l-1)T+)u((n+l-1)T+))=(1-δ10001-δ20001-δ3)(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)).
Note that the eigenvalues of
(4.5)S=(1-δ10001-δ20001-δ3)(100010001)Φ(T)
are μ1=(1-δ1)exp(aT), μ2=(1-δ2)exp(-∫0Td1+(c2z*(s)/(β2x*(s)+α2))ds) and μ3=(1-δ3)exp(-d2T). Clearly, μ2<1 and μ3<1. If aT+ln(1-δ1)>0, then μ1>1. Therefore, by Floquet theory [24], the periodic solution (0,0,z*(t)) is unstable.
To prove a global stability of the periodic solution(0,0,z*(t)), first, we assume that aT+ln(1-δ1)<0. Then μ1=(1-δ1)exp(aT)<1. It means that the periodic solution (0,0,z*(t)) is locally asymptotically stable.
Let(x(t),y(t),z(t)) be any solution of (1.1). Take a number ϵ1 with 0<ϵ1<d1α1/k1c1 and let ξ=(1-δ1)exp((-d1+(k1c1/α1)ϵ1)T). Note that 0<ξ<1. From the first equation in (1.1), we get
(4.6)x′(t)=x(t)(a-bx(t))-c1x(t)y(t)α1+x(t)+β1y(t)≤x(t)(a-bx(t)),
for t≠(n+l-1)T and t≠nT. By Lemma 2.2, we obtain x(t)≤x̃(t) for t≥0, where x̃(t) is the solution of (2.4). Using Lemma 2.3, we also get that x̃(t)→0 as t→∞. It implies that there exists a number T1>0 satisfying x(t)≤ϵ1 for t≥T1. Without loss of generality, we may assume that x(t)≤ϵ1 for all t>0. From the second equation in (1.1), we obtain that for t≠(n+l-1)T and t≠nT,
(4.7)y′(t)=-d1y(t)+k1c1x(t)y(t)α1+x(t)+β1y(t)-c2y(t)z(t)α2+y(t)+β2z(t)≤-d1y(t)+k1c1α1x(t)y(t)≤y(t)(-d1+k1c1α1ϵ1).
Integrating both sides of the inequality (4.7) on ((n+l-1)T,(n+l)T], we get
(4.8)y((n+l)T)≤y((n+l-1)T+)exp⁡((-d1+k1c1α1ϵ1)T)=y((n+l-1)T)ξ.
It implies that y((n+l)T)≤y(lT)ξn. Therefore y((n+l)T)→0 as n→∞. We also obtain that for t∈((n+l-1)T,(n+l)T],
(4.9)y(t)≤y((n+l-1)T+)exp⁡((-d1+k1c1α1ϵ1)(t-(n+l-1)T))≤y((n+l-1)T).
It thus implies that y(t)→0 as t→∞.
Now, take0<ϵ2<(d2α2/k2c2) in order to prove that z(t)→z*(t) as t→∞. Since limt→∞y(t)=0, there is a T2>0 such that y(t)≤ϵ2 for t≥T2. For the sake of simplicity, we assume that y(t)≤ϵ2 for all t≥0. From the third equation in (1.1), we get that, for t≠(n+l-1)T and t≠nT,
(4.10)-d2z(t)≤z′(t)=-d2z(t)+k2c2y(t)z(t)α2+y(t)+β2z(t)≤(-d2+k2c2α2ϵ2)z(t).
Thus, by Lemma 2.2, we induce that z̃1(t)≤z(t)≤z̃2(t), where z̃1(t) is the solution of (2.6) and z̃2(t) is also the solution of (2.6) with d2 changed into d2-(k2c2/α2)ϵ2. Using Lemma 2.4 and letting ϵ2→0, z̃1(t) and z̃2(t) tend to z*(t) as t→∞. We thus prove that |z(t)-z*(t)|→0 as t→∞.


## 4.2. Proof of Theorem3.2

To determine the stability of the periodic solution(x*(t),0,z*(t)), we will use the Floquet theory. First, we construct the monodromy matrix and calculate its eigenvalues:
(4.11)S=(1-δ10001-δ20001-δ3)(100010001)Φ(T),
where Φ(t) satisfies
(4.12)dΦdt=(a-2bx*(t)-c1x*(t)α1+x*(t)00-d1+k1c1x*(t)α1+x*(t)-c2z*(t)α2+β2z*(t)00k2c2z*(t)α2+β2z*(t)-d2)Φ(t),
and Φ(0)=I is the identity matrix. Then all eigenvalues of the matrix S are
(4.13)μ1=(1-δ1)exp⁡(∫0Ta-2bx*(t)dt)μ2=(1-δ2)exp⁡(∫0T-d1+k1c1x*(t)α1+x*(t)-c2z*(t)α2+β2z*(t)dt)μ3=(1-δ3)exp⁡(-d2T)<1.
Note that
(4.14)∫0Tx*(t)dt=1b(ln⁡(1-δ1)+aT),∫0Tx*(t)α1+x*(t)dt=1a+bα1ln⁡(bα1(1-η1)+η1(a+bα1)exp⁡(aT)aη1+bα1),∫0Tz*(t)α2+β2z*(t)dt=1β2d2ln⁡((η2+β2p)(η2+β2p(1-δ3)exp⁡(-d2lT))(η2+β2pexp⁡(-d2lT))(η2+β2p(1-δ3)exp⁡(-d2T))),
where η1=((1-δ1)exp(aT)-1)/(exp(aT)-1) and η2=α2(1-(1-δ3)exp(-d2T)).From (4.14) and aT+ln(1-δ1)>0, we get that μ1<1 and μ2 is equivalent to (3.1) in the statement of Theorem 3.2. By the hypothesis of Theorem 3.2, we obtain μ2<1. Finally, based on the Floquet theory [24], we get that (x*(t),0,z*(t)) is locally asymptotically stable.Suppose thataT+ln(1-δ1)>0 and (3.2) hold.Let(x(t),y(t),z(t)) be any solution of (1.1). The condition (3.2) implies
(4.15)μ2≤(1-δ2)exp⁡(∫0T-d1+k1c1x*(t)α1+x*(t)dt)<1.
Thus the periodic solution (x*(t),0,z*(t)) is locally asymptotically stable. Further, we can choose ϵ3>0 such that
(4.16)0<ζ≡(1-δ2)exp⁡(∫0T-d1+k1c1(x*(t)+ϵ3)α1+x*(t)+ϵ3dt)<1.
As the proof of Theorem 3.1, by Lemma 2.2, we obtain x(t)≤x̃2(t) for t≥0, where x̃2(t) is the solution of (2.4). It follows from Lemma 2.3 that there exists a T3>0 such that x(t)≤x*(t)+ϵ3 for t≥T3. Without loss of generality, we may assume that x(t)≤x*(t)+ϵ3 for t≥0. From the second equation in (1.1), we get that for t≠(n+l-1)T and t≠nT,
(4.17)y′(t)=-d1y(t)+k1c1x(t)y(t)α1+x(t)+β1y(t)-c2y(t)z(t)α2+y(t)+β2z(t)≤y(t)(-d1+k1c1(x*(t)+ϵ3)α1+x*(t)+ϵ3).
By integrating both sides of (4.17) on ((n+l-1)T,(n+l)T], we obtain that
(4.18)y((n+l)T)≤y((n+l-1)T+)exp⁡(∫(n+l-1)T(n+l)T-d1+k1c1(x*(t)+ϵ3)α1+x*(t)+ϵ3dt)=y((n+l-1)T)ζ.
It implies that y((n+l)T)≤y(lT)ζn. Finally, we get that y((n+l)T)→0 as n→∞.From the inequality(4.19)y′(t)≤k1c1α1(x*(t)+ϵ3)y(t)≤k1c1α1(abexp⁡(aT)+ϵ3)y(t),
we get
(4.20)y(t)≤y((n+l-1)T)(1-δ2)exp⁡(k1c1α1[abexp⁡(aT)+ϵ3]T),
for t∈((n+l-1)T,(n+l)T]. Consequently y(t)→0 as t→∞.In order to show that|x(t)-x*(t)|→0 as t→∞, we take ϵ4 such that 0<ϵ4<aα1/c1. Since limt→∞y(t)=0, there exists a T5>0 such that y(t)<ϵ4 for t>T5. For the sake of simplicity, we may suppose that y(t)<ϵ4 for all t≥0. Therefore, for t≠(n+l-1)T and t≠nT, we obtain
(4.21)x′(t)=x(t)(a-bx(t))-c1x(t)y(t)α1+x(t)+β1y(t)≥x(t)([a-c1y(t)α1]-bx(t))≥x(t)([a-c1ϵ4α1]-bx(t)).
Thus, from Lemma 2.2, we obtain that x̃1(t)≤x(t), where x̃1(t) is the solution of (2.4) with a changed into a-c1ϵ4/α1. From Lemma 2.3 and taking sufficiently small ϵ4, it is seen that x̃1(t) and x̃2(t) tend to x*(t) as t→∞. Thus, we get |x(t)-x*(t)|→0 as t→∞.Note that-d2z(t)≤z′(t)=-d2z(t)+k2c2y(t)z(t)/(α2+y(t)+β2z(t))≤-d2z(t)+(k2c2/α2)ϵ4 for t≠(n+l-1)T and t≠nT. By using the same process as the proof of Theorem 3.1, we can show that |z(t)-z*(t)|→0 as t→∞.


## 5. Conclusions

We have considered the impulsively controlled three species food chain system with the Beddington-DeAngelis functional response proposed by the authors in [11]. To control the food chain system with three species, two control terms, biological and chemical controls, are employed. Here, a biological control means an impulsive and periodic releasing of top-predator with a fixed proportion and a chemical control means that, for instance, as a result of pesticide spreading fixed proportions of prey, mid-predator, and top-predator, their population will be impulsively degraded. Under controlling environment, we first show the conditions for extinction and growing of the prey and mid-predator using Floquet theory and comparison method. In addition, a suffcient condition for local and global stability of the mid-predator free solution is established, which means that if the mid-predator is regarded as the pest we can control the pest population under some conditions. These results will correct the misleading results shown in [11].



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*Source: 418564-2012-01-19.xml*