# Extinction and Permanence of a Three-Species Lotka-Volterra System with Impulsive Control Strategies

Authors: Hunki Baek
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

A three-species Lotka-Volterra system with impulsive control strategies containing
the biological control (the constant impulse) and the chemical control (the proportional
impulse) with the same period, but not simultaneously, is investigated. By
applying the Floquet theory of impulsive differential equation and small amplitude
perturbation techniques to the system, we find conditions for local and global stabilities
of a lower-level prey and top-predator free periodic solution of the system. In
addition, it is shown that the system is permanent under some conditions by using
comparison results of impulsive differential inequalities. We also give a numerical
example that seems to indicate the existence of chaotic behavior.


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## Body

## 1. Introduction

The mathematical study of a predator-prey system in
population dynamics has a long history starting with the work of Lotka and
Volterra. The principles of Lotka-Volterra models have remained valid until
today and many theoretical ecologists adhere to their principles [1–9]. Thus, we need to consider
a Lotka-Volterra-type food chain model which can be described by the following
differential equations:(1.1)x′(t)=x(t)(a−bx(t)−cy(t)),y′(t)=y(t)(−d1+c1x(t)−e1z(t)),z′(t)=z(t)(−d2+e2y(t)),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; a>0 is called intrinsic growth rate of the prey; b>0 is the coefficient of intraspecific
competition; c>0 and e1>0 are the per-capita rate of predation of the
predator; c1>0 and e2>0 denote the product of the per-capita rate of
predation and the conversion rate; d1>0 and d2>0 denote the death rate of the predators.Now, we regardx(t) as a pest to establish a new system dealing
with impulsive pest control strategies from system (1.1). There are many ways
to control pest population. One of the most important methods for pest control
is chemical control. A principal substance in chemical control is pesticide.
Pesticides are often useful because they quickly kill a significant portion of
pest population. However, there are many deleterious effects associated with the
use of chemicals that need to be reduced or eliminated. These include human
illness associated with pesticide applications, insect resistance to
insecticides, contamination of soil and water, and diminution of biodiversity.
As a result, we should combine pesticide efficacy tests with other ways of
control like biological control. Biological control is another important
strategy to control pest population. It is defined as the reduction of the pest
population by natural enemies and typically involves an active human role.
Natural enemies of insect pests, also known as biological control agents,
include predators, parasites, and pathogens. Virtually, all pests have some
natural enemies, and the key to successful pest control is to identify the pest
and its natural enemies and release them at fixed time for pest control. Such
different pest control tactics should work together rather than against each
other to accomplish successful pest population control [10–12]. Thus, in this paper, we
consider the following Lotka-Volterra-type food chain model with periodic
constant releasing natural enemies (mid-level predator) and spraying pesticide
at different fixed time:(1.2)x′(t)=x(t)(a−bx(t)−cy(t)),t≠nT,t≠(n+τ−1)T,y′(t)=y(t)(−d1+c1x(t)−e1z(t)),t≠nT,t≠(n+τ−1)T,z′(t)=z(t)(−d2+e2y(t)),t≠nT,t(n+τ−1)T,x(t+)=(1−p1)x(t),t=(n+τ−1)T,y(t+)=(1−p2)y(t),t=(n+τ−1)T,z(t+)=(1−p3)z(t),t=(n+τ−1)T,x(t+)=x(t),t=nT,y(t+)=y(t)+q,t=nT,z(t+)=z(t),t=nT,(x(0+),y(0+),z(0+))=(x0,y0,z0),where 0<τ<1, T is the period of the impulsive immigration or
stock of the mid-level predator, 0≤p1,p2,p3<1 present the fraction of the prey and the
predator which die due to the harvesting or pesticides, and q is the size of immigration or stock of the
predator.
Such system is an impulsive differential equation whose theories and applications were greatly
developed by the efforts of Bainov and Simeonov [13] and Lakshmikantham et al.
[14]. Also, Nieto and O'Regan. [18]
presented a new approach to obtain the existence of solutions to some impulsive problems.
Moreover [15–17],
the theory of impulsive differential equations is being recognized to be not only richer
than the corresponding theory of differential equations without impulses, but also to represent a more
natural framework for mathematical modeling of real-world phenomena [18–22].In recent years, many authors have studied
two-dimensional predator-prey systems with impulsive perturbations [23–29]. Moreover, three-species food chain systems with
sudden perturbations have been intensively researched, such as those of
Holling-type [30, 31] and Beddington-type
[32, 33]. However, most researches
about food chain systems mentioned above have just dealt with biological
control and have only given conditions for extinction of the lowest-level prey
and top predator by observing the local stability of lower-level prey and
top-predator free periodic solution. For this reason, the main purpose of this
paper is to investigate the conditions for the extinction and the permanence of
system (1.2).The organization of the paper is as follows. In the
next section, we introduce some notations and lemmas which are used in this
paper. In Section3, we find conditions for local and global stabilities of a
lower-level prey and top-predator free periodic solution by applying the
Floquet theory and for permanence of system (1.2) by using the comparison
theorem. In Section 4, we give some numerical examples including chaotic phase
portrait. Finally, we have a conclusion in Section 5.


## 2. Preliminaries

Now, we will introduce a few notations and definitions
together with a few auxiliary results relating to comparison theorem, which
will be useful for our main results.Letℝ+=[0,∞) and ℝ+3={x=(x(t),y(t),z(t))∈ℝ3:x(t),y(t),z(t)≥0}. Denote ℕ as the set of all nonnegative integers, ℝ+*=(0,∞),
and f=(f1,f2,f3)T as the right-hand side
of the first three equations in (1.2). We first give the definition of a
solution for (1.2).Definition 2.1 (see [14]).
LetΩ⊂ℝ3 be an open set and let D=ℝ+×Ω. A function x=(x(t),y(t),z(t)):(0,a)→ℝ3, a>0, is said to be a solution of (1.2) if(1)
x(0+)=(x0,y0,z0) and (t,x(t))∈D for t∈[0,a),(2)
x(t) is continuously differentiable and satisfies the first three equations
in (1.2) for t∈[0,a), t≠nT, and t≠(n+τ−1)T,(3)
0≤t<a; then x(t) is left continous at t=(n+τ−1)T and nT, and(2.1)x(t+)−x(t)={(−p1x(t),−p2y(t),−p3z(t))ift=(n+τ−1)T,(0,q,0)ift=nT.Now, we introduce another definition to formulate the comparison result. LetV:ℝ+×ℝ+3→ℝ+, then V is said to be in a class V0 if(1)
V is continuous on (nT,(n+1)T]×ℝ+3, and lim(t,y)→(nT,x),t>nTV(t,y)=V(nT+,x) exists;(2)
V is locally Lipschitzian in x.Definition 2.2 (see [14]).
ForV∈V0,
one defines the upper-right Dini derivative of V with respect to the
impulsive differential system (1.2) at (t,x)∈(nT,(n+1)T]×ℝ+3 by
(2.2)D+V(t,x)=limsuph→0+1h[V(t+h,x+hf(t,x))−V(t,x)].Remark 2.3.
The smoothness properties off guarantee the global existence and uniqueness
of solutions of system (1.2). (See [13, 14] for details.)We will use a comparison result of impulsive differential inequalities. We suppose thatg:ℝ+×ℝ+→ℝ satisfies the following hypothesis.(H)g is continuous on (nT,(n+1)T]×ℝ+ and the limit lim(t,y)→(nT+,x)g(t,y)=g(nT+,x) exists and is finite for x∈ℝ+ and n∈ℕ.Lemma 2.4 (see [14]).
SupposeV∈V0 and
(2.3)D+V(t,x)≤g(t,V(t,x)),t≠(n+τ−1)T,nT,V(t,x(t+))≤ψn1(V(t,x)),t=(n+τ−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(2.4)u′(t)=g(t,u(t)),t≠(n+τ−1)T,nT,u(t+)=ψn1(u(t)),t=(n+τ−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 (2.3).We now indicate a
special case of Lemma2.4 which provides estimations for the solution of a
system of 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 2.5 (see [14]).
Let the functionu(t)∈PC1(ℝ+,ℝ) satisfy the inequalities(2.5)dudt≤f(t)u(t)+h(t),t≠τk,t>0,u(τk+)≤αku(τk)+βk,k≥0,u(0+)≤u0,where f,h∈PC(ℝ+,ℝ) and αk≥0, βk,u0 are constants, and (τk)k≥0 is a strictly increasing sequence of positive
real numbers. Then, for t>0,(2.6)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)βk.Similar results
can be obtained when all conditions of the inequalities in Lemmas2.4 and 2.5
are reversed. Using Lemma 2.5, it is possible to prove that the solutions of
the Cauchy problem (2.4) with strictly positive initial value remain strictly
positive.Lemma 2.6.
The positive octant(ℝ+*)3 is an invariant region for system (1.2).Proof.
Let(x(t),y(t),z(t)):[0,t0)→ℝ3 be a solution of system (1.2) with a strictly
positive initial value (x0,y0,z0).
By Lemma 2.5, we can obtain that, for 0≤t<t0,(2.7)x(t)=x(0)(1−p1)[t/T]exp(∫0tf1(s)ds),y(t)=y(0)(1−p2)[t/T]exp(∫0tf2(s)ds),z(t)=z(0)(1−p3)[t/T]exp(∫0tf3(s)ds),where f1(s)=a−bx(s)−cy(s), f2(s)=−d1+c1x(s)−e1z(s),
and f3(s)=−d2+e2y(s).
Thus, x(t),y(t),
and z(t) remain strictly positive on [0,t0).Lemma 2.7.
IfaT+ln(1−p1)≤0,
then x(t)→0 as t→∞ for any solution x(t) of the following impulsive differential
equation:(2.8)x′(t)=x(t)(a−bx(t)),t≠nT,t≠(n+τ−1)T,x(t+)=(1−p1)x(t),t=(n+τ−1)T,x(t+)=x(t),t=nT.Proof.
It is easy to see that for a given
initial conditionx(0+),(2.9)x(t)=aexp(a(t−t0))x(t0)a+bx(t0)(exp(a(t−t0)−1),(n+τ−1)T≤t0<t≤(n+τ)T,for a solution x(t) of (2.8). It follows from aT+ln(1−p1)≤0 and (2.9) that(2.10)x((n+τ)T)=a(1−p1)exp(aT)x((n+τ−1)T)a+b(1−p1)x((n+τ−1)T)(exp(aT)−1)≤x((n+τ−1)T)1+(b/a)(1−p1)x((n+τ−1)T)(exp(aT)−1)≤x((n+τ−1)T).Thus, we know that the sequence {x((n+τ)T)}n≥0 is monotonically decreasing and bounded from
below by 0, and so it converges to some L≥0.
From (2.10), we obtain that(2.11)L=a(1−p1)exp(aT)La+b(1−p1)L(exp(aT)−1).Since ln(1−p1)+aT≤0,
we get L=0.
It is from (2.9) that, for t∈((n+τ−1)T,(n+τ)T],(2.12)x(t)≤x((n+τ−1)T)exp(aT).Therefore, we have limt→∞x(t)=0.Now, we give the basic properties of another impulsive
differential equation as follows:(2.13)y′(t)=−d1y(t),t≠nT,t≠(n+τ−1)T,y(t+)=(1−p2)y(t),t=(n+τ−1)T,y(t+)=y(t)+q,t=nT,y(0+)=y0>0.System (2.13) is a periodically
forced linear system. It is easy to obtain that(2.14)y*(t)={qexp(−d1(t−(n−1)T))1−(1−p2)exp(−d1T),(n−1)T<t≤(n+τ−1)T,q(1−p2)exp(−d1(t−(n−1)T))1−(1−p2)exp(−d1T),(n+τ−1)T<t≤nT,y*(0+)=y*(nT+)=q/(1−(1−p2)exp(−d1T)),
and y*((n+τ−1)T+)=q(1−p2)exp(−d1τT)/(1−(1−p2)exp(−d1T)) is a positive periodic solution of (2.13).
Moreover, we can obtain that(2.15)y(t)={(1−p2)n−1(y(0+)−q(1−p2)e−T1−(1−p2)exp(−d1T))exp(−d1t)+y*(t),(n−1)T<t≤(n+τ−1)T,(1−p2)n(y(0+)−q(1−p2)e−T1−(1−p2)exp(−d1T))exp(−d1t)+y*(t),(n+τ−1)T<t≤nT,is a solution of (2.13). From
(2.14) and (2.15), we get easily the following result.Lemma 2.8.
All solutionsy(t) of (2.13) tend to y*(t). That is, |y(t)−y*(t)|→0 as t→∞.It is from Lemma2.8 that the general solution y(t) of (2.13) can be synchronized with the positive
periodic solution y*(t) of (2.13).


## 3. Extinction and Permanence

Firstly, we
show that all solutions of (1.2) are uniformly ultimately bounded.Theorem 3.1.
There is anR>0 such that x(t)≤R,y(t)≤R, and z(t)≤R for all t large enough, where (x(t),y(t),z(t)) is a solution of system (1.2).Proof.
Let(x(t),y(t),z(t)) be a solution of (1.2) with x0,y0,z0≥0 and let u(t)=(c1/c)x(t)+y(t)+(e1/e2)z(t) for t≥0.
Then, if t≠nT, t≠(n+τ−1)T, and t>0,
then we obtain that(3.1)dudt=−c1bcx2(t)+c1acx(t)−d1y(t)−e1d2e2z(t).Choosing 0<β0<min{d1,d2},
we get(3.2)dudt+β0u(t)≤−c1bcx2(t)+c1c(a+β0)x(t),t≠nT,t≠(n+τ−1)T,t>0.As the right-hand side of (3.2)
is bounded from above by R0=c1(a+β0)2/4bc,
it follows that(3.3)du(t)dt+β0u(t)≤R0,t≠nT,t≠(n+τ−1)T,t>0.If t=nT,
then u(t+)=u(t)+q and if t=(n+τ−1)T,
then u(t+)≤(1−p)u(t),
where p=min{p1,p2,p3}. From Lemma 2.5, we get that(3.4)u(t)≤u0(1−p)[t/kT]exp(∫0t−β0ds)+∫0t(1−p)[(t−s)/kT]exp(∫st−β0dγ)R0ds+∑j=1[t/kT](1−p)[(t−kT)/jT]exp(∫kTt−β0dγ)q≤u0exp(−β0t)+R0β0(1−exp(−β0t)+qexp(β0T)exp(β0T)−1).Since the limit of the
right-hand side of (3.4) as t→∞ is(3.5)R0β0+cqexp(β0T)exp(β0T)−1<∞,it easily follows that u(t) is bounded for sufficiently large t.
Therefore, x(t),y(t), and z(t) are bounded by a constant for sufficiently
large t.
Hence, there is an R>0 such that x(t)≤R,y(t)≤R, and z(t)≤R for a solution (x(t),y(t),z(t)) with all t large enough.Theorem 3.2.
The periodic solution(0,y*(t),0) is locally asymptotically stable
if(3.6)aT+ln(1−p1)c<Γ<d2T−ln(1−p3)e2,where(3.7)Γ=q((1−(1−p2)exp(−d1T)−p2exp(−d1τT)))d1(1−(1−p2)exp(−d1T)).Proof.
The local stability of the periodic solution(0,y*(t),0) of system (1.2) 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 system (1.2). Define u(t)=x(t),v(t)=y(t)−y*(t),w(t)=z(t).
Then they may be written as(3.8)(u(t)v(t)w(t))=Φ(t)(u(0)v(0)w(0)),where Φ(t) satisfies(3.9)dΦdt=(a−cy*(t)00c1y*(t)−d1−e1y*(t)00−d2+e2y*(t))Φ(t)and Φ(0)=I,
the identity matrix. So the fundamental solution matrix is(3.10)Φ(t)=(exp(∫0ta−cy*(s)ds)00exp(c1∫0ty*(s)ds)exp(−d1t)exp(−e1∫0ty*(s)ds)00exp(∫0t−d2+e2y*(s)ds)).The resetting impulsive conditions
of system (1.2) become(3.11)(u((n+τ−1)T+)v((n+τ−1)T+)u((n+τ−1)T+))=(1−p10001−p20001−p3)(u((n+τ−1)T)v((n+τ−1)T)w((n+τ−1)T)),(u(nT+)v(nT+)w(nT+))=(100010001)(u(nT)v(nT)w(nT)).Note that all eigenvalues
of(3.12)S=(1−p10001−p20001−p3)(100010001)Φ(T)are μ1=(1−p1)exp(∫0Ta−cy*(t)dt), μ2=(1−p2)exp(−d1T)<1, and μ3=(1−p3)exp(∫0T−d2+e2y*(t)dt).
Since(3.13)∫0Ty*(t)dt=q(1−(1−p2)exp(−d1T)−p2exp(−d1τT))d1(1−(1−p2)exp(−d1T)),the conditions μ1<1 and μ3<1 are equivalent to the equation (3.7). By
Floquet theory [13,
Chapter 2], we obtain that (0,y*(t),0) is locally asymptotically stable.Theorem 3.3.
The periodic solution(0,y*(t),0) is globally stable if
(3.14)aT+ln(1−p1)≤0,Γ<d2T−ln(1−p3)e2.Proof.
Suppose thataT+ln(1−p1)≤0 and Γ<(d2T−ln(1−p3))/e2.
Then there are sufficiently small numbers ϵ1,ϵ2>0 such that(3.15)ϕ≡(1−p3)exp(−d2T+Λ(d1−c1ϵ1)(1−(1−p2)exp(−(d1−c1ϵ1)T)+e2ϵ2T)<1,where Λ=e2q(1−(1−p2)exp(−(d1−c1ϵ1)T)−p2exp(−(d1−c1ϵ1)τT).
It follows from the first equation in (1.2) that x′(t)=x(t)(a−bx(t)−cy(t))≤x(t)(a−bx(t)) for t≠nT,t≠(n+τ−1)T.
By Lemma 2.4, x(t)≤x˜(t) for t>0,
where x˜(t) is the solution of (2.8). By Lemma 2.7, we get x˜(t)→0 as t→∞, which implies that there is a T1>0 such that x(t)≤ϵ1 for t≥T1.
For the sake of simplicity, we suppose that x(t)≤ϵ1 for all t>0.
We can infer from the second equation in (1.2) that y′(t)=y(t)(−d1+c1x(t)−e1z(t))≤y(t)(−d1+c1x(t))≤y(t)(−d1+c1ϵ1) for t≠nT,t≠(n+τ−1)T.
Let y˜1(t) be the solution of the following
equation:(3.16)y˜1′(t)=−(d1−c1ϵ1)y˜1(t),t≠nT,t≠(n+τ−1)T,y˜1(t+)=(1−p2)y˜1(t),t=(n+τ−1)T,y˜1(t+)=y˜1(t)+q,t=nT,y˜1(0+)=y0.Then we know that y(t)≤y˜1(t) by Lemma 2.4. Thus, from the third equation in
(1.2) and Lemma 2.8, we obtain that(3.17)z′(t)≤z(t)(−d2+e2y˜1(t))≤z(t)(−d2+e2y˜1*(t)+e2ϵ2),where(3.18)y˜1*(t)={qexp(−(d1−c1ϵ1)(t−(n−1)T))1−(1−p2)exp(−(d1−c1ϵ1)T),(n−1)T<t≤(n+τ−1)T,q(1−p2)exp(−(d1−c1ϵ1)(t−(n−1)T))1−(1−p2)exp(−(d1−c1ϵ1)T),(n+τ−1)T<t≤nT,is the periodic solution of
(3.16). Integrating (3.17) on ((n+τ−1)T,(n+τ)T],
we obtain(3.19)z((n+τ)T)≤z((n+τ−1)T+)exp(∫(n+τ−1)T(n+τ)T−d2+e2y˜1*(t)+e2ϵ2dt)=z((n+τ−1)T)ϕ.Therefore, we have z((n+τ)T)≤z(τT)ϕn→0 as n→∞.
Also, we obtain, for t∈((n+τ−1)T,(n+τ)T],(3.20)z(t)≤z((n+τ−1)T+)exp(∫(n+τ−1)Tt−d2+e2y˜1*(t)+e2ϵ1dt)≤z((n+τ−1)T)exp(qe21−(1−p2)exp(−(d1−c1ϵ1)T)+e2ϵ1T),which implies that z(t)→0 as t→∞.
Thus, we may assume that z(t)≤ϵ2 for all t>0.
It is from the second equation in (1.2) that(3.21)y′(t)≥y(t)(−d1+c1x(t)−e1ϵ2)≥y(t)(−d1−e1ϵ2).Let y˜2(t) and y˜2*(t) be the solution and the periodic solution of
(2.13), respectively, with d1 changed into d1+e1ϵ2 and the same initial value y0.
Then we infer from Lemmas 2.4 and 2.7 that y˜2(t)≤y(t)≤y˜1(t) and y˜1(t) and y˜2(t) become close to y˜1*(t) and y˜2*(t) as t→∞,
respectively. Note that y˜1*(t) and y˜2*(t) are close to y*(t) as ϵ1,ϵ2→0.
Therefore, we obtain y(t)→y*(t) as t→∞.Definition 3.4.
System (1.2) is permanent if
there exist M≥m>0 such that, for any solution (x(t),y(t),z(t)) of system (1.2) with x0,y0,z0>0,(3.22)m≤limt→∞infx(t)≤limt→∞supx(t)≤M,m≤limt→∞infy(t)≤limt→∞supy(t)≤M,m≤limt→∞infz(t)≤limt→∞supz(t)≤M.To prove the
permanence of system (1.2), we consider the following two subsystems. If the
top predator is absent, that is, z(t)=0,
then system (1.2) can be expressed as(3.23)x′(t)=x(t)(a−bx(t)−cy(t)),t≠nT,t≠(n+τ−1)T,y′(t)=y(t)(−d1+c1x(t)),t≠nT,t≠(n+τ−1)T,x(t+)=(1−p1)x(t),t≠(n+τ−1)T,y(t+)=(1−p2)y(t),t≠(n+τ−1)T,x(t+)=x(t),t=nT,y(t+)=y(t)+p,t=nT,(x(0+),y(0+))=(x0,y0).If the prey is extinct, then
system (1.2) can be expressed as(3.24)y′(t)=y(t)(−d1−e1z(t)),t≠nT,t≠(n+τ−1)T,z′(t)=z(t)(−d2+e2y(t)),t≠nT,t≠(n+τ−1)T,y(t+)=(1−p2)y(t),t≠(n+τ−1)T,z(t+)=(1−p3)z(t),t≠(n+τ−1)T,y(t+)=y(t)+p,t=nT,z(t+)=z(t),t=nT,(y(0+),z(0+))=(y0,z0).Especially, Liu et al. [23] have given a condition for permanence of subsystem
(3.23).Theorem 3.5 (see [23]).
Subsystem (3.23) is
permanent if Γ<(aT+ln(1−p1))/c,
where(3.25)Γ=q((1−(1−p2)exp(−d1T)−p2exp(−d1τT)))d1(1−(1−p2)exp(−d1T)).Theorem 3.6.
Subsystem (3.24) is permanent if (d2T−ln(1−p3))/e2<Γ,
where(3.26)Γ=q((1−(1−p2)exp(−d1T)−p2exp(−d1τT)))d1(1−(1−p2)exp(−d1T)).Proof.
Let(y(t),z(t)) be a solution of subsystem (3.24) with y(0)>0 and z(0)>0. From Theorem 3.1, we may assume that y(t)≤R with d1+R>0 and z(t)≤R/e1.
Then y′(t)≥−(d1+R)y(t).
From Lemmas 2.4 and 2.8, we have y(t)≥u*(t)−ϵ for sufficiently small ϵ>0,
where(3.27)u*(t)={qexp(−(d1+R)(t−(n−1)T))1−(1−p2)exp(−(d1+R)T),(n−1)T<t≤(n+τ−1)T,q(1−p2)exp(−(d1+R)(t−(n−1)T))1−(1−p2)exp(−(d1+R)T),(n+τ−1)T<t≤nT.Thus, we obtain that y(t)≥(q(exp(−(d1+R)T)/(1−(1−p2)exp(−(d1+R)T)))−ϵ≡m0 for sufficiently large t.
Therefore, we only need to find an m2>0 such that z(t)≥m2 for large enough t. We will do this in the following two steps.
Step 1.
From (3.26), we can
choose m1>0, ϵ1>0 small enough such that   (3.28)Φ≡(1−p3)exp(−d2T+Δ(d1+e1m1)(1−(1−p2)exp(−(d1+e1m1)T))−e2ϵ1T)>1,where Δ=e2q(1−(1−p2)exp(−(d1+e1m1)T)−p2exp(−(d1+e1m1)τT)). In this step, we will show that z(t1)≥m1 for some t1>0.
Suppose that z(t)<m1 for t>0.
Consider the following system:(3.29)v′(t)=−(d1+e1m1)v(t),t≠(n+τ−1)T,nT,w′(t)=−(d2−e2v(t))w(t),t≠(n+τ−1)T,nT,v(t+)=(1−p2)v(t),t=(n+τ−1)T,w(t+)=(1−p3)w(t),t=(n+τ−1)T,v(t+)=v(t)+p,t=nT,w(t+)=w(t),t=nT,(v(0+),w(0+))=(y0,z0).Then, by Lemma 2.4, we obtain z(t)≥w(t).
By Lemma 2.8, we have v(t)≥v*(t)−ϵ1,
where, for t∈((n−1)T,nT],(3.30)v*(t)={qexp(−(d1+e1m1)(t−(n−1)T))1−(1−p2)exp(−(d1+e1m1)T),(n−1)T<t≤(n+τ−1)T,q(1−p2)exp(−(d1+e1m1)(t−(n−1)T))1−(1−p2)exp(−(d1+e1m1)T),(n+τ−1)T<t≤nT.Thus, for t≠(n+τ−1)T,t≠nT,(3.31)w′(t)≥(−d2+e2(v*(t)−ϵ1))w(t).Integrating (3.31) on ((n+τ−1)T,(n+τ)T], we get
(3.32)w((n+τ)T)≥w((n+τ−1)T+)exp(∫(n+τ−1)T(n+τ)T−d2+e2(v*(t)−ϵ1)dt)=w((n+τ−1)T)Φ.Therefore, z((n+τ+k)T)≥w((n+τ+k)T)≥w((n+τ)T)Φk→∞ as k→∞ which is a contradiction to the boundedness of z(t).
Step 2.
Without loss of generality, we may letz(t1)=m1.
If z(t)≥m1 for all t>t1,
then subsystem (3.24) is permanent. If not, we may let t2=inft>t1{z(t)<m1}.
Then z(t)≥m1 for t1≤t≤t2 and, by continuity of z(t),
we have z(t2)=m1 and t1<t2.
There exists a t′(>t2) such that z(t′)≥m1 by Step 1. Set t3=inft>t2{z(t)≥m1}.
Then z(t)<m1 for t2<t<t3 and z(t3)=m1. We can continue this process by using Step 1. If the process is stopped in
finite times, we complete the proof. Otherwise, there exists an interval
sequence [t2k,t2k+1],k∈N, which has the following
properties: z(t)<m1,t∈(t2k,t2k+1), t2k−1<t2k≤t2k+1,
and z(tn)=m1, where k,n∈N.
Let T0=sup{t2k+1−t2k∣k∈N}.
If T0=∞,
then we can take a subsequence {t2ki} satisfying t2ki+1−t2ki→∞ as ki→∞.
As in the proof of Step 1, this will lead to a contradiction to the boundedness
of z(t).
Then we obtain T0<∞.
Note that
(3.33)z(t)≥z(t2k)exp(∫t2kt−d2+e2(v*(s)−ϵ1)ds)≥m1exp(−d2T0)≡m2,t∈(t2k,t2k+1],k∈N.Thus we obtain that liminft→∞z(t)≥m2. Therefore, we complete the proof.Theorem 3.7.
System (1.2) is permanent if(3.34)d2T−ln(1−p3)e2<Γ<aT+ln(1−p1)c.Proof.
Consider the following two
subsystems of system (1.2):(3.35)x1′(t)=x1(t)(a−bx1(t)−cy1(t)),t≠nT,t≠(n+τ−1)T,y1′(t)=y1(t)(−d1+c1x1(t)),t≠nT,t≠(n+τ−1)T,x1(t+)=(1−p1)x1(t),t=(n+τ−1)T,y1(t+)=(1−p2)y1(t),t=(n+τ−1)T,x1(t+)=x1(t),t=nT,y1(t+)=y1(t)+p,t=nT,(x1(0+),y1(0+))=(x0,y0);(3.36)y2′(t)=y2(t)(−d1−e1z2(t)),t≠nT,t≠(n+τ−1)T,z2′(t)=z2(t)(−d2+e2y2(t)),t≠nT,t≠(n+τ−1)T,y2(t+)=(1−p2)y2(t),t=(n+τ−1)T,z2(t+)=(1−p3)z2(t),t=(n+τ−1)T,y2(t+)=y2(t)+p,t=nT,z2(t+)=z2(t),t=nT,(y2(0+),z2(0+))=(y0,z0).It follows from Lemma 2.4 that x1(t)≤x(t), y1(t)≥y(t), y2(t)≤y(t),
and z2(t)≤z(t).
If Γ<(aT+ln(1−p1))/c,
by Theorem 3.5, subsystem (3.35) is permanent. Thus
we can take T1>0 and m1>0 such that x(t)≥m1 for t≥T1.
Further, if (d2T−ln(1−p3))/e2<Γ,
by Theorem 3.6, subsystem (3.36) is also permanent. Therefore, there exist a T2>0 and m2,m3>0 such that y(t)≥m2 and z(t)≥m3 for t≥T2. The proof is complete.


## 4. Numerical Examples

In this section,
we are concerned with the numerical investigation of some situations covered by
Theorems3.2 and 3.7 which may lead to a chaotic behavior of system (1.2). It
is easy to see that the unperturbed three-species food chain system (1.1) has
four nonnegative equilibria:(1)
the trivial
equilibriumA(0,0,0);(2)
the
mid-predator and top-predator free equilibriumB(d1/b,0,0);(3)
the
top-predator free equilibriumC(d1/c1,(ac1−d1b)/cc1,0)⋅(ac1−d1b>0);(4)
the
positive equilibriumE*=(x*,y*,z*) if and only if ae2c1−d2cc1−d1be2>0,
where(4.1)x*=ae2−d2cbe2,y*=d2e2,z*=ae2c1−d2cc1−d1be2be1e2.  The stability of
equilibrium of system (1.1) has been studied by Zhang and Chen [30].Lemma 4.1 (see [30]).
(1) If positive equilibriumE* exists, then E* is globally stable.
(2) If positive equilibriumE* does not exist and C exists, then C is globally stable.
(3) If positive equilibriumE* and C do not exist, then B is globally stable.Throughout this section, we chose(x(0),y(0),z(0))=(5,2,4) as an initial point.Fora=2.0, b=0.001, c=0.5, c1=0.01, d1=0.3, d2=0.2, e1=0.01, e2=0.02, p1=0.3, p2=0.1, p3=0.01, τ=0.2,
and T=5, it follows from Theorem 3.2 that the periodic
solution (0,y*(t),0) is locally stable if 6.3771<q<16.6987.
The unperturbed system (1.1) has a globally stable top-predator free
equilibrium C(20,3.94,0),
but no positive equilibria. The behavior of the trajectories of system (1.2)
when q=10 is depicted in Figure 1. Another behavior is illustrated in
Figure 2 for a=4.0, b=0.001, c=0.5, c1=0.01, d1=0.3, d2=0.02, e1=0.01, e2=0.02, p1=0.3, p2=0.1, p3=0.01, τ=0.2, and T=5. In this case, the trajectory of system (1.2) tends to a periodic orbit of
period T. We know from Theorem 3.7 that system (1.2) is
permanent when 1.8194<q<12.9901. The unperturbed system (1.1) has a global stable positive equilibrium E*=(3500,1,3470).
An example of
chaotic behavior is exhibited in Figure 3 for a=4.0, b=0.0002, c=1.0, c1=0.3, d1=0.3, d2=0.01, e1=0.05, e2=0.0005, p1=0.3, p2=0.1, p3=0.01, τ=0.2, and T=5. In this case, the unperturbed system (1.1)
also has a globally stable top-predator free equilibrium C(2/3,3.99,0),
but no positive equilibria. By Theorem 3.2, we can figure out that system (1.2)
is also locally stable if 6.4951<q<39.7113.
Figure 3 indicates that a trajectory may have chaotic behavior.a=2.0, b=0.001, c=0.5, c1=0.01, d1=0.3, d2=0.2, e1=0.01, e2=0.02, p1=0.3, p2=0.1, p3=0.01, τ=0.2, and T=5. (a)–(c) Time series of system (1.2) when q=10.
(a)(b)(c)a=4.0, b=0.001, c=0.5, c1=0.01, d1=0.3, d2=0.02, e1=0.01, e2=0.02, p1=0.3, p2=0.1, p3=0.01, τ=0.2, and T=5. (a) The trajectory of system (1.2) when q=5. (b)–(d) Time series.
(a)(b)(c)(d)a=4.0, b=0.0002, c=1.0, c1=0.3, d1=0.3, d2=0.01, e1=0.05, e2=0.0005, p1=0.3, p2=0.1, p3=0.01, τ=0.2, and T=5. (a) The trajectory of system (1.2) when q=3. (b) The two-dimensional plot x versus y.
(a)(b)


## 5. Conclusion

In this paper,
we have studied dynamical properties of a food chain system with Lotka-Volterra
functional response and impulsive perturbations. We have found sufficient
conditions for extinction and permanence of the system by means of the Floquet theory
and a comparison theorem. We also have given numerical examples that exhibit a
periodic trajectory and a chaotic behavior.Now, assume thatΓ<(d2T−ln(1−p3))/e2. It follows from Theorem 3.3 that if p1 is large enough to make aT+ln(1−p1)≤0 negative (in other words, if we choose strong
pesticide to eradicate pests), then the lowest-level prey and top-predator free
periodic solution is globally stable, which means that we succeed in
controlling pest population. Further, if we only consider biological control in
system (1.2), that is, if we take τ=0, p1=p2=p3=0, then we obtain with the help of Theorems 3.2 and 3.7 the following results.Theorem 5.1.
Suppose thatτ=p1=p2=p3=0. Then the following statements hold.
(1)
The
periodic solution(0,y*(t),0)is locally asymptotically stable ifad1T/c<q<d1d2T/e2.(2)
System
(1.2) is permanent ifd1d2T/e2<q<ad1T/c.Especially, we get Theorem 3.1 in [30] as corollary of Theorem 5.1(1). From Theorem 5.1, we
note that if there is no chemical control, global stability of a lower-level
prey and top-predator free periodic solution of system (1.2) is not guaranteed.
In other words, it is possible to fail to control pest population by using just
one control strategy. Theoretically speaking, we need to use more than two
different pest control tactics simultaneously to succeed in pest population
control.



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*Source: 752403-2008-10-28.xml*