# An Impulsive Two-Prey One-Predator System with Seasonal Effects

Authors: Hunki Baek
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

In recent years, the impulsive population systems have been studied by many researchers. However, seasonal effects on prey are rarely discussed. Thus, in this paper, the dynamics of the Holling-type IV two-competitive-prey one-predator system with impulsive perturbations and seasonal effects are analyzed using the Floquet theory and comparison techniques. It is assumed that the impulsive perturbations act in a periodic fashion, the proportional impulses (the chemical controls)
for all species and the constant impulse (the biological control) for the predator at different fixed time but,  the same period. In addition, the intrinsic growth rates of prey population are regarded as a periodically varying function of time due to seasonal variations. Sufficient conditions for the local and global stabilities of the two-prey-free periodic solution are established. It is proven that the system is permanent under some conditions. Moreover, sufficient conditions, under which one of the
two preys is extinct and the remaining two species are permanent, are also found. Finally, numerical
examples and conclusion are given.


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## Body

## 1. Introduction

Recently, it is of great interest to study dynamical properties for impulsive perturbations in
population dynamics. Impulsive prey-predator population systems have been
discussed by a number of researchers [1–8] and, what is more, there are also many literatures on
simple multispecies systems consisting of a three-species food chain with
impulsive perturbations [7, 9–18]. Especially, two-prey and
one-predator impulsive systems are drawing notice. For examples, Song and Li [13] studied
dynamical behavior of a Holling type II two-prey one-predator system with
impulsive effect concerning biological control and chemical control strategies
at fixed time.  Zhang  et al. [17, 18] studied a Lotka-Volterra type two-prey one-predator
system with impulsive effect on the predator of a fixed moment.It is necessary and important to consider systems with
periodic ecological parameters which might be quite naturally exposed such as
those due to seasonal effects of weather or food supply [19]. Indeed, it has been studied
that dynamical systems with simple dynamical behavior may display complex
dynamical behavior when they have periodic perturbations[20–22]. For this reason, in this
paper, we consider the intrinsic growth rates A of prey
population as a periodically varying function of time due to seasonal
variations. The seasonality is superimposed as follows[19–22]:(1.1)A0=A(1+ϵsin(ωt)),where the parameter ϵ(i=1,2) represents the degree of seasonality, λ=Aϵ≥0 is the
magnitude of the perturbation in A0, and ω is the angular
frequency of the fluctuation caused by seasonality. It is pertinent to point
out that the forced ecosystem we are studying in this paper is similar to
forced nonlinear oscillators in physics such as the Duffing oscillator.Thus, we develop the Holling-type IV
two-competitive-prey one-predator system with seasonality by introducing a
proportional periodic impulsive poisoning(spraying pesticide) for all species
and a constant periodic releasing, or immigrating, for the predator at
different fixed time as follows:(1.2)x1′(t)=x1(t)(a1+λ1sin(ω1t)−b1x1(t)−γ1x2(t)−e1y(t)1+c1x12(t)),x2′(t)=x2(t)(a2+λ2sin(ω2t)−b2x2(t)−γ2x1(t)−e2y(t)1+c2x22(t)),y′(t)=y(t)(−a3+β1x1(t)1+c1x12(t)+β2x2(t)1+c2x22(t)),t≠nT,t≠(n+τ−1)T,Δx1(t)=−p1x1(t),Δx2(t)=−p2x2(t),Δy(t)=−p3y(t),t=(n+τ−1)T,Δx1(t)=0,Δx2(t)=0,Δy(t)=q,t=nT,(x1(0+),x2(0+),y(0+))=(x01,x02,y0),where ai(i=1,2) are intrinsic
rates of increase, bi(i=1,2) are the
coefficients of intra-specific competition, γi(i=1,2) are parameters
representing competitive effects between two preys, ei(i=1,2) are the
per-capita rates of predation of the predator, ci(i=1,2) are the
half-saturation constants, a3>0 denotes the
death rate of the predator, βi(i=1,2) are the rates
of conversing prey into predator, λi(i=1,2) are the
magnitude, ωi(i=1,2) are the angular
frequency, τ,T are the period
of spaying pesticides (harvesting) and the impulsive immigration or stock of the
predator, respectively, 0≤p1,p2,p3<1 present the
fraction of the preys and the predator which die due to the harvesting or
pesticides, and q is the size of
immigration or stock of the predator.In Section2, we give some notations and lemmas. In
Section 3, we show the boundedness of the system and take into account the local
and global stabilities of two-prey-free periodic solutions by using Floquet
theory for the impulsive equation, small amplitude perturbation skills and
comparison techniques, and finally, prove that the system is permanent under
some conditions. Moreover, we give the sufficient conditions under which one of
the two prey extinct and the remaining two species are permanent. Numerical
examples are given in Section 4.


## 2. Preliminaries

Letℝ+=[0,∞), ℝ+*=(0,∞),  and ℝ+3={x=(x(t),y(t),z(t))∈ℝ3:x(t),y(t),z(t)≥0}. Denote ℕ the set of all
of nonnegative integers and f=(f1,f2,f3)T the right hand
of the first three equations in (1.2). Let V:ℝ+×ℝ+3→ℝ+, then V is said to
belong to class V0 if
(1)
V is continuous on ((n−1)T,(n+τ−1)T]×ℝ+3∪((n+τ−1)T,nT]×ℝ+3, and  lim(t,y)→(t0,x)V(t,y)=V(t0,x) exists, where  t0=(n+τ−1)T+ and nT+,(2)
V is locally Lipschitzian in x.Definition 2.1.
LetV∈V0. For (t,x)∈((n−1)T,(n+τ−1)T]×ℝ+3∪((n+τ−1)T,nT]×ℝ+3, the upper right derivative of V with respect to
the impulsive differential system (1.2) is defined as(2.1)D+V(t,x)=limsuph→0+1h[V(t+h,x+hf(t,x))−V(t,x)].The solution of
system (1.2) is a piecewise continuous function X(t):ℝ+→R+3, X(t) is continuous
on ((n−1)T,(n+τ−1)T)∪((n+τ−1)T,nT)(n∈ℕ,0≤τ≤1). Obviously,  the smoothness properties of f guarantee the
global existence and uniqueness of solutions of system (1.2) [23, 24].Definition 2.2.
The system (1.2) is permanent if there exist M≥m>0 such that, for
any solution (x1(t),x2(t),y(t)) of system (1.2)
with (x01,x02,y0)>0,(2.2)m≤limt→∞infx1(t)≤limt→∞supx1(t)≤M,m≤limt→∞infx2(t)≤limt→∞supx2(t)≤M,m≤limt→∞infy(t)≤limt→∞supy(t)≤M.We
will use a comparison result of impulsive differential inequalities. Suppose
that g:ℝ+×ℝ+→ℝ satisfies the
following hypotheses.(H)g is continuous on ((n−1)T,(n+τ−1)T]×ℝ+∪((n+τ−1)T,nT]×ℝ+ and the limit lim(t,y)→(t0,x)g(t,y)=g(t0,x) exists, where t0=(n+τ−1)T+ and nT+, and is finite for x∈ℝ+ and n∈ℕ.Lemma 2.3 (see [24]).
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≥0,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 Lemma 2.3 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 2.4 (see [24]).
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)+θk,k≥0,u(0+)≤u0,where f,h∈PC(ℝ+,ℝ) and αk≥0,θk and u0 are constants
and (τk)k≥0 is a strictly
increasing sequence of positive real numbers. Then, for t>0,(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)θk. Similar result
can be obtained when all conditions of the inequalities in the Lemmas 2.3 and
2.4 are reversed. Using Lemma 2.4, it is possible to prove that the solutions
of the Cauchy problem (2.4) with strictly positive initial value remain
strictly positive.Lemma 2.5.
The positive octant(ℝ+*)3 is an invariant
region for system (1.2).Proof.
Let(x1(t),x2(t),y(t)):[0,t0)→ℝ3 be a solution
of system (1.2) with a strictly positive initial value (x01,x02,y0). By Lemma 2.4, we can obtain that, for 0≤t<t0,(2.7)x1(t)≥x01(1−p1)[t/T]exp(∫0tg1(s)ds),x2(t)≥x02(1−p2)[t/T]exp(∫0tg2(s)ds),y(t)≥y0(1−p3)[t/T]exp(∫0tg3(s)ds),where g1(s)=a1−λ1−b1x1(s)−γ1x2(s)−e1y(s), g2(s)=a2−λ2−b2x2(s)−γ2x1(s)−e2y(s), and g3(s)=−a3. Thus, x1(t),x2(t) and y(t) remain strictly
positive on [0,t0).Now, we give the basic properties of an impulsive
differential equation as follows:(2.8)y′(t)=−a3y(t),t≠nT,t≠(n+τ−1)T,Δy(t)=−p3y(t),t=(n+τ−1)T,Δy(t)=q,t=nT.System (2.8) is a periodically
forced linear system. It is easy to obtain that(2.9)y*(t)={qexp(−a3(t−(n−1)T))1−(1−p3)exp(−a3T),(n−1)T<t≤(n+τ−1)T,q(1−p3)exp(−a3(t−(n−1)T))1−(1−p3)exp(−a3T),(n+τ−1)T<t≤nT,y*(0+)=y*(nT+)=q/(1−(1−p3)exp(−a3T)), y*((n+τ−1)T+)=(q(1−p3)exp(−a3τT))/(1−(1−p3)exp(−a3T)) is a positive
periodic solution of (2.8). Moreover, we can obtain that(2.10)y(t)={(1−p3)n−1(y(0+)−q(1−p3)e−T1−(1−p3)exp(−a3T))exp(−a3t)+y*(t),(n−1)T<t≤(n+τ−1)T,(1−p3)n(y(0+)−q(1−p3)e−T1−(1−p3)exp(−a3T))exp(−a3t)+y*(t),(n+τ−1)T<t≤nTis a solution of (2.8). From
(2.9) and (2.10), we get easily the following result.Lemma 2.6.
limt→∞|y(t)−y*(t)|=0 for all
solutions y(t) of (2.8) with y(0+)≥0.Therefore, system (1.2) has a two-prey-free
periodic solution(2.11)(0,0,y*(t)).


## 3. Main Results

Theorem 3.1.
The periodic solution(0,0,y*(t)) of system (1.2)
is globally asymptotically stable if for i=1,2,(3.1)(ai+λi)T−bi2eiqΨbi2+ci(ai+λi)2<ln11−pi,where Ψ=(1−(1−p3)exp(−a3T)−p3exp(−a3τT))/a3(1−(1−p3)exp(−a3T)).Proof.
First,
we will prove the local stability of the periodic solution(0,0,y*(t)). For this, consider the following impulsive
differential equation:(3.2)x11′(t)=x11(t)(a1+λ1−b1x11(t)−γ1x12(t)−e1y1(t)1+c1x112(t)),x12′(t)=x12(t)(a2+λ2−b2x12(t)−γ2x11(t)−e2y1(t)1+c2x122(t)),y1′(t)=y1(t)(−a3+β1x11(t)1+c1x112(t)+β2x12(t)1+c2x122(t)),t≠nT,t≠(n+τ−1)T,Δx11(t)=−p1x11(t),Δx12(t)=−p2x12(t),Δy1(t)=−p3y1(t),t=(n+τ−1)T,Δx11(t)=0,Δx12(t)=0,Δy1(t)=q,t=nT,(x11(0+),x12(0+),y1(0+))=(x01,x02,y0).Then, 0≤x1(t)≤x11(t), 0≤x2(t)≤x12(t), and 0≤y(t)≤y1(t) by Lemma 2.3,
where (x1(t),x2(t),y(t)) and (x11(t),x12(t),y1(t)) are solutions
of systems (1.2) and (3.2), respectively. Thus we will show the local stability
of the solution (0,0,y1*(t)) of system
(3.2), where y1*(t)=y*(t). The local stability of the two-pest-free periodic
solution (0,0,y1*(t)) may be
determined by considering the behavior of small amplitude perturbations of the
solution. Let (x11(t),x12(t),y1(t)) be any solution
of system (3.2). Define u(t)=x11(t),v(t)=x12(t),w(t)=y(t)−y1*(t). Then, they may be written as(3.3)(u(t)v(t)w(t))=Φ(t)(u(0)v(0)w(0)),where Φ(t) satisfies(3.4)dΦdt=(a1+λ1−e1y1*(t)000a2+λ2−c2y1*(t)0β1y1*(t)β2y1*(t)−a3)Φ(t)and Φ(0)=I, the identity matrix. So the fundamental solution
matrix is(3.5)Φ(t)=(exp(∫0ta1+λ1−e1y1*(s)ds)000exp(∫0ta2+λ2−e2y1*(s)ds)0exp(∫0tβ1y1*(s)ds)exp(∫0tβ2y1*(s)ds)exp(∫0t−a3ds)).The resetting impulsive
conditions of system (3.2) become(3.6)(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(3.7)S=(1−p10001−p20001−p3)(100010001)Φ(T)are μ1=(1−p1)exp(∫0Ta1+λ1−e1y1*(t)dt), μ2=(1−p2)exp(∫0Ta2+λ2−e2y1*(t)dt), and μ3=(1−p3)exp(−a3T)<1. Since(3.8)(a1+λ1)T−e1qΨ<(a1+λ1)T−b12e1qΨb12+c1(a1+λ1)2,(a2+λ2)T−e2qΨ<(a2+λ2)T−b22e2qΨb22+c2(a2+λ2)2,∫0Ty1*(t)dt=q(1−(1−p3)exp(−a3T)−p3exp(−a3τT))a3(1−(1−p3)exp(−a3T)),we obtain from (3.1) that the
conditions |μ1|<1 and |μ2|<1 hold.
Therefore, from the Floquet theory [23], we obtain (0,0,y*(t)) is locally
stable.
Now, to prove the global stability of the
two-prey-free periodic solution, let(x1(t),x2(t),y(t)) be a solution
of system (1.2). From (3.1), we can take a sufficiently small number ϵ1>0 satisfying(3.9)ζ=(1−p1)exp((a1+λ1)T+b12e1(ϵ1T−qΨ)b12+c1((a1+λ1)2+2(a1+λ1)b1ϵ1+b12ϵ12))<1.It follows from the first
equation in (1.2) that x1′(t)≤x1(t)(a1+λ1−b1x1(t)) for t≠nT,t≠(n+τ−1)T and x1(t+)=(1−p1)x1(t)≤x1(t) for t=(n+τ−1)T. Then, from Lemma 2.3, we have x1(t)≤u(t), where u(t) is a solution
of the following impulsive differential equation:(3.10)u′(t)=u(t)(a1+λ1−b1u(t)),t≠nT,t≠(n+τ−1)T,Δu(t)=0,t=nT,t=(n+τ−1)T,u(0+)=x1(0+).Since u(t)→(a1+λ1)/b1 as t→∞, x1(t)≤(a1+λ1)b1+ϵ for any ϵ>0 with t large enough.
For simplicity, we may assume that x1(t)≤(a1+λ1)/b1+ϵ1 for all t>0. Similarly, we get x2(t)≤(a2+λ2)/b2+ϵ2 for any ϵ2>0 and t>0. Since y′(t)≥−a3y(t) for t≠nT,(n+τ−1)T, we can obtain from Lemmas 2.3 and 2.6
that(3.11)y(t)>y*(t)−ϵ1,for t sufficiently
large. Without loss of generality, we may suppose that (3.11) holds for all t≥0. From (1.2), and (3.11) we obtain(3.12)x1′(t)≤x1(t)(a1+λ1−e1(y*(t)−ϵ1)1+c1(((a1+λ1)/b1)+ϵ1)2),t≠nT,t≠(n+τ−1)T,Δx1(t)=−p1x1(t),t=(n+τ−1)T,Δx1(t)=0,t=nT.Integrating (3.12) on ((n+τ−1)T,(n+τ)T], we get(3.13)x1((n+τ)T)≤x1((n+τ−1)T+)exp(∫(n+τ−1)T(n+τ)Ta1+λ1−e1(y*(t)−ϵ1)1+c1(((a1+λ1)/b1)+ϵ1)2dt)=x1((n+τ−1)T)ζ,and thus x1((n+τ)T)≤x1(τT)ζn which implies
that x1((n+τ)T)→0 as n→∞. Further, we obtain, for t∈((n+τ−1)T,(n+τ)T],(3.14)x1(t)≤x1((n+τ−1)T+)exp(∫(n+τ−1)Tta1+λ1−e1(y*(t)−ϵ1)1+c1(((a1+λ1)/b1)+ϵ1)2dt)≤x1((n+τ−1)T)exp((a1+λ1+e1ϵ1)T),which implies that x1(t)→0 as t→∞. Similarly, we obtain x2(t)→0 as t→∞. Now, take sufficiently small positive numbers ϵ3 and ϵ4 satisfying β1ϵ3+β2ϵ4<a3 to prove that y(t)→y*(t) as t→∞. Without loss of generality, we may assume that x1(t)≤ϵ3 and x2(t)≤ϵ4 for all t≥0. It follows from the third equation in (1.2) that,
for t≠(n+τ−1)T and t≠nT,(3.15)y′(t)≤y(t)(−a3+β1ϵ3+β2ϵ4).Thus, by Lemma 2.3, we induce
that y(t)≤y˜*(t), where y˜*(t) is the solution
of (2.8) with a3 changed into a3−β1ϵ3−β2ϵ4. Therefore, by taking sufficiently small ϵ3 and ϵ4, we obtain from Lemma 2.6 and (3.11) that y(t) tends to y*(t) as t→∞.Let Fi(T)=(ai+λi)T−(bi2eiqΨ/(bi2+ci(ai+λi)2))+ln(1−pi)  for i=1,2. Then Fi(0)=ln(1−pi)<0, Fi(T)→∞, and Fi′′(T)>0. Thus Fi(T) has a unique
positive solution Ti.Corollary 3.2.
The periodic solution(0,0,y*(t)) of system (1.2)
is globally asymptotically stable if T<min{T1,T2}.From the proof of Theorem 3.1, we can easily
get the following corollary.Corollary 3.3.
The periodic solution(0,0,y*(t)) of system (1.2)
is locally stable if(3.16)(a1+λ1)T−e1qΨ<ln11−p1,(a2+λ2)T−e2qΨ<ln11−p2. Furthermore,
the periodic solution of system (1.2) may remain globally stable even if there
are no the seasonal effects on system (1.2).Corollary 3.4.
Suppose thatλ1=λ2=0. Then, the periodic solution (0,0,y*(t)) of system (1.2)
is globally asymptotically stable if(3.17)a1T−b12e1qΨb12+c1a12<ln11−p1,a2T−b22e2qΨb22+c2a22<ln11−p2.Now, we show that all solutions of system
(1.2) are uniformly bounded.Theorem 3.5.
There is anM>0 such that x1(t)≤M,x2(t)≤M, and y(t)≤M for all t large enough,
where (x1(t),x2(t),y(t)) is a solution
of system (1.2).Proof.
Let(x1(t),x2(t),y(t)) be a solution
of (1.2) with x01,x02,y0≥0 and let F(t)=(β1/e1)x1(t)+(β2/e2)x2(t)+y(t) for t>0. Then, if t≠nT and t≠(n+τ−1)T, we obtain that (dF(t)/dt)+δF(t)=−(b1β1/e1)x12(t)+(β1/e1)(a1+λ1sin(ω1t)+δ)x1(t)−(β1γ1/e1)x1(t)x2(t)−(b2β2/e2)x22(t)+(β2/e2)(a2+λ2sin(ω2t)+δ)x2(t)−(β2γ2/e2)x1(t)x2(t)+(δ−a3)y(t). From choosing 0<δ0<a3, we have, for t≠nT,t≠(n+τ−1)T and t>0,(3.18)dF(t)dt+δ0F(t)≤−b1β1e1x12(t)+β1e1(a1+λ1+δ0)x1(t)−b2β2e2x22(t)+β2e2(a2+λ2+δ0)x2(t).As the right-hand side of (3.18)
is bounded from above by M0=(β1(a1+λ1+δ0)2/4b1e1)+(β2(a2+λ2+δ0)2/4b2e2), it follows that(3.19)dF(t)dt+δ0F(t)≤M0,t≠nT,n≠(n+τ−1)T,t>0.If t=nT, then ΔF(t)=q and if t=(n+τ−1)T, then ΔF(t)≤−pF(t), where p=min{p1,p2,p3}. From Lemma 2.4, we get that(3.20)F(t)≤F0(1−p)[t/kT]exp(∫0t−δ0ds)+∫0t(1−p)[(t−s)/kT]exp(∫st−δ0dγ)M0ds+∑j=1[t/kT](1−p)[(t−kT)/jT]exp(∫kTt−δ0dγ)q≤F0exp(−δ0t)+M0δ0(1−exp(−δ0t))+qexp(δ0T)exp(δ0T)−1,where F0=(β1/e1)x01+(β2/e2)x02+y0. Since the limit of the right-hand side of (3.20) as t→∞ is(3.21)M0δ0+qexp(δ0T)exp(δ0T)−1<∞,it easily follows that F(t) is bounded for
sufficiently large t. Therefore, x1(t),x2(t) and y(t) are bounded by
a constant M for
sufficiently large t.Theorem 3.6.
System (1.2) is permanent if a3>max{((a1−λ1)β1/b1),((a2−λ2)β2/b2)},(3.22)(a1−λ1−γ1(a2−λ2)b2)T−e1qΘ2>ln11−p1,(a2−λ2−γ2(a1−λ1)b1)T−e2qΘ1>ln11−p2,where (3.23)Θi=1−(1−p3)exp((−a3+(βi(ai−λi)/bi))T)−p3exp((−a3+(βi(ai−λi)/bi))τT)(a3−(βi(ai−λi)/bi))(1−(1−p3)exp((−a3+(βi(ai−λi)/bi))T)),i=1,2.Proof.
Let(x1(t),x2(t),y(t)) be a solution
of system (1.2) with (x01,x02,y0)>0. From Theorem
3.5, we may assume that x1(t),x2(t),y(t)≤M and M>max{(a1/e1),(a2/e2)}. Thus, we only need to prove the existence of the
lower bound m. For this, we consider the following impulsive differential
equation:(3.24)x21′(t)=x21(t)(a1−λ1−b1x21(t)−γ1x22(t)−e1y2(t)1+c1x212(t)),x22′(t)=x22(t)(a2−λ2−b2x22(t)−γ2x21(t)−e2y2(t)1+c2x222(t)),y2′(t)=y2(t)(−a3+β1x21(t)1+c1x112(t)+β2x22(t)1+c2x222(t)),t≠nT,t≠(n+τ−1)T,Δx21(t)=−p1x21(t),Δx22(t)=−p2x22(t),Δy2(t)=−p3y2(t),t=(n+τ−1)T,Δx21(t)=0,Δx22(t)=0,Δy2(t)=q,t=nT,(x21(0+),x22(0+),y2(0+))=(x01,x02,y0).Then, x1(t)≥x21(t), x2(t)≥x22(t) and y(t)≥y2(t) by Lemma 2.3,
where (x1(t),x2(t),y(t)), and (x21(t),x22,y2(t)) are solutions
of systems (1.2) and (3.24), respectively. So, we will show that x21(t)≥m1,x22(t)≥m1, and y2(t)≥m1. As in the proof of Theorem 3.1, we can show that x21(t)≤((a1−λ1)/b1)+ϵ1 and x22(t)≤((a2−λ2)/b2)+ϵ2 for t>0. Let m=(q(1−p3)exp(−a3T)/1−(1−p3)exp(−a3T))−ϵ for ϵ>0. Since y2(t)≥−a3y(t) for t≠nT,t≠(n+τ−1)T, it follows from Lemmas 2.3 and 2.6 that y2(t)>y*(t)−ϵ and hence y2(t)>m for
sufficiently large t. Thus we only need to find m¯1 and m¯2 such that x21(t)≥m¯1 and x22(t)≥m¯2 for t large enough.
We will do this in the following two steps.
Step 1.
First, take sufficiently small positive numbersm1 and m2 such that m1<(1/β1)(a3−β2(((a2−λ2)/b2)+ϵ2)), m2<(1/β2)(a3−β1(((a1−λ1)/b1)+ϵ1)) and β1m1+β2m2<a3. We will prove, there exist t1,t2∈(0,∞) such that x21(t1)≥m1 and x22(t2)≥m2. Suppose not. Then that, we have only the following three
cases:
(i)
there exists at2>0 such that x22(t2)≥m2, but x21(t)<m1, for all t>0;(ii)
there exists at1>0 such that x21(t1)≥m1, but x22(t)<m2, for all t>0;(iii)
x21(t)<m1 and x22(t)<m2 for all t>0.
Case (i): from
(3.22) we can take η1>0 small enough
such that(3.25)ϕ1=(1−p1)exp((a1−λ1−b1m1−γ1(a2−λ2b2+ϵ2)−e1η1)T−e1qΘ2)>1. We obtain from the condition of
case (i) that y′(t)≤y2(t)(−a3+β1x21(t)+β2x22(t))≤y2(t)(−a3+β1m1+β2(((a2−λ2)/b2)+ϵ2))≡E1y2(t) for t≠nT,t≠(n+τ−1)T, where E1=−a3+β1m1+β2(((a2−λ2)/b2)+ϵ2)<0. Thus we have y2(t)≤u(t) and u(t)→u*(t) as t→∞, where u(t) is a solution
of system(3.26)u′(t)=E1u(t),t≠nT,t≠(n+τ−1)T,Δu(t)=−p3u(t),t=(n+τ−1)T,Δu(t)=q,t=nT,u(0+)=y(0+),u*(t)={qexp(E1(t−(n−1)T))1−(1−p3)exp(E1T),(n−1)T<t≤(n+τ−1)T,q(1−p3)exp(E1(t−(n−1)T))1−(1−p3)exp(E1T),(n+τ−1)T<t≤nT.Therefore, we can take a T1>0 such that y(t)≤u(t)<u*(t)+η1 for t>T1. Thus we get(3.27)x21′(t)≥x21(t)(a1−λ1−b1m1−γ1(a2−λ2b2+ϵ2)−e1(u*(t)+η1)),t≠nT,t≠(n+τ−1)T,Δx21(t)=−p1x21(t),t=(n+τ−1)T,Δx21(t)=0,t=nT,for t>T1. Let N1∈ℕ be such that (N1+τ−1)T≥T1. Integrating (3.17) on ((n+τ−1)T,(n+τ)T],n≥N1, we can obtain that x21((n+τ)T)≥x21((n+τ−1)T)(1−p1)exp(∫(n+τ−1)T(n+τ)Ta1−λ1−b1m1−γ1(((a2−λ2)/b2)+ϵ2)−e1(v*(t)+η1)dt=x21((n+τ−1)T)ϕ1. Thus x21((N1+k+τ)T)≥x21((N1+τ)T)ϕ1k→∞ as k→∞, which is a contradiction to the boundedness of x21(t).
Case (ii): the
same argument as the case (i) can be applied. So we omit it.
Case (iii): we
chooseη2>0 sufficiently
small so that(3.28)ϕ2=(1−p1)exp((a1−λ1−b1m1−γ1m2)T−e1(qΨ+η2T))>1. Then we obtain y2′(t)≤y2(t)(−a3+β1m1+β2m2)≡E2y2(t) for t≠nT,t≠(n+τ−1)T, where E2=−a3+β1m1+β2m2<0. It follows from
Lemmas 2.3 and 2.6 that y2(t)≤w(t) and w(t)→w*(t) as t→∞, where w(t) is a solution
of the following system:(3.29)w′(t)=E2w(t),t≠nT,t≠(n+τ−1)T,Δw(t)=−p3w(t),t=(n+τ−1)T,Δw(t)=q,t=nT,w(0+)=y(0+),w*(t)={qexp(E2(t−(n−1)T))1−(1−p3)exp(E2T),(n−1)T<t≤(n+τ−1)T,q(1−p3)exp(E2(t−(n−1)T))1−(1−p3)exp(E2T),(n+τ−1)T<t≤nT.Thus there exists a T2>0 such that y2(t)≤w(t)<w*(t)+η2 for t>T2 and(3.30)x21′(t)≥x21(t)(a1−λ1−b1m1−γ1m2−e1(w*(t)+η2)),t≠nT,t≠(n+τ−1)T,Δx21(t)=−p1x21(t),t=(n+τ−1)T,Δx21(t)=0,t=nT,for t>T2. Let N2∈ℕ be such that (N2+τ−1)T≥T2. Integrating (3.30) on ((n+τ−1)T,(n+τ)T],n≥N2, we can obtain that x21((n+τ)T)≥x21((n+τ−1)T)(1−p1)exp(∫(n+τ−1)T(n+τ)a1−λ1−b1m1−γ1m2−e1(w*(t)+η2)dt)=x21((n+τ−1)T)ϕ2. Similarly, we have x21((N2+k+τ)T)≥x21((N2+τ)T)ϕ2k→∞ as k→∞, which is a contradiction to the boundedness of x21(t). Therefore, there exist t1>0 and t2>0 such that x21(t1)≥m1 and x22(t2)≥m2.
Step 2.
Ifx21(t)≥m1 for all t≥t1, then we are done. If not, we may let t*=inft>t1{x21(t)<m1}. Then,  x21(t)≥m1 for t∈[t1,t*] and, by the
continuity of x21(t), we have x21(t*)=m1. In this step, we have only to consider two possible cases.
(i) Suppose thatt*=(n1+τ−1)T for some n1∈ℕ. Then,  (1−p1)m1≤x21(t*+)=(1−p1)x21(t*)<m1. Select n2,n3∈ℕ such that (n2−1)T>ln(η1/(M+q))/E1 and (1−p1)n2ϕ1n3exp(n2σT)>(1−p1)n2ϕ1n3exp((n2+1)σT)>1, where σ=a1−λ1−b1m1−γ1(((a2−λ2)/b2)+ϵ2)−e1M<0. Let T′=n2T+n3T. In this case, we will show that there exists t3∈(t*,t*+T′] such that x21(t3)≥m1. Otherwise, by (2.10) and (3.26) with v(n1T+)=y(n1T+), we have(3.31)v(t)={(1−p3)n−(n1+1)(v(n1T+)−q(1−p3)exp(−T)1−(1−p3)exp(E1T))exp(E1(t−n1T))+v*(t),(n−1)T<t≤(n+τ−1)T,(1−p3)(n−n1)(v(n1T+)−q(1−p3)exp(−T)1−(1−p3)exp(E1T))exp(E1(t−n1T))+v*(t),(n+τ−1)T<t≤nT,and n1+1≤n≤n1+1+n2+n3. So we get |v(t)−v*(t)|≤(M+q)exp(E1(t−n1T))<η1 and y2(t)≤v(t)≤v*(t)+η1 for n1T+(n2−1)T≤t≤t*+T′, which implies that (3.27) holds for t∈[t*+n2T,t*+T′]. As in step 1, we have(3.32)x21(t*+T′)≥x21(t*+n2T)ϕ1n3.Since y2(t)≤M, we have(3.33)x21′(t)≥x21(t)(a1−λ1−b1m1−γ1(a2−λ2b2+ϵ2)−e1M)=σx21(t),t≠nT,t≠(n+τ−1)T,Δx21(t)=−p1x21(t),t=(n+τ−1)T,Δx21(t)=0,t=nT,for t∈[t*,t*+n2T]. Integrating
(3.33) on [t*,t*+n2T], we
have(3.34)x21((t*+n2T))≥m1exp(σn2T)>m1.Thus x21(t*+T′)≥m1(1−p1)n2exp(σn2T)ϕ1n3>m1 which is a
contradiction. Now, let t¯ = inft>t*{x21(t)≥m1}. Then, x21(t)≤m1 for t*≤t<t¯ and x21(t¯)=m1. So, we have, for t∈[t*,t¯), x21(t)≥m1(1−p1)n2+n3exp(σ(n2+n3)T)≡m¯1. For t>t* the same
argument can be continued since x21(t¯)≥m1. Hence x21(t)≥m¯1 for all t>t1.
(ii)t*≠(n+τ−1)T,n∈ℕ. Suppose that t*∈((n1′+τ−1)T,(n1′+τ)T) for some n1′∈ℕ. There are two possible cases for t∈(t*,(n1′+τ)T). Firstly, if x21(t)≤m1 for all t∈(t*,(n1′+τ)T), similar to case (i), we can prove there must be a t3′∈[(n1′+τ)T,(n1′+τ)T+T′] such that x21(t3′)≥m1. Here we omit it. Let t^=inft>t*{x21(t)≥m1}. Then,  x21(t)≤m1 for t∈(t*,t^) and x21(t^)=m1. For t∈(t*,t^), we have x21(t)≥m1(1−p1)n2+n3exp(σ(n2+n3+1)T)=m1. So, m1<m¯1 and x21(t)≥m1 for t∈(t*,t^). For t>t* the same
argument can be continued since x21(t^)≥m1. Hence x21(t)≥m¯1 for all t>t1. Secondly, if there exists a t∈(t*,(n1′+τ)T) such  that x21(t)≥m1. Let t˘=inft>t*{x21(t)≥m1}. Then x21(t)≤m1 for t∈(t*,t˘) and x21(t˘)=m1. For t∈(t*,t˘), we have x21(t)≥x21(t*)exp(σ(t−t*))≥m1exp(σT)>m1. This process can be continued since x21(t˘)≥m1, and have x21(t)≥m¯1 for all t>t1. Similarly, we can show that x22(t)≥m¯2 for all t>t2. This completes the proof.Corollary 3.7.
Suppose thatλ1=λ2=0. Then, system (1.2) is permanent if a3>max{(a1β1/b1),(a2β2/b2)}, (a1−(γ1a2/b2))T−e1qΘ2>ln1/(1−p1),and(a2−(γ2a1/b1))T−e2qΘ1>ln1/(1−p2).It follows from
Theorems3.1 and 3.6 that one of the two prey extinct and the remaining two
species are permanent under some conditions.
Corollary 3.8.
Let(x1(t),x2(t),y(t)) be any solution
of system (1.2). Then, x1 and y(t) are permanent,
and x2(t)→0 as t→∞ provided that a3>(a2−λ2)β2/b2,  (a1−λ1−(γ1(a2−λ2)/b2))T−e1qΘ2>ln(1/(1−p1)), and (a2+λ2)T−(b22e2qΨ/(b22+c2(a2+λ2)2))<ln(1/(1−p2)).Corollary 3.9.
Let(x1(t),x2(t),y(t)) be any solution
of system (1.2). Then, x2 and y(t) are permanent,
and x1(t)→0 as t→∞ provided that a3>(a1−λ1)β1/b1,  (a1+λ1)T−(b12e1qΨ/(b12+c1(a1+λ1)2))<ln1/(1−p1) and (a2−λ2−(γ2(a1−λ1)/b1))T−e2qΘ1>ln1/(1−p2).


## 4. Numerical Examples

In this
section, we will give some examples.Case 1.
If we takea1=1.1,a2=1.0,a3=0.7,b1=1.0,b2=1.2,c1=0.9,c2=1.0,e1=0.3,e2=0.2,γ1=0.1,γ2=0.1,β1=0.8,β2=1.0,p1=0.7,p2=0.8,p3=0.001,τ=0.6,T=1,q=5,λ1=0.5, and λ2=0.6, then these parameters satisfy the condition of
Theorem 3.1. Thus the periodic solution (0,0,y*(t)) is globally
asymptotically stable. (See Figure 1.)(a)–(c) Time series of system (1.2) with an initial
value (2,3,1),  (d)–(f) time series of system (1.2) with an initial
value (100,100,100).
(a)(b)(c)(d)(e)(f)Case 2.
Leta1=5,a2=4,a3=0.7,b1=1,b2=1.2,c1=0.9,c2=1,e1=0.3,e2=0.2,γ1=0.1,γ2=0.1,β1=0.1,β2=0.2,p1=0.3,p2=0.4,p3=0.001,τ=0.6,T=4,q=2,λ1=4.0 and λ2=3. Then, from Theorem 3.6, we know that system (1.2) is
permanent, (see Figure 2).(a)–(c) Time series. (d) The trajectory of system
(1.2) with an initial value (2,3,1).
(a)(b)(c)(d)Case 3.
From Corollary3.8, we obtain that x1 and y(t) are permanent,
and x2(t)→0 as t→∞, where (x1(t),x2(t),y(t)) is a solution
of system (1.2) with a1=5,a2=1.3,a3=0.7,b1=1,b2=0.8,c1=0.9,c2=1,e1=0.3,e2=1,γ1=0.1,γ2=0.1,β1=0.8,β2=0.9,p1=0.5,p2=0.8,p3=0.001,τ=0.6,λ1=1,λ2=0.5,T=1, and q=2. Figure 3 exhibits this phenomenon.(a)–(c) Time series. (d) The trajectory of system
(1.2) with an initial value (2,3,1).
(a)(b)(c)(d)Case 4.
It follows from Corollary3.9 that x2 and y(t) are permanent,
and x1(t)→0 as t→∞, where (x1(t),x2(t),y(t)) is a solution
of system (1.2) with a1=1,a2=5,a3=0.8,b1=0.8,b2=0.8,c1=0.8,c2=1,e1=1,e2=0.2,γ1=0.1,γ2=0.1,β1=0.9,β2=0.8,p1=0.8,p2=0.5,p3=0.001,τ=0.6,T=1,q=3,λ1=0.5 and λ2=1. (see Figure 4.)(a)–(c) Time series. (d) The trajectory of system
(1.2) with an initial value (2,3,1).
(a)(b)(c)(d)


## 5. Conclusion

In this paper, based on a Holling-type IV two-prey
one-predator system, we established an impulsive differential equation to model
the seasonal effects, the process of a proportional periodic impulsive
harvesting, and a constant periodic releasing of the predator at different fixed
time. Using the Floquet theory of impulsive differential equation and small
amplitude perturbation skills, we proved that there exists a globally (locally)
stable two-prey-free periodic solution if the conditions of Theorem3.1 (Corollary 3.3) are satisfied. Also, we established permanence conditions of
system (1.2) via the method of comparison including multiple Liapunov functions
when the conditions of Theorem 3.6 are satisfied. From main Theorems, we easily
obtained the sufficient conditions under which one of the two preys is extinct
and whic of the remaining two species are permanent. In addition, we gave some examples
by using numerical simulations.



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*Source: 793732-2009-03-29.xml*