# Stability and Permanence of a Pest Management Model with Impulsive Releasing and Harvesting

Authors: Jiangtao Yang; Zhichun Yang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We formulate a pest management model with periodically releasing infective pests, immature and mature natural enemies, and harvesting pests and crops at two different fixed moments. Sufficient conditions ensuring the locally and globally asymptotical stability of the susceptible pest-eradication period solution are found by means of Floquet theory, small amplitude perturbation techniques, and multicomparison results. Furthermore, the permanence of system is also derived. By numerical analysis, we also show that impulsive releasing and harvesting at two different fixed moments can bring obvious effects on the dynamics of system, which also corroborates our theoretical results.


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## Body

## 1. Introduction

As is known to all, pest outbreaks often cause serious ecological and economic problems. Therefore, how to effectively control insects and other arthropods has become an increasingly complex issue. Usually, chemical pesticides were taken as a relatively simple way to solve the pest-related problems, and some mathematical models on pest management with toxin (pesticide) input were studied in [1–4]. However, the overuse of chemical pesticides may create new ecological and sociological harm such as pesticide pollution and pesticide-resistant pest varieties and inflicts harmful effects on humans and so forth. Therefore, nonchemical use instead for pest control has become a hot topic in order to reduce pest density to tolerable levels and minimize the damage caused. For instance, biological control methods by periodically releasing infective pests or their natural enemies are often taken due to their advantage in the aspects of self-sustainable mechanism, lower environmental impact, and cost effectiveness.Recently, some biocontrol models on pest management described by impulsive differential equation were proposed and the dynamics such as stability, permanence, periodicity, and bifurcation are deeply investigated (see also, e.g., [2–12]). In [5], an impulsive system to model the process of periodic releasing natural enemies and harvesting pest at different fixed time for pest control is considered, and the sufficient conditions on the existence and global stability of the periodic solution are derived for the given model. Georgescu et al. [6, 7] construct an integrated pest management model which relies on the simultaneous periodic release of infective pest individuals and of natural predators with age structure and obtain some sufficient conditions on the local and global stability, permanence, and bifurcation of the systems. However, most of the existing models on pest management scarcely take into account the factor on the relation between pest and its food (e.g., crop). In fact, farmers may harvest crops several times in process of its growth, which should cause a great impact on the density of the pest.Motived by the above discussion, we construct a model of pest control by periodically releasing infective pests, immature and mature natural enemies, and harvesting pests and crops. To account for the discontinuity of release and harvest at different fixed moments, our model is based on impulsive differential equations. We analyze the dynamical behavior of the system by using the theory of impulsive differential equation introduced in [13–15].The rest of this paper is organized as follows. A pest management model with impulsive releasing and harvesting is introduced in Section2 and some useful preliminaries are given in Section 3. Section 4 deals with stability and permanence analysis of system. In this section, two sufficient conditions are deduced including the locally and globally asymptotical stability of the susceptible pest-eradication period solution, the permanence of system is also discussed. A simple example and conclusions are given in Section 5.


## 2. Model Description

In the following, to establish our pest management model, we rely on the following biological assumptions.(A1)
The pest population is divided into two classes, the susceptible and infective. The infective pests neither recover nor reproduce and infective pests cannot damage crops. The disease is transmitted from infective pests to susceptible pests and does not propagate to predators.(A2)
In the absence of susceptible pests, the crops have a logistic growth rate with intrinsic birth rater and carrying capacity K.(A3)
The predators (natural enemies) have an age structure, that is, immature and mature. Only the mature predators have the ability to feed on susceptible pests, but do not prey on infective pests and crops.(A4)
The functional response of the susceptible pest is described by the abstract functionP1, the functional response of the mature predator is described by the abstract function P2, and the infection rate is described by the abstract function g, where P1, P2, and g satisfy certain assumptions outlined below.On the basis of the above assumptions, we establish the following impulsively controlled system:(1)x′(t)=rx(t)(1-x(t)K)-P1(x(t))S(t),S′(t)=βP1(x(t))S(t)-g(I(t))S(t)-P2(S(t))yM(t)-dSS(t),I′(t)=g(I(t))S(t)-dII(t),yJ′(t)=λP2(S(t))yM(t)-dJyJ(t)-myJ(t),yM′(t)=myJ(t)-dMyM(t),t≠(n+τ-1)T,t≠nT,Δx(t)=-δx(t),ΔS(t)=-PSS(t),ΔI(t)=-PII(t),ΔyJ(t)=-PJyJ(t),ΔyM(t)=-PMyM(t),t=(n+τ-1)T,Δx(t)=0,ΔS(t)=0,ΔI(t)=δI,ΔyJ(t)=δJ,ΔyM(t)=δM,t=nT,n∈ℕ,
where x(t) represents the density of the crop at time t, S(t) represents the density of the susceptible pest at time t, I(t) represents the density of the infective pest at time t, yJ(t) and yM(t) represent the density of the immature and mature predator at time t, respectively; r is the logistic intrinsic growth rate of the crop in the absence of the susceptible pest, K is its carrying capacity; 0<β, λ≤1 represent the conversion rate at which ingested preys in excess of what is needed for maintenance is translated into predator population increase; m is the rate at which the immature predators become the mature predators. dS, dI, dJ, dM>0 are the death rates of the susceptible pest population, infective pest population, and of the immature and mature predator population, respectively; Δx(t)=x(t+)-x(t), ΔS(t)=S(t+)-S(t), ΔI(t)=I(t+)-I(t), ΔyJ(t)=yJ(t+)-yJ(t), ΔyM(t)=yM(t+)-yM(t); T is the period of the impulsive effect; δ(0≤δ<(1-e-rT)/2) is the harvesting rate of crop population; 0≤PS, PI, PJ, PM<1 denote the transfer rate of susceptible pest population, infective pest population, immature and mature predator population at every impulsive period (n+τ-1)T(n∈ℕ,0<τ<1), respectively; δI, δJ, δM>0 represents the amount of infective pests, immature and mature predators, respectively, which are released at every impulsive period nT(n∈ℕ), respectively; Also, P1(·), P2(·), g(·)∈H, here H={f:R→R∣f(0)=0, f′(x)>0 and f′′(x)≤0 for all x>0}.Some familiar examples of functionsf∈H in the biological literature include (F1)
f1(x)=ax, with a>0;(F2)
f2(x)=ax/(1+bx), with a,b>0;(F3)
f3(x)=a(1-e-bx), with a,b>0,where functions (F1) and (F2) are known as Holling type functional responses (see, [16–26]), and (F3) belongs to Ivlev type functional responses (see, [27–30]).


## 3. Preliminaries

In this section, we will give some definitions and lemmas, which will be useful for our main results. Letℝ+=[0,∞) and ℝ+5={X=(x(t),S(t),I(t),yJ(t),yM(t))∈ℝ5∣x(t),S(t),I(t),yJ(t),yM(t)≥0}. Denote f=(fx,fS,fI,fJ,fM)T  the map defined by the right hand of the first five equations in system (1). Let V:ℝ+×ℝ+5→ℝ+, then V∈V0 if (1)
V is continuous in ((n-1)T,(n+τ-1)T]×ℝ+5, ((n+τ-1)T,nT]×ℝ+5 and for each x∈R+5,n∈ℕ, lim(t,y)→((n+τ-1)T+,x)V(t,y)=V((n+τ-1)T+,x) and lim(t,y)→(nT+,x)V(t,y)=V(nT+,x) exist.(2)
V is locally Lipschitzian x.Definition 1.
LettingV∈V0, one defines the upper right derivative of V with respect to the impulsive differential system (1) at (t,x)∈((n-1)T,(n+τ-1)T]×ℝ+5 and ((n+τ-1)T,nT]×ℝ+5 by
(2)D+V(t,x)=limsuph→0+1h[V(t+h,x+hf(t,x))-V(t,x)].Definition 2.
The system (1) is said to be permanent if there are positive constants m,M>0 and a finite time T0 such that all solutions of (1) with initial values x(0+), S(0+), I(0+), yJ(0+), yM(0+), m≤x(t), S(t), I(t), yJ(t), yM(t)≤M hold for all t≥T0, where m and M  are independent of initial value, T0 may depend on initial value.Remark 3.
The global existence and uniqueness of system (1) is guaranteed by the smoothness properties of f (for details, see [13, 14]).Lemma 4 (see [15]).
LetV:ℝ+×ℝn→ℝ+m satisfy Vi∈V0,i=1,2,…,m, and assume that
(3)D+V(t,x(t))≤(≥)g(t,V(t,x(t))),t≠(k+τ-1)T,kT,V(t,x(t+))≤(≥)ψkτ(V(t,x(t))),t=(k+τ-1)T,V(t,x(t+))≤(≥)ψk(V(t,x(t))),t=kT,k∈ℕ,x(0+)=x0,
where g:ℝ+×ℝ+m→ℝ+m is continuous in ((k-1)T,(k+τ-1)T]×ℝm and ((k+τ-1)T,kT]×ℝm, for each p∈ℝm,k=1,2,…, the limit lim(t,q)→((k+τ-1)T+,p)g(t,q)=g((k+τ-1)T+,p) and lim(t,q)→((k-1)T+,p)g(t,q)=g((k-1)T+,p) exists. g(t,q) is quasimonotone nondecreasing in q. ψkτ,ψk:ℝ+m→ℝ+m is nondecreasing for all k∈ℕ. Let θ(t) be the maximal  (minimal) solution of the following impulsive differential equation on [0,∞):
(4)w′(t)=g(t,w(t)),t≠(k+τ-1)T,kT,w(t+)=ψkτ(w(t)),t=(k+τ-1)T,w(t+)=ψk(w(t)),t=kT,k∈ℕ,w(0+)=w0.
Then for any solutionx(t) of the system (3), V(0+,x0)≤  (≥)w0 implies that V(t,x(t))≤(≥)θ(t) for all t≥0.Lemma 5 (see [13, 15]).
Consider the following system:(5)v′(t)≤(≥)p(t)v(t)+q(t),t≠tk,v(tk+)≤(≥)dkv(tk)+bk,t=tk,k∈ℕ,v(0+)≤(≥)v0,
where p,q∈PC(ℝ+,ℝ) and dk≥0, v0 and bk are constants. Suppose that (A1)
the sequencetk satisfies 0≤t1≤t2<⋯, with limt→∞tk=∞;(A2)
v∈PC′(ℝ+,ℝ) and v(t) is left-continuous at tk,k∈ℕ.
Then, fort>0,
(6)v(t)≤(≥)v0e∫0t‍p(s)ds∏0<tk<t‍dk+∑0<tk<t‍(∏tk<tj<t‍dje∫tkt‍p(s)ds)bk+∫0t‍(∏s<tk<t‍dk)e∫st‍p(τ)dτq(s)ds.Lemma 6.
There exists a constantM=max{(1/λ)((L/d)+(ρedT/(edT-1))),K}>0, such that x(t), S(t), I(t), yJ(t), yM(t)≤M for each solution of (1) with t large enough.Proof.
Sincex′(t)≤rx(1-(x(t)/K)),  then x′(t)|x(t)=K≤0, and x((n+τ-1)T+)≤x((n+τ-1)T)(0<δ<1), so x(t)≤K for t large enough. Let us define V(t)∈V0 by V(t)=λβx(t)+λS(t)+λI(t)+yJ(t)+yM(t) and denote d=min{dS,dI,dJ,dM}. Then, it is obvious that
(7)dV(t)dt+dV(t)≤λβ(r+d)x(t)-λβrx2(t)K,t≠(n+τ-1)T,t≠nT.
Since the right-hand side (7) is bounded from above by L=Kλβ(r+d)2/4r, it follows that
(8)dV(t)dt+dV(t)≤L,t≠(n+τ-1)T,t≠nT.
When t=(n+τ-1)T and t=nT, it is easy to obtain that
(9)V((n+τ-1)T+)≤V((n+τ-1)T),V(nT+)=V(nT)+(λδI+δJ+δM).
Then, by Lemma 5, we can obtain that
(10)V(t)≤V(0)e-dt+∫0t‍Le-d(t-s)ds+∑0<kT<t‍ρe-d(t-kT)⟶Ld+ρedTedT-1,t⟶∞,
where ρ=λδI+δJ+δM. So it follows that V(t) is uniformly bounded on [0,∞). The proof is completed.Lemma 7 (see [31]).
Let one consider the following impulsive control subsystem:(11)x′(t)=rx(t)(1-x(t)K),t≠(n+τ-1)T,Δx(t)=-δx(t),t=(n+τ-1)T.
Supposeδ0*=1-e-rT. Then one has the following results. (1)
Ifδ>δ0*, then the trivial periodic solution of system (11) is locally asymptotically stable.(2)
Ifδ<δ0*, then the system (11) has a unique positive periodic solution x*(t), which is globally asymptotically stable, where
(12)x*(t)=K(1-δ-e-rT)1-δ-e-rT+δe-r(t-(n+τ-1)T),t∈((n+τ-1)T,(n+τ)T],n∈ℕ,x*(0+)=x*(nT+)=K((1-δ)erT-1)(erτT-1)+(1-δ)(erT-erτT).Remark 8.
From Lemma7, we have (1a)
ifδ0*>2δ, then x*(t)>K/2 for all t≥0;(2a)
ift∈((n-1)T,nT], n∈ℕ, then the periodic solution x*(t) can be rewritten in the form
(13)x*(t)={K(1-δ-e-rT)1-δ-e-rT+δe-r(T-τT)e-r(t-(n-1)T),t∈((n-1)T,(n+τ-1)T],K(1-δ-e-rT)1-δ-e-rT+δe-r(t-(n+τ-1)T),t∈((n+τ-1)T,nT],n∈ℕ.Lemma 9.
Let one consider the following impulsive control subsystem:(14)z′(t)=a(t)-dz(t),t≠(n+τ-1)T,t≠nT,Δz(t)=-pz(t),t=(n+τ-1)T,Δz(t)=δ,t=nT,n∈ℕ,z(0+)=z0,
where a(t) is a T-periodic PC(ℝ+,ℝ) function. p,d are the positive real constants and p<1. Then system (14) has a unique T-periodic solution z*(t), and for each solution z(t) of (14), z(t)→z*(t) as t→∞, where
(15)z*(t)=e-d(t-(n-1)T)(z*(0+)+∫0t-(n-1)T‍a(s)edsds),t∈((n-1)T,(n+τ-1)T],z*(t)=e-d(t-(n-1)T)(z*(τT+)edτT+∫τTt-(n-1)T‍a(s)edsds),t∈((n+τ-1)T,nT],z*(0+)=[(1-p)∫0τT‍a(s)edsds+∫τTT‍a(s)edsds]e-dT+δ1-(1-p)e-dT,z*(τT+)=(1-p)[∫0τT‍a(s)edsds+e-dT∫τTT‍a(s)edsds+δ]e-dτT1-(1-p)e-dT.Proof.
First, it is easy to obtain that(16)z(t)=e-dt(z(0+)+∫0t‍a(s)edsds),t∈(0,τT],z(t)=e-d(t-τT)z(τT+)+e-dt∫τTt‍a(s)edsds,t∈(τT,T].
Since the T-periodicity requirement, we have
(17)z*(τT+)=e-dτT(z*(0+)+∫0τT‍a(s)edsds)(1-p),z*(0+)=e-d(T-τT)z*(τT+)+e-dT∫τTT‍a(s)edsds+δ.
By (17), we can obtain that
(18)z*(0+)=[(1-p)∫0τT‍a(s)edsds+∫τTT‍a(s)edsds]e-dT+δ1-(1-p)e-dT,z*(τT+)=(1-p)[∫0τT‍a(s)edsds+e-dT∫τTT‍a(s)edsds+δ]e-dτT1-(1-p)e-dT.
So, we will obtain the T-periodic solution of (14):
(19)z*(t)=e-d(t-(n-1)T)(z*(0+)+∫0t-(n-1)T‍a(s)edsds),t∈((n-1)T,(n+τ-1)T],z*(t)=e-d(t-(n-1)T)(z*(τT+)edτT+∫τTt-(n-1)T‍a(s)edsds),t∈((n+τ-1)T,nT].
Let Z(t)=z(t)-z*(t). Substituting Z(t) into (14), we have
(20)Z′(t)=-dZ(t),t≠(n+τ-1)T,t≠nT,ΔZ(t)=-pZ(t),t=(n+τ-1)T,ΔZ(t)=0,t=nT,n∈ℕ,Z(0+)=z0-z*(0+).
Then, Z(t)=Z(0+)e-dt∏0<(n+τ-1)T<t‍(1-p)→0, as t→∞. The proof is completed.


## 4. Main Results

### 4.1. Local and Global Stability

In this section, we will study the existence and stability of the system (1) susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)). To this purpose, it is seen first that when S(t)=0, system (1) can be rewritten in the form
(21)x′(t)=rx(t)(1-x(t)K),I′(t)=-dII(t),yJ′(t)=-(dJ+m)yJ(t),yM′(t)=myJ(t)-dMyM(t),t≠(n+τ-1)T,t≠nT,Δx(t)=-δx(t),ΔI(t)=-PII(t),ΔyJ(t)=-PJyJ(t),ΔyM(t)=-PMyM(t),t=(n+τ-1)T,Δx(t)=0,ΔI(t)=δI,ΔyJ(t)=δJ,ΔyM(t)=δM,t=nT,
which describes the dynamics of system (1) in the absence of the susceptible pest population. So, when t∈((n-1)T,nT](n∈ℕ), we can calculate the T-periodic solution of (21) by Lemmas 7 and 9. It is seen that
(22)x*(t)={K(1-δ-e-rT)1-δ-e-rT+δe-r(T-τT)e-r(t-(n-1)T),t∈((n-1)T,(n+τ-1)T],K(1-δ-e-rT)1-δ-e-rT+δe-r(t-(n+τ-1)T),t∈((n+τ-1)T,nT],(23)I*(t)={δIe-dI(t-(n-1)T)1-(1-PI)e-dIT,t∈((n-1)T,(n+τ-1)T],δI(1-PI)e-dI(t-(n-1)T)1-(1-PI)e-dIT,t∈((n+τ-1)T,nT],(24)yJ*(t)={δJe-(m+dJ)(t-(n-1)T)1-(1-PJ)e-(m+dJ)T,t∈((n-1)T,(n+τ-1)T],δJ(1-PJ)e-(m+dJ)(t-(n-1)T)1-(1-PJ)e-(m+dJ)T,t∈((n+τ-1)T,nT],(25)yM*(t)={e-dM(t-(n-1)T)(yM*(0+)+A(t-(n-1)T)),t∈((n-1)T,(n+τ-1)T],e-dM(t-(n-1)T)(yM*(τT+)edMτT+B(t-(n-1)T)),t∈((n+τ-1)T,nT],(26)yM*(0+)=[(1-PM)A(τT)+B(T)]e-dMT+δM1-(1-PM)e-dMT,(27)yM*(τT+)=(1-PM)[A(τT)+e-dMTB(T)+δM]e-dMτT1-(1-PM)e-dMT,
where
(28)A(t)=mδJ(e(dM-(m+dJ))t-1)(1-(1-PJ)e-(m+dJ)T)(dM-(m+dJ)),t∈(0,τT],B(t)=mδJ(1-PJ)(e(dM-(m+dJ))t-e(dM-(m+dJ))τT)(1-(1-PJ)e-(m+dJ)T)(dM-(m+dJ)),t∈(τT,T].
To discuss the locally asymptotical stability of the susceptible pest-eradication periodic solution, we now introduce the Floquet theory for a linear impulsive control system:
(29)ω′(t)=A(t)ω(t),t≠τk,Δω(t)=Bkω(t),t=τk,k∈ℕ,
under the following conditions: H1:
A(·)∈PC(ℝ,Mn(ℝ)) and A(t+T)=A(t) for t≥0.H2:
Bk∈Mn,  det(In+Bk)≠0, τk<τk+1 for k∈ℕ, and In denotes the n×n real identity matrix.H3:
There is aq∈ℕ such that Bk+q=Bk,  τk+q=τk+T for k∈ℕ.LetΨ(t) be a fundamental matrix of (29), then there is a unique reversible matrix M∈Mn(ℝ) such that Ψ(t+T)=Ψ(t)M for all t∈ℝ, which is called the monodromy matrix of (29) corresponding to Ψ. All monodromy matrices of (29) are similar and they have the same eigenvalues λ1,λ2,…,λn, which are called the Floquet multipliers of (29).Lemma 10 (see [13] (Floquet theory)).
Let the conditionsH1–H3 hold. Then system (29) have the following properties (1)
stable if and only if all Floquet multipliersλi(1≤i≤n) of (29) satisfy |λi|≤1 and moreover, to those λi for which |λi|=1, there correspond simple elementary divisors;(2)
asymptotically stable if and only if all Floquet multipliersλi(1≤i≤n) of (29) satisfy |λi|<1;(3)
unstable if there is a Floquet multipliersλi(1≤i≤n) of (29) such that |λi|>1.In the following, we present two main results with the locally and globally asymptotical stability of the susceptible pest-eradication periodic solution(x*(t), 0, I*(t), PJ*(t), PM*(t)).Theorem 11.
If(30)β∫0T‍P1(x*(t))dt-∫0T‍g(I*(t))dt-P2′(0)∫0T‍yM*(t)dt-dST<ln11-PS,
then the susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)) of system (1) is locally asymptotically stable.Proof.
Let(x(t),S(t),I(t),yJ(t),yM(t)) be any solution of system (1). We define error e1(t)=x(t)-x*(t),e2(t)=S(t),e3(t)=I(t)-I*(t),e4(t)=yJ(t)-yJ*(t),e5(t)=yM(t)-yM*(t). The linearized system of (1) at (x*(t),0,I*(t),yJ*(t),yM*(t)) is
(31)e1′(t)=(r-2rx*(t)K)e1(t)-P1(x*(t))e2(t),e2′(t)=(βP1(x*(t))-g(I*(t))-P2′(0)yM*(t)-dS)e2(t),e3′(t)=g(I*(t))e2(t)-dIe3(t),e4′(t)=λP2′(0)yM*(t)e2(t)-(dJ+m)e4(t),e5′(t)=me4(t)-dMe5(t),t≠(n+τ-1)T,t≠nT,Δe1(t)=-δe1(t),Δe2(t)=-PSe2(t),Δe3(t)=-PIe3(t),Δe4(t)=-PJe4(t),Δe5(t)=-PMe5(t),t=(n+τ-1)T,Δe1(t)=Δe2(t)=Δe3(t)=Δe4(t)=Δe5(t)=0,t=nT.
Let Ψ(t) be the fundamental matrix of (31), then Ψ(t) satisfies (32)dΨ(t)dt=(r-2rx*(t)KP1(x*(t))0000βP1(x*(t))-g(I*(t))-P2′(0)yM*(t)-dS0000g(I*(t))-dI000λP2′(0)yM*(t)0-(dJ+m)0000m-dM)Ψ(t).Then, a fundamental matrix Ψ(t)(Ψ(0)=I4) of (31) is(33)Ψ(t)=(e∫0t‍(r-2r(x*(s)/K))dsϕ12(t)0000e∫0t‍(βP1(x*(s))-g(I*(s))-P2′(0)yM*(s)-dS)ds0000ϕ32(t)e-dIt000ϕ42(t)0e-(dJ+m)t00ϕ52(t)0ϕ54(t)e-dMt),where
(34)ϕ12(t)=-e-∫0t‍(r-2r(x*(s)/K))ds×∫0t‍P1(x*(s))e∫0s‍(βP1(x*(ξ))-g(I*(ξ))-P2′(0)yM*(ξ)-dS)dξ×e∫0s‍(r-2r(x*(ξ)/K))dξds,ϕ32(t)=e-dIt∫0t‍g(I*(s))e∫0s‍(βP1(x*(ξ))-g(I*(ξ))-P2′(0)yM*(ξ)-dS)dξ×edIsds,ϕ42(t)=e-dJt∫0t‍λP2′(0)yM*(s)e∫0s‍(βP1(x*(ξ))-g(I*(ξ))-P2′(0)yM*(ξ)-dS)dξ×edJsds,ϕ52(t)=e-dMt∫0t‍mϕ42(s)edMsds,ϕ54(t)=m(e-(dJ+m)t-e-dMt)dM-(dJ+m).
The resetting impulsive condition of (31) becomes
(35)(e1((n+τ-1)T+)e2((n+τ-1)T+)e3((n+τ-1)T+)e4((n+τ-1)T+)e5((n+τ-1)T+))=(1-δ000001-PS000001-PI000001-PJ000001-PM)×(e1((n+τ-1)T)e2((n+τ-1)T)e3((n+τ-1)T)e4((n+τ-1)T)e5((n+τ-1)T)),(e1(nT+)e2(nT+)e3(nT+)e4(nT+)e5(nT+))=(1000001000001000001000001)(e1(nT)e2(nT)e3(nT)e4(nT)e5(nT)).
Then, it is easy to obtain all eigenvalues of
(36)M=(1-δ000001-PS000001-PI000001-PJ000001-PM)×(1000001000001000001000001)Ψ(T).
We have λ1=(1-δ)e∫0T‍(r-2r(x*(s)/K))ds, λ2=(1-PS)e∫0T‍(βP1(x*(s))-g(I*(s))-P2′(0)yM*(s)-dS)ds, λ3=(1-PI)e-dIT<1, λ4=(1-PJ)e-(dJ+m)T<1 and λ5=(1-PM)e-dMT<1. Since x*(t)>(K/2), so λ1<1. By the condition (30), we have λ2<1. Therefore, according to Lemma 10, the susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)) of system (1) is locally asymptotically stable. The proof is completed.Theorem 12.
If(37)β∫0T‍P1(x*(t))dt-∫0T‍g(I*(t))dt-min0≤ϖ≤USP2′(ϖ)∫0T‍yM*(t)dt-dST<ln11-PS,
where US is an ultimate boundedness constant for S, then the susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)) of system (1) is globally asymptotically stable.Proof.
Since(38)β∫0T‍P1(x*(t))dt-∫0T‍g(I*(t))dt-min0≤ϖ≤USP2′(ϖ)∫0T‍yM*(t)dt-dST<ln11-PS,
we can choose an ε small enough such that
(39)β∫0T‍P1(x*(t)+ε)dt-∫0T‍g(I*(t)-ε)dt-min0≤ϖ≤USP2′(ϖ)∫0T‍(yM*(t)¯-ε)dt-dST-ln11-PS=ϵ<0,
where yM*(t)¯ is defined in the following. According to system (1), we have
(40)x′(t)≤rx(t)(1-x(t)K),I′(t)≥-dII(t),yJ′(t)≥-dJyJ(t)-myJ(t),yM′(t)=myJ(t)-dMyM(t),t≠(n+τ-1)T,t≠nT,Δx(t)=-δx(t),ΔI(t)=-PII(t),ΔyJ(t)=-PJyJ(t),ΔyM(t)=-PMyM(t),t=(n+τ-1)T,Δx(t)=0,ΔI(t)=δI,ΔyJ(t)=δJ,ΔyM(t)=δM,t=nT,n∈ℕ.
From the first equation of system (40), we obtain the following comparison system:
(41)ν′(t)=rν(t)(1-ν(t)K),t≠(n+τ-1)T,Δν(t)=-δν(t),t=(n+τ-1)T.
By Lemma 7, system (41) has a positive periodic solution ν*(t), and for any solution ν(t) of (41), ν(t)→ν*(t) as t large enough, where ν*(t)=x*(t). Then, according to Lemmas 4 and 7, there exists a positive constant n* such that for all t≥n*T(42)x(t)≤x*(t)+ε.
Let us define V(t)=(V1(t),V2(t))T∈C[ℝ+×ℝ2,ℝ+2] and Vi(t)∈V0,(i=1,2), where V1(t)=I(t), V2(t)=yJ(t). Then, we have
(43)V′(t)≥(-dII(t)-(dJ+m)yJ(t))=(-dI00-(dJ+m))V(t),t≠(n+τ-1)T,t≠nT,(44)V((n+τ-1)T+)=(1-PI001-PJ)V((n+τ-1)T),V(nT+)=(1001)V(nT)+(δIδJ),V(0+)=(I(0+),yJ(0+)).
Then, the multicomparison system of (43) is
(45)w′(t)=Aw(t),t≠(n+τ-1)T,nT,w(t+)=Bw(t),t=(n+τ-1)T,w(t+)=I2w(t)+C,t=nT,w(0+)=V(0+),
where A=(-dI00-(dJ+m)),B=(1-PI001-PJ), and C=(δIδJ).
By Lemma9, it is easy to obtain a periodic solution (I*(t),yJ*(t))T of system (45). Then, according to Lemmas 4 and 9, one may find n0*(n0*>n*) such that for all t≥n0*T(46)I(t)≥I*(t)-ε,yJ(t)≥yJ*(t)-ε.
From the fourth equation of system (40), we have yM′(t)≥m(yJ*(t)-ε)-dMyM(t), by Lemmas 4 and 9, there exists n1*(n1*>n0*) such that for all t≥n1*T(47)yM(t)≥yM*(t)¯-ε,
where
(48)yM*(t)¯={e-dM(t-(n-1)T)×(yM*(0+)¯+m∫0t-(n-1)T‍(yJ*(t)-ε)edMsds),t∈((n-1)T,(n+τ-1)T],e-dM(t-(n-1)T)×(yM*(τT+)¯edMτT+m∫τTt-(n-1)T‍(yJ*(t)-ε)edMsds),t∈((n+τ-1)T,nT],(49)yM*(0+)¯=(m[(1-PM)∫0τT‍(yJ*(t)-ε)edMsds+∫τTT‍(yJ*(t)-ε)edMsds]e-dMT+δM)×(1-(1-PM)e-dMT)-1,yM*(τT+)¯=((1-PM)[m∫0τT‍(yJ*(t)-ε)edMsds+me-dMT∫τTT‍(yJ*(t)-ε)edMsds+δM]×e-dMτT∫τTT‍(yJ*(t)-ε))×(1-(1-PM)e-dMT)-1.
Therefore,
(50)S′(t)≤[min0≤ϖ≤USP2′(ϖ)(yM*(t)¯-ε)βP1(x*(t)+ε)-g(I*(t)-ε)-min0≤ϖ≤USP2′(ϖ)(yM*(t)¯-ε)-dS]S(t),t≠(n+τ-1)T,t≠nT,ΔS(t)=-PSS(t),t=(n+τ-1)T,ΔS(t)=0,t=nT,
for t≥n1*T. Let N∈ℕ and (N+τ-1)≥n1*. Integrating (50) on ((n+τ-1)T,(n+τ)T],n≥N, we have
(51)S((n+τ)T)≤S((n+τ-1)T)(1-PS)×e∫(n+τ-1)T(n+τ)T‍(βP1(x*(t)~-ε)-g(I*(t)~+ε)-min0≤ϖ≤USP2′(ϖ)(yM*(t)~+ε)-dS)dt=S((n+τ-1)T)eϵ.
Then S(t)≤S((n+τ)T)ekϵ for t∈((n+τ+k)T,(n+τ+k+1)T]. Since ϵ<0, we can easily get S(t)→0, as t→∞. In the following, we prove x(t)→x*(t),I(t)→I*(t),yJ(t)→yJ*(t),yM(t)→yM*(t), as t→∞. Give ε0>0 small enough (ε0<(r/P′1(0))), there must exist n2*(n2*>n1*) such that S(t)<ε0, for t≥n2*T. Then, we have
(52)x′(t)≥(r-P1′(0)ε0)x(t)(1-rx(t)K(r-P1′(0)ε0)),I′(t)≤-(dI-g′(0)ε0)I(t),yJ′(t)≤λP2(ε0)M-(dJ+m)yJ(t),yM′(t)=myJ(t)-dMyM(t),t≠(n+τ-1)T,t≠nT,Δx(t)=-δx(t),ΔI(t)=-PII(t),ΔyJ(t)=-PJyJ(t),ΔyM(t)=-PMyM(t),t=(n+τ-1)T,Δx(t)=0,ΔI(t)=δI,ΔyJ(t)=δJ,ΔyM(t)=δM,t=nT.
Analyzing (52) with similarity as (40), there exists n3*(n3*>n2*) such that for all t≥n3*T(53)x(t)≥x*(t)^-ε,I(t)≤I*(t)^+ε,yJ(t)≤yJ*(t)^+ε,yM(t)≤yM*(t)^+ε,
where
(54)x*(t)^={K(r-P1′(0)ε0)(1-δ-e-(r-P1′(0)ε0)T)×(r[1-δ-e-(r-P1′(0)ε0)T+δe-(r-P1′(0)ε0)(T-τT)e-(r-P1′(0)ε0)(t-(n-1)T)])-1,t∈((n-1)T,(n+τ-1)T],K(r-P1′(0)ε0)(1-δ-e-(r-P1′(0)ε0)T)r[1-δ-e-(r-P1′(0)ε0)T+δe-(r-P1′(0)ε0)(t-(n+τ-1)T)],t∈((n+τ-1)T,nT],I*(t)^={δIe-(dI-g′(0)ε0)(t-(n-1)T)1-(1-PI)e-(dI-g′(0)ε0)T,t∈((n-1)T,(n+τ-1)T],δI(1-PI)e-(dI-g′(0)ε0)(t-(n-1)T)1-(1-PI)e-(dI-g′(0)ε0)T,t∈((n+τ-1)T,nT],yJ*(t)^={e-(m+dJ)(t-(n-1)T)×(yJ*(0+)^+λP2(ε0)M∫0t-(n-1)T‍e(m+dJ)sds),t∈((n-1)T,(n+τ-1)T],e-(m+dJ)(t-(n-1)T)×(∫τTt-(n-1)T‍e(m+dJ)sdsyJ*(τT+)^e(m+dJ)τT+λP2(ε0)M∫τTt-(n-1)T‍e(m+dJ)sds),t∈((n+τ-1)T,nT],yJ*(0+)^=(λP2(ε0)M[(1-PJ)∫0τT‍e(m+dJ)sds+∫τTT‍e(m+dJ)sds]×e-(m+dJ)T+δJ∫τTT‍e(m+dJ)s)(1-(1-PJ)e-(m+dJ)T)-1,yJ*(τT+)^=((1-PJ)[λP2(ε0)M∫0τT‍e(m+dJ)sds+e-(m+dJ)TλP2(ε0)M∫τTT‍e(m+dJ)sds+δJ]×e-(m+dJ)τT1-PJ[λP2(ε0)M∫0τT‍e(m+dJ)sds)(1-(1-PJ)e-(m+dJ)T)-1,yM*(t)^={e-dM(t-(n-1)T)×(∫0t-(n-1)T‍(yJ*(t)^+ε)yM*(0+)^+m∫0t-(n-1)T‍(yJ*(t)^+ε)edMsds),t∈((n-1)T,(n+τ-1)T],e-dM(t-(n-1)T)×(∫τTt-(n-1)T‍(yJ*(t)^+ε)yM*(τT+)^edMτT+m∫τTt-(n-1)T‍(yJ*(t)^+ε)edMsds),t∈((n+τ-1)T,nT],yM*(0+)^=(m[(1-PM)∫0τT‍(yJ*(t)^+ε)edMsds+∫τTT‍(yJ*(t)~+ε)edMsds]e-dMT+δM)×(1-(1-PM)e-dMT)-1,yM*(τT+)^=(×∫τTT‍(yJ*(t)~+ε)edMsds+δM]e-dMτT(1-PM)×[m∫0τT‍(yJ*(t)^+ε)edMsds+me-dMT×∫τTT‍(yJ*(t)~+ε)edMsds+δM]e-dMτT)×(1-(1-PM)e-dMT)-1.
Letting ε,ε0→0, we have yM*(t)¯→yM*(t),x*(t)^→x*(t),I*(t)^→I*(t),yJ*(t)^→yJ*(t),yM*(t)^→yM*(t). Together with (42), (46), (47), and (53), we get x(t)→x*(t),I(t)→I*(t),yJ(t)→yJ*(t),yM(t)→yM*(t) as t→∞. Therefore, the susceptible pest-eradication periodic solution (x*(t), 0, I*(t), yJ*(t), yM*(t)) is globally attractive. Since (37) implies (30), it follows from Theorem 11 that (x*(t), 0, I*(t), yJ*(t)) is locally asymptotically stable. So, the susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)) of system (1) is globally asymptotically stable. The proof is completed.


### 4.2. Permanence

Next, we will discuss the permanence of system (1). In order to facilitate discussion, we give one lemma.Lemma 13.
There exists a constantm4>0, such that x(t),I(t),yJ(t),yM(t)≥m4  for each solution of (1) with t large enough.Proof .
First, we discussx(t). Since S(t)≤M, by the first equation of system (1), we have
(55)x′(t)≥(r-P1′(0)M)x(t)(1-rx(t)K(r-P1′(0)M)),t≠(n+τ-1)T,t≠nT,Δx(t)=-δx(t),t=(n+τ-1)T,Δx(t)=0,t=nT.
Then, we obtain the following comparison system:
(56)χ′(t)=(r-P1′(0)M)χ(t)(1-rχ(t)K(r-P1′(0)M)),t≠(n+τ-1)T,t≠nT,Δχ(t)=-δχ(t),t=(n+τ-1)T,Δχ(t)=0,t=nT.
Letting r>P1′(0)M and (1-δ)e(r-P1′(0)M)T>1, by Lemma 7, the system (56) has a positive periodic solution χ*(t), and for any solution χ(t) of (56), χ(t)→χ*(t) as t large enough, where
(57)χ*(t)={K(r-P1′(0)M)(1-δ-e-(r-P1′(0)M)T)×(r[1-δ-e-(r-P1′(0)M)T+δe-(r-P1′(0)M)(T-τT)×e-(r-P1′(0)M)(t-(n-1)T)])-1,t∈((n-1)T,(n+τ-1)T],K(r-P1′(0)M)(1-δ-e-(r-P1′(0)M)T)×(r[1-δ-e-(r-P1′(0)M)T+δe-(r-P1′(0)M)(t-(n+τ-1)T)])-1,t∈((n+τ-1)T,nT].
According to Lemmas 4 and 7, one may find n4*∈ℕ such that x(t)≥χ*(t)-ε for t≥n4*T. Since χ*(t)-ε≥(K(r-P1′(0)M)(1-δ-e-(r-P1′(0)M)T)/r(1-e-(r-P1′(0)M)T))-ε=m0>0, so x(t)≥m0 for t≥n4*T. Next, we will discuss the rest of parts.
From (46) and (47), we know that there exists n5*(n5*>max{n1*,n4*}) such that I(t)≥I*(t)-ε, yJ(t)≥yJ*(t)-ε, and yM(t)≥yM*(t)¯-ε for all t≥n5*T. By (22), (23), and (48), we have I(t)≥(δI(1-PI)e-dIT/(1-(1-PI)e-dIT))-ε=m1>0, yJ(t)≥(δJ(1-PJ)e-(m+dJ)T/(1-(1-PJ)e-(m+dJ)T))-ε=m2>0, and yM(t)≥mM-ε=m3>0, where mM=min0<t≤TyM*(t)¯. Let m4=min{m0,m1,m2,m3}, then x(t),I(t),yJ(t),yM(t)≥m4 for t≥n5*T. The proof is completed.Theorem 14.
If(58)β∫0T‍P1(x*(t))dt-∫0T‍g(I*(t))dt-P2′(0)×∫0T‍yM*(t)dt-dST>ln11-PS,
then system (1) is permanent.Proof.
By Lemmas6 and 13, we have already known that there exist two constants m4,M>0, such that x(t),I(t),yJ(t),yM(t)≥m4 and x(t),S(t),I(t),yJ(t),yM(t)≤M for t large enough. Thus, we only need to find m*>0 such that S(t)≥m* for t large enough. We will do this in the following two steps.
Step  1. Let m5>0 and ε1>0 small enough, so that m5<min{(r/P1′(0)),(dI/g′(0)),M} and
(59)β∫0T‍P1(x*(t)~-ε1)dt-∫0T‍g(I*(t)~+ε1)dt-P2′(0)×∫0T‍(yM*(t)~+ε1)dt-dST-ln11-PS=η>0,
where
(60)x*(t)~={K(r-P1′(0)m5)(1-δ-e-(r-P1′(0)m5)T)×(r[1-δ-e-(r-P′1(0)m5)T+δe-(r-P1′(0)m5)(T-τT)×e-(r-P1′(0)m5)(t-(n-1)T)])-1,t∈((n-1)T,(n+τ-1)T],K(r-P1′(0)m5)(1-δ-e-(r-P1′(0)m5)T)×(r[1-δ-e-(r-P1′(0)m5)T+δe-(r-P1′(0)m5)(t-(n+τ-1)T)])-1,t∈((n+τ-1)T,nT],I*(t)~={δIe-(dI-g′(0)m5)(t-(n-1)T)1-(1-PI)e-(dI-g′(0)m5)T,t∈((n-1)T,(n+τ-1)T],δI(1-PI)e-(dI-g′(0)m5)(t-(n-1)T)1-(1-PI)e-(dI-g′(0)m5)T,t∈((n+τ-1)T,nT],yJ*(t)~={e-(m+dJ)(t-(n-1)T)×(yJ*(0+)~+λP2(m5)M∫0t-(n-1)T‍e(m+dJ)sds),t∈((n-1)T,(n+τ-1)T],e-(m+dJ)(t-(n-1)T)×(yJ*(τT+)~e(m+dJ)τT+λP2(m5)M∫τTt-(n-1)T‍e(m+dJ)sds),t∈((n+τ-1)T,nT],yJ*(0+)~=(λP2(m5)M[(1-PJ)∫0τT‍e(m+dJ)sds+∫τTT‍e(m+dJ)sds]×e-(m+dJ)T+δJ∫0τT‍e(m+dJ)sds)(1-(1-PJ)e-(m+dJ)T)-1,yJ*(τT+)~=([λP2(m5)M∫0τT‍e(m+dJ)sds+e-(m+dJ)T(1-PJ)×[λP2(m5)M∫0τT‍e(m+dJ)sds+e-(m+dJ)T×λP2(m5)M∫τTT‍e(m+dJ)sds+δJ]×e-(m+dJ)τT[λP2(m5)M∫0τT‍e(m+dJ)sds+e-(m+dJ)T)(1-(1-PJ)e-(m+dJ)T)-1,yM*(t)~={e-dM(t-(n-1)T)×(yM*(0+)~+m∫0t-(n-1)T‍×(yJ*(t)~+ε1)edMsds),t∈((n-1)T,(n+τ-1)T],e-dM(t-(n-1)T)×(yM*(τT+)~edMτT+m∫τTt-(n-1)T‍(yJ*(t)~+ε1)edMsds),t∈((n+τ-1)T,nT],yM*(0+)~=(m[(1-PM)∫0τT‍(yJ*(t)~+ε1)edMsds+∫τTT‍(yJ*(t)~+ε1)edMsds]×e-dMT+δM∫0τT‍(yJ*(t)~+ε1))(1-(1-PM)e-dMT)-1,yM*(τT+)~=(∫0τT‍(yJ*(t)~+ε1)(1-PM)×[m∫0τT‍(yJ*(t)~+ε1)edMsds+me-dMT∫τTT‍(yJ*(t)~+ε1)edMsds+δM]×e-dMτT∫0τT‍(yJ*(t)~+ε1))(1-(1-PM)e-dMT)-1.
We shall prove that one cannot have S(t)<m5 for all t>0, otherwise
(61)x′(t)≥(r-P1′(0)m5)x(t)(1-rx(t)K(r-P1′(0)m5)),I′(t)≤-(dI-g′(0)m5)I(t),yJ′(t)≤λP2(m5)M-(dJ+m)yJ(t),yM′(t)=myJ(t)-dMyM(t),t≠(n+τ-1)T,t≠nT,Δx(t)=-δx(t),ΔI(t)=-PII(t),ΔyJ(t)=-PJyJ(t),ΔyM(t)=-PMyM(t),t=(n+τ-1)T,Δx(t)=0,ΔI(t)=δI,ΔyJ(t)=δJ,ΔyM(t)=δM,t=nT.
Analyzing (61) with similarity as (40), it is easy to obtain that there exists a positive constant n6*, such that x(t)≥x*(t)~-ε1,I(t)≤I*(t)~+ε1,yJ(t)≤yJ*(t)~+ε1,yM(t)≤yM*(t)~+ε1 for t≥n6*T. Therefore,
(62)S′(t)≥βP1(x*(t)~-ε1)S(t)-g(I*(t)~+ε1)S(t)-P2′(0)(yM*(t)~+ε1)S(t)-dSS(t),t≠(n+τ-1)T,t≠nT,ΔS(t)=-PSS(t),t=(n+τ-1)T,ΔS(t)=0,t=nT,
for t≥n6*T. Let N0∈ℕ and (N0+τ-1)≥n6*. Integrating (62) on ((n+τ-1)T,(n+τ)T],n≥N0, we have
(63)S((n+τ)T)≥S((n+τ-1)T)(1-PS)×e∫(n+τ-1)T(n+τ)T‍(βP1(x*(t)~-ε1)-g(I*(t)~+ε1)-P2′(0)(yM*(t)~+ε1)-dS)dt=S((n+τ-1)T)eη.
Then S((N0+τ+k)T)≥S((N0+τ)T)ekη→∞ as k→∞, which is a contradiction. So there exists a t1(t1>n6*T) such that S(t1)≥m5.
Step  2. If S(t)≥m5 for all t≥t1, then Our purpose is obtained. If not, let t2=inf{t>t1∣S(t)<m5}. Then S(t)≥m5 for t∈[t1,t2) and S(t2)=m5. In this step, we consider two possible cases for t2.
Case  1. t2=(n1+τ-1)T,n1∈ℕ. Then S(t2+)=(1-PS)S(t2)<m5. Select n2,n3∈ℕ such that (n2-1)≥n6* and (1-PS)n2en3η+n2σT>(1-PS)n2en3η+(n2+1)σT>1, where σ=βP1(m4)-g(M)-P2′(0)M-dS<0. Let T~=(n2+n3)T, then we have the claim: there exists t3∈(t2,t2+T~] such that S(t3)≥m5. If the claim is false, we will get a contradiction in the following.
According to Step 1, we havex(t)≥x*(t)~-ε1,I(t)≤I*(t)~+ε1,yJ(t)≤yJ*(t)~+ε1,yM(t)≤yM*(t)~+ε1 for t≥(n1+n2-1)T. Then, we have
(64)S′(t)≥βP1(x*(t)~-ε1)S(t)-g(I*(t)~+ε1)S(t)-P2′(0)(yM*(t)~+ε1)S(t)-dSS(t),t≠(n+τ-1)T,t≠nT,ΔS(t)=-PSS(t),t=(n+τ-1)T,ΔS(t)=0,t=nT,
for t∈[t2+n2T,t2+T~]. As in Step 1, we have
(65)S(t2+T~)≥S(t2+n2T)en3η.
Sincex(t)≥m4,I(t)≤M,yM(t)≤M and P2(S(t))<P2′(0)S(t), we have
(66)S′(t)≥(βP1(m4)-g(M)-P2′(0)M-dS)S(t)=σS(t),t≠(n+τ-1)T,t≠nT,ΔS(t)=-PSS(t),t=(n+τ-1)T,ΔS(t)=0,t=nT,
for t∈[t2,t2+n2T]. Integrating (66) on [t2,t2+n2T], we have
(67)S(t2+n2T)≥S(t2+)en2σT=(1-PS)m5en2σT≥(1-PS)n2m5en2σT.
Thus, by (65) and (67), we have S(t2+T~)≥(1-PS)n2m5en3η+n2σT>m5, which is a contradiction. Let t4=inf{t>t2∣S(t)≥m5}, then for t∈[t2,t4),S(t)<m5 and S(t4)=m5. So, S(t)≥S(t2+)eσ(t-t2)=(1-PS)m5eσ(t-t2)≥(1-PS)n2+n3m5eσ(n2+n3)T=m1~ for t∈[t2,t4).
Case  2
(
t
2
≠
(
n
1
+
τ
-
1
)
T
,
n
1
∈
ℕ
). Suppose that t2∈((n1′+τ-1)T,(n1′+τ)T),n1′∈ℕ.  S(t)≥m5 for t∈[t1,t2) and S(t2)=m5. There are two possible cases for t∈(t2,(n1′+τ)T).
Case  2a. If S(t)≤m5 for all t∈(t2,(n1′+τ)T), similar to Case 1, we can prove there exists a t3′∈((n1′+τ)T,(n1′+τ)T+T~] such that S(t3′)≥m5. Let t4′=inf{t>t2∣S(t)≥m5}, then for t∈[t2,t4′),S(t)<m5 and S(t4′)=m5. So, S(t)≥S(t2)eσ(t-t2)=m5eσ(t-t2)≥(1-PS)n2+n3m5eσ(n2+n3+1)T=m*<m1~ for all t∈[t2,t4′).
Case  2b. If there exists a t∈(t2,(n1′+τ)T) such that S(t)≥m5. Let t4′¯=inf{t>t2∣S(t)≥m5}, then for t∈[t2,t4′¯),S(t)<m5 and S(t4′¯)=m5. So, S(t)≥S(t2)eσ(t-t2)=m5eσ(t-t2)≥m5eσT>m* for all t∈[t2,t4′¯).
SinceS(t)≥m5 for some t≥t1, in both cases a similar discussion can be continued. The proof is completed.


## 4.1. Local and Global Stability

In this section, we will study the existence and stability of the system (1) susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)). To this purpose, it is seen first that when S(t)=0, system (1) can be rewritten in the form
(21)x′(t)=rx(t)(1-x(t)K),I′(t)=-dII(t),yJ′(t)=-(dJ+m)yJ(t),yM′(t)=myJ(t)-dMyM(t),t≠(n+τ-1)T,t≠nT,Δx(t)=-δx(t),ΔI(t)=-PII(t),ΔyJ(t)=-PJyJ(t),ΔyM(t)=-PMyM(t),t=(n+τ-1)T,Δx(t)=0,ΔI(t)=δI,ΔyJ(t)=δJ,ΔyM(t)=δM,t=nT,
which describes the dynamics of system (1) in the absence of the susceptible pest population. So, when t∈((n-1)T,nT](n∈ℕ), we can calculate the T-periodic solution of (21) by Lemmas 7 and 9. It is seen that
(22)x*(t)={K(1-δ-e-rT)1-δ-e-rT+δe-r(T-τT)e-r(t-(n-1)T),t∈((n-1)T,(n+τ-1)T],K(1-δ-e-rT)1-δ-e-rT+δe-r(t-(n+τ-1)T),t∈((n+τ-1)T,nT],(23)I*(t)={δIe-dI(t-(n-1)T)1-(1-PI)e-dIT,t∈((n-1)T,(n+τ-1)T],δI(1-PI)e-dI(t-(n-1)T)1-(1-PI)e-dIT,t∈((n+τ-1)T,nT],(24)yJ*(t)={δJe-(m+dJ)(t-(n-1)T)1-(1-PJ)e-(m+dJ)T,t∈((n-1)T,(n+τ-1)T],δJ(1-PJ)e-(m+dJ)(t-(n-1)T)1-(1-PJ)e-(m+dJ)T,t∈((n+τ-1)T,nT],(25)yM*(t)={e-dM(t-(n-1)T)(yM*(0+)+A(t-(n-1)T)),t∈((n-1)T,(n+τ-1)T],e-dM(t-(n-1)T)(yM*(τT+)edMτT+B(t-(n-1)T)),t∈((n+τ-1)T,nT],(26)yM*(0+)=[(1-PM)A(τT)+B(T)]e-dMT+δM1-(1-PM)e-dMT,(27)yM*(τT+)=(1-PM)[A(τT)+e-dMTB(T)+δM]e-dMτT1-(1-PM)e-dMT,
where
(28)A(t)=mδJ(e(dM-(m+dJ))t-1)(1-(1-PJ)e-(m+dJ)T)(dM-(m+dJ)),t∈(0,τT],B(t)=mδJ(1-PJ)(e(dM-(m+dJ))t-e(dM-(m+dJ))τT)(1-(1-PJ)e-(m+dJ)T)(dM-(m+dJ)),t∈(τT,T].
To discuss the locally asymptotical stability of the susceptible pest-eradication periodic solution, we now introduce the Floquet theory for a linear impulsive control system:
(29)ω′(t)=A(t)ω(t),t≠τk,Δω(t)=Bkω(t),t=τk,k∈ℕ,
under the following conditions: H1:
A(·)∈PC(ℝ,Mn(ℝ)) and A(t+T)=A(t) for t≥0.H2:
Bk∈Mn,  det(In+Bk)≠0, τk<τk+1 for k∈ℕ, and In denotes the n×n real identity matrix.H3:
There is aq∈ℕ such that Bk+q=Bk,  τk+q=τk+T for k∈ℕ.LetΨ(t) be a fundamental matrix of (29), then there is a unique reversible matrix M∈Mn(ℝ) such that Ψ(t+T)=Ψ(t)M for all t∈ℝ, which is called the monodromy matrix of (29) corresponding to Ψ. All monodromy matrices of (29) are similar and they have the same eigenvalues λ1,λ2,…,λn, which are called the Floquet multipliers of (29).Lemma 10 (see [13] (Floquet theory)).
Let the conditionsH1–H3 hold. Then system (29) have the following properties (1)
stable if and only if all Floquet multipliersλi(1≤i≤n) of (29) satisfy |λi|≤1 and moreover, to those λi for which |λi|=1, there correspond simple elementary divisors;(2)
asymptotically stable if and only if all Floquet multipliersλi(1≤i≤n) of (29) satisfy |λi|<1;(3)
unstable if there is a Floquet multipliersλi(1≤i≤n) of (29) such that |λi|>1.In the following, we present two main results with the locally and globally asymptotical stability of the susceptible pest-eradication periodic solution(x*(t), 0, I*(t), PJ*(t), PM*(t)).Theorem 11.
If(30)β∫0T‍P1(x*(t))dt-∫0T‍g(I*(t))dt-P2′(0)∫0T‍yM*(t)dt-dST<ln11-PS,
then the susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)) of system (1) is locally asymptotically stable.Proof.
Let(x(t),S(t),I(t),yJ(t),yM(t)) be any solution of system (1). We define error e1(t)=x(t)-x*(t),e2(t)=S(t),e3(t)=I(t)-I*(t),e4(t)=yJ(t)-yJ*(t),e5(t)=yM(t)-yM*(t). The linearized system of (1) at (x*(t),0,I*(t),yJ*(t),yM*(t)) is
(31)e1′(t)=(r-2rx*(t)K)e1(t)-P1(x*(t))e2(t),e2′(t)=(βP1(x*(t))-g(I*(t))-P2′(0)yM*(t)-dS)e2(t),e3′(t)=g(I*(t))e2(t)-dIe3(t),e4′(t)=λP2′(0)yM*(t)e2(t)-(dJ+m)e4(t),e5′(t)=me4(t)-dMe5(t),t≠(n+τ-1)T,t≠nT,Δe1(t)=-δe1(t),Δe2(t)=-PSe2(t),Δe3(t)=-PIe3(t),Δe4(t)=-PJe4(t),Δe5(t)=-PMe5(t),t=(n+τ-1)T,Δe1(t)=Δe2(t)=Δe3(t)=Δe4(t)=Δe5(t)=0,t=nT.
Let Ψ(t) be the fundamental matrix of (31), then Ψ(t) satisfies (32)dΨ(t)dt=(r-2rx*(t)KP1(x*(t))0000βP1(x*(t))-g(I*(t))-P2′(0)yM*(t)-dS0000g(I*(t))-dI000λP2′(0)yM*(t)0-(dJ+m)0000m-dM)Ψ(t).Then, a fundamental matrix Ψ(t)(Ψ(0)=I4) of (31) is(33)Ψ(t)=(e∫0t‍(r-2r(x*(s)/K))dsϕ12(t)0000e∫0t‍(βP1(x*(s))-g(I*(s))-P2′(0)yM*(s)-dS)ds0000ϕ32(t)e-dIt000ϕ42(t)0e-(dJ+m)t00ϕ52(t)0ϕ54(t)e-dMt),where
(34)ϕ12(t)=-e-∫0t‍(r-2r(x*(s)/K))ds×∫0t‍P1(x*(s))e∫0s‍(βP1(x*(ξ))-g(I*(ξ))-P2′(0)yM*(ξ)-dS)dξ×e∫0s‍(r-2r(x*(ξ)/K))dξds,ϕ32(t)=e-dIt∫0t‍g(I*(s))e∫0s‍(βP1(x*(ξ))-g(I*(ξ))-P2′(0)yM*(ξ)-dS)dξ×edIsds,ϕ42(t)=e-dJt∫0t‍λP2′(0)yM*(s)e∫0s‍(βP1(x*(ξ))-g(I*(ξ))-P2′(0)yM*(ξ)-dS)dξ×edJsds,ϕ52(t)=e-dMt∫0t‍mϕ42(s)edMsds,ϕ54(t)=m(e-(dJ+m)t-e-dMt)dM-(dJ+m).
The resetting impulsive condition of (31) becomes
(35)(e1((n+τ-1)T+)e2((n+τ-1)T+)e3((n+τ-1)T+)e4((n+τ-1)T+)e5((n+τ-1)T+))=(1-δ000001-PS000001-PI000001-PJ000001-PM)×(e1((n+τ-1)T)e2((n+τ-1)T)e3((n+τ-1)T)e4((n+τ-1)T)e5((n+τ-1)T)),(e1(nT+)e2(nT+)e3(nT+)e4(nT+)e5(nT+))=(1000001000001000001000001)(e1(nT)e2(nT)e3(nT)e4(nT)e5(nT)).
Then, it is easy to obtain all eigenvalues of
(36)M=(1-δ000001-PS000001-PI000001-PJ000001-PM)×(1000001000001000001000001)Ψ(T).
We have λ1=(1-δ)e∫0T‍(r-2r(x*(s)/K))ds, λ2=(1-PS)e∫0T‍(βP1(x*(s))-g(I*(s))-P2′(0)yM*(s)-dS)ds, λ3=(1-PI)e-dIT<1, λ4=(1-PJ)e-(dJ+m)T<1 and λ5=(1-PM)e-dMT<1. Since x*(t)>(K/2), so λ1<1. By the condition (30), we have λ2<1. Therefore, according to Lemma 10, the susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)) of system (1) is locally asymptotically stable. The proof is completed.Theorem 12.
If(37)β∫0T‍P1(x*(t))dt-∫0T‍g(I*(t))dt-min0≤ϖ≤USP2′(ϖ)∫0T‍yM*(t)dt-dST<ln11-PS,
where US is an ultimate boundedness constant for S, then the susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)) of system (1) is globally asymptotically stable.Proof.
Since(38)β∫0T‍P1(x*(t))dt-∫0T‍g(I*(t))dt-min0≤ϖ≤USP2′(ϖ)∫0T‍yM*(t)dt-dST<ln11-PS,
we can choose an ε small enough such that
(39)β∫0T‍P1(x*(t)+ε)dt-∫0T‍g(I*(t)-ε)dt-min0≤ϖ≤USP2′(ϖ)∫0T‍(yM*(t)¯-ε)dt-dST-ln11-PS=ϵ<0,
where yM*(t)¯ is defined in the following. According to system (1), we have
(40)x′(t)≤rx(t)(1-x(t)K),I′(t)≥-dII(t),yJ′(t)≥-dJyJ(t)-myJ(t),yM′(t)=myJ(t)-dMyM(t),t≠(n+τ-1)T,t≠nT,Δx(t)=-δx(t),ΔI(t)=-PII(t),ΔyJ(t)=-PJyJ(t),ΔyM(t)=-PMyM(t),t=(n+τ-1)T,Δx(t)=0,ΔI(t)=δI,ΔyJ(t)=δJ,ΔyM(t)=δM,t=nT,n∈ℕ.
From the first equation of system (40), we obtain the following comparison system:
(41)ν′(t)=rν(t)(1-ν(t)K),t≠(n+τ-1)T,Δν(t)=-δν(t),t=(n+τ-1)T.
By Lemma 7, system (41) has a positive periodic solution ν*(t), and for any solution ν(t) of (41), ν(t)→ν*(t) as t large enough, where ν*(t)=x*(t). Then, according to Lemmas 4 and 7, there exists a positive constant n* such that for all t≥n*T(42)x(t)≤x*(t)+ε.
Let us define V(t)=(V1(t),V2(t))T∈C[ℝ+×ℝ2,ℝ+2] and Vi(t)∈V0,(i=1,2), where V1(t)=I(t), V2(t)=yJ(t). Then, we have
(43)V′(t)≥(-dII(t)-(dJ+m)yJ(t))=(-dI00-(dJ+m))V(t),t≠(n+τ-1)T,t≠nT,(44)V((n+τ-1)T+)=(1-PI001-PJ)V((n+τ-1)T),V(nT+)=(1001)V(nT)+(δIδJ),V(0+)=(I(0+),yJ(0+)).
Then, the multicomparison system of (43) is
(45)w′(t)=Aw(t),t≠(n+τ-1)T,nT,w(t+)=Bw(t),t=(n+τ-1)T,w(t+)=I2w(t)+C,t=nT,w(0+)=V(0+),
where A=(-dI00-(dJ+m)),B=(1-PI001-PJ), and C=(δIδJ).
By Lemma9, it is easy to obtain a periodic solution (I*(t),yJ*(t))T of system (45). Then, according to Lemmas 4 and 9, one may find n0*(n0*>n*) such that for all t≥n0*T(46)I(t)≥I*(t)-ε,yJ(t)≥yJ*(t)-ε.
From the fourth equation of system (40), we have yM′(t)≥m(yJ*(t)-ε)-dMyM(t), by Lemmas 4 and 9, there exists n1*(n1*>n0*) such that for all t≥n1*T(47)yM(t)≥yM*(t)¯-ε,
where
(48)yM*(t)¯={e-dM(t-(n-1)T)×(yM*(0+)¯+m∫0t-(n-1)T‍(yJ*(t)-ε)edMsds),t∈((n-1)T,(n+τ-1)T],e-dM(t-(n-1)T)×(yM*(τT+)¯edMτT+m∫τTt-(n-1)T‍(yJ*(t)-ε)edMsds),t∈((n+τ-1)T,nT],(49)yM*(0+)¯=(m[(1-PM)∫0τT‍(yJ*(t)-ε)edMsds+∫τTT‍(yJ*(t)-ε)edMsds]e-dMT+δM)×(1-(1-PM)e-dMT)-1,yM*(τT+)¯=((1-PM)[m∫0τT‍(yJ*(t)-ε)edMsds+me-dMT∫τTT‍(yJ*(t)-ε)edMsds+δM]×e-dMτT∫τTT‍(yJ*(t)-ε))×(1-(1-PM)e-dMT)-1.
Therefore,
(50)S′(t)≤[min0≤ϖ≤USP2′(ϖ)(yM*(t)¯-ε)βP1(x*(t)+ε)-g(I*(t)-ε)-min0≤ϖ≤USP2′(ϖ)(yM*(t)¯-ε)-dS]S(t),t≠(n+τ-1)T,t≠nT,ΔS(t)=-PSS(t),t=(n+τ-1)T,ΔS(t)=0,t=nT,
for t≥n1*T. Let N∈ℕ and (N+τ-1)≥n1*. Integrating (50) on ((n+τ-1)T,(n+τ)T],n≥N, we have
(51)S((n+τ)T)≤S((n+τ-1)T)(1-PS)×e∫(n+τ-1)T(n+τ)T‍(βP1(x*(t)~-ε)-g(I*(t)~+ε)-min0≤ϖ≤USP2′(ϖ)(yM*(t)~+ε)-dS)dt=S((n+τ-1)T)eϵ.
Then S(t)≤S((n+τ)T)ekϵ for t∈((n+τ+k)T,(n+τ+k+1)T]. Since ϵ<0, we can easily get S(t)→0, as t→∞. In the following, we prove x(t)→x*(t),I(t)→I*(t),yJ(t)→yJ*(t),yM(t)→yM*(t), as t→∞. Give ε0>0 small enough (ε0<(r/P′1(0))), there must exist n2*(n2*>n1*) such that S(t)<ε0, for t≥n2*T. Then, we have
(52)x′(t)≥(r-P1′(0)ε0)x(t)(1-rx(t)K(r-P1′(0)ε0)),I′(t)≤-(dI-g′(0)ε0)I(t),yJ′(t)≤λP2(ε0)M-(dJ+m)yJ(t),yM′(t)=myJ(t)-dMyM(t),t≠(n+τ-1)T,t≠nT,Δx(t)=-δx(t),ΔI(t)=-PII(t),ΔyJ(t)=-PJyJ(t),ΔyM(t)=-PMyM(t),t=(n+τ-1)T,Δx(t)=0,ΔI(t)=δI,ΔyJ(t)=δJ,ΔyM(t)=δM,t=nT.
Analyzing (52) with similarity as (40), there exists n3*(n3*>n2*) such that for all t≥n3*T(53)x(t)≥x*(t)^-ε,I(t)≤I*(t)^+ε,yJ(t)≤yJ*(t)^+ε,yM(t)≤yM*(t)^+ε,
where
(54)x*(t)^={K(r-P1′(0)ε0)(1-δ-e-(r-P1′(0)ε0)T)×(r[1-δ-e-(r-P1′(0)ε0)T+δe-(r-P1′(0)ε0)(T-τT)e-(r-P1′(0)ε0)(t-(n-1)T)])-1,t∈((n-1)T,(n+τ-1)T],K(r-P1′(0)ε0)(1-δ-e-(r-P1′(0)ε0)T)r[1-δ-e-(r-P1′(0)ε0)T+δe-(r-P1′(0)ε0)(t-(n+τ-1)T)],t∈((n+τ-1)T,nT],I*(t)^={δIe-(dI-g′(0)ε0)(t-(n-1)T)1-(1-PI)e-(dI-g′(0)ε0)T,t∈((n-1)T,(n+τ-1)T],δI(1-PI)e-(dI-g′(0)ε0)(t-(n-1)T)1-(1-PI)e-(dI-g′(0)ε0)T,t∈((n+τ-1)T,nT],yJ*(t)^={e-(m+dJ)(t-(n-1)T)×(yJ*(0+)^+λP2(ε0)M∫0t-(n-1)T‍e(m+dJ)sds),t∈((n-1)T,(n+τ-1)T],e-(m+dJ)(t-(n-1)T)×(∫τTt-(n-1)T‍e(m+dJ)sdsyJ*(τT+)^e(m+dJ)τT+λP2(ε0)M∫τTt-(n-1)T‍e(m+dJ)sds),t∈((n+τ-1)T,nT],yJ*(0+)^=(λP2(ε0)M[(1-PJ)∫0τT‍e(m+dJ)sds+∫τTT‍e(m+dJ)sds]×e-(m+dJ)T+δJ∫τTT‍e(m+dJ)s)(1-(1-PJ)e-(m+dJ)T)-1,yJ*(τT+)^=((1-PJ)[λP2(ε0)M∫0τT‍e(m+dJ)sds+e-(m+dJ)TλP2(ε0)M∫τTT‍e(m+dJ)sds+δJ]×e-(m+dJ)τT1-PJ[λP2(ε0)M∫0τT‍e(m+dJ)sds)(1-(1-PJ)e-(m+dJ)T)-1,yM*(t)^={e-dM(t-(n-1)T)×(∫0t-(n-1)T‍(yJ*(t)^+ε)yM*(0+)^+m∫0t-(n-1)T‍(yJ*(t)^+ε)edMsds),t∈((n-1)T,(n+τ-1)T],e-dM(t-(n-1)T)×(∫τTt-(n-1)T‍(yJ*(t)^+ε)yM*(τT+)^edMτT+m∫τTt-(n-1)T‍(yJ*(t)^+ε)edMsds),t∈((n+τ-1)T,nT],yM*(0+)^=(m[(1-PM)∫0τT‍(yJ*(t)^+ε)edMsds+∫τTT‍(yJ*(t)~+ε)edMsds]e-dMT+δM)×(1-(1-PM)e-dMT)-1,yM*(τT+)^=(×∫τTT‍(yJ*(t)~+ε)edMsds+δM]e-dMτT(1-PM)×[m∫0τT‍(yJ*(t)^+ε)edMsds+me-dMT×∫τTT‍(yJ*(t)~+ε)edMsds+δM]e-dMτT)×(1-(1-PM)e-dMT)-1.
Letting ε,ε0→0, we have yM*(t)¯→yM*(t),x*(t)^→x*(t),I*(t)^→I*(t),yJ*(t)^→yJ*(t),yM*(t)^→yM*(t). Together with (42), (46), (47), and (53), we get x(t)→x*(t),I(t)→I*(t),yJ(t)→yJ*(t),yM(t)→yM*(t) as t→∞. Therefore, the susceptible pest-eradication periodic solution (x*(t), 0, I*(t), yJ*(t), yM*(t)) is globally attractive. Since (37) implies (30), it follows from Theorem 11 that (x*(t), 0, I*(t), yJ*(t)) is locally asymptotically stable. So, the susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)) of system (1) is globally asymptotically stable. The proof is completed.


## 4.2. Permanence

Next, we will discuss the permanence of system (1). In order to facilitate discussion, we give one lemma.Lemma 13.
There exists a constantm4>0, such that x(t),I(t),yJ(t),yM(t)≥m4  for each solution of (1) with t large enough.Proof .
First, we discussx(t). Since S(t)≤M, by the first equation of system (1), we have
(55)x′(t)≥(r-P1′(0)M)x(t)(1-rx(t)K(r-P1′(0)M)),t≠(n+τ-1)T,t≠nT,Δx(t)=-δx(t),t=(n+τ-1)T,Δx(t)=0,t=nT.
Then, we obtain the following comparison system:
(56)χ′(t)=(r-P1′(0)M)χ(t)(1-rχ(t)K(r-P1′(0)M)),t≠(n+τ-1)T,t≠nT,Δχ(t)=-δχ(t),t=(n+τ-1)T,Δχ(t)=0,t=nT.
Letting r>P1′(0)M and (1-δ)e(r-P1′(0)M)T>1, by Lemma 7, the system (56) has a positive periodic solution χ*(t), and for any solution χ(t) of (56), χ(t)→χ*(t) as t large enough, where
(57)χ*(t)={K(r-P1′(0)M)(1-δ-e-(r-P1′(0)M)T)×(r[1-δ-e-(r-P1′(0)M)T+δe-(r-P1′(0)M)(T-τT)×e-(r-P1′(0)M)(t-(n-1)T)])-1,t∈((n-1)T,(n+τ-1)T],K(r-P1′(0)M)(1-δ-e-(r-P1′(0)M)T)×(r[1-δ-e-(r-P1′(0)M)T+δe-(r-P1′(0)M)(t-(n+τ-1)T)])-1,t∈((n+τ-1)T,nT].
According to Lemmas 4 and 7, one may find n4*∈ℕ such that x(t)≥χ*(t)-ε for t≥n4*T. Since χ*(t)-ε≥(K(r-P1′(0)M)(1-δ-e-(r-P1′(0)M)T)/r(1-e-(r-P1′(0)M)T))-ε=m0>0, so x(t)≥m0 for t≥n4*T. Next, we will discuss the rest of parts.
From (46) and (47), we know that there exists n5*(n5*>max{n1*,n4*}) such that I(t)≥I*(t)-ε, yJ(t)≥yJ*(t)-ε, and yM(t)≥yM*(t)¯-ε for all t≥n5*T. By (22), (23), and (48), we have I(t)≥(δI(1-PI)e-dIT/(1-(1-PI)e-dIT))-ε=m1>0, yJ(t)≥(δJ(1-PJ)e-(m+dJ)T/(1-(1-PJ)e-(m+dJ)T))-ε=m2>0, and yM(t)≥mM-ε=m3>0, where mM=min0<t≤TyM*(t)¯. Let m4=min{m0,m1,m2,m3}, then x(t),I(t),yJ(t),yM(t)≥m4 for t≥n5*T. The proof is completed.Theorem 14.
If(58)β∫0T‍P1(x*(t))dt-∫0T‍g(I*(t))dt-P2′(0)×∫0T‍yM*(t)dt-dST>ln11-PS,
then system (1) is permanent.Proof.
By Lemmas6 and 13, we have already known that there exist two constants m4,M>0, such that x(t),I(t),yJ(t),yM(t)≥m4 and x(t),S(t),I(t),yJ(t),yM(t)≤M for t large enough. Thus, we only need to find m*>0 such that S(t)≥m* for t large enough. We will do this in the following two steps.
Step  1. Let m5>0 and ε1>0 small enough, so that m5<min{(r/P1′(0)),(dI/g′(0)),M} and
(59)β∫0T‍P1(x*(t)~-ε1)dt-∫0T‍g(I*(t)~+ε1)dt-P2′(0)×∫0T‍(yM*(t)~+ε1)dt-dST-ln11-PS=η>0,
where
(60)x*(t)~={K(r-P1′(0)m5)(1-δ-e-(r-P1′(0)m5)T)×(r[1-δ-e-(r-P′1(0)m5)T+δe-(r-P1′(0)m5)(T-τT)×e-(r-P1′(0)m5)(t-(n-1)T)])-1,t∈((n-1)T,(n+τ-1)T],K(r-P1′(0)m5)(1-δ-e-(r-P1′(0)m5)T)×(r[1-δ-e-(r-P1′(0)m5)T+δe-(r-P1′(0)m5)(t-(n+τ-1)T)])-1,t∈((n+τ-1)T,nT],I*(t)~={δIe-(dI-g′(0)m5)(t-(n-1)T)1-(1-PI)e-(dI-g′(0)m5)T,t∈((n-1)T,(n+τ-1)T],δI(1-PI)e-(dI-g′(0)m5)(t-(n-1)T)1-(1-PI)e-(dI-g′(0)m5)T,t∈((n+τ-1)T,nT],yJ*(t)~={e-(m+dJ)(t-(n-1)T)×(yJ*(0+)~+λP2(m5)M∫0t-(n-1)T‍e(m+dJ)sds),t∈((n-1)T,(n+τ-1)T],e-(m+dJ)(t-(n-1)T)×(yJ*(τT+)~e(m+dJ)τT+λP2(m5)M∫τTt-(n-1)T‍e(m+dJ)sds),t∈((n+τ-1)T,nT],yJ*(0+)~=(λP2(m5)M[(1-PJ)∫0τT‍e(m+dJ)sds+∫τTT‍e(m+dJ)sds]×e-(m+dJ)T+δJ∫0τT‍e(m+dJ)sds)(1-(1-PJ)e-(m+dJ)T)-1,yJ*(τT+)~=([λP2(m5)M∫0τT‍e(m+dJ)sds+e-(m+dJ)T(1-PJ)×[λP2(m5)M∫0τT‍e(m+dJ)sds+e-(m+dJ)T×λP2(m5)M∫τTT‍e(m+dJ)sds+δJ]×e-(m+dJ)τT[λP2(m5)M∫0τT‍e(m+dJ)sds+e-(m+dJ)T)(1-(1-PJ)e-(m+dJ)T)-1,yM*(t)~={e-dM(t-(n-1)T)×(yM*(0+)~+m∫0t-(n-1)T‍×(yJ*(t)~+ε1)edMsds),t∈((n-1)T,(n+τ-1)T],e-dM(t-(n-1)T)×(yM*(τT+)~edMτT+m∫τTt-(n-1)T‍(yJ*(t)~+ε1)edMsds),t∈((n+τ-1)T,nT],yM*(0+)~=(m[(1-PM)∫0τT‍(yJ*(t)~+ε1)edMsds+∫τTT‍(yJ*(t)~+ε1)edMsds]×e-dMT+δM∫0τT‍(yJ*(t)~+ε1))(1-(1-PM)e-dMT)-1,yM*(τT+)~=(∫0τT‍(yJ*(t)~+ε1)(1-PM)×[m∫0τT‍(yJ*(t)~+ε1)edMsds+me-dMT∫τTT‍(yJ*(t)~+ε1)edMsds+δM]×e-dMτT∫0τT‍(yJ*(t)~+ε1))(1-(1-PM)e-dMT)-1.
We shall prove that one cannot have S(t)<m5 for all t>0, otherwise
(61)x′(t)≥(r-P1′(0)m5)x(t)(1-rx(t)K(r-P1′(0)m5)),I′(t)≤-(dI-g′(0)m5)I(t),yJ′(t)≤λP2(m5)M-(dJ+m)yJ(t),yM′(t)=myJ(t)-dMyM(t),t≠(n+τ-1)T,t≠nT,Δx(t)=-δx(t),ΔI(t)=-PII(t),ΔyJ(t)=-PJyJ(t),ΔyM(t)=-PMyM(t),t=(n+τ-1)T,Δx(t)=0,ΔI(t)=δI,ΔyJ(t)=δJ,ΔyM(t)=δM,t=nT.
Analyzing (61) with similarity as (40), it is easy to obtain that there exists a positive constant n6*, such that x(t)≥x*(t)~-ε1,I(t)≤I*(t)~+ε1,yJ(t)≤yJ*(t)~+ε1,yM(t)≤yM*(t)~+ε1 for t≥n6*T. Therefore,
(62)S′(t)≥βP1(x*(t)~-ε1)S(t)-g(I*(t)~+ε1)S(t)-P2′(0)(yM*(t)~+ε1)S(t)-dSS(t),t≠(n+τ-1)T,t≠nT,ΔS(t)=-PSS(t),t=(n+τ-1)T,ΔS(t)=0,t=nT,
for t≥n6*T. Let N0∈ℕ and (N0+τ-1)≥n6*. Integrating (62) on ((n+τ-1)T,(n+τ)T],n≥N0, we have
(63)S((n+τ)T)≥S((n+τ-1)T)(1-PS)×e∫(n+τ-1)T(n+τ)T‍(βP1(x*(t)~-ε1)-g(I*(t)~+ε1)-P2′(0)(yM*(t)~+ε1)-dS)dt=S((n+τ-1)T)eη.
Then S((N0+τ+k)T)≥S((N0+τ)T)ekη→∞ as k→∞, which is a contradiction. So there exists a t1(t1>n6*T) such that S(t1)≥m5.
Step  2. If S(t)≥m5 for all t≥t1, then Our purpose is obtained. If not, let t2=inf{t>t1∣S(t)<m5}. Then S(t)≥m5 for t∈[t1,t2) and S(t2)=m5. In this step, we consider two possible cases for t2.
Case  1. t2=(n1+τ-1)T,n1∈ℕ. Then S(t2+)=(1-PS)S(t2)<m5. Select n2,n3∈ℕ such that (n2-1)≥n6* and (1-PS)n2en3η+n2σT>(1-PS)n2en3η+(n2+1)σT>1, where σ=βP1(m4)-g(M)-P2′(0)M-dS<0. Let T~=(n2+n3)T, then we have the claim: there exists t3∈(t2,t2+T~] such that S(t3)≥m5. If the claim is false, we will get a contradiction in the following.
According to Step 1, we havex(t)≥x*(t)~-ε1,I(t)≤I*(t)~+ε1,yJ(t)≤yJ*(t)~+ε1,yM(t)≤yM*(t)~+ε1 for t≥(n1+n2-1)T. Then, we have
(64)S′(t)≥βP1(x*(t)~-ε1)S(t)-g(I*(t)~+ε1)S(t)-P2′(0)(yM*(t)~+ε1)S(t)-dSS(t),t≠(n+τ-1)T,t≠nT,ΔS(t)=-PSS(t),t=(n+τ-1)T,ΔS(t)=0,t=nT,
for t∈[t2+n2T,t2+T~]. As in Step 1, we have
(65)S(t2+T~)≥S(t2+n2T)en3η.
Sincex(t)≥m4,I(t)≤M,yM(t)≤M and P2(S(t))<P2′(0)S(t), we have
(66)S′(t)≥(βP1(m4)-g(M)-P2′(0)M-dS)S(t)=σS(t),t≠(n+τ-1)T,t≠nT,ΔS(t)=-PSS(t),t=(n+τ-1)T,ΔS(t)=0,t=nT,
for t∈[t2,t2+n2T]. Integrating (66) on [t2,t2+n2T], we have
(67)S(t2+n2T)≥S(t2+)en2σT=(1-PS)m5en2σT≥(1-PS)n2m5en2σT.
Thus, by (65) and (67), we have S(t2+T~)≥(1-PS)n2m5en3η+n2σT>m5, which is a contradiction. Let t4=inf{t>t2∣S(t)≥m5}, then for t∈[t2,t4),S(t)<m5 and S(t4)=m5. So, S(t)≥S(t2+)eσ(t-t2)=(1-PS)m5eσ(t-t2)≥(1-PS)n2+n3m5eσ(n2+n3)T=m1~ for t∈[t2,t4).
Case  2
(
t
2
≠
(
n
1
+
τ
-
1
)
T
,
n
1
∈
ℕ
). Suppose that t2∈((n1′+τ-1)T,(n1′+τ)T),n1′∈ℕ.  S(t)≥m5 for t∈[t1,t2) and S(t2)=m5. There are two possible cases for t∈(t2,(n1′+τ)T).
Case  2a. If S(t)≤m5 for all t∈(t2,(n1′+τ)T), similar to Case 1, we can prove there exists a t3′∈((n1′+τ)T,(n1′+τ)T+T~] such that S(t3′)≥m5. Let t4′=inf{t>t2∣S(t)≥m5}, then for t∈[t2,t4′),S(t)<m5 and S(t4′)=m5. So, S(t)≥S(t2)eσ(t-t2)=m5eσ(t-t2)≥(1-PS)n2+n3m5eσ(n2+n3+1)T=m*<m1~ for all t∈[t2,t4′).
Case  2b. If there exists a t∈(t2,(n1′+τ)T) such that S(t)≥m5. Let t4′¯=inf{t>t2∣S(t)≥m5}, then for t∈[t2,t4′¯),S(t)<m5 and S(t4′¯)=m5. So, S(t)≥S(t2)eσ(t-t2)=m5eσ(t-t2)≥m5eσT>m* for all t∈[t2,t4′¯).
SinceS(t)≥m5 for some t≥t1, in both cases a similar discussion can be continued. The proof is completed.


## 5. Numerical Simulations and Conclusions

In this section, we will give an example and its simulations to show the efficiency of the criteria derived in Section4.In system (1), let P1(x(t))=ax(t),  g(I(t))=bI(t), and P2(S(t))=h(1-e-cS(t)), a,b,c,h>0. Namely, P1(x(t)) describes an Holling type-I functional response of the pest, P2(S(t)) describes a Ivlev-type functional response of the pest's natural predator. Therefore, we consider the pest management model with impulsive releasing and harvesting at two different fixed moments:
(68)x′(t)=rx(t)(1-x(t)K)-ax(t)S(t),S′(t)=βax(t)S(t)-bI(t)S(t)-h(1-e-cS(t))yM(t)-dSS(t),I′(t)=bI(t)S(t)-dII(t),yJ′(t)=λh(1-e-cS(t))yM(t)-dJyJ(t)-myJ(t),yM′(t)=myJ(t)-dMyM(t),t≠(n+τ-1)T,t≠nT,Δx(t)=-δx(t),ΔS(t)=-PSS(t),ΔI(t)=-PII(t),ΔyJ(t)=-PJyJ(t),ΔyM(t)=-PMyM(t),t=(n+τ-1)T,Δx(t)=0,ΔS(t)=0,ΔI(t)=δI,ΔyJ(t)=δJ,ΔyM(t)=δM,t=nT.
So, by (22), (23), and (25), we have
(69)β∫0T‍P1(x*(t))dt=βaKr(rT+ln(1-δ))=θ1,∫0T‍g(I*(t))dt=bδI[1-e-dIτT+(1-PI)(e-dIτT-e-dIT)]dI(1-(1-PI)e-dIT)=θ2,∫0T‍yM*(t)dt=yM*(0+)dM(1-e-dMτT)+mδJ(1-(1-PJ)e-(m+dJ)T)(dM-(m+dJ))×(1-e-(m+dJ)τTm+dJ-1-e-dMτTdM)+yM*(τT+)edMτTdM(e-dMτT-edMT)+mδJ(1-PJ)(1-(1-PJ)e-(m+dJ)T)(dM-(m+dJ))×(e-(m+dJ)τT-e-(m+dJ)Tm+dJ-e-dMτT-e-dMTdMe(dM-(m+dJ))τT)=θ3,
where
(5)yM*(0+)=[(1-PM)A(τT)+B(T)]e-dMT+δM1-(1-PM)e-dMT,yM*(τT+)=(1-PM)[A(τT)+e-dMTB(T)+δM]e-dMτT1-(1-PM)e-dMT,A(τT)=mδJ(e(dM-(m+dJ))τT-1)(1-(1-PJ)e-(m+dJ)T)(dM-(m+dJ)),B(T)=mδJ(1-PJ)(e(dM-(m+dJ))T-e(dM-(m+dJ))τT)(1-(1-PJ)e-(m+dJ)T)(dM-(m+dJ)).
Then, by Theorems 11 and 14, we have the following. (T1)
Ifθ1-θ2-θ3-dST<ln(1/(1-PS)), then the susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)) of system (68) is locally asymptotically stable.(T2)
Ifθ1-θ2-hce-cMθ3-dST<ln(1/(1-PS)), then the susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)) of system (68) is globally asymptotically stable, where M is defined in Lemma 6.(T3)
Ifθ1-θ2-hcθ3-dST>ln(1/(1-PS), then system (68) is permanent.In the following, we analyze the locally asymptotical stability of the susceptible pest-eradication periodic solution and permanence of system (68).Assume thatx(0)=20, S(0)=2, I(0)=2, yJ(0)=0.5, yM(0)=0.5, r=8, K=10, a=0.8, β=0.5, b=0.3, h=8, c=0.2, λ=0.6, m=2, dS=0.2, dI=0.5, dJ=0.4, dM=0.2, δ=0.4, PS=PI=PJ=PM=0.2, δI=0.2, δJ=0.3, δM=0.5, τ=0.3, T=0.5. Obviously, the condition of (T1) is satisfied, then the susceptible pest-eradication periodic solution of system (68) is locally asymptotically stable, which can be seen from the numerical simulation in Figures 1 and 2.Time series of the system (68) with r=8, K=10, a=0.8, β=0.5, b=0.3, h=8, c=0.2, λ=0.6, m=2, dS=0.2, dI=0.5, dJ=0.4, dM=0.2, δ=0.4, PS=PI=PJ=PM=0.2, δI=0.2, δJ=0.3, δM=0.5, τ=0.3, T=0.5, x(0)=20, S(0)=2, I(0)=2, yJ(0)=0.5, yM(0)=0.5.
(a)
(b)
(c)
(d)
(e)Phase portrait of the system (68) with r=8, K=10, a=0.8, β=0.5, b=0.3, h=8, c=0.2, λ=0.6, m=2, dS=0.2, dI=0.5, dJ=0.4, dM=0.2, δ=0.4, PS=PI=PJ=PM=0.2, δI=0.2, δJ=0.3, δM=0.5, τ=0.3, T=0.5, x(0)=20, S(0)=2, I(0)=2, yJ(0)=0.5, yM(0)=0.5.
(a)
(b)Assume thatx(0)=20, S(0)=0.2, I(0)=2, yJ(0)=2, yM(0)=2, r=8, K=10, a=0.8, β=0.5, b=0.3, h=8, c=0.2, λ=0.6, m=2, dS=0.2, dI=0.5, dJ=0.4, dM=0.2, δ=0.4, PS=PI=PJ=PM=0.2, δI=0.2, δJ=0.3, δM=0.5, τ=0.3, T=1. Obviously, the condition of (T3) is satisfied. Then, system (68) is permanent, which can also be seen from Figures 3 and 4.Time series of the system (68) with r=8, K=10, a=0.8, β=0.5, b=0.3, h=8, c=0.2, λ=0.6, m=2, dS=0.2, dI=0.5, dJ=0.4, dM=0.2, δ=0.4, PS=PI=PJ=PM=0.2, δI=0.2, δJ=0.3, δM=0.5, τ=0.3, T=1, x(0)=20, S(0)=0.2, I(0)=2, yJ(0)=2, yM(0)=2.
(a)
(b)
(c)
(d)
(e)Phase portrait of the system (68) with r=8, K=10, a=0.8, β=0.5, b=0.3, h=8, c=0.2, λ=0.6, m=2, dS=0.2, dI=0.5, dJ=0.4, dM=0.2, δ=0.4, PS=PI=PJ=PM=0.2, δI=0.2, δJ=0.3, δM=0.5, τ=0.3, T=1, x(0)=20, S(0)=0.2, I(0)=2, yJ(0)=2, yM(0)=2.
(a)
(b)
(c)From results of the numerical simulation, we know that there exists an impulsive harvesting(or releasing) periodic thresholdT*, which satisfies 0.5<T*<1. If T<T* and the other parameters are fixed (r=8, K=10, a=0.8, β=0.5, b=0.3, h=8, c=0.2, λ=0.6, m=2, dS=0.2, dI=0.5, dJ=0.4, dM=0.2, δ=0.4, PS=PI=PJ=PM=0.2, δI=0.2, δJ=0.3, δM=0.5, τ=0.3, T=0.5.), then the susceptible pest-eradication periodic solution (x*(t),0,I*(t),yJ*(t),yM*(t)) of system (68) is locally asymptotically stable. If T>T* and the other parameters are fixed (r=8, K=10, a=0.8, β=0.5, b=0.3, h=8, c=0.2, λ=0.6, m=2, dS=0.2, dI=0.5, dJ=0.4, dM=0.2, δ=0.4, PS=PI=PJ=PM=0.2, δI=0.2, δJ=0.3, δM=0.5, τ=0.3, T=0.5), then system (68) is permanent. The same discussion can be applied to other parameters.In this paper, we proposed a pest management model with impulsive releasing (periodic infective pests, immature and mature natural enemies releasing) and harvesting (periodic crops harvesting) at two different fixed moments. By means of Floquet theory and multicomparison results for impulsive differential equations, two sufficient conditions ensuring the locally and globally asymptotical stability of the susceptible pest-eradication period solution and permanence of the system are derived.



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*Source: 832701-2013-04-17.xml*