# The Study of a Class of Pest Control Pollution Model with Stage-Structure and Time Delay

Authors: Jianwen Jia; Bo Wu
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

We construct a pest control pollution model with stage-structure for pests and with epidemic in the predator by spraying pesticides and
releasing susceptible predators together. We assume that only the pests and infective predators are affected by pesticide. We show that there exists a globally attractive
pest-extinction periodic solution and we get the condition of global attractiveness of the pest-extinction periodic solution by applying comparison theorem of impulsive differential equation. Further, the condition for the permanence of the system is also given.


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## Body

## 1. Introduction

With the rapid development of modern technology, industry, and agriculture, it is of great interest to consider the effects of toxicant on ecological communities from both an environmental and conservational point of view. Qualitatively estimating the effect of a toxicant on a population by mathematical models is a relatively new field that began only in the early 1980s [1–3]. Population toxicant coupling has been applied in several contexts including Lotka-Volterra and chemostat-like environments, resulting in ordinary, integrodifferential, and impulsive differential equation and stochastic models. So in this paper, we consider the above effects and introduce the pollution model to model the process of pest control problems and study its dynamics, and this is different from the previous pest control model which assumed that pests were reduced proportionally by spraying pesticides [4–6].In the natural world, many species have a life history that takes their individual members through two stages: immature and mature; the authors of [7] studied an ecological model with stage-structure for predator. A general functional response was considered, and the authors analyzed the stability and the permanence of the system. The authors [8, 9] analyzed prey-predator models with age structure and impulsive control. The authors [10] investigated the dynamics of a pest control model with age structure for pests by introducing a constant periodic pesticide input and releasing natural enemies at different fixed moments. they analyzed the conditions for the global attractivity of the pest-extinction periodic solution and the permanence of the system.Modeling studies on disease-dominated ecological populations have addressed issues like disease-related mortality, reduction in reproduction, change in population sizes, and disease-induced oscillation of population states. Chattopadhyay and Arino [11] formulated a prey-predator model with prey infection and observed destabilization due to infection. Venturino analyzed prey-predator models with disease in the prey [12] and the predator [13].Motivated by the above, in this paper, we construct a pest control model with epidemic in the predator by spraying pesticides and releasing susceptible predators at the same time. The pest is stage-structured, and the effects of spraying pesticides into the environment and into the organism are considered. So the pollution model provides a natural description of such a system and should be introduced to our model.The organization of this paper is as follows: in Section2, we introduce a pest control pollution model with stage-structure for pest and with epidemic in the predator by introducing a constant periodic pesticide input and releasing susceptible predators together. In Section 3, we will introduce some definitions and lemmas which will be used in the paper. In Section 4, sufficient conditions are obtained for the global attractiveness of pest-extinction periodic solution. In Section 5, sufficient conditions are obtained for the permanence of the system. We give a brief conclusion of our results in the last section.


## 2. Model Formulation

In this paper, we suppose that pesticides hardly have influence on the susceptible predators, and the susceptible predators only feed on mature pests. Now we consider the following impulsive differential equation:(2.1)ẋ1(t)=αx2(t)-d1x1(t)-αe-d1τx2(t-τ)-r1co(t)x1(t)ẋ2(t)=αe-d1τx2(t-τ)-fx22(t)-β1x2(t)S(t)-r2co(t)x2(t)Ṡ(t)=β1cx2(t)S(t)1+hβ1x2(t)-β2S(t)I(t)1+kI(t)-d2S(t)İ(t)=β2S(t)I(t)1+kI(t)-d3I(t)-r4co(t)I(t)ċo(t)=kce(t)-gco(t)-mco(t)ċe(t)=-hce(t),t≠nT,n∈N,x1(t+)=x1(t),x2(t+)=x2(t),S(t+)=S(t)+pI(t+)=I(t),co(t+)=co(t),ce(t+)=ce(t)+q,t=nT,n∈N.Herex1=x1(t) and x2=x2(t) represent the density of the immature and mature pest (the prey) at time t, respectively; S=S(t) and I=I(t) represent the density of susceptible predator and infective predator at time t, respectively; ce(t) represents the concentration of pesticide in the environment at time t; co(t) represents the concentration of pesticide in the organism at time t; we use a special functional response, that is, when the number of the prey captured is less, the digestive capacity of the predator will increase with the density of the prey. Here α is the birth rate of the immature pest; d1 is the death rate of the mature pest; d2 and d3 are the death rates of the susceptible predator and the infective predator, respectively; τ is the mean length of the juvenile period; β1 represents the capturing rate of the susceptible predator; h represents digestive time of the susceptible predator; c is the transformation rate of the susceptible predator; f represents the intraspecific competition coefficient of mature pest; β2S(t)I(t)/(1+kI(t)) represents saturation incidence rate; T is the period of the impulsive effect; p is the releasing amount of the susceptible predator at t=nT; q is the amount of the pesticides spraying at every impulsive period nT; r1,r2,andr4 represent the decreasing rate of the intrinsic growth rate associated with the uptake of the pesticide in the organism for the immature pest, mature pest, susceptible predator, and infective predator, respectively; kce(t) represents an organism’s net uptake of toxin from the environment; gco(t) and mco(t) represent the digestion and depuration rates of pesticide in an organism, respectively; hce(t) represents the loss of pesticide in the environment due to natural degradation. All the coefficients are positive constants.The initial conditions of system (2.1) are(2.2)(x1(t),x2(t),S(t),I(t),co(t),ce(t))∈C([-τ,0],R+6),x1(0)>0,x2(0)>0,S(0)>0,I(0)>0,co(0)>0,ce(0)>0,
where R+6={(x1,x2,S,I,co,ce):x1≥0,x2≥0,S≥0,I≥0,co≥0,ce≥0}. To assure the continuity of the initial values, we assume that x1(0)=∫-τ0αeγθx2(θ)dθ. This suggests that if we know the properties of x2(t), then the properties of x1(t) can be obtained.Note that the variablex1(t) does not appear in the second, third, forth, fifth, and sixth equations of system (2.1), hence we only need to consider the subsystem of (2.1) as follows:(2.3)ẋ2(t)=αe-d1τx2(t-τ)-fx22(t)-β1x2(t)S(t)-r2co(t)x2(t)Ṡ(t)=β1cx2(t)S(t)1+hβ1x2(t)-β2S(t)I(t)1+kI(t)-d2S(t)İ(t)=β2S(t)I(t)1+kI(t)-d3I(t)-r4co(t)I(t)ċo(t)=kce(t)-gco(t)-mco(t)ċe(t)=-hce(t),t≠nT,n∈N,x2(t+)=x2(t),S(t+)=S(t)+p,I(t+)=I(t)co(t+)=co(t),ce(t+)=ce(t)+q,t=nT,n∈N.The initial conditions for system (2.3) are(2.4)(x2(t),S(t),I(t),co(t),ce(t))∈C([-τ,0],R+5),x2(0)>0,S(0)>0,I(0)>0,co(0)>0,ce(0)>0,
where R+5={(x2,S,I,co,ce):x2≥0,S≥0,I≥0,co≥0,ce≥0}.


## 3. Definitions and Lemmas

LetR+=[0,∞],R+5={X=(x2(t),S(t),I(t),co(t),ce(t)):x2≥0,S≥0,I≥0,co≥0,ce≥0}, and N be the set of all nonnegative integers, let f=(f1,f2,f3,f4,f5)T the map defined by the right of system (2.3). The solution of system (2.3), denoted by X=(x2(t),,S(t),I(t),co(t),ce(t)):R+→R+5, is continuously differentiable on (nT,(n+1)T). Let V:R+×R+5→R+, then V is said to belong to class V0 if;(1)V is continuous on (nT,(n+1)T]×R+5, and for each X(t)∈R+5,n∈N andlim(t,y)→(nT+,x)V(t,y)=V(nT+,x) exist;(2)V is locally Lipschitzian inX.Definition 3.1.
LetV∈V0, then for (t,X)∈(nT,(n+1)T]×R+5, the upper right derivative of V(t,X) with respect to impulsive differential system (2.3) is defined as
(3.1)D+V(t,X)=lim⁡h→0+sup⁡1h[V(t+h,X+hf(t,X))-V(t,X)].Definition 3.2.
System (2.3) is said to be permanent if there are constants m,M>0(independent of initial value) and a finite time T0 such that for every positive solution (x2(t),S(t),I(t),co(t),ce(t))∈R+5 with initial conditions of system (2.3) satisfies m≤x2(t)≤M, m≤S(t)≤M, m≤I(t)≤M, m≤co(t)≤M,m≤ce(t)≤M for all t≥T0, here T0 may depend on the initial condition of system (2.3).Lemma 3.3 (the comparison theorem of impulsive differential equation [14]).
LetV∈V0. Assume that
(3.2)D+V(t,x)≤g(t,V(t,x)),t≠nT,V(t,X(t+))≤φn(V(t,x)),t=nT,where g:R+×R+→R is continuous in (nT,(n+1)T]×R+, and for each u∈R+,n∈N,lim(t,v)→(nT+,u)g(t,v)=g(nT+,u) exists and is finite; φn:R+→R+ is nondecreasing.
Letr(t) be the maximal solution of the scalar impulsive differential equation defined on [0,+∞), then(3.3)u̇(t)=g(t,u(t)),t≠nT,u(t+)=φn(u(t)),t=nT,u(0+)=u0.
So V(0+,X0)≤u0 implies that
(3.4)V(t,X(t))≤r(t),t≥0,
where X(t) is any solution of system (2.3).Lemma 3.4 (see [15]).
Consider the following equation:(3.5)dxdt=ax(t-τ)-bx(t),
where a,b,τ>0, x(t)>0 for t∈[-τ,0], one has(i)
ifa<b, then limt→∞x(t)=0,(ii)
ifa>b, then limt→∞x(t)=+∞.Remark 3.5 (see [16]).
co(t),ce(t) are the concentration of toxicant. To assure 0≤co(t)≤1,0≤ce(t)≤1, it is necessary that g≤k≤g+m.Remark 3.6 (see [16]).
From the point of the biological meaning, we assume thatk<h.Lemma 3.7.
Consider the following subsystem of system (2.3)
(3.6)ċo(t)=kce(t)-gco(t)-mco(t)ċe(t)=-hce(t),t≠nT,n∈N,co(t+)=co(t)ce(t+)=ce(t)+q,t=nT,n∈N.
Then, system (3.6) has a unique positive T-periodic solution (co*(t),ce*(t)) and for each solution (co(t),ce(t)) of system (3.6), co(t)→co*(t) and ce(t)→ce*(t) as t→+∞. Where
(3.7)co*(t)=co*(0)e-(g+m)(t-nT)+kq(e-(g+m)(t-nT)-e-h(t-nT))(h-g-m)(1-e-hT),ce*(t)=qe-h(t-nT)1-e-hT,co*(0)=kq(e-(g+m)T-e-hT)(h-g-m)(1-e-hT)(1-e-(g+m)T),ce*(0)=q1-e-hT,
for t∈(nT,(n+1)T] and n∈N.Lemma 3.8.
There exists a constantM>0 such that x1(t)≤M,x2(t)≤M,S(t)≤M,I(t)≤M,co(t)≤M,ce(t)≤M for each solution X(t)=(x1(t),x2(t),S(t),I(t),co(t),ce(t)) of system (2.1) with all t large enough.Proof.
DefineV(t)=c(x1+x2)+S+I+co+ce. Choose 0<l<min{d1,d2,d3,g+m,h-k}, we have V∈V0 and
(3.8)D+V(t)+lV(t)≤-cfx22+c(l+α)x2+(l-d2)S+(l-d3)I+(l-g-m)co+(l+k-h)ce+c(l-d1)x1,t≠nT,n∈N,≤-cfx22+c(l+α)x2,V(nT+)=V(nT)+p+q,t=nT,n∈N.
Hence there exists a positive constant K such that
(3.9)D+V(t)≤K-lV(t),t≠nT,n∈N,V(nT+)=V(nT)+p+q,t=nT,n∈N,
by Lemma 3.3, for t≥0, we have
(3.10)V(t)≤(V(0+)-Kl)e-lt+(p+q)(1-e-nlT)e-l(t-nT)1-e-lT+Kl,t∈(nT,(n+1)T],n∈N.
Then limt→∞V(t)≤(K/l)+(p+q)elT/(elT-1).
SoV(t) is uniformly ultimately bounded. By the definition of V(t), there exists a constant M>0 such that x1(t)≤M,x2(t)≤M,S(t)≤M,I(t)≤M,co(t)≤M,ce(t)≤M for t large enough. The proof is complete.


## 4. The Global Attractivity of Periodic Solution

In this section, the sufficient conditions are obtained for the global attractivity of the pest-extinction periodic solution.We first demonstrate the expression of the pest-extinction solution of system (2.3), in which the pest individual and infective predator individual are entirely absent from the model, that is, x2(t)=0,I(t)=0 for all t≥0.Whenx2(t)=0 and I(t)=0, S(t) satisfies the following system:(4.1)Ṡ(t)=-d2S,t≠nT,S(t+)=S(t)+p,t=nT,S(0+)=S0. Clearly, we can obtain the unique positive periodic solution with the form(4.2)S*(t)=pe-d2(t-nT)1-e-d2T,t∈(nT,(n+1)T],n∈N.Therefore,S(t)=(S(0+)-p/1-e-d2T)e-d2t+S*(t),t∈(nT,(n+1)T], n∈N is the solution of system (4.1) with initial value S0.Denoteη1=e-(β2/k+d2)T/(1-e-(β2/k+d2)T),η2=k(e-(g+m)T-e-hT)e-(g+m)T/(h-g-m)(1-e-(g+m)T)(1-e-hT).Theorem 4.1.
LetX(t)=(x2(t),S(t),I(t)co(t),ce(t)) be any solution of system (2.3) with positive initial values, if
(4.3)αe-d1τβ1η1p+r2qη2<1,β2pe-d2Td3(1-e-d2T)<1,
then (0,S*(t),0,co*(t),ce*(t)) is globally attractive.Proof.
From system (2.3), we have
(4.4)Ṡ(t)≥-(d2+β2k)S,t≠nT,S(t+)=S(t)+p,t=nT.
Consider the following comparison system
(4.5)u̇(t)=-(d2+β2k)u,t≠nT,u(t+)=u(t)+p,t=nT,u(0+)=S(0+).
Obviously, system (4.5) has a positive periodic solution
(4.6)u*(t)=pe(-d2-β2/k)(t-nT)1-e(-d2-β2/k)T,t∈(nT,(n+1)T],
which is globally asymptotically stable. By Lemma 3.3, we conclude that for an arbitrary positive constant ε1 small enough, there exists an N1∈Z such that
(4.7)S(t)≥Z(t)>Z*(t)-ε1,t∈(N1T,(N1+1)T].
From which, we get
(4.8)S(t)>pe(-d2-β2/k)T1-e(-d2-β2/k)T-ε1,t∈(N1T,(N1+1)T].
That is,
(4.9)S(t)>pη1-ε1,t∈(N1T,(N1+1)T].
By Lemma 3.7, we conclude that for a sufficiently small ε2>0, there exists an N2∈Z such that
(4.10)c0(t)>c0*(t)-ε2,t∈(N2T,(N2+1)T].
that is,
(4.11)c0(t)>qη2-ε2,t∈(N2T,(N2+1)T].
Let T̅=max{N1T,N2T}, from the first equation of systems (2.3), (4.9), and (4.11). We have
(4.12)ẋ2(t)<αe-d1τx2(t-τ)-fx22(t)-β1x2(t)(pη1-ε1)-r2x2(t)(qη2-ε2),t>T̅+τ.
Now consider the following comparison equation:
(4.13)Ṗ(t)=αe-d1τP(t-τ)-fP2(t)-[β1(pη1-ε1)+r2(qη2-ε2)]P(t).
Since the first condition of the theorem holds, we can choose the above ε1,ε2 small enough such that
(4.14)αe-d1τ<β1(pη1-ε1)+r2(qη2-ε2).
By (4.14) and Lemma 3.4, we have limt→∞P(t)=0.
By Lemma3.3, we get
(4.15)lim⁡t→∞⁡x2(t)≤lim⁡t→∞⁡P(t)=0.
Incorporating the positivity of x2(t), we get limt→∞x2(t)=0.
Then for a sufficiently smallε3∈(0,d2) and t large enough, we have 0<x2(t)<ε3/cβ1, without loss of generality, we may assume 0<x2(t)<ε3/cβ1 as t≥0.
From the second equation of system (2.3), we have(4.16)Ṡ(t)≤(ε3-d2)S,t≠nT,S(t+)=S(t)+p,t=nT.
Consider the following system:
(4.17)Ġ(t)=(ε3-d2)G(t),t≠nT,G(t+)=G(t)+p,t=nT,G(0+)=S(0+).
Obviously, system (4.17) has a positive periodic solution
(4.18)G*(t)=pe(-d2+ε3)(t-nT)1-e(-d2+ε3)T,nT<t≤(n+1)T,
which is globally asymptotically stable. Thus, for a sufficiently small ε4>0, when t is large enough, we have
(4.19)S(t)≤G(t)<V*(t)+ε4.
By the second condition of the theorem, when t is large enough, we have
(4.20)β2(pe-d2T1-e-d2T+ε4)-d3<0.
By (4.19), when t is large enough,
(4.21)S(t)<pe-d2T1-e-d2T+ε4.
Combining the third equation of system (2.3) with (4.21), we obtain
(4.22)İ(t)≤[β2(pe-d2T1-e-d2T+ε4)-d3]I(t).
By (4.20) and I≥0, we have I(t)→0 as t→∞.
Further, sinceI(t)→0, for an arbitrary positive constant ε5 small enough, we have I≤ε5 as t is large enough. Then(4.23)Ṡ(t)≥-(d2+β2ε5)S,t≠nT,S(t+)=S(t)+p,t=nT.
Consider the following system:
(4.24)U̇(t)=-(d2+β2ε5)U,t≠nT,U(t+)=U(t)+p,t=nT,U(0+)=S(0+).
So system (4.24) has a positive periodic solution
(4.25)U*(t)=pe-(d2+β2ε5)(t-nT)1-e-(d2+β2ε5)T,nT<t≤(n+1)T,
which is globally asymptotically stable. Therefore, for an arbitrary positive constant ε6 small enough, when t is large enough, we have
(4.26)S(t)≥U(t)>U*(t)-ε6.
Combining (4.26) with (4.19), we obtain U*(t)-ε6<S(t)<G*(t)+ε4, since ε4,ε5,ε6 are all sufficient small constants, we know
(4.27)lim⁡t→∞S(t)=S*(t).
By Lemma 3.7, we get
(4.28)lim⁡t→∞c0(t)=c0*(t),lim⁡t→∞ce(t)=ce*(t).
The proof is complete.Remark 4.2.
Obviously, we know that the global attractiveness of pest-eradication periodic solution(0,0,S*(t),0,co*(t),ce*(t)) of system (2.2) is equivalent to the global attractiveness of mature pest-eradication periodic solution (0,S*(t),0,co*(t),ce*(t)) of system (2.3).


## 5. Permanence

Theorem 5.1.
System (2.3) is permanent provided that
(5.1)αe-d1τ-β1pe-d2T1-e-d2T-r2kq(e-(g+m)T-e-hT)(h-g-m)(1-e-hT)(1-e-(g+m)T)>0,β2pe-d2T(1-e-d2T)d3>1.Proof.
By Lemma3.8, we know that there exists an M>0,and M>(αe-d1τ)/f such that x2(t)≤M, S(t)≤M, I(t)≤M, co(t)≤M,ce(t)≤M for all t>0, we will prove the theorem through the following five steps.Step 1.
From (4.9), we know that there exists an m2=pη1-ε1 such that S(t)≥m2 for t large enough.Step 2.
From (4.11), we know that there exists an m3=qη2-ε2 such that c0(t)≥m3 for t large enough.Step 3.
From Lemma3.7, for an arbitrary positive constant ε0 small enough, when t is large enough, ce(t)≥qe-hT/(1-e-hT)-ε0≜m4.Step 4.
We will prove that there exists anm1>0 such that x2(t)≥m1 for t large enough, we will do it in the following two steps.
(i) By the condition of the theorem, we can select positive constantsε7 and m5 small enough such that(5.2)m5<αe-d1τf,δ=β1cm5<d2,αe-d1τ-fm5-β1pe(-d2+δ)T1-e(-d2+δ)T-β1ε7-r2η-r2ε2>0.
Now we will prove that x2(t)<m5 cannot hold for all t≥0, otherwise,
(5.3)Ṡ(t)≤(-d2+δ)S.
Let
(5.4)Ṙ(t)=(-d2+δ)R,t≠nT,R(t+)=R(t)+p,t=nT,R(0+)=S(0+).
By Lemma3.3, S(t)≤R(t) and R(t)→R*(t),t→∞, where R*(t)=pe(-d2+δ)(t-nT)/(1-e(-d2+δ)T),t∈(nT,(n+1)T]. So there exists a T1>0 such that(5.5)S(t)≤R(t)<R*(t)+ε7,
for t>T1. From Lemma 3.7, we know that when t is large enough,
(5.6)c0(t)<co*(0)e-(g+m)T+kq(e-(g+m)T-e-hT)(h-g-m)(1-e-hT)+ε2≜η+ε2,
for a sufficiently small ε2>0, then
(5.7)ẋ2(t)≥αe-d1τx2(t-τ)-[fm5+β1(pe(-d2+δ)T1-e(-d2+δ)T+ε7)+r2(η+ε2)]x2(t),
by Lemma 3.4, x2(t)→∞, as t→∞. This is a contradiction to the boundedness of x2(t). So there exists a t1>0 such that x2(t1)≥m5.
(ii) Ifx2(t)≥m5 for all t≥t1, our aim is obtained. Otherwise, we consider that x2(t) is oscillating about m5.
Lett2=inft≥t1{x2(t)<m5}, then x2(t)≥m5 for t∈[t1,t2), since x2(t) is continuous, x2(t2)=m5; and because x2(t) is oscillating about m5, we know that there exists a t3=inft≥t2{x2(t)>m5}, then x2(t)≤m5 for t∈[t2,t3), from the continuity of x2(t), we get x2(t3)=m5; and continue we can obtain the time sequence t1≤t2<t3<⋯<t2k<t2k+1<⋯, which satisfies
(a)
wheni=2,3,4,…,x2(ti)=m5,(b)
whent∈(t2k,t2k+1),k=1,2,…,x2(ti)<m5,(c)
whent∈(t2k+1,t2k+2),k=1,2,…,x2(ti)>m5.
We claim that there exists aT0=sup{t2k+1-t2k, k∈N}, otherwise, there exists a subsequence {Tj∣Tj=t2kj+1-t2kj, j∈N} such that limj→∞Tj=+∞; obviously, (5.7) holds when t∈(t2kj,t2kj+1).
By Lemma3.4, we get limj→∞x2(t2kj+1)=+∞; this is a contradiction to x2(t2kj+1)=m5.
By the boundedness of the system, we have(5.8)S(t)<M,c0(t)<M,
for t≥t1. It is clear that
(5.9)ẋ2(t)≥-(fm5+β1M+r2M)x2(t),x2(t2k)=m5,
for t∈(t2k,t2k+1), k=1,2,… Let m1=m5e-(fm5+β1M+r2M)T0, then we have x2(t)≥m1 for all t≥t1.Step 5.
We will prove that there exists anm6>0 such that I(t)≥m6 for t large enough, we will do it in the following two steps.
(i) From the condition of the theorem, letm7>0 and ε8>0 be small enough such that(5.10)β21+km7(pe-(β2m7+d2)T1-e-(β2m7+d2)T-ε8)-d3-r4M>0.
We will prove I(t)<m7 cannot hold for all t≥0. Otherwise, we have
(5.11)Ṡ(t)≥-(β2m7+d2)S(t).
Consider the following system:
(5.12)Q̇(t)=-(d2+β2m7)Q,t≠nT,Q(t+)=Q(t)+p,t=nT,Q(0+)=S(0+).
By Lemma 3.3, we have S(t)≥Q(t) and Q(t)→Q*(t), as t→∞, where Q*(t)=pe-(d2+β2m7)(t-nT)/(1-e-(d2+β2m7)T),t∈(nT,(n+1)T].
So there exists aT2>0, when t>T2S(t)≥Q(t)>Q*(t)-ε8.
Hence,(5.13)İ(t)≥[β21+km7(pe-(β2m7+d2)T1-e-(β2m7+d2)T-ε8)-d3-r4M]I(t),
from (5.10), we get I(t)→∞, as t→∞, this is a contradiction to the boundedness of I(t).So there exists a t0>0 such that I(t0)≥m7.
(ii) Similar to the method of step (ii) of Step4, we can find an m6=m7e-d3T0′ such that I(t)≥m6 for all t≥t0.
Therefore,m≤x1(t),x2(t),S(t),I(t),co(t),ce(t)≤M for t large enough, where m=min{m1,m2,m3,m4,m6}. The proof is complete.


## 6. Conclusion

In this paper, we propose and analyze a pest control model with age structure for pest and pulse spraying pesticides and pulse releasing infective predators. By Lemma3.8, we know that any solution of system (2.1) is bounded for t large enough and get the specific form of the upper boundedness. From Theorem 4.1, we get the sufficient condition of global attractiveness of the pest-extinction periodic solution:(6.1)αe-d1τβ1η1p+r2qη2<1,β2pe-d2Td3(1-e-d2T)<1.By Theorem5.1, we get the sufficient condition for the permanence of the system:(6.2)αe-d1τ-β1pe-d2T1-e-d2T-r2kq(e-(g+m)T)-e-hT)(h-g-m)(1-e-hT)(1-e-(g+m)T)>0,β2pe-d2T(1-e-d2T)d3>1.



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*Source: 598539-2012-05-28.xml*