# Dynamic Analysis of General Integrated Pest Management Model with Double Impulsive Control

Authors: Yingke Li; Zhidong Teng; Kai Wang; Ahmadjan Muhammadhaji
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

A general predator-prey model with disease in the prey and double impulsive control is proposed and investigated for the purpose of integrated pest management. By using the Floquet theory, the comparison theorem of impulsive differential equations, and the persistence theory of dynamical systems, we obtain that if threshold valueR0<1, then the susceptible pest eradication periodic solution is globally asymptotically stable and if R0>1, then the model is permanent. The numerical examples not only illustrate the theoretical results, but also show that when the model is permanent, then it may possess a unique globally attractive T-periodic solution.


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## Body

## 1. Introduction

Integrated pest management (IPM) is a long term management tactic that uses a combination of chemical, biological strategies to reduce pests to tolerable level or below the threshold, with less cost to the farmers and minimal effect on the environment (see [1, 2]). Such techniques include mechanical methods (erecting pest barriers or using pest traps) and biological methods (breeding natural predators of the pest or using biological insecticides). Some successful biological control examples contain the use of the predatory arthropodOrius sauteri against the pestThrips palmi Karny to protect eggplant crops in greenhouses (see [3]) and the use of the predatory mitesPhytoseiulus persimilis andNeoseiulus californicus to regulate the red spider miteTetranychus urticae Koch in field-grown strawberries (see [4]).The discontinuity of human activities and the abrupt variation in the amount of the pest population, which occurs immediately after successful control measures (such as spraying pesticides, releasing natural enemies of the pest, and freeing infective pest individuals), may be described mathematically through making use of impulsive differential equations (see [5–16]).Many scholars have been devoted to the analysis of impulsive differential equation models describing IPM strategies and some rich results have been obtained (see [6–15, 17]). They assumed that the disease incidence rate should be distinguished; as far as disease transmission is concerned, nonlinear, bilinear, and standard incidence rates have often been used in establishing ecoepidemic models, which depends on different infective disease and environment. Georgescu and Zhang (see [10]) investigated a predator-pest model with incidence rate given by g(I)S, Pang and Chen (see [12]) discussed an SI model with bilinear incidence rate βSI, Wang et al. (see [13]) analyzed an SI model with incidence rate given by f(S)I, and so forth. Main results of these theses have focused on conditions of pest eradication and permanence of the system. According to the authors’ knowledge, at present stage, there are few studies of general incidence rate. So one of the goals of this paper is to generalize the incidence rate.The functional response between pests and natural enemies plays an important role in assessing dynamical behavior of the system. People use natural enemy, as in some sense like a pesticide, to control pest via augmentation or releasing natural enemy once the quantity of pest has reached or exceeded the economic threshold (see [9, 10, 14, 15]). Shi et al. (see [14]) analyzed a predator-pest model with disease in the pest and functional response given by Holling-II type and the time-dependent impulsive strategy including release of infective pest individuals and those natural predators at different point in time; the threshold on pest eradication was obtained. However, little of paper has been devoted to analysis of models which combine release of infective pest individuals and those natural predators. The approach to biological control which we adopted is to release both infective pest individuals and natural predators periodically with constant amount at different point in time; what is more, the functional response is also a more general form. Motivated by the valuable contributions of Georgescu and Zhang [10], Wang et al. [13], and Shi et al. [14], in this paper general IPM model will be considered as follows: (1)S′t=StfSt,It-gSt,It-hSt,yt,I′t=gSt,It-ωIt,y′t=δhSt,yt-dyt,tn≠n+l-1T,tn≠nT,Itn+=Itn+p1,tn=n+l-1T,0<l<1,ytn+=ytn+p2,tn=nT.The model is based on the following assumptions:(H1)
The pest is divided into the susceptible and the infective, and the infective cannot produce offsprings as a result of the disease, but the infective still consume crop. The incidence rate of the infective is given by functiong(S,I). The growth rate of the susceptible is assumed to function Sf(S,I).(H2)
Parameterp1>0 represents the amount of infective pest released periodically at time tn=(n+l-1)T, where n∈Z+={1,2,…} and 0<l<1. Parameter p2>0 represents the amount of natural enemy released periodically at time tn=nT, n∈Z+.(H3)
The natural enemy only hunts the susceptible and the functional response is given by functionhS,y;δ>0 is the conversion rate.(H4)
Positive constantsω and d are the death rates of the infective pest and the natural enemy, respectively.In model (1), S(t) and I(t) denote the density of the susceptible pest and the infective pest (prey) population, respectively. y(t) is the density of natural enemy (predator) population. For model (1), in this paper we will investigate global stability of the susceptible pest eradication periodic solution and the permanence of model (1). In Section 2, the positivity and boundedness of solutions are presented. In Section 3, by using the Floquet theory for impulsive differential equations, the theorem on the global asymptotic stability of the susceptible pest eradication periodic solution is established. In Section 4, by using the persistence theory of dynamical systems, the theorem on the permanence of model (1) is established. In Section 5, we will give the numerical simulations to illustrate the main results obtained in this paper. Finally, in last section a brief discussion and some possible future researches are proposed.


## 2. Preliminaries

DenoteR+=[0,+∞) and R+n={(x1,x2,…,xn):xi∈R+,i=1,2,…,n}. For model (1), we introduce the following assumptions.(A1)
Functionf(x,y) is continuous on R+2 and is nonincreasing for x and y, respectively. Consider supx∈R+(xf(x,0)+γx)<∞ with γ=min{ω,d}.(A2)
Functiong(x,y) is continuously differentiable for x and y on R+2, g(0,y)=0, and ∂g(x,y)/∂x≥0, ∂g(x,y)/∂y≥0 for all (x,y)∈R+2.(A3)
Functionh(x,y) is continuously differentiable for x and y on R+2, h(0,y)=0, and ∂h(x,y)/∂x≥0, ∂h(x,y)/∂y≥0 for all (x,y)∈R+2.The solution of model (1), denoted by x(t)=(S(t),I(t),y(t)):R+→R+3, is piecewise continuous on ((n-1)T,(n+l-1)T] and ((n+l-1)T,nT], n∈Z+, x((n+l-1)T+)=limt→(n+l-1)T+x(t), and x(nT+)=limt→nT+x(t) exist. The global existence and uniqueness of solution for model (1) with any initial value x(0+)≥0 are guaranteed by the smoothness of the right-hand functions of model (1) (see [18]). Firstly, the following results are obtained easily.Lemma 1.
Assume thatx(t) is the solution of model (1) with x(0+)≥0, and then x(t)≥0 for all t≥0. Furthermore, if x(0+)>0, then x(t)>0 for all t>0.Lemma 2.
Letα be a positive constant. Then system (2)u′t=-αut,tn≠n+l-1T,0≤l≤1,n∈Z+,utn+=utn+u0,tn=n+l-1T,has a positive periodic solution (3)u∗t=u∗0+exp⁡-αt-n-1T,n-1T<t≤n+l-1T,u∗0+exp⁡-αlT+u0exp⁡-αt-n+l-1T,n+l-1T<t≤nT,where u∗(0+)=u0/(1-exp-αT). Furthermore, any solution u(t) of system (2) with initial value u(0+) can be expressed as (4)ut=u0+-u01-exp⁡-αTexp⁡-αt+u∗t,n-1T<t≤nT,and satisfies u(t)→u∗(t) as t→+∞.If susceptible pestS(t) is absent, then model (1) reduces to (5)I′t=-ωIt,y′t=-dyt,tn≠n+l-1T,tn≠nT,Itn+=Itn+p1,tn=n+l-1T,0<l<1,ytn+=ytn+p2,tn=nT.By Lemma 2, positive periodic solution (I∗(t),y∗(t)) of system (5) is (6)I∗t=I∗0+exp⁡-ωt-n-1T,n-1T<t≤n+l-1T,I∗0+exp⁡-ωlT+p1exp⁡-ωt-n+l-1T,n+l-1T<t≤nT,y∗t=y∗0+exp⁡-dt-n-1T,n-1T<t≤nT,where I∗(0+)=p1/(1-exp(-ωT)) and y∗(0+)=p2/(1-exp(-dT)). Furthermore, any solution (I(t),y(t)) of system (5) with initial values I(0+) and y(0+) can be expressed as (7)It=I0+-I∗0+exp⁡-ωt+I∗t,n-1T<t≤nT,yt=y0+-y∗0+exp⁡-dt+y∗t,n-1T<t≤nT.It is easy to get the following conclusion.Lemma 3.
System (5) has positive periodic solutions z∗(t)=(I∗(t),y∗(t)). For any solution z(t)=(I(t),y(t)) of system (5) with initial value z(0+)=(I(0+),y(0+)), one has z(t)→z∗(t) as t→∞.On the ultimate boundedness of solutions for model (1), we have the following conclusion.Lemma 4.
There exists a constantM>0 such that, for any solution x(t)=(S(t),I(t),y(t)) of model (1) with initial value x(0+)=(S(0+),I(0+),y(0+))∈R+3, one has S(t)≤M, I(t)≤M, and y(t)≤M with t large enough.Proof.
Define(8)Vt=Vxt=St+It+1δyt.By calculating the derivative of V(t) with respect to model (1), when tn≠(n+l-1)T and tn≠nT,(9)D+Vt=StfSt,It-ωIt-dδyt≤StfSt,0-ωIt-dδyt≤M~-γVt,where γ=min{ω,d} and M~=supS≥0{Sf(S,0)+γS}<∞. When tn=(n+l-1)T,V((n+l-1)T+)≤V((n+l-1)T)+p1, and when tn=nT,V(nT+)≤V(nT)+p2. By Lemma 2.2 given in [18], it is obvious that (10)limsupt→∞⁡Vt≤M~γ+p1+p2exp⁡γTexp⁡γT-1.From this, there exists a constant M>0 such that S(t)≤M,I(t)≤M, and y(t)≤M for t large enough. This completes the proof.In the following, we introduce some necessary definitions and lemma on the persistence of dynamical systems, which will be used for the discussion of permanence of model (1). For more details, see [19, 20].LetX be a metric space with metric d and let f:X→X be a continuous map. For any x∈X, we represent fn(x)=f(fn-1(x)) for any integer n>1 and f1(x)=f(x).f is said to be compact in X if, for any bounded set H⊂X, set f(H)={f(x):x∈H} is precompact in X.f is said to be point dissipative if there is a bounded set B0⊂X such that, for any x∈X,(11)limn→∞⁡dfnx,B0=0.For any x0∈X, the positive semiorbit through x0 is defined by γ+(x0)={fn(x0)=xn,n=1,2,…}, the negative semiorbit through x0 is defined as a sequence γ-(x0)={xk} satisfying f(xk-1)=xk for integers k≤0, the omega limit set of γ+(x0) is defined by ω(x0)={y∈X: there is a sequence nk→∞ such that limk→∞xnk=y}, and the alpha limit set of γ-(x0) is defined by α(x0)={y∈X: there is a sequence nk→-∞ such that limk→∞xnk=y}.A nonempty setB⊂X is said to be invariant if f(B)⊆B. A nonempty invariant set M of X is called isolated in X if it is the maximal invariant set in a neighborhood of itself. For a nonempty set M of X, set Ws(M)≔{x∈X:limn→∞d(fn(x),M)} is called the stable set of M.LetM1 and M2 be two isolated invariant sets and set M1 is said to be chained to set M2, usually expressed as M1→M2, if there exists a full orbit though some x∉M1∪M2 such that ω(x)⊂M2 and α(x)⊂M1. A finite sequence M={M1,…,Mn} of isolated invariant sets is called a chain if M1→M2→⋯→Mn, and if Mn=M1 the chain is called a cycle.LetX0 be a nonempty open set of X. We denote (12)∂X0≔X∖X0,M∂≔x∈∂X0:fnx∈∂X0,∀n≥0.Lemma 5 (see [19, 20]).
Letf:X→X be a continuous map. Assume that the following conditions hold:(C1)
Mapf is compact and point dissipative and f(X0)⊆X0.(C2)
There exists a finite sequenceM={M1,…,Mn} of compact and isolated invariant sets in ∂X0 such that(1)
Mi∩Mj=⌀ for any i,j=1,2,…,n and i≠j;(2)
ΩM∂≔∪x∈M∂ω(x)⊂∪i=1nMi;(3)
no subset ofM forms a cycle in ∂X0;(4)
Ws(Mi)∩X0=⌀ for each 1≤i≤n.
Then mapf is uniformly persistent with respect to X0,∂X0; that is, there exists a constant η>0 such that liminfn→∞d(fn(x),∂X0)≥η for all x∈X0.


## 3. Global Stability of Susceptible Pest Eradication Periodic Solution

From Lemma3, we know that model (1) has a susceptible pest eradication periodic solution (0,I∗(t),y∗(t)). On the global asymptotic stability of this periodic solution, we have the following theorem.Theorem 6.
Assume that(13)∫0Tf0,I∗t-∂g∂S0,I∗t-∂h∂S0,y∗tdt<0.Then periodic solution (0,I∗(t),y∗(t)) of model (1) is globally asymptotically stable.Proof.
To investigate the local stability of susceptible pest eradication periodic solution(0,I∗(t),y∗(t)), let S(t)=u1(t),I(t)=u2(t)+I∗(t),y(t)=u3(t)+y∗(t). We have (14)Δu2tn=0,tn=n+l-1T,0<l<1,Δu3tn=0,tn=nT.The corresponding linearized system is (15)u1′t=f0,I∗t-∂g∂S0,I∗t-∂h∂S0,y∗t·u1t,u2′t=∂g∂S0,I∗tu1t-ωu2t,u3′t=δ∂h∂S0,y∗tu1t-du3t.Let Φ(t) be the fundamental matrix of system (15); then (16)dΦtdt=f0,I∗t-∂g∂S0,I∗t-∂h∂S0,y∗t00∂g∂S0,I∗t-ω0δ∂h∂S0,y∗t0-dΦt,with Φ(0)=I, a 3×3 identity matrix. By calculating, we get (17)Φt=exp⁡∫0tρ0sds00a21exp⁡∫0t-ωds0a310exp⁡∫0t-dds,where(18)ρ0s=f0,I∗s-∂g∂S0,I∗s-∂h∂S0,y∗s,a21=∫0t∂g∂S0,I∗sexp⁡ωs-t+∫0sρ0τdτds,a31=∫0tδ∂h∂S0,y∗sexp⁡ds-t+∫0sρ0τdτds.Obviously, the eigenvalues of matrix M=Φ(T) are (19)λ2=exp⁡∫0T-ωds<1,λ3=exp⁡∫0T-dds<1,λ1=exp⁡∫0Tf0,I∗t-∂g∂S0,I∗t-∂h∂S0,y∗tdt.If (13) holds, then λ1<1. By the Floquet theory (see [21]), (0,I∗(t),y∗(t)) is locally asymptotically stable.
From (13), we can choose a small enough constant ϵ1>0 such that (20)ρ=exp⁡∫0Tf0,I∗t-ϵ1-min0≤ξ1≤ϵ1⁡∂gξ1,I∗t-ϵ1∂S-min0≤ξ2≤ϵ1⁡∂hξ2,y∗t-ϵ1∂Sdt=exp⁡∫0Tρ0t,ϵ1dt<1.From the second equation of model (1) and assumption (A2), we obtain (21)I′t≥-ωIt,tn≠n+l-1T.Consider the following impulsive differential equations: (22)v2′t=-ωv2t,tn≠n+l-1T,v2tn+=v2tn+p1,tn=n+l-1T,and, by the comparison theorem of impulsive differential equations (see [18]) and Lemma 3, it follows that I(t)≥v2(t) and v2(t)→I∗(t) as t→∞. Therefore, (23)It≥v2t≥I∗t-ϵ1for t large enough. Similarly, (24)yt≥y∗t-ϵ1for t large enough.
From assumptions(A2) and (A3), there exist 0<ξ1<S and 0<ξ2<S such that g(S,I)=(∂g(ξ1,I)/∂S)S and h(S,I)=(∂h(ξ2,I)/∂S)S. From (23), (24), and the first equation of model (1), there exists a T0>0 such that (25)S′t=SfS,I-gS,IS-hS,yS≤Sf0,I∗t-ϵ1-∂gξ1,I∗t-ϵ1∂S-∂hξ2,y∗t-ϵ1∂S≤Sf0,I∗t-ϵ1-min0≤ξ1≤ϵ1⁡∂gξ1,I∗t-ϵ1∂S-min0≤ξ2≤ϵ1⁡∂hξ2,y∗t-ϵ1∂S,for all t≥T0. For any t>T0 we can choose an integer n≥0 such that t=T0+nT+t~, where t~∈[0,T). Integrating the above inequality from T0 to t, we obtain (26)St≤ST0exp⁡∫T0tρ0s,ϵ1ds=ST0·exp⁡∫T0T0+nTρ0s,ϵ1ds·exp⁡∫T0+nTT0+nT+t~ρ0s,ϵ1ds≤ST0eM0Tρn⟶0asn⟶∞,where M0=supt≥0ρ0(t,ϵ1)>0, which implies S(t)→0 as t→∞.
Next, we show thatI(t)→I∗(t) and y(t)→y∗(t) as t→∞. Choose constant ϵ2>0 small enough such that η=sup0≤I≤M[g(ϵ2,I)/I]-ω<0. From limt→∞S(t)=0, it follows that S(t)≤ϵ2 for t large enough. From assumption (A3) and the second equation in model (1), (27)-ωIt≤I′t≤ηIt,tn≠n+l-1T.Consequently, v2(t)≤I(t)≤w2(t) and v2(t)→I∗(t), w2(t)→w2∗(t) as t→∞, where v2(t) is the solution of (22) and w2(t) is the solution of the following system: (28)w2′t=ηw2t,tn≠n+l-1T,w2tn+=w2tn+p1,tn=n+l-1T,w∗t=p11-exp⁡ηTexp⁡ηt-n-1T,n-1T<t≤n+l-1T,p1exp⁡ηlT1-exp⁡ηT+p1exp⁡ηt-n+l-1T,n+l-1T<t≤nT.Since limϵ2→0η=-ω, then (29)limϵ2→0⁡w2∗t=I∗t.Therefore, for any ϵ3>0 small enough, we have I∗(t)-ϵ3≤I(t)≤I∗(t)+ϵ3 for t large enough, which implies I(t)→I∗(t) as t→∞.
Lastly, a similar argument as from (27) to (29), we also can obtain that y(t)→y∗(t) as t→∞. This completes the proof.Corollary 7.
When the right-hand functions in model (1) are (30)fS,I=r1-S+θIK,gS,I=βSIq,hS,y=aSy,then condition (13) is equivalent to the following form: (31)rT<rθp1Kω+βp1qqω1-exp⁡-qωT1-exp⁡-ωTq+ap2d,where r,θ,K,β,q, and a are positive constants.Remark 8.
In (31), if p1=0, then p2>drT/a, which means that if only natural enemies are released periodically, then the release amount must be larger than drT/a to ensure the eradication of the pest. If p2=0, then the release amount must satisfy the inequality (32)rT<rθp1Kω+βp1qqω1-exp⁡-qωT1-exp⁡-ωTqto ensure the eradication of the pest.


## 4. Permanence of the Model

Theorem 9.
Assuming that(33)∫0Tf0,I∗t-∂g∂S0,I∗t-∂h∂S0,y∗tdt>0,then model (1) is permanent.Proof.
Since the impulsive effects in model (1) are periodic, model (1) can be regarded as periodic model with period T. Therefore, we can use the persistence theory of dynamical systems to discuss the permanence of model (1). Define (34)X=S,I,y:S≥0,I≥0,y≥0,X0=S,I,y∈X:S>0,I≥0,y≥0.Thus (35)∂X0=X∖X0=S,I,y∈X:S=0.From Lemma 1, we claim that X and X0 are positively invariant with respect to model (1). ∂X0 is a relatively closed set in X.
LetP:X→X be a Poincaré map associated with model (1); that is, (36)PS0,I0,y0=uT,S0,I0,y0,S0,I0,y0∈X,where u(t,S0,I0,y0) is the unique solution of model (1) with initial value u(0+,S0,I0,y0)=(S0,I0,y0). By Lemma 4, Poincaré map P is compact and point dissipative on X. Therefore, condition (C1) of Lemma 5 holds.
Let(37)M∂=S0,I0,y0∈∂X0:PnS0,I0,y0∈∂X0,n=1,2,…,where Pn=P(Pn-1), n>1, and P1=P.
Firstly, we will testify(38)M∂=∂X0.Clearly, M∂⊆∂X0. For any (0,I0,y0)∈∂X0, by S0=0, the solution (S(t),I(t),y(t)) of model (1) with initial value (S(0+),I(0+),y(0+))=(0,I0,y0) satisfies S(t)=0,I(t)≥0, and y(t)≥0 for all t≥0. Therefore, for any integer n>0, we obtain (39)Pn0,I0,y0∈∂X0.This implies (0,I0,y0)∈M∂. Therefore, (38) holds.
Model (1) can be simplified as model (5) in ∂X0. By Lemma 3, model (1) has globally attractive periodic solution (0,I∗(t),y∗(t)) in ∂X0. This shows that map P has a global attractor M1={(0,I∗(0),y∗(0))} in ∂X0. It is clear that, in ∂X0,{M1} is isolated, invariant, and does not form a cycle. Therefore, conditions 1–(3) of (C2) in Lemma 5 hold.
Secondly, letx0=(S0,I0,y0)∈X0. By the continuity of solutions with respect to the initial value, for any ε>0, there is a δ1>0; when x0-M1≤δ1, we have (40)ut,x0-ut,M1<ε,∀t∈0,T.Now, we claim that (41)limsupn→∞⁡dPnx0,M1≥δ1.Suppose the conclusion is not true; then we have(42)limsupn→∞⁡dPnx0,M1<δ1for some x0∈X0. For the sake of simplicity, one may assume that (43)dPnx0,M1<δ1∀n≥0.Also, from (40) we obtain (44)ut,Pnx0-ut,M1<ε∀n≥0,t∈0,T.Then, for any t≥0, let t=nT+t~, where t~∈[0,T) and n=[t/T] is the greatest integer less than or equal to t/T; we can get (45)ut,x0-ut,M1=ut~,Pnx0-ut~,M1<ε.Since u(t,x0)=(S(t),I(t),y(t)) and u(t,M1)=(0,I∗(t),y∗(t)), (45) signifies that (46)0<St≤ε,It≤I∗t+ε,yt≤y∗t+εfor all t≥0. By condition (33), we can choose constant ε>0 such that (47)ϱ=exp⁡∫0Tfε,I∗t+ε-max0≤θ1≤ε⁡∂gε,I∗t+ε∂S-max0≤θ1≤ε⁡∂hε,y∗t+ε∂Sdt=exp⁡∫0Tϱ0t,εdt>1.Further, from assumptions (A1) and (A2) and inequalities (46), we have (48)S˙t=SfS,I-gS,IS-hS,yS≥Sfε,I∗t+ε-∂gθ1,I∗t+ε∂S-∂hθ2,y∗t+ε∂S≥Sfε,I∗t+ϵ-max0≤θ1≤ϵ⁡∂gθ1,I∗t+ε∂S-max0≤θ2≤ϵ⁡∂hθ2,y∗t+ε∂S,where 0<θ1, θ2<S. For any t≥0, choose an integer k≥0 such that t=kT+t^, where t^∈[0,T). Integrating (48) from 0 to t and noticing (47), then (49)St≥S0+exp⁡∫0tϱ0s,εds=S0+exp⁡k∫0Tϱ0s,εds+∫kTkT+t^ϱ0s,εds≥S0+exp⁡TN0ϱk,where N0=supt≥0ϱ0(t,ϵ)>0. Thus, limt→∞S(t)=∞, which is a contradiction with 0≤S(t)≤ε for all t≥0. Hence, claim (41) holds. This shows Ws(M1)∩X0=∅. Therefore, condition (4) of (C2) holds. Consequently, by Lemma 5, P is uniformly persistent with respect to (X0,∂X0).
Lastly, since model (1) is periodic, we obtain that model (1) is uniformly persistent. From Lemma 4, model (1) also is permanent. This completed the proof.Corollary 10.
When functionsf(S,I),g(S,I), and h(S,y) are given in (30), then condition (33) is equivalent to the following condition: (50)rT>rθp1Kω+βp1qqω1-exp⁡-qωT1-exp⁡-ωTq+ap2d.Remark 11.
In (50), if p1=0, then we have p2<drT/a, which means that if only natural enemies are released periodically and the amount is less than drT/a, then the system is permanent and the pest will not be eradicated. If p2=0, then the release amount satisfies (51)rT>rθp1Kω+βp1qqω1-exp⁡-qωT1-exp⁡-ωTqto ensure the system is permanent and the pest will not be eradicated.Remark 12.
Applying Theorem1 given in [16], it is clear that when condition (33) holds, model (1) at least has one positive T-periodic solution.Remark 13.
Taking the functions in model (1) as (52)fS,I=r1-S+IK,gS,I=βSI,hS,y=aSy1+ωS,the model has been discussed in [14]. Furthermore, noticing (30), we have reason to confirm that our study makes the model in [14] more general.Corollary 14.
Define constant(53)R0=exp⁡∫0Tf0,I∗t-∂g∂S0,I∗t-∂h∂S0,y∗tdt.If R0<1, then susceptible pest eradication periodic solution (0,I∗(t),y∗(t)) of model (1) is globally asymptotically stable, and if R0>1, then model (1) is permanent.


## 5. Numerical Example

In this section, we give some examples and numerical simulations to confirm the above theoretical analysis. Let us consider model (1) with functions f(S,I),g(S,I), and h(S,y) given in (30) and parameters as follows.Example 1.
Taker=1.5, K=20, θ=0.5, β=0.8, q=2, δ=0.5, d=0.2, ω=0.5, a=0.7, p1=0.5, p2=0.6, and T=1.5. By computing, we have (54)rT=2.25<rθp1Kω+βp1qqω1-exp⁡-qωT1-exp⁡-ωTq+ap2d≈5.31.Therefore, inequality (31) holds; from Theorem 6, periodic solution (0,I∗(t),y∗(t)) of model (1) is globally asymptotically stable, which is illustrated in Figure 1.Figure 1
Time sires ofS(t) (a), I(t) (b), y(t) (c), and phase portrait (d) in model (1) with the initial values S(0+)=0.1+0.5k, I(0+)=2+0.4k, and y(0+)=3+0.4k, k = 1, 2, 3, 4, 5.
(a)
(b)
(c)
(d)Example 2.
Taker=6, K=13, θ=0.5, β=0.3, q=2, δ=0.8, d=0.4, ω=0.6, a=1, p1=2, p2=2, and T=2. By computing, M≈63, m≈1.5 in Lemma 4, and (55)rT=12>rθp1Kω+βp1qqω1-exp⁡-qωT1-exp⁡-ωTq+ap2d≈9.57.Therefore, inequality (50) holds; from Theorem 9, model (1) is permanent. Numerical simulation (see Figure 2) shows that there exists a unique positive T-periodic solution (S~(t),I~(t),y~(t)) of model (1) which is globally attractive.
From the above example, we can guess that only inequality (33) holds; then model (1) has a unique positiveT-periodic solution which is globally attractive.Figure 2
Time sires ofS(t) (a), I(t) (b), y(t) (c), and phase portrait (d) in model (1) with the initial values S(0+)=0.1+0.5k, I(0+)=2+0.4k, and y(0+)=3+0.4k, k= 1, 2, 3, 4, 5.
(a)
(b)
(c)
(d)


## 6. Discussion

In this paper, a general ecoepidemic model with impulsive control strategy is proposed and its dynamical behavior is analyzed for the purpose of integrated pest management. Meanwhile, the model which the researchers obtained in [14] was generalized. By using Floquet theorem and theory of persistence of dynamical systems, we show that if condition (13) holds, the susceptible pest eradication periodic solution (0,I∗(t),y∗(t)) is globally asymptotically stable (see Figure 1), which means that the pest has been eradicated; when condition (33) holds, model (1) is permanent (see Figure 2), which means the pest and their natural enemy can coexist in the area. Furthermore, from Remark 12, model (1) has at least one positive T-periodic solution.Further study includes investigating whether or not a nontrivial periodic solution emerges when the thresholdR0=1 holds and analyzing dynamical behavior about model with delay.



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*Source: 839097-2015-06-28.xml*