# An Impulsively Controlled Three-Species Prey-Predator Model with Stage Structure and Birth Pulse for Predator

Authors: Yanyan Hu; Mei Yan; Zhongyi Xiang
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

We investigate the dynamic behaviors of a two-prey one-predator system with stage structure and birth pulse for predator.
By using the Floquet theory of linear
periodic impulsive equation and small amplitude perturbation method, we show that there exists a
globally asymptotically stable two-prey eradication
periodic solution when the impulsive period is less than
some critical value. Further, we study the permanence of the investigated model. Our
results provide valuable strategy for biological economics
management. Numerical analysis is also inserted to illustrate the results.


---

## Body

## 1. Introduction

In the natural world, the predator-prey relationship is one of the important interactions among species, and it has been extensively studied by many authors [1–7] because of its universal existence. The following two-prey, one-predator model was studied by many works [8–10]:(1)x˙1=x1b1-x1-αx2-r1y,x˙2=x2b2-βx1-x2-r2y,y˙=y1-b3+dr1x1+dr2x2,where xit(i=1,2) and y(t) are densities of the prey population and the predator population, respectively. ri>0(i=1,2) are coefficients of decrease of prey due to predation; bi>0(i=1,2,3) are intrinsic rates of increase or decrease. α>0, β>0 are parameters representing competitive effects between two prey species; d>0 is an equal transformation rate of the predator.As we know, life history often occurs in natural ecological environments which has significant morphological and behavioral differences between immature and mature species; the dynamics of stage-structured prey-predator system has been widely studied [11–13]. Recently, many impulsive differential equation models or hybrid dynamical systems have been proposed to model the introduction of a periodic IPM strategy [14, 15]; Xiang and Song [16] proposed impulsive prey-dependent consumption two-prey one-predator models with stage structure for the predator, which combined the biological control and chemical control:(2)x˙1t=b1x1t1-x1t-αx1tx2t-r1x1ty2t,x˙2t=b2x2t1-x2t-βx1tx2t-r2x2ty2t,y˙1t=λ1r1x1ty2t1+h1r1x1t+λ2r2x2ty2t1+h2r2x2t-m+μy1t,y˙2t=my1t-μy2t,t≠nT,Δx1t=-E1x1t,Δx2t=-E2x2t,Δy1t=p1,Δy2t=p2,t=nT.The biological meanings of the parameters in (2) can be seen in [16]. As far as the population dynamics is concerned, most models often considered that the population reproduces throughout all year. However, many species give birth seasonally or in regular pules. Particular examples of birth pulse type are ΔN=B(N)N. The model discussed in [17] considered the birth pulse ΔN=B(N)N, B(N)=b-CNθ. A birth pulse type ΔN=B(N)N with B(N)=beN and B(N)=p/(q+Nn) are assumed in [18]. Comparatively, we consider birth pulse ΔN=N(a-bN).In view of birth pulse and impulsive control strategy, we formulate the following two prey-predator models with stage-structure and birth pulse for predator:(3)x˙1t=x1tb1-x1t-αx1tx2t-r1x1ty1t-r2x1ty2t,x˙2t=x2tb2-x2t-βx1tx2t-r3x2ty1t-r4x2ty2t,y˙1t=λ1r1x1ty1t+λ3r3x2ty1t-m+μy1t,y˙2t=λ2r2x1ty2t+λ4r4x2ty2t+my1t-μy2t,t≠n+lT,t≠n+1T,Δx1t=-p1x1t,Δx2t=-p2x2t,Δy1t=-p3y1t,Δy2t=-p4y2t,t=n+lT,n=1,2,…Δx1t=0,Δx2t=0,Δy1t=y2ta-by2t,Δy2t=0,t=n+1T,n=1,2,…,where xi(t)(i=1,2) are densities of the prey population and y1(t) and y2(t) represent the densities of the immature and the mature predator population, respectively. ri>0(i=1,2,3,4) are coefficients of decrease of prey due to predation, α>0, β>0 are parameters representing competitive effects between two prey species, and λi>0(i=1,2,3,4) represent the rate of conversion of nutrients into the predators. The maturity is m, which determines the mean length of the juvenile period; μ is the natural death rate of the predator population. Δy1(t)=y2(t)(a-by2(t)) is birth pulse at t=(n+1)T as intrinsic rate of natural increase and density dependence rate of predator population are denoted by a, b, respectively. a/b is the carrying capacity of the predator population. 0≤l≤1, 0≤pi≤1(i=1,2,3,4) represent the fraction of prey and predators which die due to the pesticide at t=(n+l)T, n∈Z+.The organization of this paper is as follows. In the next section, some important lemmas are presented. Sections3 and 4 give the global asymptotical stability of the prey eradication periodic solution and permanence for system (3). Numerical analysis is displayed in Section 5. Finally, a brief discussion is given to conclude this work.


## 2. Preliminaries

In this section, we will give some definitions, notations, and some lemmas which will be useful for our main results.LetR+=[0,∞), R+4={x∈R4:x>0}. Thus Ω=int⁡R+4; let Z+ be the set of all nonnegative integers. Denote f=(f1,f2,f3,f4), the map defined by the right hand side of the first four equations of system (3). Let V:R+×R+4↦R+; then V is said to belong to class V∈V0 if(i)
V is continuous in (nT,(n+l)T]×R+4 and ((n+l)T,(n+1)T]×R+4 for each, x∈R+4, n∈Z+, lim(t,y)→((n+l)T+,x)V(t,y)=V((n+l)T+,x) and lim(t,y)→((n+1)T+,x)V(t,y)=V((n+1)T+,x) existing;(ii)
V is locally Lipschitzian in x.Definition 1.
ConsiderV∈V0; then, for (t,x)∈(nT,(n+l)T]×R+4 and (t,x)∈((n+l)T,(n+1)T]×R+4, the upper right derivative of V(t,x) with respect to the impulsive differential system (3) is defined as D+V(t,x)=limh→0+sup⁡(1/h)[V(t+h,x+hf(t,x))-V(t,x)].
The solution of system (3), denoted by x(t)=(x1(t),x2(t),y1(t),y2(t)):R+↦R+4, is continuously differentiable on (nT,(n+l)T] and ((n+l)T,(n+1)T], n∈Z+, 0≤l≤1.
Obviously the smoothness properties off guarantee the global existence and uniqueness of solutions of system (3); for details see [19]. The following lemma is obvious.Lemma 2.
Supposex(t) is a solution of (3) with x(0+)≥0; then x(t)≥0 for all t≥0. And further x(t)>0, t≥0, if x(0+)>0.
Make a notation as(4)p4∗=e-mT1-p3+p3e-mT-e-mlTa-eμT+1+a1-e-mT-1-p3e-m+μT1+a1-e-mT-1-p3e-m+μT.
Ifx1(t)=0, x2(t)=0, then system (3) can be rewriten to (5)y˙1t=-m+μy1,y˙2t=my1t-μy2t,t≠n+lT,t≠n+1T,Δy1t=-p3y1t,Δy2t=-p4y2t,t=n+lT,Δy1t=y2ta-by2t,Δy2t=0,t=n+1T.
It is easy to obtain the analytic solution of system (5) between pulses: (6)y1t=y1nT+e-m+μt-nT,t∈nT,n+lT,y1n+lT+e-m+μt-n+lT,t∈n+lT,n+1T,y2t=e-μt-nTy2nT++y1nT+1-e-mt-nT,t∈nT,n+lT,e-μt-n+lTy2n+lT++y1n+lT+1-e-mt-n+lT,t∈n+lT,n+1T.
Considering (5), we have the stroboscopic map of system(7)y1n+1T+=ae-μT1-p4y2nT++a1-p4e-μT1-e-mlT+1-p3e-μ+mlT1-e1-lmT+1-p3·e-m+μTy1nT+-be-μT1-p4y2nT++1-p4e-μT1-e-mlT+1-p3e-μ+mlT1-e1-lmTy1nT+,y2n+1T+=e-μT1-p4y2nT++1-p4·e-μT1-e-mlT+1-p3e-μ+mlT1-e1-lmTy1nT+.
For convenience, we chooseA=e-μT(1-p4)>0, B=(1-p3)e-(μ+ml)T>0, and C=1-p4e-μT1-e-mlT+1-p3e-μ+mlT1-e1-lmT>0; the following two equivalence relations are calculated to be (8)p4>p4∗⟺B+aC-1-AB+A<0,p4<p4∗⟺B+aC-1-AB+A>0.The two fixed points of (5) are obtained as G0(0,0) and G(y1∗,y2∗), where (9)y1∗=B+aC-1-AB+A1-AbC2,p4<p4∗,y2∗=B+aC-1-AB+AbC,p4<p4∗.Lemma 3.
(i) Ifp4>p4∗, the fixed point G0(0,0) is globally asymptotically stable.
(ii) Ifp4<p4∗, the fixed point G(y1∗,y2∗) is globally asymptotically stable.Proof.
For convenience, we make a notation as(y1n,y2n)=(y1∗,y2∗). Linear form (5) can be written as (10)y1n+1y2n+1=Jy1ny2n.
Obviously, the near dynamics ofG0(0,0) and G(y1∗,y2∗) are determined by linear system (10). The stabiles of G0(0,0) and G(y1∗,y2∗) are determined by the eigenvalue of J less than 1. If J satisfies the Jury criteria [20], we can know that the eigenvalue of J is less than 1: (11)1-trJ+det⁡J>0,where(12)J=B+aC-2bC2y1∗-2bACy2∗aA-2bA2y2∗-2bACy1∗CA.(i) If p4>p4∗, then B+aC-1-AB+A<0,G0(0,0) is the unique fixed point of system (5); we have (13)J1=J0,0B+aCaACA.Calculating 1-trJ1+det⁡J1=1-A-B+AB-aC>0, and from the Jury criteria G0(0,0) is locally stable, and then it is globally asymptotically stable.
(ii) Ifp4<p4∗, then B+aC-1-AB+A>0,G0(0,0) is unstable. For B+aC-1-AB+A>0,G(y1∗,y2∗) exists, we have(14)J2=Jy1∗,y2∗·B+aC-2bCCy1∗+Ay2∗aA-2bACy1∗+Ay2∗CA.Consider 1-trJ+det⁡J=1-A-B+AB-aC>0. From the Jury criteria G(y1∗,y2∗) is locally stable, and then it is globally asymptotically stable. The proof is completed.Summarizing the above results, we have the following theorems.Theorem 4.
The triviality periodic solutionG0(0,0) is stable if p4>p4∗; otherwise, if p4<p4∗, the triviality periodic solution G0(0,0) is unstable, and the periodic solution G(y1∗,y2∗) is globally asymptotically stable.
Correspondingly, system (5) has a globally asymptotically stable positive periodic solution (y~1(t),y~2(t)), where (15)y~1t=y1∗e-m+μt-nT,t∈nT,n+lT,y1∗e-m+μlTe-m+μt-n+lT,t∈n+lT,n+1T,y~2t=e-μt-nTy2∗+y1∗1-e-mt-nT,t∈nT,n+lT,e-μt-n+lTe-μlTy2∗+y1∗1-e-mlT+y1∗e-m+μlT1-e-mt-n+lT,t∈n+lT,n+1T.Lemma 5.
There exists a constantM>0 such that xit≤M, yit≤M(i=1,2), for each positive solution x(t)=x1t,x2t,y1t,y2t of (5) with all t large enough.Proof.
Define a functionV(t,x) such that V(t,x)=∑i=12(λxi(t)+yi(t)), where λ=max{λ1,λ2,λ3,λ4}. When t≠(n+l)T,  t≠(n+1)T, we have(16)D+Vt,x+μVt,x≤-b1x1-b1+μ2b12+b1+μ24b1-b2x1-b2+μ2b22+b2+μ24b2≤b1+μ24b1+b2+μ24b2≜M1.When t=(n+l)T,(17)Vn+lT+=λx1n+lT++λx2n+lT++y1n+lT++y2n+lT+≤Vn+lT,and when t=(n+1)T,(18)Vn+1T+=λx1n+1T++λx2n+1T++y1n+1T++y2n+1T+≤Vn+1T+a24b.
It follows from the comparison theorem of impulsive differential equations (see Lemma2.2 [21], page 23) that, for t∈(nT,(n+1)T), we have (19)Vt,x≤V0+e-μt+M1μ1-e-μt+a24be-μt-T1-eμT+a24beμTeμT-1⟶M1μ+a24beμTeμT-1,t⟶∞.
SoV(t,x) is ultimately bounded. Hence, by the definition of V(t,x), there exists a constant M>0, such that xi(t)≤M,  yi(t)≤M,  i=1,2, for t large enough. The proof is complete.Definition 6.
System (3) is said to be permanent if there are constants m,M>0 and a finite time T0 such that all solutions xi(t),yi(t) with all initial values xi(0+)>0, yi(0+)>0, m≤xi(t)≤M, and m≤xi(t)≤M hold for all t≥T0.


## 3. Extinction

In this section, we investigate the stability of the two-pest prey eradication periodic solution as a solution of system (3). We give the condition which assures the asymptotical stability of the prey eradication periodic solution (0,0,y~1(t),y~2(t)).Theorem 7.
Let(x1(t),x2(t),y1(t),y2(t)) be any solution of (3); then (0,0,y~1(t),y~2(t)) is globally asymptotically stable if (20)p4<p4∗,T<min⁡ln⁡1/1-p1+r1y1∗/m+μ+r2y1∗+y2∗/μb1,ln⁡1/1-p2+r3y1∗/m+μ+r4y1∗+y2∗/μb2hold true.Proof.
Firstly, we prove the local stability ofT-period solution (0,0,y~1(t),y~2(t)) which may be determined by considering the behavior of small-amplitude perturbations (u1(t),u2(t),v1(t),v2(t)) of the solution.
Define(21)x1t=u1t,x2t=u2t,y1t=v1t+y~1t,y2t=v2t+y~2t,where u1(t), u2(t), v1(t), and v2(t) are small perturbations; they may be written as (22)u1tu2tv1tv2t=Φtu10u20v10v20,where Φ(t) satisfy (23)dΦtdt=b1-r1y~1t-r2y~2t0000b2-r3y~1t-r4y~2t00λ1r1y~1tλ3r3y~1t-m+μ0λ2r2y~2tλ4r4y~2tm-μΦt,with Φ(0)=I, where I is the identity matrix.
Hence the fundamental solution matrix is(24)Φt=exp∫0tH1ds000∗exp⁡∫0tH2ds00∗∗exp⁡-m+μt0∗∗exp⁡-μt-exp⁡-μ+mtexp⁡-μt,where H1=b1-r1y~1(s)-r2y~2(s) and H2=b2-r3y~1(s)-r4y~2(s). There is no need to calculate the exact form of (∗) as it is not required in the analysis that follows. The resetting impulsive conditions of (3) become (25)u1n+lT+u2n+lT+v1n+lT+v2n+lT+=1-p100001-p200001-p300001-p4u1n+lTu2n+lTv1n+lTv2n+lT,(26)u1n+1T+u2n+1T+v1n+1T+v2n+1T+=10000100001a0001u1n+1T+u2n+1T+v1n+1T+v2n+1T+.Hence, the stability of the periodic solution (0,0,y~1(t),y~2(t)) is determined by the eigenvalues of (27)M=1-p100001-p200001-p300001-p410000100001a0001ΦT
which are(28)λ1=1-p1exp⁡∫0Tb1-r1y~1s-r2y~2sds,λ2=1-p2exp⁡∫0Tb2-r3y~1s-r4y~2sds,λ3=1-p3e-μ+mT+ae-μT1-e-mT<1,λ4=1-p4e-μT<1,λ1<1ifT<ln⁡1/1-p1+r1y1∗/m+μ+r2y1∗+y2∗/μb1,λ2<1ifT<ln⁡1/1-p2+r3y1∗/m+μ+r4y1∗+y2∗/μb2.
According to the Floquet theory (see [21]), the two-pest eradication solution (0,0,y~1(t),y~2(t)) is locally asymptotically stable.
In the following, we prove the global attractivity. Chooseε1,ε2>0 such that (29)η=1-p3·exp⁡∫0Tb1-r1y~1t-ε1-r2y~2t-ε2ds<1.Noting that y˙1(t)≥-(m+μ)y1(t), y˙2(t)≥my1(t)-μy2(t), consider the following impulsive differential equation: (30)z˙1t=-m+μz1t,z˙2t=mz1t-μz2t,t≠n+lT,t≠n+1T,Δz1t=-p3z1t,Δz2t=-p4z2t,t=n+lT,Δz1t=z2ta-bz2t,Δz2t=0,t=n+1T.It follows from impulsive differential equation (30) that y1(t)≥z1(t), y2(t)≥z2(t) and z1(t)→y~1(t), z2(t)→y~2(t) as t→∞. This is (31)y1t≥z1t≥y~1t-ε1,y2t≥z2t≥y~2t-ε2,for all t large enough. For convenience, we may assume that (31) holds. From (3) and (31) we have (32)x˙1t≤x1tb1-r1y~1t-ε1-r2y~2t-ε2,t≠n+lT,x1t+=1-p1x1t,t=n+lT,holding for all t large enough. Integrate (32) on ((n+l)T,(n+1)T], which yields x1((n+l)T)≤x1nT+1-p1exp⁡(∫nT(n+l)T(b1-r1(y~1(s)-ε1)-r2(y~2(s)-ε2))ds). Thus x1((n+l)T)≤x1(lT+)ηn and x1((n+l)T)→0 as t→∞. Therefore x1(t)→0 as t→∞, since 0<x1(t)<x1n+lT1-p1exp⁡(b1T) for (n+l)T≤t≤(n+1)T. By the same method we can prove x2(t)→0, as t→∞, so we omit it.
Next, we will prove thaty1(t)→y~1(t),y2(t)→y~2(t), as t→∞. If limt→∞x1(t)=0 and limt→∞x2(t)=0, for 0<ɛ≤(m+μ)/(λ1r1+λ3r3), there exists t0>0 such that 0<x1(t)<ε1 and 0<x2(t)<ɛ, for all t>t0; then we have (33)-m+μy1t≤y˙1t≤-m+μ-λ1r1ɛ-λ3r3ɛy1t,my1t-μy2t≤y˙2t≤my1t-μ-λ2r2ɛ-λ4r4ɛy2t.
From the left hand inequality of (33), it follows from impulsive differential equation (5) that y1(t)→y~1(t) and y2(t)→y~2(t) as t→∞. For the right hand inequality, we consider the following impulsive differential equation: (34)z˙3t=-m+μ-λ1r1ɛ-λ3r3ɛz3t,z˙4t=mz3t-μ-λ2r2ɛ-λ4r4ɛz4t,t≠n+lT,t≠n+1T,Δz3t=-p3z3t,Δz4t=-p4z4t,t=n+lT,Δz3t=z4ta-bz4t,Δz4t=0,t=n+1T.System (34) has a globally asymptotically stable positive periodic solution (z~3(t),z~4(t)), where(35)z~3t=z3∗e-m+μ-λ1r1ɛ-λ3r3ɛt-nT,t∈nT,n+lT,z3∗e-m+μ-λ1r1ɛ-λ3r3ɛlTe-m+μ-λ1r1ɛ-λ3r3ɛt-n+lT,t∈n+lT,n+1T,z~4t=e-μ-λ2r2ɛ-λ4r4ɛt-nTz4∗+z3∗1-e-mt-nT,t∈nT,n+lT,e-μ-λ2r2ɛ-λ4r4ɛt-n+lTe-μ-λ2r2ɛ-λ4r4ɛlTz4∗+z3∗1-e-mlT+z3∗e-m+μ-λ1r1ɛ-λ3r3ɛlT1-e-mt-n+lT,t∈n+lT,n+1T,where(36)z3∗=B1+aC1-1-A1B1+A11-A1bC12,p4<p4⋆,z4∗=B1+aC1-1-A1B1+A1bC1,p4<p4⋆,and A1=e-(μ-λ2r2ɛ-λ4r4ɛ)T(1-p4)>0, B1=(1-p3)e-(m+μ-λ1r1ɛ-λ3r3ɛ)T>0, and C1=(1-p4)e-(μ-λ2r2ɛ-λ4r4ɛ)T(1-e-mlT)+(1-p3)e-(m+μ-λ1r1ɛ-λ3r3ɛ)T(1-e(1-l)mT)>0.
Set(37)p4⋆=e-θ1Teθ2T1-p3+1-μ11-e-mT-mlTa-eθ2T+1+a1-e-mT-1-p3e-θ1T1+a1-e-mT-1-p3e-θ1T,where θ1=m+μ-λ1r1ɛ-λ3r3ɛ and θ2=μ-λ2r2ɛ-λ4r4ɛ.
Therefore, there existsε1>0 such that (38)z~1t-ε1<y1t<z~3t+ε1for t large enough; let ɛ→0; then we get z~3(t)→y~1(t), z~1(t)→y~1(t). Hence y1(t)→y~1(t) as t→∞. Similarly, we can prove y2(t)→y~2(t) as t→∞, so we omit it. This proof is complete.


## 4. Permanence

In this section, we will investigate the permanence of system (3). In biological terms, the permanence implies that preys and predators will coexist, none of them facing extinction or growing indefinitely.Theorem 8.
System (3) is permanent if p4<p4⋇,b1>αb2, and b2>βb1,(39)-r1y1∗e-m+μ-r3λ3b2lTe-m+μ-r3λ3b2T-1-m+μ-r3λ3b2-r2y1∗e-m+μ-r3λ3b2lTe-μ-r4λ4b2+mT-1-μ-r4λ4b2-m-ln11-p1-r2e-μ-r4λ4b2T-1e-μ-r4λ4b2lTy1∗1-e-mlT+y2∗+y1∗e-m+μ-r3λ3b2lT-μ-r4λ4b2+b1-αb2T>0,-r3y1∗e-m+μ-r3λ3b2lTe-m+μ-r3λ3b2T-1-m+μ-r3λ3b2-r4y1∗e-m+μ-r3λ3b2lTe-μ-r4λ4b2+mT-1-μ-r4λ4b2-m-ln11-p2-r4e-μ-r4λ4b2T-1e-μ-r4λ4b2lTy1∗1-e-mlT+y2∗+y1∗e-m+μ-r3λ3b2lT-μ-r4λ4b2+b2-βb1T>0.Proof.
Let us suppose a solutionx(t)=(x1(t),x2(t),y1(t),y2(t)) of (3) with initial value x(t)>0. From Lemma 5, we have proved that there exists a constant M>0 such that xit≤M, yit≤M(i=1,2), and M>max{b1/(r1+r2),b2/(r3+r4)} for t>0. Note that (40)x˙1t≤x1tb1-x1t,t≠n+lT,Δx1t=-p1x1t,t=n+lT,x10+≥0.Consider the following comparison equation:(41)ω˙1t=ω1tb1-ω1t,t≠n+lT,Δω1t=-p1ω1t,t=n+lT,ω10+=ω10+≥0.We have x1(t)≤ω1(t) and ω1(t)→b1 as t→∞; thus there exists ε3′ such that x1(t)<b1+ε3′ for t large enough. Without loss of generality, we may assume x1(t)<b1+ε3′ for t>0. Similarly, we may set x2(t)<b2+ε4′. From (31), we know y1(t)>z~1(t)-ɛ,y2(t)>z~2(t)-ɛ for all t large enough (ɛ>0 is enough small), so that y1(t)≥y1∗e-(m+μ)T-ɛ≜m3,y2(t)≥e-μT(y2∗+y1∗(1-e-mlT))+y1∗e-(μ+ml)T(1-e-m(1-l)T)-ɛ≜m4 for t large enough. Thus we only need to find m¯1>0,m¯2>0 such that x1(t)≥m¯1,x2(t)≥m¯2 for t large enough. We will do it in the following two steps.
Step
1. Denote (42)0<m1<min⁡m+μ-r3λ3b2+ε4′r1λ1,μ-m-r4λ4b2+ε4′r2λ2,0<m2<min⁡m+μ-r1λ1b1+ε4′r3λ3,μ-m-r2λ2b1+ε4′r4λ4,and r1λ1m1+r3λ3m2>m+μ. We will prove that there exist t1,t2∈(0,∞), such that x1(t1)≥m1 and x2(t2)≥m2. Otherwise there will be three cases:(i)
There existst2>0, such that x2(t2)≥m2 and x1(t)<m1 for all t>0.(ii)
There existst1>0, such that x1(t1)≥m1 and x2(t)<m2 for all t>0.(iii)
Considerx1(t)<m1, x2(t)<m2, for all t>0.Firstly, consider case (i). Let ε1>0 be small enough, so that (43)σ=b1-m1-αb2+ε4′-r1ε1-r2ε1T-r1z6∗e-ϖ1lTe-ϖ1T-1-ϖ1-ln11-p1>0-e-ϖ2T-1e-ϖ2lTz6∗1-e-mlT+z7∗+z6∗e-ϖ1lT-ϖ2-z6∗e-ϖ1lTe-ϖ2+mT-1-ϖ2-m,where (44)ϖ1=m+μ-r1λ1m1-r3λ3b2+ε4′,ϖ2=μ-r2λ2m1-r4λ4b2+ε4′.According to the above assumption, it is easy to get that y˙1(t)≤-y1(t)ϖ1 and y˙2(t)≤my1(t)-ϖ2y2(t). Then we consider the following impulsive differential equation:(45)z˙6t=-ϖ1z6t,z˙7t=mz7t-ϖ2z7t,t≠n+lT,t≠n+1T,Δz6t=-p3z6t,Δz7t=-p4z7t,t=n+lT,Δz6t=z7ta-bz7t,Δz7t=0,t=n+1T,z~6t=z6∗e-ϖ1t-nT,t∈nT,n+lT,z6∗e-ϖ1lTe-ϖ1t-n+lT,t∈n+lT,n+1T,z~7t=e-ϖ2t-nTz7∗+z6∗1-e-mt-nT,t∈nT,n+lT,e-ϖ2t-n+lTe-ϖ2lTz7∗+z6∗1-e-mlT+z6∗e-ϖ1lT1-e-mt-n+lT,t∈n+lT,n+1T,where(46)z6∗=B2+aC2-1-A2B2+A21-A2bC22,p4<p4⋇,z7∗=B2+aC2-1-A2B2+A2bC2,p4<p4⋇,and A2=e-ϖ1T(1-p4)>0, B2=(1-p3)e-ϖ1T>0, C2=(1-p4)e-ϖ2T(1-e-mlT)+(1-p3)e-ϖ1T(1-e(1-l)mT)>0, and (47)p4⋇=e-ϖ1Teϖ2T1-p3+1-μ11-e-mT-mlTa-eϖ2T+1+a1-e-mT-1-p3e-ϖ1T1+a1-e-mT-1-p3e-ϖ1T.Therefore, there exists T1>0 such that(48)y1t≤z6t≤z~6t+ε1,y2t≤z7t≤z~7t+ε1.Thus we can easily get (49)x˙1t≥x1tb1-m1-αb2+ε4′-r1z~6t+ε1-r2z~7t+ε1,t≠n+lT,Δx1t=-p1x1t,t=n+lT,for t>T1. Set (N+l-1)T≥T1, N∈Z+. Integrating (49) on ((n+l)T,(n+1)T],  n≥N, then we obtain (50)x1n+l+1T≥x1n+lT+1-p1exp⁡∫n+lTn+l+1Tb1-m1-αb2+ε4′-r1z~6t+ε1-r2z~7t+ε1dt=x1n+lTeσ;consequently x1((N+n+l)T)≥x1((N+l)T)enσ→∞ as n→∞, which contradicts to the boundedness of x1(t).
Similarly, cases (ii) and (iii) can be analyzed as in case (i). Here we omit it. From the above three cases, we conclude that there existt1>0,  t2>0 such that x1(t1)≥m1,  x2(t2)≥m2.
Step
2. If x1(t)≥m1 for all t≥t1, then our aim is obtained. Otherwise, x1(t)<m1 for any t≥t1. Let t~=inft>t1{x1(t)<m1}; there are two possible cases for t~.
Case
1. Consider t~=(n1+l)T,  n1∈Z+. Then x1(t)≥m1 for t∈[t1,t~] and (1-p1)m1≤x1(t~+)=(1-p1)x1(t~)<m1. Choose n2,n3∈Z+ such that (51)n2-1T>max⁡ln⁡ε1/M-ϖ1,ln⁡ε1/M-ϖ2,1-p1n2eσn3en2η3T>1.Set T′=n2T+n3T; we claim that there must exist t2∈(t~,t~+T′) such that x1(t2)>m1. In view of (45), with z6(n1T+)=y1(n1T+), z7(n1T+)=y2(n1T+), we have (52)z6t=z6n1T+e-m+μ+r1λ1m1+r3λ3b2+ε4′t-nT+z~6t,t∈nT,n+lT,z6n1+lT+e-m+μ+r1λ1m1+r3λ3b2+ε4′t-n+lT+z~6t,t∈n+lT,n+1T,z7t=e-μ-r2λ2m1-r4λ4b2+ε4′t-nTz7n1T++z6n1T+1-e-mt-nT+z~7t,t∈nT,n+lT,e-μ-r2λ2m1-r4λ4b2+ε4′t-n+lTz7n1+lT++z6n1+lT+1-e-mt-n+lT+z~7t,t∈n+lT,n+1T.And n1+1≤n≤n1+n2+n3. Then (53)z6t-z~6t<M·exp⁡-m+μ+r1λ1m1+r3λ3b2+ε4′·t-nT<ε1,z7t-z~7t<M·exp⁡-μ-r2λ2m1-r4λ4b2+ε4′<ε1;thus y1(t)≤z6(t)<z~6(t)+ε1 and y2(t)≤z7(t)<z~7(t)+ε1 for n1T+n2-1T≤t≤t~+T′, which implies that (49) holds for t~+n2T≤t≤t~+T′. So as in step 1, we get (54)x1t~+T′≥x1t~+n2Teσn3.From system (3), we get (55)x˙1t≥x1tb1-m1-αb2+ε4′-r1M-r2M,t≠n+lT,Δx1t=-p1x1t,t=n+lT.Integrating (55) on [t~,t~+n2T], we have (56)x1t~+n2T≥m11-p1n2exp⁡n2η3T.Therefore, we deduce that (57)x1t~+T′≥m11-p1n2eσn3en2η3T>m1,which is a contradiction.
Lett^=inft>t~x1t≥m1; then for t∈(t~,t^), x1(t)<m1 and x1(t^)=m1. For t∈(t~,t^), set t∈(t~+(r-1)T,t~+rT], r∈Z+, r≤n2+n3; from (55), we have (58)x1t≥x1t~+1-p1r-1exp⁡r-1η3T·exp⁡η3t-t~+r-1T≥m11-p1r·exp⁡rη3T≥m11-p1n2+n3·exp⁡n2+n3η3T.Let m^1=m1(1-p1)n2+n3exp((n2+n3)η3T), so we have x1(t)≥m^1 for t∈(t~,t^). For t>t^, the same argument can be continued since x1(t^)≥m1.
Case
2. Consider t~≠(n+l)T, n∈Z+. Then x1(t)≥m1 for t∈[t1,t~) and x1(t~)=m1. Let t~∈((n1′+l)T,(n1′+1)T), n1′∈Z+. There are two possible cases for t∈(t~,(n1′+l)T).
Ifx1(t)<m1 for all t∈(t~,(n1′+1)T), there must be t2′∈[(n1′+1)T,(n1′+1)T+T′] such that x1(t2′)≥m1. Here, we omit it.
Lettˇ=inft>t~x1t≥m1; then for t∈(t~,tˇ),x1(t)<m1 and x1(tˇ)=m1. For t∈(t~,tˇ), let t∈((n1′+l)T+(r′-1)T,(n1′+l)T+r′T],  r′∈Z+,  r′≤n2+n3; from (55), we have (59)x1t≥m11-p1r′exp⁡r′+1η3T≥m11-p1n2+n3exp⁡n2+n3+1η3T.Let m¯1=m1(1-p1)n2+n3exp((n2+n3+1)η3T), so we have x1(t)≥m¯1 for t∈(t~,tˇ). For t>tˇ, the same argument can be continued since x1(tˇ)≥m1.
If there existst∈(t~,(n1′+l)T) such that x1(t)≥m1, let t¯=inft>t~{x1(t)≥m1}; then x1(t)<m1 for t∈(t~,t¯) and x1(tˇ)=m1. For t∈(t~,t¯), (55) holds; integrating (55) on (t~,t¯), we have (60)x1t≥x1t~exp⁡η3t-t~≥m1exp⁡η3T>m¯1.Since x1(t¯)≥m1 for t>t¯, the same argument can be continued. Hence, we have x1(t)≥m¯1 for all t>t1. Similarly, we can prove x2(t)≥m¯2 for all t≥t2. The proof is complete.From the proof of Theorems7 and 8, we can derive the following results.Corollary 9.
Let(x1(t),x2(t),y1(t),y2(t)) be any solution of (3); then yi, i=1,2, are permanent; x1(t)→0 as t→∞ provided, b1>αb2, b2>βb1,(61)T<ln⁡1/1-p2+r3y1∗/m+μ+r4y1∗+y2∗/μb2,-r1y1∗e-m+μ-r3λ3b2lTe-m+μ-r3λ3b2T-1-m+μ-r3λ3b2-r2y1∗e-m+μ-r3λ3b2lTe-μ-r4λ4b2+mT-1-μ-r4λ4b2-m-ln⁡11-p1-r2e-μ-r4λ4b2T-1e-μ-r4λ4b2lTy1∗1-e-mlT+y2∗+y1∗e-m+μ-r3λ3b2lT-μ-r4λ4b2+b1-αb2T>0.Corollary 10.
Let(x1(t),x2(t),y1(t),y2(t)) be any solution of (3); then x1 and yi, i=1,2, are permanent; x2(t)→0 as t→∞ provided, b1>αb2, b2>βb1,(62)T<ln⁡1/1-p1+r1y1∗/m+μ+r2y1∗+y2∗/μb1,-r3y1∗e-m+μ-r3λ3b2lTe-m+μ-r3λ3b2T-1-m+μ-r3λ3b2-r4y1∗e-m+μ-r3λ3b2lTe-μ-r4λ4b2+mT-1-μ-r4λ4b2-m-ln⁡11-p2-r4e-μ-r4λ4b2T-1e-μ-r4λ4b2lTy1∗1-e-mlT+y2∗+y1∗e-m+μ-r3λ3b2lT-μ-r4λ4b2+b2-βb1T>0.


## 5. Numerical Analysis

In this section, we carry out numerical simulations of system (3) to illustrate and support our analytical results.(1) Consider the following choice of parametric values:α=0.3, β=0.2, b1=1, b2=1, r1=0.2, r2=0.5, r3=0.3, r4=0.4, λ1=0.1, λ2=0.2, λ3=0.3, λ4=0.4, m=0.2, μ=0.1, d=0.4, p1=0.01, p2=0.01, p3=0.01, p4=0.01, a=2, b=1, l=0.6, and T=2.Correspondingly,min⁡⁡{(ln⁡(1/(1-p1))+r1y1∗/m+μ+r2y1∗+y2∗/μ)/b1,(ln⁡(1/(1-p2))⁡+r3y1∗/(m+μ)+r4(y1∗+y2∗)/μ)/b2}≈5.07>T=2 and p4=0.01<p4∗≈0.4. According to Theorem 7, we know that the prey eradication periodic solution of system (3) is globally asymptotically stable for this case (see Figure 1).Figure 1
Globally asymptotically stable prey eradication solution of system (3). (a, b) Time series of prey populations x1(t), x2(t) and (c, d) time series of predator populations y1(t), y2(t) with initial values x1(0)=0.5, x2(0)=0.5, y1(0)=0.5, and y2(0)=0.5.
(a)
(b)
(c)
(d)(2) Consider the following set of parameters:α=0.3, β=0.2, b1=1.5, b2=1.5, r1=0.2, r2=0.5, r3=0.3, r4=0.4, λ1=0.1, λ2=0.2, λ3=0.3, λ4=0.4, m=0.2, μ=0.1, d=0.4, p1=0.01, p2=0.01, p3=0.01, p4=0.01, a=2.45, b=1, l=0.6, and T=2.Since conditions of Theorem8 hold for this choice, system (3) is permanent (see Figure 2).Figure 2
The permanence of system (3). (a, b) Time series of prey populations x1(t), x2(t) and (c, d) time series of predator populations y1(t), y2(t) with initial values x1(0)=0.5, x2(0)=0.5, y1(0)=0.5, and y2(0)=0.5.
(a)
(b)
(c)
(d)


## 6. Discussion

In this paper, we study an impulsively controlled three-species prey-predator model with stage structure and birth pulse for predator. Using the comparison theorems, we have shown that there exists a globally asymptotically stable two-pest eradication periodic solution when the impulsive periodT<min⁡{(ln(1/(1-p1))+r1y1∗/(m+μ)+r2(y1∗+y2∗)/μ)/b1,(ln⁡(1/(1-p2))+r3y1∗/m+μ)/b2+r4(y1∗+y2∗)/μb2}. In biological terms, the pest population is eradicated totally and the predator population will tend to a stable level. However, the prey population is eradicated totally, which is not most desired for us from the biological point and saving resources; we hope to control the species under the economic threshold level or drive the target prey population to extinction and let the nontarget pest (or harmless) be permanent. Therefore, we give the sufficient conditions for the permanence of the system. What is more, the sufficient conditions for the extinction of one of the two prey species and permanence of the remaining other preys and predators are given in Corollaries 9 and 10.Compared to earlier modeling studies on the prey-predator concerning chemical control for pest control at different fixed time, our model considers the predator given birth in regular pules att=(n+1)T, n=1,2,…. To avoid the fact that pesticides impact on the newborn predator, pesticide applications and birth pulse in predator occur at different time in this paper. All these results show that the dynamical behaviors of system (3) become more complex under different time periodically impulsive effect. Our results suggest a new approach in pest control. We are more interested in the following: how different patterns of insecticide applications affect the results, for instance, applying pesticide after predator is born and applying pesticide when predator is born. We also can study the results of two cases and find the better control way. These works will be done in our next consideration.



---

*Source: 380492-2015-07-14.xml*