Periodic Solution of a Two-species Competitive Model with State-Dependent Impulsive Replenish the Endangered Species∗

HE Zhi-long,NIE Lin-fei

(College of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830046,China)

0 Introduction

Competition is a universal phenomenon in ecological communities due to the limitation of resources.In population dynamics,Lotka-Volterra competitive systems are ecological models that describe the interaction among various competing species and have been extensively investigated in recent years[1,2].By the principle of competitive exclusion,we can understand that only the dominant species survives when the competition between species is sufficiently strong.In order to prevent the the weaker species face extinction,the population of the weaker species should not under the reasonable value.Therefore,it is necessary to control the population of the weaker species should be remained the reasonable level.

It is well known that the ecological system is often affected by environmental changed and other human activities,such as vaccination,chemotherapeutic treatment of disease,chemostat,birth pulse,control and optimization.These discrete nature of human actions or environmental changes lead to population densities changing very rapidly in a short space of time.These short-time perturbations are often assumed to be in the form of impulses in the modeling process.Recently,the state pulse control has been widely applied in the predator-prey model due to its economic,high efficiency,and feasibility nature[3–6].Particular,Tang[7]introduced the predator-prey model with state dependent impulsive effects by poisoning the prey and releasing the predator,and analyzed the existence and stability of a positive period-1 solution by using the Poincar´e map,properties of the LambertWfunction or analogue of Poincar´e criterion.

Based on the above discussion,the main purposes of this paper is to present a dynamic model of two-species competitive with state dependent impulsive control strategy,and discuss the effect of the state dependent impulsive control to protect the endangered species.

The paper is organized as follows,some de finitions and lemmas are provided in the next section.In Section 3,the existence,uniqueness and stability of a positive order-1 periodic solution of this model for two cases are stated and proved.Additional,we also discuss the transcritical bifurcation.

1 Model and Preliminaries

A classical autonomous Lotka-Volterra two-species competition model

wherex(t)andy(t)represent the population densities of the first and second species at timet,respectively.riis the natural growth rates,aiiis the co-efficients of intraspeci fic competition,aij(i,j)denotes the degree to which the presence ofj−thspecies affects the growth ofi−thspecies.

It is easy to calculate that model(1)always has an unstable trivial equilibriumE0(0,0),a saddleE1(r1/a11,0)and a stable boundary equilibriumE2(0,r2/a22),and not positive equilibrium under the conditionr1/r2

(A1)When the density of speciesxdrops to the threshold valueHat timetk(H),we replenish(for example,arti ficial reproduction)some endangered speciesxat ratep∈(0,+∞).

Based on the above assumption,we propose the following model

From the biological background of model(2),we only consider the dynamical behavior of model(2)in region R2+={(x,y):x≥0,y≥0}.We assume,throughout this paper,thatr1/r2

To discuss the dynamic behaviors of model(2),we de fine a impulsive set Σ1={(x,y)∈ R:x=H,y≥ 0}and a phase set Σ2:=I(Σ1)={(x,y)∈R:x=(1+p)H,y≥0},whereI:(x,y)∈Σ1→((1+p)H,y)∈Σ2is continuous function for anyP∈Σ1.HereIis called the impulse mapping.

The following de finitions and lemmas of model(2)are necessary throughout papers[3,6,8,9].

De finition 1For any pointP∈R,ΠP:R →Xde fined as ΠP(t)= Π(P,t)is a continuous map such that:(a)Π(P,0)=Pfor allP∈X;(b)Π(Π(P,t),s)=Π(P,t+s)for allP∈Xandt,s∈R,where R=[0,+∞),X=R.Then model(2)is called a semi-continuous dynamical system which is denoted by(X,Π,Σ1,I),and call ΠP(t)the trajectory passing through pointP.

De finition 2Trajectory Π(P0,t)is called order-1 periodic solution with periodTif there is a pointP0∈ Σ2andT>0 such that Π(P0,t)=P∈Σ1andI(P)=I(Π(P0,T))=P0∈Σ2.

De finition 3Suppose Π is an order-1 periodic solution of model(2).If for any ε >0,there must exists δ >0 andt0≥ 0,such that for any pointwe have the distanced(Π(P1,t),Π(P0,t))< ε fort>t0,then Π(P0,t)is orbitally asymptotically stable.

De finition 4Supposeg:Σ2→Σ2be a map.For anyP((1+p)H,yP)∈Σ2,there exists at1>0 such thatis called the successor function of pointP,and the pointP+1is called the successor point ofP.

By the zero point theorem of continuous function in the closed interval,we have the following lemma.

Lemma 1[9]The successor functiong(P)is continuous.Further,model(2)exist a positive order-1 periodic solution if there exist two pointP1,P2∈Σ2satisfyingg(P1)g(P2)<0.

Remark 1Obviously,ifg(P)=0,then the trajectory Π(P0,t)with initial pointP0is an order-1 periodic solution of model(2).

Lemma 2[9]TheT-periodic solution(φ(t),ψ(t))of the system

is orbitally asymptotically stable if the Floquet multiplierµ satis fies the condition|µ|<1,where

andf,g,∂ξ/∂x,∂ξ/∂y,∂η/∂x,∂η/∂y,∂ϕ/∂xand ∂ϕ/∂yare calculated at the pointand τj(j∈N)is the time of the j-th jump.

Lemma3[10]LetF:R×R→Rbeaone-parameterfamilyofC2mapsatisfying:(i)F(0,µ)=0;(ii)(∂F/∂x)(0,0)=1;(iii)(∂2F/∂x∂µ)(0,0)>0;(iv)(∂2F/∂x2)(0,0)<0.ThenFhas two branches of fixed points forµ near zero.The first branch isx(µ)=0 for allµ.The second bifurcating branchy(µ)=0.The fixed points of the first branch are stable ifµ<0 and unstable ifµ>0,while those of the bifurcating branch having the opposite stability.

On non-negative solutions of model(2),we have the following lemma.

Lemma 4Suppose that(x(t),y(t))is a solution of model(2)with initial valuesx(0+)≥0,y(0+)≥0,thenx(t)≥0 andy(t)≥0.Moreover,ifx(0+)>0,y(0+)>0,thenx(t)>0 andy(t)>0 for allt≥0.

The proof of Lemma 4 is obvious,hence we omit here.

2 Main Results

In this section we will take into account the existence and stability of periodic solutions of model(2)in different case.Obviously,regionis divided into three different domains with isoclines dx/dt=0 and dy/dt=0 of model(2),where D1:=?(x,y)∈:dx/dt>0,dy/dt>0?,D2:=?(x,y)∈R2+:dx/dt<0,dy/dt>0?and D3:=?(x,y)∈R2+:dx/dt<0,dy/dt<0?.

2.1 The case of(1+p)Hr2/a21

The following theorem is on existence and stability of positive order-1 periodic solutions with(1+p)Hr2/a21.

Theorem 1For anyp∈(0,+∞),we have

(i)if(1+p)H

(ii)ifHr2/a21,then model(2)has a unique positive order-1 periodic solution and which is orbitally asymptotically stable.

ProofWe first prove(i).Suppose that isocline dy/dt=0 intersects Σ1and Σ2at pointAandBrespectively.The trajectory ΠB(t)from the initial pointB((1+p)H,yB)reaches pointB1(H,yB1)on Σ1,next jumps to pointon Σ2,so pointis the success point ofB.However,from the geometrical construction of the phase region D2,it is easy to see that pointis located above pointB,that is,yB+1>yB.Therefore,the success functiong(B)=yB+1−yB>0.

On the other hand,trajectory ΠA(t)starting from the initial pointA(H,yA)jumps to pointA+((1+p)H,yA)on Σ2due to the control strategies and reaches Σ1at pointA1(H,yA1)again.Further,ΠA(t)jumps to point((1+p)H,yA1)at Σ2.By the geometrical construction of the phase regions D2and D3,obviously,pointis below pointA+,that is,

Therefore,from Lemma 1,there exists a point∈Σ2between pointsBandA+such thatg()=0.So,there exists a positive order-1 periodic solutionP+0P0P+0of model(2),initial pointof which is between pointsBandA+in set Σ2.

Next,we discuss the uniqueness of positive order-1 periodic solution of model(2)for this case.Since the trajectories starting from pointsB0((1+p)H,yB0)(0yA)will enter setBA+={(x,y)|x=(1+p)H,yB

The setBA+is mapped to setB1A1={(x,y)|x(1+p)H,yB1

and

Therefore,the sequenceis convergent monotonously and,which implies that there exists a unique pointsuch thatg()=0.Further,model(2)has a unique positive order-1 periodic solution for(1−p)H

Now,we discuss its orbital stability.Fromit follows that limn→∞yA+n=limn→∞yB+n=yP+0.For any pointC((1+p)H,yC)∈Σ2,where

Without loss of generality,lettrajectory ΠC(t)starting from the initial pointCintersects Σ1at point,next jumps to poindue to the control strategies,whereRepeating the process,we obtain a sequenceof Σ2,wherSincethenSimilarly,ifwe also can get.Thus trajectory from initiating any pointCof Σ2which is between pointsBandA+ultimately tends to be the unique positive order-1 periodic solution

From the above,we obtain that model(2)with(1+p)H

Now we turn to(ii),that is,Hr2/a21.Let pointsJ(H,yJ)andK(H,yK)be the intersection of Σ1with isocline dx/dt=0 and isocline dy/dt=0,respectively.Suppose pointKis subject to impulsive effects to pointK+((1+p)H,yK)∈Σ2,trajectory ΠK+(t)starting from the initial pointK+(H,yK+)reaches pointK1(H,yK1)∈Σ1,and jumps to pointon Σ2due to control strategies.By the geometrical construction of the phase regions D2and D3,we have,thus the successor function of pointK+isg(K+)=yK1+−yK+<0.

Take another any pointS((1+p)H,yS)∈ Σ2satisfying,trajectory ΠS(t)from the initial pointS((1+reaches pointon Σ1,next jumps to pointon Σ2,so pointis success point ofSandyS+1>yS,so we haveg(S)=yS+1−yS>0.

Therefore,we conclude that there exists a positive order-1 periodic solution of model(2),whose initial point is between pointsJ+andK+on Σ2.Similar to the proof of case(i),model(2)has a unique positive order-1 periodic solution and which is orbitally asymptotically stable forHr2/a21.The proof is complete.

2.2 The case of r1/a11

Suppose thaty(t)=0 fort∈[0,∞),then model(2)is reduced to the following subsystem:

Setx(0)=(1+p)H,then,by the first equation of(4),we obtainx(t)=r1exp(r1t)/(a11exp(r1t)+C),whereC=[r1−(1+p)a11H]/[(1+p)H].Letx(T)=H,thenx(T+)=(1+p)HandT=ln{[r1−(1+p)a11H]/[(1+p)(r1−a11H)]}.This means that model(2)withr1/a11

On the stability of this semi-trivial periodic solution,we have the following result.

Theorem 2For anyp∈(0,+∞)andr1/a11p∗,then periodic solution(5)of model(2)is orbitally asymptotically stable.Wherep∗satis fied the following equation

ProofComparing with model(3),it is known thatf(x,y)=x(r1−a11x−a12y),g(x,y)=y(r2−a21x−a22y),ξ(x,y)=px,η(x,y)=0,ϕ(x,y)=x−H,(φ(T),ψ(T))=(H,0),and(φ(T+),ψ(T+))=((1+p)H,0).Therefore,

byG(p)inpthe first derivative

therefore,G(p)is a monotonically decreasing function inp.Thus,|µ|<1 whenp>p∗.Hence,(5)be a orbitally asymptotically stable.This completes the proof.

Remark 2ForG(p∗)=1,a bifurcation may occur atp=p∗for|µ|=1,and a positive periodic solution may appear forr2/a21H−1

Now,we will discuss the bifurcation near the semi-trivial periodic solution.Consider the Poincar´e map of phase set Σ2as follows:

wherethefunctionL(,p)iscontinuouslydifferentiablewithrespecttobothandp,L(0,p)=0,thenlimy+k→0+L(,p)=L(0,p)=0.

From the bifurcation of map(6),we can obtain the following Theorem 3.

Theorem 3A transcritical bifurcation occurs whenp=p∗.Therefore,a stable positive fixed point appears when parameterpchanges throughp∗from right to left.Correspondingly,model(2)has a unique stable positive order-1 periodic solution ifp∈(r2/(a21H)−1,p∗).

ProofFrom the geometrical construction of the regions D1and D2,any trajectory ΠP(t)through the initial point((1+p)H,yP))∈ Σ1,whereyP:= θ≥ 0,always reach Σ1at point(H,L(yP,p)).So,model(2)can be transformed as follows:,

Let(x,y(x;x0,y0))be an orbit of system(7),and setx0=(1+p)H,y0:=θ≥0,then

By(8)and the differentiability theorem of solution on initial condition,we have

Next,we check whether the following conditions are satis fied.(a)Obviously,

(b)From(9),we have

which yields

(c)From(10),we have

Fromr1/a11

This means that(0,p∗)is a fixed point with eigenvalue 1 of the Poincar´e map(6).

(d)Finally,by(11),we have

By the assumptionr1/r2

It is easy to see that conditions(12)-(15)satisfy all conditions of Lemma 3.Thus,a transcritical bifurcation occurs whenp=p∗.Therefore,a stable positive fixed point appears when parameterpchanges throughp∗from right to left,and model(2)has a positive order-1 periodic solution forr2/(a21H)−1

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