Analysis for a Delayed Predator-prey System with Dissymmetric Impulse Dispersal between Two Patches∗

XU Gao,ZHANG Long

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

Abstract: In this paper,we study a delayed predator-prey system with dissymmetric impulse dispersal between two patches,by using a comparison theorem of impulsive differential equation and some analysis skills,the sufficient conditions ensuring the global attractivity of predator-extinction positive periodic solution and the permanence of a delayed predator-prey system with dissymmetric impulse dispersal are established.

Key words:Dissymmetric impulse dispersal;Periodic solution;Global attractivity;Permanence;Delay

0 Introduction

Species dispersal is a ubiquitous prevalent phenomena in nature,because it is important for us to know the ecosystem,in recent years,there are a large number of mathematical models focus on the dynamical behaviors of the system with diffusion(see[1-5,13,14,16-21,27],and the references cited therein),but in practice,diffusion often occurs in regular pulse.For example,birds often migrate between patches to search for a suitable environments.Animal movements between patches occurs at fixed periods of populations between regions or patches occurs at the impulsive instants.Thus,impulsive dispersal seems more accurately to model dispersal behavior in real ecological systems(see[6-12,15,22,23]and the references cited therein).

As we well know,predator-prey system with impulsive diffusion is very important mathematical model in the theory of mathematical biology.Many important and interesting population dynamical system have been investigated[4,6,8,10,13,14,16,17,21,23,24,25,26,28,29].Take the system in[8]into considered the authors studied the following delayed differential equation with impulsive diffusion:

Whereri,ai(i=1,2,)are the intrinsic rate of natural increase and density-dependent rate of population in theith habitat fori=1,2,andr3is the death rate of the predator,a3is the density-dependent rate of the predator,b1is the capturing rate of the predator,a3/b1is the conversion rate of nutrient into the production rate of the predator,τ1≥0 is a constant delay due to the gestation of the predator.And,we have included the term −b2y(t−τ2)in the dynamics of predator to incorporate the negative feedback of predator crowding,diis the dispersal rate in theithpatch,and 0

However,in the above models(1)it is too idealized that the dispersal occurs between homogeneous patches.In other words,the dispersal rates between two patches are symmetrical.In practice,dispersal between patches is usually not the same rate in both directions.And there are traveling losses in the process of movement.Hence,in[7],the authors proposed the following single species model with logistic growth and dissymmetric impulsive bi-directional dispersal:

whereDi(i=1,2)denote the rate of populationNiemigrating from theith patch fori=1,2,anddi(i=1,2)denote the rate of populationNiimmigrating from theith patch.Anddi,Disatisfying 0≤di≤Di≤1,(i=1,2).

Based on the above considerations,we consider a delayed predator-prey system with dissymmetric impulse dispersal between two patches as following:

with the initial conditions

N1(s)= φ1(s),N2(s)= φ2(s),y(s)=φ3(s),φ =(φ1,φ2,φ3)T∈C([−τ,0],R3+),φi(0)>0,i=1,2,3,τ =max{τ1,τ2}.

Where we assume that the system is composed of two patches connected by diffusion and occupied by a single species.The impulsive dispersal occurs every period T(T is a positive constant).The system evolves from its initial condition without being further affected by diffusion until the next pulse appears,N(t)=(N1(t),N2(t),y(t))Tdenote the solution of system(3)and what is piecewise continuous on(nT,(n+1)T].It is obvious that the global existence and uniqueness of the solution of system(3)is guaranteed by the smoothness properties of the right-hand side of(3),see[30]we can get thatN1(t)≥0,N2(t)≥0,y(t)≥0.

This paper is organized as follows.In the next section,we will introduce the important lemmas and the definition of permanence.In section 3,the main results that the global attractivity of the predator-extinction positive periodic solution and the permanence of the system(3)are established by using a comparison theorem of impulsive differential equation and some analysis methods.

1 Preliminaries

We first give a definition of permanence.

Definition 1System(3)is said to be permanent if there are constantsm,M>0,and a finite timeT0such that for all positive solutions(N1(t),N2(t),y(t))Tof(3),m≤N1(t)≤M,m≤N2(t)≤M,m≤y(t)≤Mholds for allt≥T0.

Next,we introduce the main lemma,which can be seen in[7].For system(2),we takewhich on substituting into(1)becomes

By integrating and solving the first two equations of system(4)between pulses,

we have

fornT

where 0

Define the map

Then we have the following results.

Lemma 1[7]There exists a unique positive fixed pointq=(q1,q2)of system(6),and for every(x1,x2)>(0,0),F(n)(x1,x2)→(q1,q2)asn→∞,if one of the following conditions is true

Where

Lemma 1 imply that the positive fixed point(q1,q2)of system(6)is existent and globally stable,which means that the positive periodic solutionof system(4)is existent and globally stable,furthermore,the positive periodic solution of system(2)is existent and globally stable.That is,

The following lemma will be used in the proof of the main results.

Lemma 2Considering the following system:

There exists a unique positive fixed point(ξ,η)of system(9),and for every(x1,x2)>(0,0),F(n)(x1,x2) → (ξ,η)asholds and one of the following conditions is true

ProofIn order to apply lemma 1,we need rewrite system(9),from the first equation of system(9),we have

Let

and the last two equations of(9)become

assume thatTherefore,we have

Next,we give the following lemma,which is important in the proof of the main results.

Lemma 3[8]Consider the following differential equality

Wherethen,for t large enough,we can obtain

Lastly,in order to establish the main result of the system(3),we takewhich on substituting into(3)becomes

2 Main results

Theorem 1For system(12),we assume that the conditions of lemma 1 and lemma 2 hold,in addition,(H2),(H3)hold as follows,

(H3)hereis a constant small enough,then system(12)admits a predator-extinction positive periodic solution,which is global attractivity.

ProofLetbe the predator-extinction positive periodic solution of system(12),(x1(t),x2(t),y(t))be the every positive solution,then the existence ofcan be obtained easily from lemma 1,so we only need to prove the predator-extinction periodic solution is global attractivity.

From the first equation of system(12)we have

Consider the following comparison system of the subsystem of(12)

By lemma 1,the periodic solution of(13)reads

fornT0,we haveThat is

By using the comparison theorem of impulsive differential equations,we have

for allt>T1.From the third equation of(12),we have

for allt>T1+τ1,i=1,2,thus we have,

for allfor above1>0 small enough,we have

that is

By using the comparison theorem of impulsive differential equations,we can obtain that there exists aT2>T1+τ1such thaty(t)≤0,and because of the positivity ofy(t),we havey(t)→0,ast→∞.Obviously,for any2>0 small enough,we have

for allt>T2.From the first equation of system(12)we get

Consider the following comparison system of the subsystem of(12)

By lemma 2,we can derive that the periodic solution of(19)reads

fornT0 small enough,there existsT3>0,we havefori=1,2.That is

By using the comparison theorem of impulsive differential equations,we get

Let2→0,we have

for alland we have

that is

for allTherefore,system(12)admits a predator-extinction positive periodic solution,which is global attractivity,which means that the predator-extinction positive periodic solution of system(3)is existent and globally attractivity.This completes the proof of theorem 1.

Theorem 2For system(12),we assume that the conditions of lemma 1 and lemma 2 hold,in addition,(H4),(H5)hold as follows,

(H4)ak1σ1−r3>0,

(H5)here α =b1δ3,where σ1,δ3is defined later,then system(12)is permanent.

ProofFrom system(12)we have

Similarly as the proof in theorem 1,whenwe have

From the third equation of(12),we have

for allBy lemma 3,we obtain that there exists a positive constantT4>T1+τ1large enough such that

for allt>T4,whereak1δ1−r3>0 because of(H4).LetM=max{δ1,δ2,δ3},and we can obtainx1(t)T4.

Next,from(12)and(24)we have

Consider the following comparison system of the system of(25)

Therefore,from lemma 2,we can obtain that the periodic solution of(26)reads

fornT0 small enough,there existsfori=1,2.That is

By using the comparison theorem of impulsive differential equations,we have

fort>T5.From(H4)we can choose a constant0>0 small enough such thatak1σ1−r3−b20>0.Therefore,for above4andM,there is a constantsuch that for anyt0≥ 0 we havexi(t)σi,i=1,2,for allFor above0,we now considery(t),there exist three cases as follows.

Case 1.There is a constantT∗≥T4such thaty(t)≤0for allt≥T∗.

Case 2.There is a constantT∗≥T4such thaty(t)≥0for allt≥T∗.

Case 3.There is an interval sequence{[sk,tk]}withT4≤s1

For case 1,from the third equation of(12),whenwe have

we further obtainy(t)→∞asn→∞,which leads to a contradiction.

We now consider case 3.For anyt≥T4,whenthent∈[sk,tk]for somek.

Iffor allt∈ [sk,tk],we have

Ifthen for anywe also have inequality(30),from(28)we can obtainxi(t)>σi,i=1,2,for alltherefore,we further have

for anywe firstly choose an integerp≥0 such thatintegrating inequality(31)fromtot,we can obtain

Lastly,for case 2,we directly havey(t)>0for allt≥T∗.Thus we finally obtainy(t)>σ3,for allt≥T∗.

Letm=min{σ1,σ2,σ3},and we havex1(t)>m,x2(t)>m,y(t)>m,t≥T∗.For all positive solution(x1(t),x2(t),y(t))Tof(12).m