李洪利;张龙;万海云
【摘 要】We consider a periodic single species model with impulse diffusion between two patches. By using the comparison theorem of impulsive differential equation and some analysis methods, we obtain some sufficient conditions on the permanence of the system. Furthermore, by using Brouwer fixed point theorem and constructing a suitable
Lyapunov function, some sufficient conditions on the existence and global asymptotic stability of unique positive periodic solution are established.%研究了两斑块间具有脉冲扩散的周期单种群模型。利用脉冲微分方程的比较原理和一些分析方法，得到了系统持久性的充分条件，然后利用Brouwer不动点理论和构造Lyapunov函数的方法，得到了系统正周期解全局渐近稳定的充分条件。 【期刊名称】《新疆大学学报（自然科学版）》 【年(卷),期】2014(000)001 【总页数】6页(P70-75)
【关键词】非对称脉冲扩散;持久性;周期解;全局渐近稳定性 【作 者】李洪利;张龙;万海云
【作者单位】新疆大学数学与系统科学学院，新疆乌鲁木齐830046;新疆大学数学与系统科学学院，新疆乌鲁木齐830046;新疆大学数学与系统科学学院，新疆乌鲁木齐830046
【正文语种】中 文 【中图分类】O231
0 Introduction
In recent years there has been a growing interest in the study of mathematical models of populations dispersing among patches in a heterogenous environment[1∼6].They obtained some sufficient conditions that guarantee permanence of population or stability of positive equilibria or positive periodic solutions.However,in all of above population
dispersing systems,the authors always assume that the dispersal occurs at every time.In practice,it is often the case that diffusion occurs at certain moment.For example,when winter comes,birds will migrate between patches in search for a better environment,whereas they do not diffuse in other seasons,and the excursion of foliage seeds occurs at certain moment every year.Therefore,it is not reasonable to characterize the population movements in these cases with continuous dispersal models.This short-time scale dispersal is more appropriately assumed to be in the form of impulses in the modeling process,in order to be in much better agreement with the real ecological situation.With the developments and applications of impulsive differential equations,theories of impulsive differential equations have been introduced into population dynamics,and many important studies have been performed.Stability and permanence were considered by[7∼9].Existence of periodic solutions were studied by[10,11].
Hui and Chen[10]proposed the following single species logistic model with impulse dispersal:
where the authors suppose that the system(1)is composed of two patches connected by diffusion,xi(i=1,2)is the density of species in the ith patch,ai,bi(i=1,2)are the intrinsic growth and density-dependent parameters of the population in the ith patch,diis the dispersal rate between the ith patch and jth patch(ij,i,j=1,2).∆xi=xi(nτ+)−xi(nτ),where xi(nτ+)represents the density of the population in the ith patch
immediately after the nth diffusion pulse at time t=nτ,xi(nτ)represents the density of the population in the ith patch before the nth diffusion pulse at time t=nτ,n=1,2,···,i=1,2.
However,in the above impulsive dispersal models,it is assumed that the dispersal occurs between homogeneous habitat patches and the dispersal rate between any two patches is equal or symmetrical,which is really too idealized for a real ecosystem.Actually,in the real world,due to the heterogeneity of the spatio-temporal distributions in nature,movement between fragments of patches is usually not the same rate in both directions.In addition,once the individuals leave their present habitat,they may not successfully reach a new one,due to predation,harvesting,or for other reasons,so that there are traveling losses.Therefore,the dispersal rates among these patches are not always the same.Rather,in real
ecological situations,they are different[12,13].Therefore,it is our basic goal to investigate single species model with dissymmetric impulse dispersal.
Besides,the coefficients of system(1)are assumed to be
constant.However,in the real world,the coefficients are not fixed constants owing to the periodic variation of environment.The effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment.Thus,it is reasonable to assume that coefficients of
system(1)are periodic functions.The effects of the periodic environment on the evolutionary theory have been the object of intensive analysis by numerous authors and some of these results can be
found[14∼16].However,to the best of authors knowledge,there is no published paper considering the notion of a periodic single species logistic model with dissymmetric impulse dispersal in two patches.Therefore,it is worthwhile investigating the permanence of the
species,existence,uniqueness,and global asymptotic stability of positive periodic solutions of system.
Based on the above considerations,in this paper,we consider the following periodic single species model with dissymmetric impulse dispersal between two patches.
where we suppose that the system(2)is composed of two patches connected by diffusion,xi(i=1,2)is the density of species in the ith patch,Di(i=1,2)is the rate of population xiemigrating from ith patch,and di(i=1,2)is the rate of population xiimmigrating from ith patch.Here we assume 0≤di≤Di≤1,which means that there possibly exists mortality during
migration between two patches.ai(t)and bi(t)(i=1,2)are τ-periodically continuous,and bi(t)>0(i=1,2)for all t∈R.
This paper is organized as follows.In section 1,we will give some notations and some lemmas.In section 2,some conditions for the permanence of the system(2)are obtained,we establish some conditions for the existence and global asymptotic stability of unique positive periodic solution of the system. 1 Preliminaries
In this section,we introduce some notations,and state some lemmas which will be useful in the subsequent sections.
Let PC and PC0denote two function spaces with PC={φ:[0,+∞)→R is continuous for t,τk,φ()exist with φ(=limt→τ+k φ(t),k=1,2,···}and PC0={φ ∈ PC:φ is differentiable at t τk,k=1,2,···}.
Lemma 1[17] Let the function x(t)∈ PC0([t0,∞),R)satisfy the inequalities
where a(t),b(t)∈ PC([t0,∞),R)and αk≥0,βkare constants.Then
similarly,we obtain
for all t≥t0,if all the inequalities of(3)are inverse. Consider the following nonlinear impulsive system:
where a(t)and b(t)are bounded and continuous periodic functions de fined on R+,b(t)≥0 for all t∈R+and impulsive coefficients θk∈ (0,1),θk=
θk+qand q is a fixed positive integer,and tk+q=tk+1. Lemma2[18]
Thereexistsauniquenonnegativeperiodicsolutionofsystem(4),whichisgloballyasymptotically stable. Moreover,if
then x∗(t)>0,if
then x∗(t)=0. 2 Main results
In this section,we will study the existence and global asymptotic stability of positive periodic solution of species for system(2). First,we discuss the permanence of the solutions of
system(2).Throughout,let X=(x1,x2)≥0 for x1≥0 and x2≥0 and let X=(x1,x2)>0 for x1>0 and x2>0.Note that any solution
X(t)=(x1(t),x2(t))satis fies X(t)≥0 for all t≥0 if the initial value X(0)≥0. Theorem 1 We assume that and
hold,then system(2)is permanent.
Proof First,we prove that the solutions of system(2)are ultimately bounded above.Let(x1(t),x2(t))be any solution of system(2)with initial value X(0)>0,W(t)=x1(t)+x2(t),when t,nτ,we have
where M=when t=nτ,we have
From(Theorem 3.11 of[19]),for t∈(nτ,(n+1)τ],we have
when t→∞,so W(t)is uniformly bounded.Hence,by the de finition of W(t),there exist constants M1>0,M2>0 such that x1(t)≤M1,x2(t)≤ M2for t large enough.
Finally,we prove that all the positive solutions of system(2)are ultimately bounded below.Since d1≥0,d2≥0,we have
In this system,we can see that there is no relation between x1(t)and x2(t).Thus,we can solve them independently.
Consider the impulsive comparison system of system(7)
If(5)holds,from Lemma 2,we obtain that(9)has a positive periodic solution(t),which is globally asymptotically stable,in view of Lemma 1,for any constant?>0 small enough,there is a T1>0 such that
for all t≥T1.A similar argument as in the proof of(10),for the above?>0 small enough,there exists a T2>0 such that
for all t≥T2.This completes the proof of Theorem 1.
Base on the discussion above mentioned,we know the set Ω =M2}is strictly positively invariant if conditions(5)and(6)hold.
Now,we will discuss the existence of positive periodic solution of system(2).Let X(t,X0)be unique solution of system(2)with initial value X0=(x1(0),x2(0)),where X(0,X0)=X0.Since Ω is positively invariant set of system(2).Then we can de fine a poincare map:
Obviously,the fixed point of map F is corresponding to the periodic solution of system(2).
Theorem 2 Assume that periodic system(2)satis fies the
conditions(5)and(6)of Theorem 1,then system(2)at least exists a strictly positive τ periodic solution.
Proof From Theorem 1,we can obtain that
is a bounded closed convex set,and FΩ⊂Ω.
According to the property of continuous dependence of solutions with respect to initial value,F is continuous.Furthermore,by applying the Brouwer fixed point theorem,there at least exists a fixed point of map F in Ω,homologous,system(2)at least exists a strictly positive τ-periodic solution.This completes the proof of Theorem 2.
Following,we will try to establish some conditions under which the τ-periodic solution((t),(t))is unique and globally asymptotically stable. Theorem 3 Assume that the conditions of Theorem 1 hold,and
then system(2)has a unique globally asymptotically stable positive τ-periodic solution.
Proof Let(x1(t),x2(t))be any solution of system(2),de fine the Lyapunov function as follows:
Whennτ,we have
when t=nτ,we have
From(Theorem 3.11 of[19]),for t∈(nτ,(n+1)τ],we have
Hence,from(12)and(13),we haveSince the arbitrariness of the solution(x1(t),x2(t)),we finally obtain the uniqueness and the global asymptotic stability of the positive τ-periodic solution((t),(t)),this completes the proof of the Theorem 3. References：
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