2010WCA

Positive periodic solutions of a nonautonomous Lotka-Volterra dispersal system with discrete and continuous infinite delays?

Ting Zhang Minghui Jiang Bin Huang

【abstract】In this paper,a general class of non-autonomous Lotka-Volterra dispersal system with discrete and continuous infinite delays is considered. This class of Lotka-Volte -rra systems model the diffusion of a single species into n patches by discrete dispersal. By using Schauder’s fixed point theorem,we prove the existence of positive periodic solutions of system. The global asymptotical stability of positive periodic solution is discussed and the sufficient conditions for exponential stability are aslo given. The condition we obtained are more general and it can be reduced to several special systems. 【keywords】Lotka-Volterra dispersal system; positive periodic solutions; Schauder’s fixed point theorem; global asymptotical stability; global exponential stability 1、 Introduction

Lotka-Volterra system form a significant component of the models of population dynamics.In recent years,it has been usefully used in population dynamics、epidemiology and biology and any other areas in general. Because of its theoretical and practial signific -ance,the Lotka-Volterra system has been exte -nsively and intensively studied[6-11].Now the basic questions of this model are the persistence,extinctions,global asymptotic beha -viors and existence of positive solutions and periodic solutions and so on.

There have many papers on the study of the Lotka-Volterra type systems that have been developed in[1-6]. In this paper,we considered the following case of combined effects:dispers -ion,time delays,periodicty of environment.We study the following general nonautonomous Lotka-Volterra type dispersal system with

discrete and continuous infinite time delays:

nx?i(t)?xi(t)[ri(t)??bij(t)xj(t??ij(t)) j?1nn??a0ij(t)xj(t)??t,s)xj(t?s)ds]j?1j?1???cij(n??Dij(t)(xj(t)?xi(t)),i?1,2,?,n. (1)

j?1with the nitial condition

xi(s)??i(s) for s?(??,0], (2)

wherexi(t)reprsents the density of species in

ith patch;ri(t)denotes the growth rate of

species in patch i;aij(t)represents the effect of interspecific (if i?j) or interspecific (if i?j) interaction at time t;bij(t) represents

the effect of interspecific (if i?j) or

interspecific (if i?j) interaction at time

t??ij(t);

?0??cij(t,s)xj(t?s)ds represent

the effect of all the past life history of the species on its present birth rate;Dij(t)is the dispersion rate of the species from patchjto patchi.

In this paper,for system(1),we assume that the following assumptions hold:

(H1) The bounded functionscij(t)(t,s)are defined on R?(??,0] and nonnegative and continuous with respect tot?Rand

integrable with respect toson(??,0]such that?0cij(t,s)dsis continuous anf bounded

??with respect tot?R;

(H2)?ij(t)is continuous and differentiable bounede functions onR,and??ij(t)is uniformly continuous with respect totonR and infR{1???ij(t)}?0;

t?(H3)ri(t),aij(t),bij(t),cij(t,s)are continuous periodic functions with period??0,and for any??0,there is a constanth?h(?)?0, such that in any interval[t,t?h]there exists

? such that the following inequalities hold:

ri(t??)?ri(t)??,

aij(t??)?aij(t)??, bij(t??)?bij(t)??,

?0??cij(t??,s)?cij(t,s)ds??.

2、 Preliminaries

Throughout from this paper,we suppose that h(t)is a bounede function defined onR. Definehu?limsuphlt??(t),h?limt??infh(t).

Definition 2.1. An ?-periodic solutionx?(t)

of system (1) is said to be globally asymptotically stable, if for any positive solution x(t)of system (1), we have that

limx(t)?x?t??(t)?0.

Definition 2.2. An ?-periodic solutionx?(t)

of system (1) is said to be globally exponentically stable with convergence rate ??0, if for any positive solution x(t)of

system (1), we have that

limx(t)?x?(t)??tt???0(e).

Definition 2.3. The{?,1}- norm of a vector

nx?Rnis defined by

x{?,1}???ixi,

i?1?i?0 for i?1,2,?,n.

Lemma 2.1. The solutionx(t)of system (1) is said to be a positive solution if it satisfies the initial condition (2).

Proof. From Eq.(1), we have that: nlnxi(t)?lnxi(0)??t0[ri(t)??aij(t)xj(t)

j?1n??bij(t)xj(t??ij(t))?cij(t,s)xj(t?s)dsj?1?0??n?1xij(t)(xj(t)?xi(t))]dt. (3)

i(t)?Dj?1When t is finite, then the right-hand side of Eq.(3) is aslo finite, thus lnxi(t)??? which impliesxi(t)?0for any finite t?0,

i?1,2,?,n. Then the lemma can be proved.

3、 The existence of positive periodic

solutions

We can discuss the existence of positive periodic solutions of system (1) in this section.

Lemma 3.1. Ifali?0, then there exists

two positive constants ? and ? such that for all i?1,2,?,n , we have the following

inequalities hold:

??lim?infxi(t)?limt??supxi(t)?? (4)

t?and

nn?rlu0ui?i?(?bij??ij(t,s)ds)?

j?1j?1???cn??Duij?0 (5)

j?1where?max{ru?in}, ??max{?i}.

i?1,2,i?1,2,n?.n?al?.nij?auijj?1j?1Proof.Assumex(t)?(x1(t),x2(t),?,xn(t))is any positive solution of system (1) with initial condition (2).

Define V(t)?max{xj(t)} for t?0,

j?1,2,?,n. Suppose thatV(t)?xi(t), so xi(t)?xj(t)(j?1,2,?,n). The upper

right derivative of V(t)along the solution of system (1) is:

nD?V(t)?x?i(t)?xi(t)[ri(t)??aij(t)xj(t)j?1n ??bij(t)xj(?t?ij(t ))j?1n??0s)ds]

j?1???cij(t,s)xj(t?n??Dij(t)(xj(t)?xi(t))

j?1n?xi(t)[ri(t)??aij(t)xj(t)].

j?1By using comparability theorem,we have

limt??supxj(t)??i,i?1,2,?,n, where

?ui?{rin}, taken ??max{?, then

?li?1,2,?.ni}aijj?1limsupxj(t)?limsupxi(t)??.

t??t??On the other hand, there exists a positive constant T for t?Tsuch that

nnx?(t)?xluuii(t)[ri??a

ijxj(t)??Dijj?1j?1nn ?(?bu0uij??j?1j?1???cij(t,s)ds)?]

By ysing comparability theorem,we can get that limintxj(t)??i,where

t??nnrlui?(?buij??0ncuij(t,s)ds)???Dij??j?1j?1???j?1in?0.

?auijj?1Taken??min{?i}, then liminfxj(t)

i?1,2,?.nt???liminfxi(t)??.

t?? Thus we can get that??liminfxi(t)

t???lim?supxi(t)??. The proof is complete.

t?Theorem 3.1. For the system (1) of initial

condition (2), if there exist positive constants

?,? such that conditions (4) and (5) hold, then system (1) has at least one positive

?-periodic solution.

Proof. Let C?C((??,0],Rn)be the

banach space with the norm

??maxsup?i(?).

i??????From Lemma 3.1. we know

??lim?infxi(t)?limt??supxi(t)??t?.

Let

??{x(?)?C:??xi(?)??i,i?1,2,?,n,x?(?)?L,??max{?i}}, then ? is a i?1,2,?,nconvex compact set in C.

We define a map Ffrom ? to C:

F:?(?)?x(???,?).From Lemma 3.1.

we know that ??xi(?)??i hold for all

i?1,2,?,n. Moreover, it is easy to prove

that x?(???)?L,then if ???,we have x??, it means that T???. By using

Schauder’s fixed point theorem, there exists

???? such that T?????. Thus x(t,??)

?x(t,T??) i.e. x(t,??)?x(t??,??).It

means that x(t,??)is a positive periodic solution of system (1). Then the proof is complete.

Remark 3.1. From theorem 3.1., we can know that the upper and the lower bounds of x?(t)

is ??x?(t)??i(i?1,2,?,n).Which may

be useful in the other applications. 4、 Globally asymptotical stability

In this part,we discuss about 5、 Globally exponential stability 6、 Conclusion

References

[1] Xinzhu Meng,Lansun Chen. Periodic solut -ion and almost periodic solution for a nonautonomous Lotka-Volterra dispersal syste -m with infinite delay. J.Math.Anal.Appl. 2007. 05.084.

[2] WeiLin,Tianping Chen. Positive periodic solutions of delayed periodic Lotka-Volterra systems. Physics letters A 334(2005)273-287. [3] Xinzhu Meng,Jianjun Jiao, Lansun Chen. Global dynamics behaviors for a nonautonom

-ous Lotka-Volterra .almost periodic dispersal system with delays.Nonlinear Analysis

68(2008)3633-3645.

[4] Yanlai Liang,Lijie Li,Lansun Chen.Almost periodic solutions for Lotka-Volterra systems with delay.Commun Nonlinear Sci Numer Simulat 14(2009)3660-3669.

[5] Zhidong Teng. Nonautonomous Lotka-Volt -erra systems with delays. Journal of Differerntial Equations 179(2002)538-561. [6] C.Zhu, G.Yin. On competitive Lotka-Volte -rra model in random environments. J.Math.A -nal.Appl.357(2009)154-170.

[7] Yongkun Li. Positive periodic solutions of periodic neutral Lotka-Volterra system with di -stributed delays. Chaos,Solitons and Fractals 37(2008)288-298.

[8] Haifeng Luo,Wantong Li. Periodic solutio -ns of a periodic Lotka-Volterra system with delays. Applied Mathematics and computation 156(2004)787-803.

[9] Teresa Faria. Sharp conditions for global stability of Lotka-Volterra systems with distributed delays. J.Differential Equations 246(2009)4391-4404.

[10] Xinzhu Meng,Lansun Chen. Almost periodic solution of nonautonomous Lotka-Vol -terra predator-prey dispersal system with delays. Journal of Theoretical Biology 243 (2006)562-574.

[11] Guichen Lu, Zhengyi Lu, Xinze Lian. Delay effect on the permanence for Lotka-Volt -erra cooperative systems. Nonlinear Analysis. Real Word Application 2009.10.005.