Global Stability for a Asymptotically Periodic Cooperative Lotka-Volterra System with Time Delays

In this paper a class of cooperative Lotka-Volterra population system with time delay is considered. Some sufficient conditions on the existence and globally asymptotically stability for the asymptotically periodic solution of the system are established by using the Lyapunov function method and the method given in Fengying Wei and Wang Ke (Applied Mathematics and Computation 182 (2006) 161-165).


Introduction
Since Lotka-Volterra system has been established and was accepted by many scientists, it becomes the most important means to explain the ecological phenomenon now.For many years, a lot of extensive research results were made in mathematical biology and mathematical ecology [1]- [8], during this time Lotka-Volterra system has played an important role in theses research field of mathematical biology and mathematical ecology.Still now many research work mostly discussed periodic Lotka-Volterra systems [2] [3] [4] [5] [6] and the references cited therein.In fact asymptotically periodic systems [3] [4] describe our world more realistic and more accurate than periodic ones.
As is well known, Lotka-Volterra Cooperative system is one of the most important classe of interaction model which is discussed widely in mathematical biology and mathematical ecology.
In this paper we consider the following Lotka-Volterra cooperative system with time delay: where ( ) x t are the density of two cooperative species at time t respectively, ( )( ) x , at time t respectively, and a t are the are cooperative coefficients between two species at time t respectively.In this paper we assume that system (1) satisfies the following assumption (H1) τ is a positive constant and ( )( ) From the viewpoint of mathematical biology, in this paper, for system (1) we consider the solution with the following initial condition 0, for 2 , 0 and 0 0, 0 for , 0 and 0 0 then for any ( ) with initial conditions has a unique solution denoted by ( ) 0 , , X t t φ .
For a continuous and bounded function ( ) Y. Nakata and Y. Muroya have proved in [1] that the system (1) is permanent under the following conditions ( ) which means that the system (1) had a bounded region that is In particularly, ( ) where * 1 x x = is the unique positive solution of ( ) ( ) and p is a positive constant such that, m M i = are given above, then set Γ is the ultimately bounded set of system (1) Following is the adjoin system (2) of system (1) f t is a continuous periodic function and Now, we present some useful lemmas.
Lemma 2.1 The set Proof.We can obtain for ( ) ( ) ∫ our results will be discussed in the positively invariant set 2 R + .
Let the set Lemma 2.2 Assume that ( ) ( ) ( ) ) ( ) , , , , V t x y V t x y l x x y y − ≤ − + − , l is a constant and satis- fies 0 l > ; , where δ is a constant and satisfies 0 δ > .Furthermore, system (2.7) has a solution ( ) t ξ for 0 t t ≥ and satisfies t H ξ ≤ .Then system (2.7) has a unique asymptotically periodic solution, which is uniformly asymptotically stable.
Our main purpose is to establish some sufficient conditions on the existence and globally asymptotically stability for the asymptotically periodic solution of the system (1).The method used in this paper is motivated by the work done by Fengying Wei and Wang Ke in [4] and the Lyapunov function method.

Main Results
Theorem 2.1 Assume that the condition of lemma 2.2 is hold and , then there exists a unique asymptotically periodic solution of system (1), which is uniformly asymptotically stable.(W defined in the proof) Proof.From Lemma 2.2, we know that the solution of system ( 1) is ultimately bounded.Γ is the region of ultimately bounded.We consider the adjoint sys- tem (2) of system ( , X t x t x t = , Y t y t y t = are the solution of system (2) in Γ × Γ .Let x t x t y t y t i = = = .Next we construct a Lyapunov functional as follows: Take ( ) ( ) ( ) ( ) W Am M λ λ = − .From the known condition of Theorem 2.1, we obtain that 0 W > , ( ) ( ) . Then 3) of Lemma 2.3 is satisfied.has system (1) has a unique positive asymptotically periodic solution in domain Γ , which is uniformly asymptotically stable.The proof is complete.

Conclusions
In [1] the author's discussed system (1) and derived some sufficient conditions on the permanence of system (1).However, in this paper, based on the permanence of the system (1), we further study system (1) in a asymptotically periodic environment and established conditions on the existence and globally asymptotically stability for the asymptotically periodic solution of the system (1) by using the Lyapunov function method and the method given in Fengying Wei and Wang Ke (Applied Mathematics and Computation 182 (2006) 161 -165).
We have more interesting topics deserve further investigation, such as the dynamical behaviors of n-species Lotka-Volterra cooperative systems with discrete time delays.

3 )
There exists continuous function ( ) p s , such that for 0 s > , ( ) , we can easily prove 1) and 2).To check the condition 3) of Lemma 2.3, we need to calculate upper-right derivative of system (2):