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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).

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 [

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

(H1)

From the viewpoint of mathematical biology, in this paper, for system (1) we consider the solution with the following initial condition

then for any

For a continuous and bounded function

Y. Nakata and Y. Muroya have proved in [

where

which means that the system (1) had a bounded region that is

In particularly,

where

Let the set

where

Following is the adjoin system (2) of system (1)

Now, we present a useful definition

Definition 1.1 (see [ [

where

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

Let the set

where

Lemma 2.2 Assume that

Lemma 2.3 ( [

1)

2)

3) There exists continuous function

Furthermore, system (2.7) has a solution

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 [

Theorem 2.1 Assume that the condition of lemma 2.2 is hold and

Proof. From Lemma 2.2, we know that the solution of system (1) is ultimately bounded.

For

Take

where

Then we have

By the following formula:

where

where

From the known condition of Theorem 2.1, we obtain that

In [

We have more interesting topics deserve further investigation, such as the dynamical behaviors of n-species Lotka-Volterra cooperative systems with discrete time delays.

This work was supported by the National Natural Science Foundation of China (Grant No. 11401509).

Tayir, T. and Mahemuti, R. (2017) Global Stability for a Asymptotically Periodic Cooperative Lotka- Volterra System with Time Delays. Open Journal of Applied Sciences, 7, 207-212. https://doi.org/10.4236/ojapps.2017.75018