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

Show more

1. Introduction

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:

(1)

where, are the density of two cooperative species at time t respectively, are intrinsic growth rate of two cooperative species at time t respectively, , are the intra patch restriction density of species, , at time t respectively, and, 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) is a positive constant and, , , and are continuous, asymptotically periodic, bounded and strictly positive functions on.

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

(2)

(3)

then for any, System (1) with initial conditions has a unique solution denoted by.

For a continuous and bounded function, we define

and

Y. Nakata and Y. Muroya have proved in [1] that the system (1) is permanent under the following conditions

and

where

for and

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

(4)

In particularly,

(5)

(6)

(7)

(8)

where is the unique positive solution of, and p is a positive constant such that,

Let the set

where are given above, then set is the ultimately bounded set of system (1)

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

(9)

Now, we present a useful definition

Definition 1.1 (see [ [3] Definition 1.1]) is called asymptotically periodic function, if is a continuous function mapping from to, and satisfies

, (10)

where is a continuous periodic function and.

Now, we present some useful lemmas.

Lemma 2.1 The set is the positively invariant set of system (1)

Proof. We can obtain for

our results will be discussed in the positively invariant set.

Let the set

where are given above (in Introduction).

Lemma 2.2 Assume that then system (1) is permanent, where and .

Lemma 2.3 ( [4] ) Let satisfy

1), where are are continuously positively increasing functions;

2), is a constant and satisfies;

3) There exists continuous function, such that for,. And as, , it follows that , where is a constant and satisfies.

Furthermore, system (2.7) has a solution for and satisfies. 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.

2. 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. is the region of ultimately bounded. We consider the adjoint system (2) of system (1)

For and are the solution of system (2) in. Let. Next we construct a Lyapunov functional as follows:

(11)

Take and by using the inequality

, 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):

where and we take

Then we have

By the following formula:

(12)

(13)

where lie in between and respectively, then . let, and if, where is a constant ,then we have

where.

From the known condition of Theorem 2.1, we obtain that, . Then 3) of Lemma 2.3 is satisfied. has system (1) has a unique positive asymptotically periodic solution in domain, which is uniformly asymptotically stable. The proof is complete.

3. 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.

Acknowledgements

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

References

[1] Nakata, Y. and Muroya, Y. (2010) Permanence for Nonautonomous Lotka-Volterra Cooperative Systems with Delays. Nonlinear Analysis: Real World Applications, 11, 528-534.

[2] Lu, S. (2008) On the Existence of Positive Periodic Solutions to a Lotka Volterra Cooperative Population Model with Multiple Delays. Nonlinear Analysis: Theory, Methods & Applications, 68, 1746-1753.

[3] Wei, F. and Wang, K. (2006) Asymptotically Periodic Solution of N-Species Cooperation System with Time Delay. Nonlinear Analysis: Real World Applications, 7, 591-596.

[4] Wei, F. and Wang, K. (2006) Global Stability and Asymptotically Periodic Solution for Nonautonomous Cooperative Lotka-Volterra Diffusion System. Applied Mathematics and Computation, 182, 161-165.

[5] Wei, F. and Wang, K. (2002) Almost Periodic Solution and Stability for Nonautonmous Cooperative Lotka-Volterra Diffusion System. Songliao Journal (Natural Science Edition), 3.

[6] Liu, C. and Chen, L. (1997) Periodic Solution and Global Stability for Nonautonomous Cooperative Lotka-Volterra Diffusion System. Journal of Lanzhou University (Natural Science), 33, 33-37.

[7] Zhang, J. and Chen, L. (1996) Permanence and Global Stability for Two-Species Co-Operative System with Delays in Two-Patch Environment. Mathematical and Computer Modelling, 23, 17-27.

[8] Chen, F. (2003) Persistence and Global Stability for Nonautonomous Co-Operative System with Diffusion and Time Delay. Acta Scientiarum Naturalium Universitatis Pekinensis, 39, 22-28.