ABSTRACT

The stability of a kind of cooperative models incorporating harvesting is considered in this paper. By analyzing the characteristic roots of the models and constructing suitable Lyapunov functions, we prove that nonnegative equilibrium points of the models are globally asymptotically stable. Further, the corresponding nonautonomous cooperative models have a unique asymptotically periodic solution, which is uniformly asymptotically stable. An example is given to illustrate the effectiveness of our results.

1. Introduction

Permanence, stability and periodic solution for LotkaVolterra models had been extensively investigated by many authors (see [1-8] and the references therein). Jorge Rebaza [1] had discussed the dynamic behaviors of predator-prey model with harvesting and refuge

(1)

he obtained that harvesting and refuge affected the stability of some coexistence equilibrium and periodic solutions of model (1), where was a continuous threshold policy harvesting function. Motivated by Jorge’s work, we consider the following cooperative system incorporating harvesting

(2)

where and denote the densities of two populations at time. The parameters are all positive constants.

Definition 1 [2] is called asymptotically - periodic function, if and it satisfies, where is continuous periodic function with periodic and.

We will discuss our problems in the region

where.

2. Permanence of System

Definition 2 [2] If there are positive constants such that each positive solution of system (2) satisfies

Then system (2) is persistent. If the system is not persistent, then system (2) is called non-persistent.

Lemma 1 If, then system (2) is persistent.

Proof. By the first equation of (2) and the comparison theorem, we get it implies that

.

For any there exists a, as, it then follows

Similarly, we have. By the discussion above, for any there exists a, as, it yields that

On the other hand, we have

.

By the comparison theorem, and letting, one gets that

.

By Definition 2, system (2) is persistent. □

3. Equilibrium Points and Stability

If, then the equilibrium points of (2) are

where

(3)

.

The general Jacobian matrix of (2) is given by

.

The characteristic equation of system (2) at is

, this immediately indicates that

is always unstable.

The characteristic equation of system (2) at is, by the condition, one then gets that is a saddle point.

The characteristic equation of system (2) at is, we derive that is a saddle point.

The characteristic equation of system (2) at takes the form

it is easy to check that , then , thus is locally asymptotically stable.

Theorem 1 If

then the positive equilibrium point of system (2) is globally asymptotically stable, where can be found in Lemma 1.

Proof. Define a Lyapunov function

it then yields that

by the conditions of theorem 1, thus,. The positive equilibrium point of system (2) is globally asymptotically stable.

4. Existence and Uniqueness of Solutions

Next, we will discuss a nonautonomous system

(4)

where are positive continuous bounded asymptotically periodic functions with period. The initial data of (4) is given by

. (5)

The solution of (4) with initial data (5) is denoted by

, ,.

For a continuous function defined on define

.

Definition 3 [2] If there exists a, for any, , there exists a

such that for, then the solution

is called ultimately bounded.

Let us consider the following asymptotically periodic system

(6)

where. Set

,

In order to discuss the existence and uniqueness of asymptotically periodic solution of system (6), we can consider the adjoint system

(7)

Lemma 2 If

and

then the solution of system (4) is ultimately boundedness.

Proof. By the first equation of system (4) and the comparison theorem, one gets that

it then implies that

.

Similarly, we have

.

By the same discussion, one thus gets that

,

Letting, we have

.

By the Definition 3, the solution of system (4) is ultimately bounded. □

Lemma 3 [2] If satisfies the following conditions:

1), where and are continuously positively increasing functions;

2)where is a constant;

3) there exists a continuous non-increasing function, such that for s > 0,. And as,

it then follows that

where is a constant; furthermore, system (6) has a solution for and satisfies.

Then system (6) has a unique asymptotically periodic solution, which is uniformly asymptotically stable.

Theorem 2 If conditions

and

hold, the conditions of Lemma 2 are satisfied, then system (4) has a unique asymptotically periodic solution, which is uniformly asymptotically stable.

Proof. By Lemma 2, the solutions of system (4) is ultimately bounded. We consider the adjoint system

(8)

Let

and be the solution of (8). By the fact

where lies between and, lies between and, it then follows

(9)

Define Lyapunov function, taking

By suing of the inequality, it is easy to check that 1) and 2) of Lemma 3 are valid. Computing the derivative of along the solution of system (8), by (9) and, we get that

taking, it yields, then, system (4) has a unique positive asymptotically periodic solution, which is uniformly asymptotically stable. □

5. Examples and Numerical Simulations

Now, let us consider a autonomous cooperative system incorporating harvesting

, (10)

it is easy to check that

,

,

, ,

the conditions of Theorem 1 are valid, then the positive equilibrium point of system (2) is globally asymptotically stable in Figures 1 and 2.

6. Conclusions

By analyzing the characteristic roots of a kind of cooperative models (2) incorporating harvesting, the stability of positive equilibrium point to model (2) is obtained by constructing a suitable Lyapunov function. Our results have shown that the harvesting coefficient affects the stability and the existence of equilibrium point to model (2).

The related non-autonomous asymptotically periodic cooperative model (4) has been discussed later. Under some conditions, which also depend on model parameters (see Theorem 2), model (4) has a unique asymptotically periodic solution, which is uniformly

asymptotically stable. Example model (10) shows the effectiveness of our results.

7. Acknowledgements

Our work is supported by Natural Science Foundation of China (11201075), the Natural Science Foundation of Fujian Province of China (2010J01005).

REFERENCES