Note on the Stability Property of a Cooperative System Incorporating Harvesting

The stability of a kind of cooperative model incorporating harvesting is revisited in this paper. By using an iterative method, the global attractivity of the interior equilibrium point of the system is investigated. We show that the condition which ensures the existence of a unique positive equilibrium is enough to ensure the global attractivity of the positive equilibrium. Our results significantly improve the corresponding results of Wei and Li (2013).


Introduction
In [1], Wei and Li proposed and studied the following cooperative system incorporating harvesting: where  and  denote the densities of two populations at time .The parameters  1 ,  2 ,  1 ,  2 ,  1 ,  2 ,  1 ,  2 , ,  are all positive constants.Assume that  1 > ; then, the equilibria of (1) are where Wei and Li had showed that  0 ,  1 ,  2 are unstable and concerned with the persistence and stability property of the system; by applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, they obtained the following results.Figure 1: Dynamics behaviors of system (7).Here, we take the initial conditions ( 1 (0),  2 (0)) = (0.5, 1.2), (1.5, 1), (0.2, 0.5) and (1, 0.6), respectively. where where , , ,  are defined by Theorem A, then the positive equilibrium point  3 of system (1) is globally asymptotically stable.
Now let us consider the following example.
Example 1.We have ) . ( Here we choose Hence, the conditions of Theorems A and B are not all satisfied; however, numeric simulations (Figure 1) show that the unique positive equilibrium (0.4806248475, 0.8507810594) is globally attractive.
The above example shows that it is possible to obtain some weaker conditions than those of Theorems A and B to ensure the persistent and stability of the system.The aim of this paper is to prove the following result.
Theorem 2. Assume that  1 >  holds; then, the unique positive equilibrium  * ( * ,  * ) is globally attractive; that is, Concerned with the persistent property of the system, as a direct corollary of Theorem 2, we have the following.

Proof of the Main Results
As a direct corollary of Lemma 2.2 of Chen [11], we have the following.
Let us assume now that our claim is true for ; that is, Then, From (34) and the expression of  ()  , it immediately follows that (36) Also, it follows from (34) that  ()  ≥  (−1)  ,  = 1, 2. Then,  1 From (37) and the expression of  ()  , it immediately follows that Therefore, lim Letting  → +∞ in (31), we obtain (40) shows that (, ) and (, ) are positive solutions of the equations Wei and Li [1] had already showed that, under the assumption that  1 >  holds, (41) has a unique positive solution  * ( * ,  * ).Hence, we conclude that Thus, the unique interior equilibrium  * ( * ,  * ) is globally attractive.This completes the proof of Theorem 2.

Discussion
In this paper, we revisited the stability property of a cooperative system incorporating harvesting which was proposed by Wei and Li [1]; by using the iterative method, we show that the condition which ensures the existence of a unique positive equilibrium is enough to ensure the global attractivity of the positive equilibrium.The numeric simulation of Example 1 shows the feasibility of our results.It seems interesting to investigate the stability property of the corresponding discrete type model of system (1); we leave this for future study.