Abstract
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).
1. 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 are all positive constants. Assume that ; then, the equilibria of (1) are where Wei and Li had showed that 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.
Theorem A. If , , , then the system (1) is persistent. More precisely, where
Theorem B. If , , , where are defined by Theorem A, then the positive equilibrium point of system (1) is globally asymptotically stable.
Now let us consider the following example.
Example 1. We have Here we choose , , , , , and the parameters are all positive constants. Obviously, , , . 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.
(a)
(b)
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 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.
Corollary 3. Assume that holds; then, system (1) is permanent.
Remark 4. A comparison of Theorems A, B, and 2 and Corollary 3 shows that , , and inequalities (6) are redundant. Therefore, our results significantly improve the corresponding main results of Wei and Li [1].
We will prove Theorem 2 in the next section. For more works on mutualism system, one could refer to [2–10] and the references cited therein.
2. Proof of the Main Results
As a direct corollary of Lemma 2.2 of Chen [11], we have the following.
Lemma 5. If and , when and , we have If , and , when and , we have
Proof of Theorem 2. By the first equation of system (1), we have From Lemma 5, it follows that Hence, for enough small , it follows from (12) that there exists a such that Similarly, for the above , it follows from the second equation of system (1) that there exists a such that (14) together with the first equation of system (1) implies that Therefore, by Lemma 5, we have That is, for to be defined by (12) and (13), there exists a such that It follows from (13) and the second equation of system (1) that Therefore, by Lemma 5, we have That is, for to be defined by (13) and (14), there exists a such that From the first equation of system (1) and the positivity of , we have Therefore, by Lemma 5, we have Hence, for to be defined by (12) and (13), there exists a such that Similarly, it follows from the second equation of system (1) that there exists a such that (24) together with the first equation of system (1) implies that Therefore, by Lemma 5, we have That is, for to be defined by (12) and (13), there exists a such that Similarly, by (23) and the second equation of system (1), for to be defined by (12) and (13), there exists a such that Noting that , , it immediately follows that Also, since , , it follows that , , and so Repeating the above procedure, we get four sequences , , , , such that for Obviously, We claim that sequences , are strictly decreasing, and sequences , are strictly increasing. To proof this claim, we will carry them out by induction. Firstly, from (29) and (30) we have Let us assume now that our claim is true for ; that is, Then, From (34) and the expression of , it immediately follows that Also, it follows from (34) that , . Then, From (37) and the expression of , it immediately follows that Therefore, 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 holds, (41) has a unique positive solution . Hence, we conclude that that is, Thus, the unique interior equilibrium is globally attractive. This completes the proof of Theorem 2.
Proof of Corollary 3. Noting that , , , are only dependent on the coefficients of the system (1) and independent of the solution of system (1), hence, (13), (14), (23), and (24) show that the system is permanent. This ends the proof of Corollary 3.
3. 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Natural Science Foundation of Fujian Province (2013J01011 and 2013J01010) and the Foundation of Fujian Education Bureau (JA13361).