DYNAMIC BEHAVIORS OF A NON-SELECTIVE HARVESTING MAY COOPERATIVE SYSTEM INCORPORATING PARTIAL CLOSURE FOR THE POPULATIONS

A cooperative system of May type incorporating partial closure for the populations and non-selective harvesting is proposed and studied in this paper. The locally stability property of the equilibria are determined by analyzing the Jacobian matrix of the system about the equilibria. By using the comparison theorem of the differential equation, sufficient conditions which ensure the global attractivity of the boundary equilibria are obtained. By using the iterative method, we are able to show that the conditions which ensure the existence of the unique positive equilibrium is enough to ensure its global attractivity. Our study shows that the intrinsic growth rate and the fraction of the stocks for the harvesting plays crucial role on the dynamic behaviors of the system. Numeric simulations are carried out to show the feasibility of our results.

May [2] suggested the following set of equations to describe a pair of mutualist: 1) where N 1 , N 2 are the densities of the species, respectively. r, K i , α, β , i = 1, 2 are positive constants. The system admits an unique positive equilibrium (N * 1 , N * 2 ), which is globally stable if αβ < 1, and the system will "run away", with both populations growing unboundedly large if αβ ≥ 1. To overcome the "run away" problem, May further considered the density restriction of the species and proposed the following system:

2)
where r i , K i , α i , ε i , i = 1, 2 are positive constants. He showed that system (1.2) has a global stability equilibrium point. Since then, many scholars ( [2,3,4]) also done works on this direction. Based on the model (1.1) and (1.2), Wei and Li [2] proposed the following cooperative system with harvestingẋ

3)
where x and y denote the densities of two populations at time t. The parameters r 1 , r 2 , a 1 ,a 2 , b 1 , b 2 , k 1 , k 2 , E, q are all positive constants. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, they obtained sufficient conditions which ensure the persistent and stability of the positive equilibrium, respectively. Xie, Chen and Xue [3] argued that the conditions in [2] is too complex, and by using the iterative method, they showed that is enough to ensure the system (1.3) admits a unique globally attractive positive equilibrium.
This result greatly improve the main results of [2]. Recently, Chen, Wu and Xie [4] argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations, corresponding to system (1.3), they further proposed the following discrete cooperative model incorporating harvesting: where x(k), y(k) are the population density of the species x and y at k-generation. By using the iterative method and the comparison principle of difference equations, they also obtained a set of sufficient conditions which ensure the global attractivity of the interior equilibrium of the system. It bring to our attention that all of the paper [2]- [4] are considered the harvesting of the first species, without harvesting of the second species, this seems unrealistic, since generally speaking, in the harvesting process, human being will try to obtain as many resources as possible, with as little cost as possible.
On the other hand, as was pointed out by Chakraborty, Das and Kar [36], the study of resourcemanagement including fisheries, forestry and wildlife management has great importance, it is necessary to harvest the population but harvesting should be regulated, such that both the ecological sustainability and conservation of the species can be implemented in a long run.
Recently, Lin [37] investigated the dynamic behaviors of the following two species commensal symbiosis model with non-monotonic functional response and non-selective harvesting in a partial closure where a i , b i , q i , i = 1, 2 c 1 , E, m(0 < m < 1) and d 1 are all positive constants, where E is the combined fishing effort used to harvest and m(0 < m < 1) is the fraction of the stock available for harvesting. His studied shows that depending on the range of the parameter m, the system may be collapse, or partial survival, or the two species could be coexist in a stable state. He also showed that if the system admits a unique positive equilibrium, then it is globally asymptotically stable. Recently, Chen [38] also studied the influence of non-selective harvesting to a Lotka-Volterra amensalism model incorporating partial closure for the populations, and he also founded that the dynamic behaviors of the system becomes complicated.
Stimulated by the works of [2]- [4], [36]- [38], in this paper, we will study the dynamic behaviors of the following non-selective harvesting May cooperative system incorporating partial closure for the populationsẋ where x and y denote the densities of two populations at time t. The parameters r 1 , r 2 , a 1 , a 2 , b 1 , b 2 , k 1 , k 2 , E, q i are all positive constants and have the same meaning as that of the system (1.3).
E is the combined fishing effort used to harvest and m(0 < m < 1) is the fraction of the stock available for harvesting.
We will try to give a thoroughly analysis of the dynamic behaviors of the above system.
The paper is arranged as follows. We investigate the existence and locally stability property of the equilibria of system (1.2) in the next section. In section 3, By applying the differential inequality theory and the iterative method, we are able to investigate the global stability property of the boundary equilibrium and the positive equilibrium, respectively. Section 4 presents some numerical simulations concerning the stability of our model. We end this paper by a briefly discussion.

Local stability of the equilibria
The system always admits the boundary equilibrium O(0, 0).
If r 2 > Emq 2 holds, the system admits the boundary equilibrium A(0, y 1 ), where If r 1 > Emq 1 holds, the system admits the boundary equilibrium B(x 1 , 0), where If r 1 > Emq 1 and r 2 > Emq 2 hold, then the system admits a unique positive equilibrium (x * , y * ), x * is the unique positive solution of the equation We shall now investigate the local stability property of the above equilibria.
The variational matrix of the system of Eq.
Proof. From (2.2) we could see that the Jacobian matrix of the system about the equilibrium The eigenvalues of the matrix are λ 1 = Emq 1 − r 1 , λ 2 = r 2 − Emq 2 . Under the assumption (2.4), 0) is locally stable. This ends the proof of Theorem 2.2.
Proof. From (2.2) we could see that the Jacobian matrix of the system about the equilibrium point A(0, y 1 ) is given by Under the assumption (2.6), the two eigenvalues of the matrix satisfies λ 1 = r 1 −Emq 1 < 0, λ 2 = Emq 2 − r 2 < 0. consequently A(0, y 1 ) is locally stable. This ends the proof of Theorem 2.3.
Proof. Noting that the equilibrium point C(x * , y * ) satisfies the equation

8)
The Jacobian matrix about the equilibrium C is given by The characteristic equation of (2.9) is which is equivalent to Therefore, the two eigenvalues of the above matrix satisfies Consequently, Hence, C(x * , y * ) is locally stable.
This ends the proof of Theorem 2.4.

Global attractivity
This section try to obtain some sufficient conditions which ensure the global asymptotical stability of the equilibria.
(1) It follows from m > max r 1 Eq 1 , r 2 Eq 2 that there exists enough small ε > 0 such that From the first equation of system (1.6) and the positivity of the solution, by using (3.5), we have

6)
Hence From the second equation of system (1.6) and the positivity of the solution, by using (3.5), we

8)
Hence (2) By using the condition m > r 2 Eq 2 , similarly to the analysis of (3.8)-(3.9), we have For arbitrary enough small ε > 0, it follows from (3.10) that there exists a T 1 > 0, such that For t > T 1 , from the first equation of system (1.6), we have (3.10) it follows from (3.10) and Lemma 3.1 that On the other hand, from the first equation of system (1.6), we also have

12)
it follows from (3.12) and Lemma 3.1 that

(3.13)
It follows from (3.11) and (3.13) that (3.14) Since ε is any arbitrary small positive constants, setting ε → 0 in (3.14) leads to (3) By using the condition m > r 1 Eq 1 , similarly to the analysis of (3.5)-(3.7), we have For arbitrary enough small ε > 0, it follows from (3.15) that there exists a T 2 > 0, such that For t > T 2 , from the second equation of system (1.6), we have On the other hand, from the second equation of system (1.6), we also have

It follows from (3.17) and (3.19) that
(3.20) Since ε is any arbitrary small positive constants, setting ε → 0 in (3.20) leads to (4) By the first equation of system (1.6), we havė Hence, for enough small ε > 0 ε < min (r 1 − Eq 1 m)k 1 , it follows from (3.21) that there exists a T 1 > 0 such that Similarly, for above ε > 0, it follows from the second equation of system (1.6) that there exists a T 1 > T 1 such that (3.23) together with the first equation of system (1.6) implieṡ for all t > T 1 .

(3.25)
That is, for ε > 0 be defined by (3.21)- (3.22), there exists a T 2 > T 1 such that It follows from (3.22) and the second equation of system (1.6) thaṫ Therefore, by Lemma 3.1, we have That is, for ε > 0 be defined by (3.22) and (3.23), there exists a T 2 > T 2 such that From the first equation of system (1.6) and the positivity of y(t), (3.30) Therefore, by Lemma 3.1, we have Hence, for ε > 0 be defined by (3.21)-(3.22), there exists a T 3 > T 2 such that Similarly, it follows from the second equation of system (1.6) that there exists a T 3 > T 3 such that 2 , for all t > T 3 .

From (3.44) and the expression of m (n)
i , it immediately follows that Letting n → +∞ in (3.40), we obtain (3.46) (3.46) shows that (x, y) and (x, y) are positive solutions of the equations

47)
Already, we had showed in the previous section that under the assumption r 1 > Eq 1 m, r 2 > Eq 2 m, (3.47) has a unique positive solution C(x * , y * ). Hence, we conclude that Thus, the unique interior equilibrium C(x * , y * ) is globally attractive.
This completes the proof of Theorem 3.1.

Numeric simulations
Now let's consider the following example.

Conclusion
Wei and Li [2] had considered the influence of the harvesting to the May cooperative system, however, they only considered the harvesting of the first species. In this paper, stimulated by the works of Chakraborty, Das, Kar [36], we propose the May cooperative system with both non-selective harvesting and partial closure for the populations, i. e., system (1.6).
Some interesting property about the system (1.6) and the influence of parameter m are obtained.
(1) Depending on the fraction of the stock available for harvesting, i. e., depending on the interval in which m is located, m > max r 1 Eq 1 , r 2 Eq 2 , r 2 Eq 2 < m < r 1 Eq 1 , r 1 Eq 1 < m < r 2 Eq 2 , m < min r 2 Eq 2 , r 1 Eq 1 , the two species could be coexist in the long run, or some of the species is extinct, while the other one is permanent, or two of the species are both driven to extinction. That is, the fraction of the stock available for harvesting plays crucial role on the dynamic behaviors of the system.
Obviously, those conditions are very simple and easily testified.
(2) Another amazing finding is that the conditions of Theorem 2.1 and 3.1 are independent of k i and a i , i = 1, 2. Though k i , a i , i = 1, 2 have influence on the final density of the both species, those parameters have no influence on the persistent property of the system. If the intrinsic growth rate of the species (r i , i = 1, 2) are enough large, and the harvesting is limited to suitable area, then two species could survival in the long run.