THE INFLUENCE OF PARTIAL CLOSURE FOR THE POPULATIONS TO A HARVESTING LOTKA-VOLTERRA COMMENSALISM MODEL

The aim of this paper is to investigate the dynamic behaviors of a harvesting Lotka-Volterra commensalism model incorporating partial closure for the populations. By analyzing the characteristic equation of the variational matrix, sufficient conditions which ensure the local stability of the equilibria are obtained; By applying the differential inequality theory and the Dulac criterion, sufficient conditions which ensure the globally asymptotical stability of the equilibria are obtained; Our study shows that depending on the fraction of the stock available for harvesting, the system maybe extinction, partial survival or two species coexist in a stable state. The dynamic behaviors of the system becomes complicated compared with the non-harvesting system. Numeric simulations are carried out to show the feasibility of the main results.


Introduction
Commensalism is a symbiotic interaction between two populations where one population gets benefit from while the other is neither harmed nor benefited due to the interaction with the previous species [1].There are many real-life examples of commensalism, for example, the relationship between squirrel and the oak, the clownfish and the sea anemone, etc, see [1]- [12] and the references cited therein.
During the last decade, many scholars ( [13]- [25]) investigated the dynamic behaviors of the mutualism model, where the interaction of two species is to the advantage of both side.Many interesting results are obtained, for example, Chen et al [13] showed that stage structure of the species could have great influence on the stability property of the mutualism model; Chen et al ([20]- [21]) showed that feedback control variables have no influence on the persistent property of the system; Xie et al [25] proved that if the mutualism system admits an unique positive equilibrium, it is globally asymptotically stable.However, there are still not so much works on commensalism model (see [1]- [12]).
Recently, Sun et al. [8] proposed the following commensalism system where r 1 , r 2 , K 1 , K 2 , α are all positive constants.The system admits four equilibria: Concern with the stability property of above equilibria,by linearizing the system at equilibrium,the author obtained the following results.
Theorem A.
(1) E 1 (0, 0) is unstable node; (2) It bring to our attention that the author gave no information about the global stability property of the equilibrium.
On the other hand, to obtain the resource for the development of the human being, harvest of the species is necessary, Chakraborty, Das and Kar [22] argued that 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.
Though there are many papers concerned with the harvesting of the predatorprey system ([26]- [30]), however, to this day, seldom did scholars consider the influence of harvesting to the commensalism model.Stimulated by the works of Sun et al [8] and Chakraborty, Das and Kar [27], in this paper, we propose the following non-selective harvesting Lotka-Volterra amensalism model incorporating partial closure for the populations: where r 1 , r 2 , K 1 , K 2 , α are all positive constants, and have the same meaning as that of the system (1.1).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 investigate the dynamic behaviors of the system (1.2), and to find out the influence of the harvesting and the fraction of the stock.
The paper is arranged as follows.We will investigate the local and global stability property of the equilibria of system (1.2) in section 2 and 3, respectively.Some examples together with their numeric simulations are present in Section 5 to show the feasibility of the main results.
We end this paper by a briefly discussion.

Local stability of the equilibria
The system always admits the boundary equilibrium E 1 (0, 0).
If r 1 > Emq 1 holds, the system admits the boundary equilibrium E 2 (x 0 , 0), where If r 2 > Emq 2 holds, the system admits the boundary equilibrium E 3 (0, y 0 ), where If r 1 r 2 K 1 + r 1 r 2 αK 2 > r 1 q 2 mαK 2 E + r 2 mq 1 EK 1 and r 2 > Emq 2 hold, then the system admits a unique positive equilibrium We shall now investigate the local stability property of the above equilibria.
Proof.The variational matrix of the system of Eq. (1.2) at (x, y) is The characteristic equation of the variational matrix is Obviously, if tr(J) < 0 and det(J) > 0, both eigenvalues of (2.1) have negative real parts, and the corresponding equilibrium solution is asymptotically stable.
The proof of Theorem 2.1 is finished.
(1)From r 1 < Eq 1 m there exists enough small ε > 0 such that From the second equation of (1.2) we have For above ε > 0, there exists a T 1 > 0, such that For t > T 1 , from the first equation of system (1.2), we have (2)Similarly to the analysis of (3.5)-(3.8),for arbitrary enough small ε > 0, there exists a T 2 > 0, such that For t > T 2 , from the first equation of system (1.2), we have Consider the equation It follows from Lemma 3.1 that By using the comparison theorem of differential equation, it follows from (3.6) that lim sup On the other hand, from the first equation of system (1.2), we also have Consider the equation It follows from Lemma 3.1 that By using the comparison theorem of differential equation, it follows from (3.11) that lim inf It follows from (3.10) and (3.12) that Since ε is any arbitrary small positive constants, setting ε → 0 in (3.10) leads to (3)The left hand side of (3.3) is equal to the inequality hence, for arbitrary enough small ε > 0, the following inequality holds From the second equation of (1.2) we have For above ε > 0, there exists an enough large T 3 > 0 such that For t > T 3 , from the first equation of system (1.2), we have (3.17) Hence (4)Firstly we proof that every solution of system (1.2)that starts in R 2 + is uniformly bounded.Similarly to the analysis of (3.15), we have Hence, for arbitrary small positive constant ε > 0, there exists a T 4 > 0 such that Similarly to the analysis of (3.17),For t > T 4 , from the first equation of system (1.2), we have Now consider the equation From (3.20) and (3.22), by applying the differential inequality theory, we have Hence, there exists a T 5 > T 4 such that where Then every solution of system (1.2) starts in R 2 + is uniformly bounded on D. Also, from Theorem 2.1 there is a unique local stable positive equilibrium E 4 (x * , y * ).To show that E 4 (x * , y * ) is globally stable, it's enough to show that the system admits no limit cycle in the area D, Let's consider the Dulac function u(x, y) = x −1 y −1 , then where By Dulac Theorem [32], there is no closed orbit in area D. Consequently, E 4 (x * , y * ) is globally asymptotically stable.This completes the proof of Theorem 3.1.
Remark 3.1.Theorem 2.1 and 3.1 show that if the system (1.2) admits the unique positive equilibrium, then the positive equilibrium is globally asymptotically stable.
Remark 3.2.It follows from Theorem 2.1 and 3.1 that the local stability of the equilibrium also implies the global one.
thus both x * and y * are the strictly decreasing function of m.This means that with the increasing of the fraction of the stock afford for harvesting, both species reduce their final density.
Therefore, to ensure the coexistence of the both species, the area for the harvesting should be restricted to the limited case Otherwise, at least one of the species will be driven to extinction.
Above numeric simulations show that if the fraction of stock is too large, the system will be collapse, while if the fraction of stock is limited, the two species could be coexist in a stable state.

Discussion
With the aim of the ecological sustainability and conservation of the species can be implemented in a long run, in this paper, we propose a non-selective harvesting Lotka-Volterra commensalism model incorporating partial closure for the populations, i. e., system (1.2), which can be seen as the generalization of the system (1.1).
For the system without harvesting, it follows from Theorem A that the positive equilibrium is locally asymptotically stable, while the other three equilibrium are all unstable.However, by introducing the harvesting, the dynamic behaviors of the system changes greatly.Depending  with m = 0.75, the initial conditions (x(0), y(0)) = (0.5, 0.1), (0.8, 1), (0.3, 3), and (0.7, 2), respectively.with m = 0.3, the initial conditions (x(0), y(0)) = (0.5, 0.1), (0.8, 1), (0.3, 3), and (0.7, 2), respectively.with m = 0.1, the initial conditions (x(0), y(0)) = (0.5, 0.1), (0.8, 1), (0.3, 3), and (0.7, 2), respectively.on the fraction of the stock and the harvesting effort, the system may be collapse in the sense that both species will be driven to extinction, or partial survival, in the sense that one of the species will be driven to extinction, while the other one is permanent, or both species could be coexisted in a stable state, despite the initial state of the both species.