A COMMENSAL SYMBIOSIS MODEL WITH NON-MONOTONIC FUNCTIONAL RESPONSE

A two species commensal symbiosis model with non-monotonic functional response takes the form dx dt = x ( a1−b1x+ c1y d1 + y2 ) ,

The aim of this paper is to investigate the dynamic behaviors of the following two species commensal symbiosis model with non-monotonic functional response where a i , b i , i = 1, 2 and c 1 are all positive constants.Here we make the following assumption: (1) Two species obey the Logistic type growing; (2) The commensal of the second species to the first one obey the non-monotonic functional response, i.e., y 1 + y 2 , which is a humped function and declines with the high densities of the second species.That is, with the increasing of the density of second species, the commensal between two species declined.
During the last decade, many scholars investigated the dynamic behaviors of the mutualism model ([1]- [9]), however, only recently did scholars paid attention to the commensal model, a model describes a relationship which is only favorable to the one side and have no influence to the other side([10]- [14]).[11] first time proposed a intraspecific commensal model:

Sun and Wei
(1.2) They investigated the local stability of all equilibrium points.
Han and Chen [12] proposed the following commensal symbiosis model with feedback controls: (1.3)They showed that system (1.3) admits a unique globally stable positive equilibrium.
Xie et al. [13] proposed the following discrete commensal symbiosis model They showed that the system (1.4) admits at least one positive ω-periodic solution.Xue et al [13] further incorporate the delay to system (1.4), and they investigated the almost periodic solution of the system.
It brings to our attention that all of the above works are based on the traditional Lotka-Volterra model, which suppose that the influence of the second species to the first one is linearize.This may not be suitable since the commensal between two species become infinity as the density of the species become infinity.Now, if we assume that the the functional response between two species is of non-monotonic type ( [15,16]), it then follows the system (1.1).
The aim of this paper is to investigate the local and global stability property of the possible equilibria of system (1.1).We arrange the paper as follows: In the next section, we will investigate the existence and local stability property of the equilibria of system (1.1).In Section 3, we will investigate the global stability property of the positive equilibrium; In Section 4, an example together with its numeric simulation is presented to show the feasibility of our main results.

The existence and local stability of the equilibria
The equilibria of system (1.1) is determined by the system (2.1) Hence, system (1.1) admits four nonnegative equilibria, A 0 (0, 0), A 1 b 2 and A 3 x * , y * , where Concerned with the local stability property of the above four equilibria, we have Theorem 2.1.A 0 (0, 0), A 1 ( a 1 b 1 , 0) and A 2 0, a 2 b 2 are unstable; A 3 x * , y * is locally stable.Proof.The Jacobian matrix of the system (1.1) is calculated as Then the Jacobian matrix of the system (1.1) about the equilibrium A 0 (0, 0), A 1 ( a 1 b 1 , 0) and (0, a 2 b 2 ) are given by

.5)
and respectively.One could easily see that all of the above three matrix has at least one positive eigenvalues, which means that A 0 (0, 0), A 1 ( a 1 b 1 , 0) and A 2 (0, a 2 b 2 ) are all unstable.The Jacobian matrix about the equilibrium A 3 is given by where The eigenvalues of the above matrix are Hence, A 3 (x * , y * ) is locally stable.
This ends the proof of Theorem 2.1.

Global stability of the positive equilibrium
Theorem 2.1 shows that the system always admits a positive equilibrium, and this equilibrium is locally stable.One interesting issue is to find out the conditions which ensure the global stability of the positive equilibrium.Proof.Firstly we proof that every solution of system (1.1) that starts in R 2 + is uniformly bounded.Noting that the second equation of (1.1) takes the form By applying Lemma 3.1 to system (3.2),we know that system (3.2) has a unique globally attractive positive equilibrium y * = a 2 b 2 .It follows from the first equation of system (1.1) that and so lim sup Hence, there exists a ε > 0 such that for all t > T x(t) < Then every solution of system (1.1) starts in R 2 + is uniformly bounded on D. Also, from Theorem 2.1 there is a unique local stable positive equilibrium A 3 (x * , y * ).To show that A 3 (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 B(x, y) = x −1 y −1 , then . By Dulac Theorem [18], there is no closed orbit in area D. Consequently, A 3 (x * , y * ) is globally asymptotically stable.This completes the proof of Theorem 3.1.

Numeric simulation
Now let us consider the following example.In this system, corresponding to system (1.1), we take 1, b 2 = 2. From Theorem 3.1, the unique positive equilibrium 127 246 , 1 2 is globally asymptotically stable.Numeric simulation (Fig. 1) also support this assertion.

Conclusion
We propose a two species commensal symbiosis model with non-monotonic functional response, to the best of the authors knowledge, this is the first time such kind of model is proposed.
We show that, despite the functional response, the dynamic behaviors of the system is very simple, i.e., the system admits a unique globally stable positive equilibrium.