Permanence and Globally Asymptotic Stability of Cooperative System Incorporating Harvesting

The stability of a kind of cooperative models incorporating harvesting is considered in this paper. By analyzing the characteristic roots of the models and constructing suitable Lyapunov functions, we prove that nonnegative equilibrium points of the models are globally asymptotically stable. Further, the corresponding nonautonomous cooperative models have a unique asymptotically periodic solution, which is uniformly asymptotically stable. An example is given to illustrate the effectiveness of our results.


Introduction
Permanence, stability and periodic solution for Lotka-Volterra models had been extensively investigated by many authors (see [1][2][3][4][5][6][7][8] and the references therein).Jorge Rebaza [1] had discussed the dynamic behaviors of predator-prey model with harvesting and refuge he obtained that harvesting and refuge affected the stability of some coexistence equilibrium and periodic solutions of model (1), where   H x was a continuous threshold policy harvesting function.Motivated by Jorge's work, we consider the following cooperative system incorporating harvesting where x and denote the densities of two populations at time .The parameters are all positive constants.
, , , , , , , , r r a a b b k k and it satisfies , where     0 t  We will discuss our problems in the region

Permanence of System
Definition 2 [2] If there are positive constants such that each positive solution Then system (2) is persistent.If the system is not persistent, then system (2) is called non-persistent.
Proof.By the first equation of ( 2) and the comparison theorem, we get     there exists a , as , it then follows Similarly, we have   By the comparison theorem, and letting 0 By Definition 2, system (2) is persistent.□

Equilibrium Points and Stability
If , then the equilibrium points of (2) are The general Jacobian matrix of ( 2) is given by The characteristic equation of system (2) at 3 H takes the form H is locally asymptotically stable.
Theorem 1 then the positive equilibrium point 3 H of system (2) is globally asymptotically stable, where , , , A B C D can be found in Lemma 1.
Proof.Define a Lyapunov function by the conditions of theorem 1, thus,   . The positive equilibrium point 3 H of system ( 2) is globally asymptotically stable.□

Existence and Uniqueness of Solutions
Next, we will discuss a nonautonomous system , , a t x x x r t b t x E t q t x y k t a t y y y r t b t y x k t , E t q t are positive continuous bounded asymptotically periodic functions with period .The initial data of ( 4) is given by The solution of (4) with initial data (5) is denoted by For a continuous function   Definition 3 [2] If there exists a , for any , , there exists a Let us consider the following asymptotically periodic system where In order to discuss the existence and uniqueness of asymptotically periodic solution of system (6), we can consider the adjoint system then the solution of system (4) is ultimately boundedness.
Proof.By the first equation of system (4) and the comparison theorem, one gets that Similarly, we have By the same discussion, one thus gets that By the Definition 3, the solution of system ( 4) is ultimately bounded.□ Lemma 3 [2] If satisfies the following conditions: , where   a r and are continuously positively increasing functions; where is a constant; 0 l  3) there exists a continuous non-increasing function , such that for s > 0, .And as where 0   t is a constant; furthermore, system (6) has a solution   for and satisfies Then system (6) has a unique asymptotically periodic solution, which is uniformly asymptotically stable.
hold, the conditions of Lemma 2 are satisfied, then system (4) has a unique asymptotically periodic solution, which is uniformly asymptotically stable.Proof.By Lemma 2, the solutions of system (4) is ultimately bounded.We consider the adjoint system and   , , , x y u v be the solution of (8) , .
Define Lyapunov function By suing of the inequality a b a b    , it is easy to check that 1) and 2) of Lemma 3 are valid.Computing the derivative of  

W t m x u
along the solution of system (8), by (9) and 1 , then, system (4) has a unique positive asymptotically periodic solution, which is uniformly asymptotically stable.□ Copyright © 2013 SciRes.APM

Examples and Numerical Simulations
Now, let us consider a autonomous cooperative system incorporating harvesting 0.2 0.5 3 , 2 1 1 it is easy to check that

Conclusions
By analyzing the characteristic roots of a kind of cooperative models (2) incorporating harvesting, the stability of positive equilibrium point 3 H to model ( 2) is obtained by constructing a suitable Lyapunov function.Our results have shown that the harvesting coefficient affects the stability and the existence of equilibrium point to model (2).

Eq
The related non-autonomous asymptotically periodic cooperative model (4) has been discussed later.Under some conditions, which also depend on model parameters (see Theorem 2), model (4) has a unique asymptotically periodic solution     , x t y t , which is uniformly  asymptotically stable.Example model (10) shows the effectiveness of our results.
. By the fact