The Complex Dynamics of a Stochastic Predator-Prey Model

and Applied Analysis 3 From the second equation of model 1.3 , we see that, for t > T , we have dy/dt ≤ ey 1 − bfy/ mb 1 . A standard comparison argument shows that lim sup t→∞ y t ≤ 1 bm bf . 2.2 Thus, we have the following conclusion. Lemma 2.1. Model 1.3 is dissipative. Lemma 2.2. If c < n, then model 1.3 is permanent. Proof. If c < n, from the first equation of model 1.3 , we have dx/dt ≥ ax 1 − bx − c/n . Therefore, by standard comparison argument, we have lim inf t→∞ x t ≥ n − c bn . 2.3 Hence, for any ε > 0 and large t, x t > n − c /bn − ε, and dy dt ≥ ey ( bmn nc − bnε − bfny bmn nc − bnε ) . 2.4 From the arbitrariness of ε > 0, we can get that lim inf t→∞ y t ≥ bmn n − c bfn . 2.5 2.2. Stability Analysis of the Equilibria In this section, we will focus on the existence of equilibria and their stabilities of model 1.3 . It is easy to find that model 1.3 always has three boundary equilibria E0 0, 0 , E1 1/b, 0 , E2 0, m/f . And the positive equilibria x, y satisfies the equations ax ( 1 − bx − cy x ny ) 0, ey ( 1 − fy m x ) 0, 2.6