The Asymptotic Behavior of a Stochastic Predator-Prey System with Holling II Functional Response

and Applied Analysis 3 disease. From this point, we say the stochastic model is more realistic than the deterministic model. The rest of this paper is organized as follows. In Section 2, we show that there is a unique nonnegative solution of system 1.3 . In Section 3, we show that there is a stationary distribution under small white noise. While in Section 4, we consider the situation when the white noise is large. We prove that the system will be extinct. Finally, we give an appendix containing the stationary distribution theory used in Section 3. 2. Existence and Uniqueness of the Nonnegative Solution To investigate the dynamical behavior, the first concern is the global existence of the solutions. Hence in this section we show that the solution of system 1.3 is global and nonnegative. It is not difficult to check the uniqueness and global existence of solutions if the coefficients of the equation satisfy the linear growth condition and local Lipschitz condition cf. 18 . However, the coefficients of system 1.3 do not satisfy the linear growth condition, but locally Lipschitz continuous, so the solution of system 1.3 may explode at a finite time. In this section, by changing variables, we first show that system 1.3 has a local solution, then show that this solution is global. Theorem 2.1. For any initial value x 0 , y 0 ∈ R2 , there is a unique solution x t , y t of system 1.3 on t ≥ 0, and the solution will remain in R2 with probability 1. Proof. First, consider the following system, by changing variables, x t e t , y t e t , du t ( a − σ 2 1 2 − be t − αe v t 1 βeu t ) dt σ1dB1 t , dv t ( − e − σ 2 2 2 kαe t 1 βeu t ) dt σ2dB2 t . 2.1 It is clear that the coefficients of system 2.1 are locally Lipschitz continuous for the given initial value logx 0 , logy 0 ∈ R2 there is a unique local solution u t , v t on t ∈ 0, τe , where τe is the explosion time see 18 . Hence, by Itô formula, we know e t , e t , t ∈ 0, τe is a unique positive local solution of system 1.3 . To show that this solution is global, we need to show that τe ∞ a.s. Let m0 ≥ 1 be sufficiently large so that x 0 , y 0 all lie within the interval 1/m0, m0 . For each integer m ≥ m0, define the stopping time: τm inf { t ∈ 0, τe : min { x t , y t } ≤ 1 m or max { x t , y t } ≥ m } , 2.2 Where, throughout this paper, we set inf ∅ ∞ as usual ∅ denotes the empty set . Clearly, τm is increasing as m → ∞. Set τ∞ limm→∞τm, whence τ∞ ≤ τe a.s. If we can show that τ∞ ∞ a.s., then τe ∞ and x t , y t ∈ R2 a.s. for all t ≥ 0. In other words, to complete the proof all we need to show is that τ∞ ∞ a.s. If this statement is false, then there is a pair of constants T > 0 and ∈ 0, 1 such that P{τ∞ ≤ T} > . 2.3 4 Abstract and Applied Analysis Hence there is an integer m1 ≥ m0 such that P{τm ≤ T} ≥ ∀m ≥ m1. 2.4 Define a C2-function V : R2 → R by V ( x, y ) ( x − c − c log x c ) 1 k ( y − 1 − logy), 2.5 where c is a positive constant to be determined later. The nonnegativity of this function can be seen from u − 1 − logu ≥ 0, for all u > 0. Using Itô’s formula, we get dV : LVdt σ1 x − c dB1 t σ2 k ( y − 1)dB2 t , 2.6