Existence, Stationary Distribution, and Extinction of Predator-Prey System of Prey Dispersal with Stochastic Perturbation

and Applied Analysis 3 system takes the following form: dxi ⎡ ⎣xi ( ri − bixi − eiyi ) n ∑ j 1 dij ( xj − αijxi ) ⎤ ⎦dt σ1ixidB1i t , dyi yi −γi − δiyi εixi ) dt σ2iyidB2i t , i 1, 2, . . . , n. 1.3 Throughout this paper, we assume dij are nonnegative constants, dij is irreducible, and the parameters ri, γi, bi, ei, δi, εi are positive constants. In order to obtain better dynamic properties of the SDE 1.3 , we will show that there exists a unique positive global solution with any initial positive value, and its pth moment is bounded in Section 2. In the study of a population dynamics, permanence is a very important and interesting topic regarding the survival of populations in ecological system. In a deterministic system, it is usually solved by showing the global attractivity of the positive equilibrium of the system. But, as mentioned above, it is impossible to expect stochastic system 1.3 to tend to a steady state. So we attempt to investigate the stationary distribution of this system by Lyapunov functional technique. The stationary can be considered a weak stability, which appears as the solution is fluctuating in a neighborhood of the equilibrium point of the corresponding deterministic model. In Section 3, we will show if the white noise is small, there is a stationary distribution of SDE 1.3 and it has ergodic property. Existing results on dynamics in a patchy environment have largely been restricted to extinction analysis which means that the population system will survive or die out in the future due to the increased complexity of global analysis. In Section 4, we give the sufficient conditions for extinction. In Section 5, we make numerical simulation to conform our analytical results. Finally, for the completeness of the paper, we give an appendix containing some theories which will be used in previous sections. The key method used in this paper is the analysis of Lyapunov functions 6, 13, 14, 16 . We will also use the graph theory in Section 3 and some graph definitions can be found in the appendix. Throughout this paper, unless otherwise specified, let Ω, {Ft}t≥0, P be a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions i.e., it is right continuous and F0 contains all P -null sets . Let R2n denote the positive cone of R2n, namely, R2n { x1, y1, . . . , xn, yn ∈ R2n : xi > 0, yi > 0, i 1, 2, . . . , n}. For convenience and simplicity in the following discussion, denote X t x1 t , y1 t , x2 t , y2 t , . . . , xn t , yn t . If A is a vector or matrix, its transpose is denoted by A . If A is a matrix, its trace norm is denoted by |A| √ trace ATA whilst its operator norm is denoted by ‖A‖ sup{|Ax| : |x| 1}. 2. Positive and Global Solutions In order for a stochastic differential equation to have a unique global i.e., no explosion at any finite time solution, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition see 17 . However, the coefficients of SDE 1.3 do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of SDE 1.3 may explode at a finite time. In this section, we will prove the solution of stochastic system 1.3 with any positive initial value is not only positive but also not exploive in infinity at any finite time. 4 Abstract and Applied Analysis Theorem 2.1. For any given initial value X 0 ∈ R2n , there is a unique positive solution X t of SDE 1.3 , and the solution will remain in R2n with probability 1. Proof. We define a C2-function V : R2n → R :