^{1,2}

^{1}

^{2,3}

^{4}

^{1}

^{2}

^{3}

^{4}

We analyze a predator prey model with stochastic perturbation. First, we show that this system has a unique positive solution. Then, we deduce conditions that the system is persistent in time average. Furthermore, we show the conditions that there is a stationary distribution of the system which implies that the system is permanent. After that, conditions for the system going extinct in probability are established. At last, numerical simulations are carried out to support our results.

Recently, the dynamic relationship between predator and prey has been one of the dominant themes in both ecology and mathematical ecology due to its universal importance. Especially, the predator prey model is the typical representative. Thereby it significantly changed the biology and the understanding of the existence and development of the basic law and has made the model become the research hot spot. One of the most famous models for population dynamics is the Lotka-Volterra predator prey system which has received plenty of attention and has been studied extensively; we refer the reader to [

The predator prey model is described as follows:

However, population dynamics in the real world is inevitably affected by environmental noise (see, e.g., [

Therefore, Lotka-Volterra predator prey models in random environments are becoming more and more popular. Ji et al. [

In this paper, we introduce the white noise into the intrinsic growth rate and predator death rate of system (

The aim of this paper is to discuss the long time behavior of system (

The rest of this paper is organized as follows. In Section

Throughout this paper, unless otherwise specified, let

To investigate the dynamical behavior, first, we should concern whether the solution is global existence. Moreover, for a population model, we should also consider whether the solution is nonnegative. Hence, in this section we show that the solution of system (

By the classical comparison theorem of stochastic differential equations, we could get the following.

Let

Consider the stochastic logistic equation

There exists a unique continuous positive solution

From Lemmas

Let

For any initial value

It is clear that the coefficients of system (

Define a

There is no equilibrium of system (

L. S. Chen and J. Chen in [

System (

Let

We have

If Assumption

According to Ito’s formula, the system (

Hence,

In this section we show there is a stationary distribution of system (

Let

Since

Define

From Lemma

Hence, from these arguments, we get the following result.

Assume the same conditions as in Theorem

In this section, we show the situation when the population of system (

Let

Now, we present a useful Lemma.

Assume that conditions (1) and (2) hold in (

Consider the first equation of system (

Therefore, with the condition

According to Ito’s formula and comparison principle, the second population of system (

Let

if

If

In this section, we give out the numerical experiment to support our results. Consider the equation

The solution of system (

The solution of system (

In Figure

In Figure

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work was supported by the Program for Changjiang Scholars and Innovative Research Team in University, NSFC of China, (no. 11371085), the Ph.D. Programs Foundation of Ministry of China (no. 200918), and the Natural Science Foundation of Jilin Province of China (no. 201115133).