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This paper presents an investigation of asymptotic properties of a stochastic predator-prey model with modified Leslie-Gower response. We obtain the global existence of positive unique solution of the stochastic model. That is, the solution of the system is positive and not to explode to infinity in a finite time. And we show some asymptotic properties of the stochastic system. Moreover, the sufficient conditions for persistence in mean and extinction are obtained. Finally we work out some figures to illustrate our main results.

The dynamic interaction between predators and their prey has been one of the dominant themes in mathematical biology due to its universal existence and importance. Much literature exists on the general problem of food chains in the classical Lotka-Volterra model. In [

Hsu and Huang [

Recently, [

As a matter of fact, population systems are often subject to environmental noise. Recently, more and more interest is focused on stochastic systems. Reference [

According to (

When white noise is taken into account in our model (

Throughout the paper, we use

As

For a given initial value

The proof is similar to [

Theorem

To begin our discussion, we impose the following assumption:

And we list the interesting lemma as follows.

Consider one-dimensional stochastic differential equation

To demonstrate asymptotic properties of the stochastic system (

On the one hand, by the comparison theorem of stochastic equations, it is obvious that

Denote by

Under assumption (H), for any initial value

By assumption (H) and Lemma

Under assumption (H), for any initial value

By virtue of (

Now let us continue to consider the asymptotic behavior of the species

Denote by

Under assumption (H), for any initial value

The proof is motivated by [

On the other hand, it follows from (

From the representation of

Under assumption (H), for any initial value

It follows from (

As we know, the property of persistence is more desirable since it represents the long-term survival to a population dynamics. Now we present the definition of persistence in mean proposed in [

System (

Assume that condition (H) holds. Then system (

Define the function

Moreover, define the function

Under condition (H), we show that the system is persistent in mean. To a large extent, (H) is the condition that stands for small environmental noises. That is, small stochastic perturbation does not change the persistence of the system. Here, we will consider that large noises may make the system extinct.

Assume that condition

Define the function

On the other hand, by the Itô formula again, we derive

We continue to discuss the asymptotic behaviors of the stochastic system (

Assume that condition

Define the function

In this section, some simulation figures are introduced to support the main results in our paper.

For model (

In Figure

In Figure

By comparing Figures

White noise is taken into account in our model in this paper. It tells us that, when the intensities of environmental noises are not too big, some nice properties such as nonexplosion and permanence are desired. However, Theorem

This research is supported by China Postdoctoral Science Foundation (no. 20100481000).