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A stochastic prey-predator system in a polluted environment with Beddington-DeAngelis functional response is proposed and analyzed. Firstly, for the system with white noise perturbation, by analyzing the limit system, the existence of boundary periodic solutions and positive periodic solutions is proved and the sufficient conditions for the existence of boundary periodic solutions and positive periodic solutions are derived. And then for the stochastic system, by introducing Markov regime switching, the sufficient conditions for extinction or persistence of such system are obtained. Furthermore, we proved that the system is ergodic and has a stationary distribution when the concentration of toxicant is a positive constant. Finally, two examples with numerical simulations are carried out in order to illustrate the theoretical results.

The Lotka-Volterra model [

In many ecosystems, predators tend to be omnivorous, they have wide variety of food sources. For example, the giant panda is omnivorous animal, since it can eat both meat and plant such as bamboos. In the lake ecosystem, some fishes not only prey on aquatic invertebrates, but also feed on algae and other aquatic plants. Polis and Strong in [

It is well known that the biological population is inevitably affected by environment perturbation while the stochastic population model is more in line with the actual situation. Recently, various models based on stochastic differential equations (SDEs) have extensively been paid the attention of the researchers (see, e.g., [

Furthermore, the prey-predator model may be perturbed by telegraph noise which is distinguished by factors such as rain falls and nutrition and can be represented by switching among two or more regimes of environment [

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

Throughout this paper, let

System (

Suppose that

(i) system (

(ii) system (

(iii) when

Next, we consider the following stochastic differential equation:

Suppose that the coefficients of (

(i)

(ii)

Then there exists a solution for (

Furthermore, we introduce some results from [

Let

(i) if there are two positive constants

(ii) if there are three positive constants

Finally, we give some basic properties of the following subsystem of system (

System (

For convenience and simplicity, define

In this section, we devote our attention to the investigation of the existence of periodic solutions of system (

Now, we discuss the existence of periodic solutions of system (

Define

If

From the first equation of system (

Since

Overall, when

The proof of this theorem is completed.

In order to investigate the existence of a nontrivial positive

Suppose that

Obviously, the coefficients of system (

Applying Itô’s formula, one has

Denote

Thus,

In this section, we investigate the long-term dynamic behaviors of the prey-predator system (

Given initial value

(i) if

(ii) if

(iii) if

(i) By Itô’s formula, we get

(ii) Similarly, from the second equation of system (

(iii) By the condition

This completes the proof of the theorem.

If

Next, we will discuss the persistence of system (

Furthermore, the persistent property of the predator species of system (

From the first equation of system (

holds, then

In summary, one gets the following.

Given initial value

(i) if

(ii) if conditions

(i) It can be seen from

(ii) Obviously,

In system (

System (

(i) For

(ii) For each

(iii) There exists a bounded open set

Moreover, the Markov process

Recently, the ergodicity and stationary distribution have been explored by many authors. In the following, we give sufficient conditions for the existence of stationary distribution of system (

Assume that for

hold; then the stochastic process

By the assumption

Define a

An application of the operator

It is easy to see that

This completes the proof of Theorem

In this article, we discussed the dynamics of stochastic prey-predator models with Beddington-DeAngelis functional response in polluted environment.

Firstly, for system (

Secondly, system (

To verify the correctness of the theoretical analysis, numerical simulations are employed in the following examples.

Assume that the Markov chain

Assume

We choose the density of white noise as the following:

Note that

Sample paths of

We change the density of the white noise to

From Theorem

Sample paths of

Choose parameters

Let

Next we only change the density of the white noise to

It is easy to see from Figures

Sample paths of

Sample paths of

Let

Next we only change the amount of toxicant to

Figures

Sample paths of

Sample paths of

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by Shandong Provincial Natural Science Foundation of China (no. ZR2015AQ001), the National Natural Science Foundation of China (no. 11371230), and Research Funds for Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources by Shandong Province and SDUST (2014TDJH102).

_{∞}control for stochastic systems with poisson jumps