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In this paper, we explore an impulsive stochastic infected predator-prey system with Lévy jumps and delays. The main aim of this paper is to investigate the effects of time delays and impulse stochastic interference on dynamics of the predator-prey model. First, we prove some properties of the subsystem of the system. Second, in view of comparison theorem and limit superior theory, we obtain the sufficient conditions for the extinction of this system. Furthermore, persistence in mean of the system is also investigated by using the theory of impulsive stochastic differential equations (ISDE) and delay differential equations (DDE). Finally, we carry out some simulations to verify our main results and explain the biological implications.

With the development of the economy, environmental pollution is caused by various industries and other activities of human, which has been one of the most important social problems in the world today. Many species have gone extinct due to the toxicant in the environment. Therefore, controlling the environmental pollution has been the important topics around the world. There are many researchers which have investigated the pollution models in recent years [

In the natural world, time delay often occurs in almost every situation. Thus it is significant to take time delay into consideration [

Furthermore, populations may suffer from sudden environmental fluctuations, such as floods and earthquakes, which cannot be described by Brownian motions. To explain these phenomena, introducing a jump process into the underlying population dynamics is one of the important methods. Thus, there are many scholars introduce Lévy jumps into the population system [

The rest of this paper is arranged as follows. Section

Throughout the paper, we assume that

For the sake of convenience, we introduce some notions and some lemmas which will be used for the main results. We define

Then we show some basic properties of the subsystem of system (

System (

It can be obtained from a simple calculation that

From Lemma

Now we give an assumption which will be used in the following proof.

There exist constants

For any given initial value

This proof is the same as Theorem

The stochastic comparison theorem and limit superior and limit inferior theory are given as follows.

Suppose that

(i) If there exist three positive constants

(ii) If there exist three positive constants

First, we explore the following auxiliary system:

For system (

(i) If

(ii) If

(iii) If

(iv) If

Applying Itô’s formula to system (

Integrating both sides of (

Now we are going to show our main results. By Lemma

For system (

(i) If

(ii) If

(iii) If

By stochastic comparison theorem, we have

In this section, we prove the permanence in mean of system (

Let

Computing (

When

When

This paper explores the dynamics of a stochastic predator-prey model with time delays in the polluted environment. We show some properties of the subsystem of the predator-prey system. Then by using comparison theorem and limit superior theory, we obtain the sufficient conditions for the extinction of this system. From Theorem

In (a),

Keep

Keep

Choose some parameters as follows

In Figure

In Figure

In Figure

From Figures

Large stochastic disturbance can cause the populations to go to extinction; that is, the persistent population of a deterministic system can become extinct due to the white noise stochastic disturbance.

Large impulsive input concentration of the toxicant or small impulsive period of the exogenous input of toxicant can cause the populations to go to extinction.

Therefore, the above numerical simulations illustrate the performance of the theoretical results, and the biological results show that the white noise stochastic disturbance and impulsive toxicant input are disadvantage for the permanence of system.

For the sake of convenience, we define

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

This work was supported by the National Natural Science Foundation of China (11371230 and 11561004), the SDUST Research Fund (2014TDJH102), Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province, the Open Foundation of the Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, China, and Shandong Provincial Natural Science Foundation, China (ZR2015AQ001).