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In this paper, a stochastic Lotka–Volterra predator-prey model with discrete delays and feedback control is studied. Firstly, the existence and uniqueness of global positive solution are proved. Further, we investigate the asymptotic property of stochastic system at the positive equilibrium point of the corresponding deterministic model and establish sufficient conditions for the persistence and extinction of the model. Finally, the correctness of the theoretical derivation is verified by numerical simulations.

In nature, time delays exist in many ecosystems [

Due to the environmental changes and increased human activities, many rare species are at risk of extinction. How to protect endangered species of floras and faunas and maintain the diversity of ecosystems is an important issue that needs to be solved urgently. In the process of marine fishery production, overfishing often results in the exhaustion of fishery resources. It is rewarding for humans to develop and utilize the ecological system of the population rationally, which also contributes to the sustainability of the system [

The solution

When the conditions

In fact, in nature, ecosystems are inevitably affected by various environmental noises [

Then, model (

Due to the interference of stochastic noise, system (

The stochastic differential equation is expressed as

If the Lyapunov function

For any given initial condition (

Since the coefficients of system (

To prove that this solution is global, we only need to prove

Obviously,

Define a

The nonnegativity of this function can be obtained from

Applying Itô’s formula yields

Therefore,

Integrating (

Set

It can be obtained by (

This is a contradiction; we must have

Due to the interference of white noise, the solution of system (

For any given initial condition (

Define the function

By Itô’s formula, we obtain

Similarly,

In the same way,

Therefore, we have

Let

Therefore,

Integrate both sides of (

Divide both sides by

Obviously,

We have completed the proof.

Theorem 2 shows that if the condition

The equilibrium point

In nature, whether ecosystems can survive or not is our main concern. Before discussing the persistence of stochastic system, we give the following assumption:

For any given initial condition (

According to (

As we know,

By the condition

Similarly, when

Define

For the extinction of system (

For any given initial condition (

when

when

when

Before proving Theorem 4, consider the following auxiliary system [

If

if

if

By Itô’s formula, we obtain

Dividing both sides of (

By using Lemma 2 of [

Substitute (

On the other hand, computing (_{21} + (_{11}, we have

By Lemma 2 in literature [

If

Proof of lemma is completed.

This is where we prove Theorem 4.

By using Itô’s formula for system (

We first prove (i): from equation (

By the condition

From (

Further, from the third equation of model (

Secondly, it proves (ii): comparing model (

From Lemma 1, we can conclude that

Therefore, population

Finally, we prove (iii): by Lemma 1, when

Computing (_{21} + (_{11},

Therefore,

Let

Considering the third equation of model (

Substitute (

It is concluded that predator

When

So,

The condition

Substituting (

So,

This paper proposes a stochastic Lotka–Volterra predator-prey model with discrete delays and feedback control. We firstly study the existence and uniqueness of global positive solution. By constructing appropriate Lyapunov functions and applying Itô’s formula, we discuss the asymptotic behavior of stochastic system at the positive equilibrium point of the corresponding deterministic model. Finally, this paper gives the conditions for the persistence and extinction of stochastic system. Theorem 3 shows that the system is persistent if the intensity

If the coefficient

If the intensity of random disturbance

Therefore, utilizing the feedback control measures to limit the predator quantity within a certain range is beneficial for the sustained existence of the population.

In order to verify the correctness of the theoretical analysis, we carry out the following numerical simulations. Choose the parameters in system (

Let

Set

Set

Set

Set

(a) Time series diagram of

Time series diagram of

Time series diagram of

(a) Time series diagram of

All data sets used in this study are hypothetical.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (11371230) and SDUST Research Fund (2014TDJH102).