Complexity
Volume 2019 (2019), Article ID 4873290, 15 pages
https://doi.org/10.1155/2019/4873290

Research Article

Dynamic Analysis of Stochastic Lotka–Volterra Predator-Prey Model with Discrete Delays and Feedback Control

Jinlei Liu¹ and Wencai Zhao1,2

¹College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
²State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Wencai Zhao; zhaowencai@sdust.edu.cn

Received 2 June 2019; Revised 26 September 2019; Accepted 24 October 2019; Published 16 November 2019

Academic Editor: Mahdi Jalili

Copyright © 2019 Jinlei Liu and Wencai Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

In nature, time delays exist in many ecosystems [1–5]. For example, maturity stage is a common phenomenon in biological population, and many diseases have a long incubation period. The mathematical model describing this phenomenon with time delay is called the delay differential equation. In 1999, Saito et al. [6] studied a Lotka–Volterra predator-prey model with discrete delays, which can be defined as follows:

where and stand for the population density of prey and predator at time t, respectively. represent the intrinsic growth rate of corresponding population. and are discrete time delays. , α and β are constants.

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 [7–14]. In 2003, Gopalsamy and Weng [15] studied the following population competition model with feedback control:

where and are the feedback control variables, , , , , and . They also discussed the existence of positive equilibrium point and global attraction of the model. In 2013, Li et al. [16] introduced feedback control variables into the two-species competition system and discussed the extinction and global attraction of equilibrium points. They found that if the two-species competition model is globally stable, the system retains the stable property after adding feedback controls and the position of equilibrium point is changed. If the two-species competition model is extinct, by choosing the suitable values of feedback control variables, they can make extinct species become globally stable, or still keep the property of extinction. In 2017, Shi et al. [17] discussed a Lotka–Volterra predator-prey model with discrete delays and feedback control as follows:

where is the feedback control variable, e and f denote the feedback control coefficients, denote the intraspecific competition rates, stand for the capturing rates of the prey and predator populations, is the time of catching prey, and is maturation delay of predator. Shi et al. [17] show that

(i)The solution of system (3) is ultimately bounded(ii)When the conditions are established, system (3) has a unique globally asymptotically stable positive equilibrium point , where , , and

In fact, in nature, ecosystems are inevitably affected by various environmental noises [18–28]. Mathematical models with environmental disturbances can usually be described by stochastic differential equations. Stochastic noise can generally be divided into two categories: one type is a small number of strong interference, usually called colored noise or electrical noise, which can be described by the Markov chain [29–31]; the other type is the sum of many small, independent random interference, called white noise, which is usually represented by Brownian motion [32–35]. Assume that the population’s intrinsic growth rate is disturbed by white noise:

Then, model (3) is transformed into

and satisfies the initial conditions

where denote the independent standard Brownian motion, denote the intensity of white noise, , and and are both nonnegative continuous functions on .

Due to the interference of stochastic noise, system (5) does not possess an equilibrium point. An interesting question is: Does model (5) still have stability? What is the influence of white noise on system (5)? This paper mainly studies the dynamical properties of stochastic systems (5) also satisfying initial conditions (6). The second part proves the suitability of the system. The third part discusses the oscillation of the stochastic model near the positive equilibrium point of the corresponding deterministic model. The fourth and fifth parts, respectively, obtain the conditions for the persistence and extinction of the stochastic system. Finally, the correctness of the theoretical derivation is verified by numerical simulation.

2. Existence and Uniqueness of Global Positive Solutions

The stochastic differential equation is expressed as

If the Lyapunov function , the stochastic differential equation of along system (7) is defined as [36]

where represent diffusion operator.

Theorem 1. For any given initial condition (6), model (5) has a unique global positive solution , and the solution will remain in with probability one.

Proof. Since the coefficients of system (5) satisfy the locally Lipschitz condition, for any given initial condition (6), model (5) has a unique local positive solution in interval , where is the explosion time.
To prove that this solution is global, we only need to prove a.s. Let be a sufficiently large constant for any initial value , and lying within the internal . For each integer , define the stopping time

Obviously, is increasing as . Let ; therefore, a.s. Now, we need to verify a.s. Otherwise, there are two constants and such that So, there is a positive integer , such that

Define a : by

where

The nonnegativity of this function can be obtained from

Applying Itô’s formula yields

where

Therefore,

where K is a positive constant. So, we get

Integrating (17) from 0 to and taking expectation on both sides, we have

Set , and from inequality (10), we have . Note that, for every , there is at least one of , or equaling either k or , and then, we have

It can be obtained by (18)

where is the indicator function of , and letting yields

This is a contradiction; we must have , and we have completed the proof.

3. Asymptotic Property

Due to the interference of white noise, the solution of system (5) will have stochastic oscillation. Next, we discuss the asymptotic property of stochastic system at the positive equilibrium point of the corresponding deterministic model. In order to study the problem conveniently, the hypothesis is

Theorem 2. For any given initial condition (6), if hypothesis is established, the solution of system (5) has the property that

where

where is the positive equilibrium point of the corresponding deterministic model (3).

Proof. Define the function

where and are positive constants. Define

By Itô’s formula, we obtain

where

Similarly,

where

In the same way,

Therefore, we have

Let , we obtain

Therefore,

Integrate both sides of (18) from 0 to t and take the expectation, and then we get

Divide both sides by t and take the limit superior, and then we have

Obviously,

where

We have completed the proof.
Theorem 2 shows that if the condition holds, the solution oscillates around the equilibrium point , and the amplitude of oscillation is positively correlated with the intensity of environmental noise. In particular, if , the influence of environmental noise is not taken into account:

The equilibrium point is globally asymptotically stable. This is the conclusion of reference [17].

4. Persistence

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

Theorem 3. For any given initial condition (6), if assumptions hold at the same time, the solution of system (5) is persistent that

Proof. According to (37), we have

As we know, and , from , one can get

By the condition we have

Similarly, when ,

5. Extinction

Define

For the extinction of system (5), we have the following conclusions.

Theorem 4. For any given initial condition (6), the solution of system (5) has the following properties:

(i)when , the population is extinct(ii)when , and , population is persistent and and u are extinct(iii)when , , and , population is extinct and and u are persistent

Before proving Theorem 4, consider the following auxiliary system [37]:

and it satisfies the initial condition: .

Lemma 1. If , the solution of system (47) has the following properties:

(i)(ii)if (iii)if

Proof. By Itô’s formula, we obtain

Dividing both sides of (48) and (49) by t, we have

By using Lemma 2 of [37], from equation (48), it follows that

and therefore,

Substitute (53) into (50), and from , we get

On the other hand, computing (50) × a₂₁ + (51) × a₁₁, we have

By Lemma 2 in literature [37], when ,

If

Proof of lemma is completed.

This is where we prove Theorem 4.

Proof. By using Itô’s formula for system (5), we have

We first prove (i): from equation (58), we can get

By the condition and Lemma 2 in literature [37],

From (57), (61), and the condition , we have

Further, from the third equation of model (5), we can get

Secondly, it proves (ii): comparing model (5) with auxiliary system (37), one can get By Lemma 1, when , ; therefore, So, we have . Then, the limit system of model (5) is as follows:

From Lemma 1, we can conclude that

Therefore, population is persistent and and u are extinct.
Finally, we prove (iii): by Lemma 1, when , And, by (54),

Computing (58) × a₂₁ + (59) × a₁₁,

Therefore,

Let and the condition is satisfied, so

Considering the third equation of model (5),

hence,

Substitute (71) into equation (67), and by the condition , we have

It is concluded that predator is persistent.
When and substituting (72) into equation (70), one can see that

So, is persistent.
The condition means that

Substituting (72) into equation (58) and by the condition , we have

So, is extinction. The proof of Theorem 4 is complete.

6. Conclusions and Numerical Simulations

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 of random disturbance and the coefficient c of feedback control variable satisfy the condition . Theorem 4 indicates that

(1)If the coefficient c of the feedback control remains unchanged and the intensity of random disturbance increases, the population cannot resist the disturbance of the external environment and extinct(2)If the intensity of random disturbance remains unchanged and the coefficient c of feedback control is small, and are satisfied, which will cause the continuous increase of predator number for a period of time, thus leading to the extinction of the prey population.

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 (5) as follows:

Let and the initial value

(i)Set , and the positive equilibrium point of the corresponding deterministic model is . It is proved that conditions are satisfied and Theorem 2 and Theorem 3 are valid. System (5) oscillates slightly near the point and persistently (see Figure 1).(ii)Set , one can get . According to Theorem 4 (i), system (5) is extinct (see Figure 2).(iii)Set , we have and . According to Theorem 4 (ii), population is persistent and and u are extinct (see Figure 3).(iv)Set , we obtain , , , and . According to Theorem 4 (iii), population is extinct and and u are persistent (see Figure 4).

(a)

(b)

Figure 1: (a) Time series diagram of and (b) its phase diagram.

Figure 2: Time series diagram of .

Figure 3: Time series diagram of .

(a)

(b)

Figure 4: (a) Time series diagram of . (b) Phase diagram of .

Data Availability

All data sets used in this study are hypothetical.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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

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