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

Liu, Jinlei; Zhao, Wencai

doi:https://doi.org/10.1155/2019/4873290

Complexity

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 4873290 | https://doi.org/10.1155/2019/4873290

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

Jinlei Liu¹and Wencai Zhao^1,2

Academic Editor: Mahdi Jalili

Received02 Jun 2019

Revised26 Sept 2019

Accepted24 Oct 2019

Published16 Nov 2019

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 intoand satisfies the initial conditionswhere 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 timeObviously, 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 thatDefine a : bywhereThe nonnegativity of this function can be obtained fromApplying Itô’s formula yieldswhereTherefore, where K is a positive constant. So, we getIntegrating (17) from 0 to and taking expectation on both sides, we haveSet , and from inequality (10), we have . Note that, for every , there is at least one of , or equaling either k or , and then, we haveIt can be obtained by (18)where is the indicator function of , and letting yieldsThis 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 thatwherewhere is the positive equilibrium point of the corresponding deterministic model (3).

Proof. Define the functionwhere and are positive constants. DefineBy Itô’s formula, we obtainwhereSimilarly,whereIn the same way,Therefore, we haveLet , we obtainTherefore,Integrate both sides of (18) from 0 to t and take the expectation, and then we getDivide both sides by t and take the limit superior, and then we haveObviously,whereWe 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 haveAs we know, and , from , one can getBy the condition we haveSimilarly, 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 obtainDividing both sides of (48) and (49) by t, we haveBy using Lemma 2 of [37], from equation (48), it follows thatand therefore,Substitute (53) into (50), and from , we getOn the other hand, computing (50) × a₂₁ + (51) × a₁₁, we haveBy 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 haveWe first prove (i): from equation (58), we can getBy the condition and Lemma 2 in literature [37],From (57), (61), and the condition , we haveFurther, from the third equation of model (5), we can getSecondly, 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 thatTherefore, 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, soConsidering the third equation of model (5),hence,Substitute (71) into equation (67), and by the condition , we haveIt is concluded that predator is persistent.
When and substituting (72) into equation (70), one can see thatSo, is persistent.
The condition means thatSubstituting (72) into equation (58) and by the condition , we haveSo, 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)

(a)

(b)

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).

References

Y. Muroya, “Permanence and global stability in a lotka-volterra predator-prey system with delays,” Applied Mathematics Letters, vol. 16, no. 8, pp. 1245–1250, 2003.
View at: Publisher Site | Google Scholar
X. Meng, F. Li, and S. Gao, “Global analysis and numerical simulations of a novel stochastic eco-epidemiological model with time delay,” Applied Mathematics and Computation, vol. 339, pp. 701–726, 2018.
View at: Publisher Site | Google Scholar
T. Ma, X. Meng, and Z. Chang, “Dynamics and optimal harvesting control for a stochastic one-predator-two-prey time delay system with jumps,” Complexity, vol. 2019, Article ID 5342031, p. 19, 2019.
View at: Google Scholar
Z. Jiang, X. Bi, T. Zhang, and B. S. A. Pradeep, “Global hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient,” Mathematical Biosciences and Engineering, vol. 16, no. 5, pp. 3807–3829, 2019.
View at: Google Scholar
F. Li, S. Zhang, and X. Meng, “Dynamics analysis and numerical simulations of a delayed stochastic epidemic model subject to a general response function,” Computational and Applied Mathematics, vol. 38, no. 2, p. 95, 2019.
View at: Google Scholar
Y. Saito, T. Hara, and W. Ma, “Necessary and sufficient conditions for permanence and global stability of a lotka-volterra system with two delays,” Journal of Mathematical Analysis and Applications, vol. 236, no. 2, pp. 534–556, 1999.
View at: Google Scholar
H. Hu, Z. Teng, and S. Gao, “Extinction in nonautonomous lotka-volterra competitive system with pure-delays and feedback controls,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2508–2520, 2009.
View at: Google Scholar
H. Huo and W. Li, “Positive periodic solutions of a class of delay differential system with feedback control,” Applied Mathematics and Computation, vol. 148, no. 1, pp. 35–46, 2004.
View at: Google Scholar
W. Wang, W. Ma, and Z. Feng, “Dynamics of reaction–diffusion equations for modeling CD T cells decline with general infection mechanism and distinct dispersal rates,” Nonlinear Analysis: Real World Applications, vol. 51, Article ID 102976, 2020.
View at: Google Scholar
W. Lv, F. Wang, and Y. Li, “Adaptive finite-time tracking control for nonlinear systems with unmodeled dynamics using neural networks,” Advances in Difference Equations, vol. 2018, Article ID 159, 2018.
View at: Publisher Site | Google Scholar
J. Wang, H. Cheng, H. Liu, and Y. Wang, “Periodic solution and control optimization of a prey-predator model with two types of harvesting,” Advances in Difference Equations, vol. 2018, Article ID 41, 2018.
View at: Publisher Site | Google Scholar
W. Lv and F. Wang, “Adaptive tracking control for a class of uncertain nonlinear systems with infinite number of actuator failures using neural networks,” Advances in Difference Equations, vol. 2017, no. 1, Article ID 374, 2017.
View at: Google Scholar
T. Zhang, N. Gao, T. Wang, H. Liu, and Z. Jiang, “Global dynamics of a model for treating microorganisms in sewage by periodically adding microbial flocculants,” Mathematical Biosciences and Engineering, vol. 17, no. 1, pp. 179–201, 2020.
View at: Publisher Site | Google Scholar
T. Zhang, T. Xu, J. Wang, Y. Song, and Z. Jiang, “Geometrical analysis of a pest management model in food-limited environments with nonlinear impulsive state feedback control,” Journal of Applied Analysis and Computation, vol. 9, no. 6, pp. 1–17, 2019.
View at: Publisher Site | Google Scholar
K. Gopalsamy and P. Weng, “Global attractivity in a competition system with feedback controls,” Computers and Mathematics with Applications, vol. 45, no. 4-5, pp. 665–676, 2003.
View at: Publisher Site | Google Scholar
Z. Li, M. Han, and F. Chen, “Influence of on an autonomous Lotka–Volterra competitive system with infinite delays feedback controls,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 402–413, 2013.
View at: Publisher Site | Google Scholar
C. Shi, X. Chen, and Y. Wang, “Feedback control effect on the lotka-volterra prey-predator system with discrete delays,” Advances in Difference Equations, vol. 2017, no. 1, p. 373, 2017.
View at: Publisher Site | Google Scholar
F. Bian, W. Zhao, Y. Song, and R. Yue, “Dynamical analysis of a class of prey-predator model with Beddington-DeAngelis functional response, stochastic perturbation, and impulsive toxicant input,” Complexity, vol. 2017, Article ID 3742197, p. 18, 2017.
View at: Publisher Site | Google Scholar
M. Chi and W. Zhao, “Dynamical analysis of multi-nutrient and single microorganism chemostat model in a polluted environment,” Advances in Difference Equations, vol. 2018, no. 1, Article ID 120, 2018.
View at: Publisher Site | Google Scholar
A. Miao, T. Zhang, J. Zhang, and C. Wang, “Dynamics of a stochastic sir model with both horizontal and vertical transmission,” Journal of Applied Analysis and Computation, vol. 8, no. 4, pp. 1108–1121, 2018.
View at: Google Scholar
T. Feng, Z. Qiu, X. Meng, and L. Rong, “Analysis of a stochastic hiv-1 infection model with degenerate diffusion,” Applied Mathematics and Computation, vol. 348, pp. 437–455, 2018.
View at: Google Scholar
X. Li and X. Mao, “Population dynamical behavior of non-autonomous lotka-volterra competitive system with random perturbation,” Discrete and Continuous Dynamical Systems-Series A, vol. 24, no. 2, pp. 523–593, 2009.
View at: Google Scholar
Y. Li and X. Meng, “Dynamics of an impulsive stochastic nonautonomous chemostat model with two different growth rates in a polluted environment,” Discrete Dynamics in Nature and Society, vol. 2019, Article ID 5498569, p. 15, 2019.
View at: Google Scholar
W. Zhang, X. Meng, and Y. Dong, “Periodic solution and ergodic stationary distribution of stochastic siri epidemic systems with nonlinear perturbations,” Journal of Systems Science and Complexity, vol. 32, no. 4, pp. 1104–1124, 2019.
View at: Publisher Site | Google Scholar
G. Liu, Z. Chang, and X. Meng, “Asymptotic analysis of impulsive dispersal predator-prey systems with markov switching on finite–state space,” Journal of Function Spaces, vol. 2019, Article ID 8057153, p. 18, 2019.
View at: Publisher Site | Google Scholar
H. Qi, X. Meng, and T. Feng, “Dynamics analysis of a stochastic non–autonomous one–predator–two–prey system with Beddington–DeAngelis functional response and impulsive perturbations,” Advances in Difference Equations, vol. 2019, no. 1, Article ID 235, 2019.
View at: Publisher Site | Google Scholar
M. Chi and W. Zhao, “Dynamical analysis of two-microorganism and single nutrient stochastic chemostat model with monod-haldane response function,” Complexity, vol. 2019, Article ID 8719067, p. 13, 2019.
View at: Publisher Site | Google Scholar
F. F. Zhu, X. Z. Meng, and T. H. Zhang, “Optimal harvesting of a competitive n-species stochastic model with delayed diffusions,” Mathematical Biosciences and Engineering, vol. 16, no. 3, pp. 1554–1574, 2019.
View at: Publisher Site | Google Scholar
X. Yu, S. Yuan, and T. Zhang, “Persistence and ergodicity of a stochastic single species model with allee effect under regime switching,” Communications in Nonlinear Science and Numerical Simulation, vol. 59, pp. 359–374, 2018.
View at: Publisher Site | Google Scholar
W. Zhao, J. Li, T. Zhang, X. Meng, and T. Zhang, “Persistence and ergodicity of plant disease model with markov conversion and impulsive toxicant input,” Communications in Nonlinear Science and Numerical Simulation, vol. 48, pp. 70–84, 2017.
View at: Publisher Site | Google Scholar
W. Zhao, J. Liu, M. Chi, and F. Bian, “Dynamics analysis of stochastic epidemic models with standard incidence,” Advances in Difference Equations, vol. 2019, no. 1, Article ID 22, 2019.
View at: Publisher Site | Google Scholar
T. Feng, Z. Qiu, and X. Meng, “Dynamics of a stochastic hepatitis c virus system with host immunity,” Discrete & Continuous Dynamical Systems–B, vol. 24, no. 12, pp. 6367–6385, 2019.
View at: Publisher Site | Google Scholar
T. Feng, Z. Qiu, and X. Meng, “Analysis of a stochastic recovery-relapse epidemic model with periodic parameters and media coverage,” Journal of Applied Analysis and Computation, vol. 9, no. 3, pp. 1007–1021, 2019.
View at: Google Scholar
N. Gao, Y. Song, X. Wang, and J. Liu, “Dynamics of a stochastic sis epidemic model with nonlinear incidence rates,” Advances in Difference Equations, vol. 2019, no. 1, Article ID 41, 2019.
View at: Publisher Site | Google Scholar
Y. Song, A. Miao, T. Zhang, X. Wang, and J. Liu, “Extinction and persistence of a stochastic sirs epidemic model with saturated incidence rate and transfer from infectious to susceptible,” Advances in Difference Equations, vol. 2018, no. 1, Article ID 293, 2018.
View at: Publisher Site | Google Scholar
X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, UK, 2007.
M. Liu, H. Qiu, and K. Wang, “A remark on a stochastic predator-prey system with time delays,” Applied Mathematics Letters, vol. 26, no. 3, pp. 318–323, 2013.
View at: Publisher Site | Google Scholar

Copyright

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.

PDF Download Citation

Download other formats

Order printed copies

Views

3301

Downloads

1432

Citations