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

Jinlei Liu,Wencai Zhao

doi:10.1155/2019/4873290

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.

Highlights

In nature, time delays exist in many ecosystems [1,2,3,4,5]
In 2013, Li et al [16] introduced feedback control variables into the twospecies competition system and discussed the extinction and global attraction of equilibrium points. ey 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
Eorem 2 shows that if the condition (A1) holds, the solution oscillates around the equilibrium point x∗, and the amplitude of oscillation is positively correlated with the intensity σ21 and σ

Summary

Introduction

Time delays exist in many ecosystems [1,2,3,4,5]. For example, maturity stage is a common phenomenon in biological population, and many diseases have a long incubation period. e mathematical model describing this phenomenon with time delay is called the delay di erential equation. Ey 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. 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 (x1(t), x2(t), u(t)) in interval t ∈ [0, τe), where τe is the explosion time. To prove that this solution is global, we only need to prove τe ∞ a.s. Let k0 > 0 be a sufficiently large constant for any initial value x1(0), x2(0), and u(0) lying within the internal [(1/k0), k0]. For any given initial condition (6), if hypothesis (A1) is established, the solution (x1(t), x2(t), u(t)) of system (5) has the property that

Asymptotic Property

Persistence

Conclusions and Numerical