# American Institute of Mathematical Sciences

May  2020, 25(5): 1699-1714. doi: 10.3934/dcdsb.2019247

## Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays

 School of Mathematics, Renmin University of China, Beijing 100872, China

* Corresponding author: Kexin Wang

Received  April 2019 Revised  July 2019 Published  November 2019

Fund Project: The first author is supported by NSFC grant No. 71531012.

In this work a stochastic Holling-Ⅱ type predator-prey model with infinite delays and feedback controls is investigated. By constructing a Lyapunov function, together with stochastic analysis approach, we obtain that the stochastic controlled predator-prey model admits a unique global positive solution. We then utilize graphical method and stability theorem of stochastic differential equations to investigate the globally asymptotical stability of a unique positive equilibrium for the stochastic controlled predator-prey system. If the stochastic predator-prey system is globally stable, then we show that using suitable feedback controls can alter the position of the unique positive equilibrium and retain the stable property. If the predator-prey system is destabilized by large intensities of white noises, then by choosing the appropriate values of feedback control variables, we can make the system reach a new stable state. Some examples are presented to verify our main results.

Citation: Kexin Wang. Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1699-1714. doi: 10.3934/dcdsb.2019247
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##### References:
The parabola $l_1$ (green line) and the hyperbola $l_2$(blue line) of 9 provided that the condition 7 holds
The region $R_0$ where positive equilibria $(x^{*}, y^{*})$ will occur under feedback controls
Dynamic behavior of the solution $(x(t), y(t))^{T}$ of system 20 with the initial condition $(\varphi_1(\theta), \varphi_2(\theta)) = (1.2 e^{\theta}, 0.8 e^{\theta}), \theta \in (-\infty, 0]$
The region $R_0$ where positive equilibria of system 20 will occur under feedback controls
Dynamic behavior of the solution $(x(t), y(t))^{T}$ of system 20 with small perturbations $\sigma_1 = \sigma_2 = 0.1$ and the initial condition $(\varphi_1(\theta), \varphi_2(\theta)) = (1.2 e^{\theta}, 0.8 e^{\theta}), \theta \in (-\infty, 0]$
Dynamic behavior of the solution $(x(t), y(t), u_1(t), u_2(t))^{T}$ of system 21 with the initial condition $(\varphi_1(\theta), \varphi_2(\theta), u_1(0), u_2(0)) = (1.2 e^{\theta}, 0.8 e^{\theta}, 1,1), \theta \in (-\infty, 0]$
Dynamic behavior of the solution $(x(t), y(t))^{T}$ of system 20 with big perturbations $\sigma_1 = \sigma_2 = 1$ and the initial condition $(\varphi_1(\theta), \varphi_2(\theta)) = (1.2 e^{\theta}, 0.8 e^{\theta}), \theta \in (-\infty, 0]$
Dynamic behavior of the solution $(x(t), y(t), u_1(t), u_2(t))^{T}$ of system 22 with the initial condition $(\varphi_1(\theta), \varphi_2(\theta), u_1(0), u_2(0)) = (1.2 e^{\theta}, 0.8 e^{\theta}, 1,1), \theta \in (-\infty, 0]$
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