Stationary distribution and optimal control of a stochastic population model in a polluted environment

An Ma; Shuting Lyu; Qimin Zhang; An Ma; Shuting Lyu; Qimin Zhang

doi:10.3934/mbe.2022525

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 11: 11260-11280. doi: 10.3934/mbe.2022525

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Research article

Stationary distribution and optimal control of a stochastic population model in a polluted environment

An Ma ¹,
Shuting Lyu ^{1
,
,},
Qimin Zhang ^{1,2
,
,}

1.
School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China
2.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China

Academic Editor: Frithjof Lutscher

Received: 15 June 2022 Revised: 16 July 2022 Accepted: 24 July 2022 Published: 05 August 2022

This paper is concerned with a stochastic population model in a polluted environment. First, within the framework of Lyapunov method, the existence and uniqueness of a global positive solution of the model are proposed, and the sufficient conditions are established for existence of an ergodic stationary distribution of the positive solution. Second, the control strategy is introduced into the stochastic population model in a polluted environment. By using Pontryagin's maximum principle, the first-order necessary conditions are derived for the existence of optimal control. Finally, some numerical simulations are presented to illustrate the analytical results.
- stationary distribution,
- optimal control,
- Lyapunov method,
- stochastic population model,
- polluted environment
Citation: An Ma, Shuting Lyu, Qimin Zhang. Stationary distribution and optimal control of a stochastic population model in a polluted environment[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11260-11280. doi: 10.3934/mbe.2022525

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Abstract

This paper is concerned with a stochastic population model in a polluted environment. First, within the framework of Lyapunov method, the existence and uniqueness of a global positive solution of the model are proposed, and the sufficient conditions are established for existence of an ergodic stationary distribution of the positive solution. Second, the control strategy is introduced into the stochastic population model in a polluted environment. By using Pontryagin's maximum principle, the first-order necessary conditions are derived for the existence of optimal control. Finally, some numerical simulations are presented to illustrate the analytical results.

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