Approximation of invariant measure for a stochastic population model with Markov chain and diffusion in a polluted environment

Ting Kang; Yanyan Du; Ming Ye; Qimin Zhang; Ting Kang; Yanyan Du; Ming Ye; Qimin Zhang

doi:10.3934/mbe.2020349

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 6702-6719. doi: 10.3934/mbe.2020349

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Research article

Approximation of invariant measure for a stochastic population model with Markov chain and diffusion in a polluted environment

1.
State Key Laboratory of High-efficiency Utilization of Coal and Green Chemical Engineering, Ningxia University, Yinchuan, 750021, China
2.
Xinhua College, Ningxia University, Yinchuan 750021, China
3.
Department of Earth, Ocean, and Atmospheric Science and Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, United States
4.
School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China

Received: 10 June 2020 Accepted: 03 September 2020 Published: 29 September 2020

In the paper, we propose a novel stochastic population model with Markov chain and diffusion in a polluted environment. Under the condition that the diffusion coefficient satisfies the local Lipschitz condition, we prove the existence and uniqueness of invariant measure for the model. Moreover, we also discuss the existence and uniqueness of numerical invariance measure for stochastic population model under the discrete-time Euler-Maruyama scheme, and prove that numerical invariance measure converges to the invariance measure of the corresponding exact solution in the Wasserstein distance sense. Finally, we give the numerical simulation to show the correctness of the theoretical results.
- stochastic population model,
- invariant measure,
- Markov chain,
- spatial diffusion,
- environmental pollution
Citation: Ting Kang, Yanyan Du, Ming Ye, Qimin Zhang. Approximation of invariant measure for a stochastic population model with Markov chain and diffusion in a polluted environment[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6702-6719. doi: 10.3934/mbe.2020349

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Abstract

In the paper, we propose a novel stochastic population model with Markov chain and diffusion in a polluted environment. Under the condition that the diffusion coefficient satisfies the local Lipschitz condition, we prove the existence and uniqueness of invariant measure for the model. Moreover, we also discuss the existence and uniqueness of numerical invariance measure for stochastic population model under the discrete-time Euler-Maruyama scheme, and prove that numerical invariance measure converges to the invariance measure of the corresponding exact solution in the Wasserstein distance sense. Finally, we give the numerical simulation to show the correctness of the theoretical results.

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