Stationary distribution and probability density function analysis of a stochastic Microcystins degradation model with distributed delay

Ying He; Yuting Wei; Junlong Tao; Bo Bi; Ying He; Yuting Wei; Junlong Tao; Bo Bi

doi:10.3934/mbe.2024026

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 1: 602-626. doi: 10.3934/mbe.2024026

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Research article

Stationary distribution and probability density function analysis of a stochastic Microcystins degradation model with distributed delay

1.
School of Mathematics and Statistics, Northeast Petroleum University, Daqing 163318, China
2.
International School of Public Health and One Health, Hainan Medical University, Haikou 571199, China

Received: 08 September 2023 Revised: 09 November 2023 Accepted: 10 December 2023 Published: 18 December 2023

A stochastic Microcystins degradation model with distributed delay is studied in this paper. We first demonstrate the existence and uniqueness of a global positive solution to the stochastic system. Second, we derive a stochastic critical value $ R_0^s $ related to the basic reproduction number $ R_0 $. By constructing suitable Lyapunov function types, we obtain the existence of an ergodic stationary distribution of the stochastic system if $ R_0^s > 1. $ Next, by means of the method developed to solve the general four-dimensional Fokker-Planck equation, the exact expression of the probability density function of the stochastic model around the quasi-endemic equilibrium is derived, which is the key aim of the present paper. In the analysis of statistical significance, the explicit density function can reflect all dynamical properties of a chemostat model. To validate our theoretical conclusions, we present examples and numerical simulations.
- Microcystins degradation model,
- distributed delay,
- stationary distribution,
- probability density function,
- extinction
Citation: Ying He, Yuting Wei, Junlong Tao, Bo Bi. Stationary distribution and probability density function analysis of a stochastic Microcystins degradation model with distributed delay[J]. Mathematical Biosciences and Engineering, 2024, 21(1): 602-626. doi: 10.3934/mbe.2024026

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Abstract

A stochastic Microcystins degradation model with distributed delay is studied in this paper. We first demonstrate the existence and uniqueness of a global positive solution to the stochastic system. Second, we derive a stochastic critical value $ R_0^s $ related to the basic reproduction number $ R_0 $. By constructing suitable Lyapunov function types, we obtain the existence of an ergodic stationary distribution of the stochastic system if $ R_0^s > 1. $ Next, by means of the method developed to solve the general four-dimensional Fokker-Planck equation, the exact expression of the probability density function of the stochastic model around the quasi-endemic equilibrium is derived, which is the key aim of the present paper. In the analysis of statistical significance, the explicit density function can reflect all dynamical properties of a chemostat model. To validate our theoretical conclusions, we present examples and numerical simulations.

References

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