Dynamics of a stochastic single species model with mean-reverting Ornstein-Uhlenbeck and Allee effect

Jin Kang; Rong Liu; Xuhui Shen; Jin Kang; Rong Liu; Xuhui Shen

doi:10.3934/math.2026668

AIMS Mathematics

2026, Volume 11, Issue 6: 16264-16287. doi: 10.3934/math.2026668

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Dynamics of a stochastic single species model with mean-reverting Ornstein-Uhlenbeck and Allee effect

1.
Department of General Education Courses, Shanxi Open University, Taiyuan 030027, China
2.
School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China

Received: 01 April 2026 Revised: 18 May 2026 Accepted: 02 June 2026 Published: 08 June 2026
MSC : 60H10, 92D25, 93E15

This paper studies the dynamics of a stochastic single species model with Ornstein-Uhlenbeck process and Allee effect. First, we establish the existence of a unique global solution and obtain moment estimates. Then, sufficient conditions for the existence of a unique stationary distribution of the model and for the extinction of species are derived. The results show that a weaker attack rate guarantees a unique stationary distribution, whereas a stronger attack rate leads to species extinction. Moreover, we obtain an approximate expression for the stationary density around the stochastic quasi-positive equilibrium by solving the associated Fokker-Planck equation. Finally, some numerical simulations are presented.
- Ornstein-Uhlenbeck process,
- Allee effect,
- density function,
- stationary distribution,
- extinction
Citation: Jin Kang, Rong Liu, Xuhui Shen. Dynamics of a stochastic single species model with mean-reverting Ornstein-Uhlenbeck and Allee effect[J]. AIMS Mathematics, 2026, 11(6): 16264-16287. doi: 10.3934/math.2026668

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Abstract

This paper studies the dynamics of a stochastic single species model with Ornstein-Uhlenbeck process and Allee effect. First, we establish the existence of a unique global solution and obtain moment estimates. Then, sufficient conditions for the existence of a unique stationary distribution of the model and for the extinction of species are derived. The results show that a weaker attack rate guarantees a unique stationary distribution, whereas a stronger attack rate leads to species extinction. Moreover, we obtain an approximate expression for the stationary density around the stochastic quasi-positive equilibrium by solving the associated Fokker-Planck equation. Finally, some numerical simulations are presented.

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