Analysis of a stochastic predator-prey model based on the Ornstein-Uhlenbeck process with Holling-Ⅱ and Beddington–DeAngelis functional responses

Wenyu Zhang; Xiaohui Ai; Wenyu Zhang; Xiaohui Ai

doi:10.3934/era.2026198

Electronic Research Archive

2026, Volume 34, Issue 7: 4486-4511. doi: 10.3934/era.2026198

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Research article

Analysis of a stochastic predator-prey model based on the Ornstein-Uhlenbeck process with Holling-Ⅱ and Beddington–DeAngelis functional responses

Wenyu Zhang ,
Xiaohui Ai ^,

School of Science, Northeast Forestry University, Harbin 150040, China

Received: 14 January 2026 Revised: 14 May 2026 Accepted: 18 May 2026 Published: 26 May 2026

In this paper, we investigated a predator–prey model driven by an Ornstein–Uhlenbeck process and featuring Holling-Ⅱ and Beddington–DeAngelis functional responses. To begin, the biological significance of the Ornstein-Uhlenbeck process in ecological modeling was illustrated, along with its validity in characterizing random environmental fluctuations. Subsequently, the existence and uniqueness of the global solution for this model were strictly proven and its ultimate boundedness was analyzed. By constructing Lyapunov functions and applying Itô's formula, the existence of the stationary distribution of the system was demonstrated, while sufficient conditions for population extinction were provided. Finally, numerical simulations were conducted to confirm the validity of the conclusions. This work provides a theoretical framework for understanding complex mutualistic-predatory communities under persistent environmental fluctuations.
- Ornstein-Uhlenbeck process,
- Holling-Ⅱ functional response,
- Beddington–DeAngelis functional response,
- stationary distribution,
- extinction
Citation: Wenyu Zhang, Xiaohui Ai. Analysis of a stochastic predator-prey model based on the Ornstein-Uhlenbeck process with Holling-Ⅱ and Beddington–DeAngelis functional responses[J]. Electronic Research Archive, 2026, 34(7): 4486-4511. doi: 10.3934/era.2026198

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Abstract

In this paper, we investigated a predator–prey model driven by an Ornstein–Uhlenbeck process and featuring Holling-Ⅱ and Beddington–DeAngelis functional responses. To begin, the biological significance of the Ornstein-Uhlenbeck process in ecological modeling was illustrated, along with its validity in characterizing random environmental fluctuations. Subsequently, the existence and uniqueness of the global solution for this model were strictly proven and its ultimate boundedness was analyzed. By constructing Lyapunov functions and applying Itô's formula, the existence of the stationary distribution of the system was demonstrated, while sufficient conditions for population extinction were provided. Finally, numerical simulations were conducted to confirm the validity of the conclusions. This work provides a theoretical framework for understanding complex mutualistic-predatory communities under persistent environmental fluctuations.

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