A stochastic Gilpin-Ayala mutualism model driven by mean-reverting OU process with Lévy jumps

Meng Gao; Xiaohui Ai; Meng Gao; Xiaohui Ai

doi:10.3934/mbe.2024182

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 3: 4117-4141. doi: 10.3934/mbe.2024182

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Theory article

A stochastic Gilpin-Ayala mutualism model driven by mean-reverting OU process with Lévy jumps

Meng Gao ,
Xiaohui Ai ^,

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

Academic Editor: Hermann J. Eberl

Received: 28 November 2023 Revised: 26 January 2024 Accepted: 08 February 2024 Published: 23 February 2024

By using the Ornstein-Uhlenbeck (OU) process to simulate random disturbances in the environment, and considering the influence of jump noise, a stochastic Gilpin-Ayala mutualism model driven by mean-reverting OU process with Lévy jumps was established, and the asymptotic behaviors of the stochastic Gilpin-Ayala mutualism model were studied. First, the existence of the global solution of the stochastic Gilpin-Ayala mutualism model is proved by the appropriate Lyapunov function. Second, the moment boundedness of the solution of the stochastic Gilpin-Ayala mutualism model is discussed. Third, the existence of the stationary distribution of the solution of the stochastic Gilpin-Ayala mutualism model is obtained. Finally, the extinction of the stochastic Gilpin-Ayala mutualism model is proved. The theoretical results were verified by numerical simulations.
- stochastic Gilpin-Ayala mutualism model,
- moment boundedness of solution,
- extinction,
- Ornstein-Uhlenbeck process,
- the existence of stationary distribution
Citation: Meng Gao, Xiaohui Ai. A stochastic Gilpin-Ayala mutualism model driven by mean-reverting OU process with Lévy jumps[J]. Mathematical Biosciences and Engineering, 2024, 21(3): 4117-4141. doi: 10.3934/mbe.2024182

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Abstract

By using the Ornstein-Uhlenbeck (OU) process to simulate random disturbances in the environment, and considering the influence of jump noise, a stochastic Gilpin-Ayala mutualism model driven by mean-reverting OU process with Lévy jumps was established, and the asymptotic behaviors of the stochastic Gilpin-Ayala mutualism model were studied. First, the existence of the global solution of the stochastic Gilpin-Ayala mutualism model is proved by the appropriate Lyapunov function. Second, the moment boundedness of the solution of the stochastic Gilpin-Ayala mutualism model is discussed. Third, the existence of the stationary distribution of the solution of the stochastic Gilpin-Ayala mutualism model is obtained. Finally, the extinction of the stochastic Gilpin-Ayala mutualism model is proved. The theoretical results were verified by numerical simulations.

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