A stochastic Gilpin-Ayala nonautonomous competition model driven by mean-reverting OU process with finite Markov chain and Lévy jumps

Meng Gao; Xiaohui Ai; Meng Gao; Xiaohui Ai

doi:10.3934/era.2024086

Electronic Research Archive

2024, Volume 32, Issue 3: 1873-1900. doi: 10.3934/era.2024086

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Research article

A stochastic Gilpin-Ayala nonautonomous competition model driven by mean-reverting OU process with finite Markov chain and Lévy jumps

Meng Gao ,
Xiaohui Ai ^,

School of Science, Northeast Forestry University, China

Received: 31 August 2023 Revised: 09 February 2024 Accepted: 19 February 2024 Published: 04 March 2024

The Ornstein-Uhlenbeck (OU) process was used to simulate random perturbations in the environment. Considering the influence of telegraph noise and jump noise, a stochastic Gilpin-Ayala nonautonomous competition model driven by the mean-reverting OU process with finite Markov chain and Lévy jumps was established, and the asymptotic behaviors of the stochastic Gilpin-Ayala nonautonomous competition model were studied. First, the existence of the global solution of the stochastic Gilpin-Ayala nonautonomous competition model was proven by the appropriate Lyapunov function. Second, the moment boundedness of the solution of the stochastic Gilpin-Ayala nonautonomous competition model was discussed. Third, the existence of the stationary distribution of the solution of the stochastic Gilpin-Ayala nonautonomous competition model was obtained. Finally, the extinction of the stochastic Gilpin-Ayala nonautonomous competition model was proved. The theoretical results were verified by numerical simulations.
Citation: Meng Gao, Xiaohui Ai. A stochastic Gilpin-Ayala nonautonomous competition model driven by mean-reverting OU process with finite Markov chain and Lévy jumps[J]. Electronic Research Archive, 2024, 32(3): 1873-1900. doi: 10.3934/era.2024086

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Abstract

The Ornstein-Uhlenbeck (OU) process was used to simulate random perturbations in the environment. Considering the influence of telegraph noise and jump noise, a stochastic Gilpin-Ayala nonautonomous competition model driven by the mean-reverting OU process with finite Markov chain and Lévy jumps was established, and the asymptotic behaviors of the stochastic Gilpin-Ayala nonautonomous competition model were studied. First, the existence of the global solution of the stochastic Gilpin-Ayala nonautonomous competition model was proven by the appropriate Lyapunov function. Second, the moment boundedness of the solution of the stochastic Gilpin-Ayala nonautonomous competition model was discussed. Third, the existence of the stationary distribution of the solution of the stochastic Gilpin-Ayala nonautonomous competition model was obtained. Finally, the extinction of the stochastic Gilpin-Ayala nonautonomous competition model was proved. The theoretical results were verified by numerical simulations.

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