Analysis of a stochastic two-species Schoener's competitive model with Lévy jumps and Ornstein–Uhlenbeck process

Yajun Song; Ruyue Hu; Yifan Wu; Xiaohui Ai; Yajun Song; Ruyue Hu; Yifan Wu; Xiaohui Ai

doi:10.3934/math.2024598

AIMS Mathematics

2024, Volume 9, Issue 5: 12239-12258. doi: 10.3934/math.2024598

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Research article

Analysis of a stochastic two-species Schoener's competitive model with Lévy jumps and Ornstein–Uhlenbeck process

School of Science, Northeast Forestry University, China

Received: 27 December 2023 Revised: 13 March 2024 Accepted: 21 March 2024 Published: 28 March 2024
MSC : 60H10, 60H30, 92D25

This paper studies a stochastic two-species Schoener's competitive model with Lévy jumps by the mean-reverting Ornstein–Uhlenbeck process. First, the biological implication of introducing the Ornstein–Uhlenbeck process is illustrated. After that, we show the existence and uniqueness of the global solution. Moment estimates for the global solution of the stochastic model are then given. Moreover, by constructing the Lyapunov function and applying Itô's formula and Chebyshev's inequality, it is found that the model is stochastic and ultimately bounded. In addition, we give sufficient conditions for the extinction of species. Finally, numerical simulations are employed to demonstrate the analytical results.
- Schoener's competitive model,
- Ornstein–Uhlenbeck process,
- extinction
Citation: Yajun Song, Ruyue Hu, Yifan Wu, Xiaohui Ai. Analysis of a stochastic two-species Schoener's competitive model with Lévy jumps and Ornstein–Uhlenbeck process[J]. AIMS Mathematics, 2024, 9(5): 12239-12258. doi: 10.3934/math.2024598

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Abstract

This paper studies a stochastic two-species Schoener's competitive model with Lévy jumps by the mean-reverting Ornstein–Uhlenbeck process. First, the biological implication of introducing the Ornstein–Uhlenbeck process is illustrated. After that, we show the existence and uniqueness of the global solution. Moment estimates for the global solution of the stochastic model are then given. Moreover, by constructing the Lyapunov function and applying Itô's formula and Chebyshev's inequality, it is found that the model is stochastic and ultimately bounded. In addition, we give sufficient conditions for the extinction of species. Finally, numerical simulations are employed to demonstrate the analytical results.

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