Dynamical behaviors of a Lotka-Volterra competition system with the Ornstein-Uhlenbeck process

Huili Wei; Wenhe Li; Huili Wei; Wenhe Li

doi:10.3934/mbe.2023341

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 5: 7882-7904. doi: 10.3934/mbe.2023341

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Research article

Dynamical behaviors of a Lotka-Volterra competition system with the Ornstein-Uhlenbeck process

Huili Wei ,
Wenhe Li ^,

School of Mathematics and Statistics, Northeast Petroleum University, Daqing 163318, China

Academic Editor: Shengqiang Liu

Received: 05 December 2022 Revised: 29 January 2023 Accepted: 16 February 2023 Published: 22 February 2023

The competitive relationship is one of the important studies in population ecology. In this paper, we investigate the dynamical behaviors of a two-species Lotka-Volterra competition system in which intrinsic rates of increase are governed by the Ornstein-Uhlenbeck process. First, we prove the existence and uniqueness of the global solution of the model. Second, the extinction of populations is discussed. Moreover, a sufficient condition for the existence of the stationary distribution in the system is obtained, and, further, the formulas for the mean and the covariance of the probability density function of the corresponding linearized system near the equilibrium point are obtained. Finally, numerical simulations are applied to verify the theoretical results.
- Lotka-Volterra competition system,
- Ornstein-Uhlenbeck process,
- extinction,
- stationary distribution,
- probability density function
Citation: Huili Wei, Wenhe Li. Dynamical behaviors of a Lotka-Volterra competition system with the Ornstein-Uhlenbeck process[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 7882-7904. doi: 10.3934/mbe.2023341

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Abstract

The competitive relationship is one of the important studies in population ecology. In this paper, we investigate the dynamical behaviors of a two-species Lotka-Volterra competition system in which intrinsic rates of increase are governed by the Ornstein-Uhlenbeck process. First, we prove the existence and uniqueness of the global solution of the model. Second, the extinction of populations is discussed. Moreover, a sufficient condition for the existence of the stationary distribution in the system is obtained, and, further, the formulas for the mean and the covariance of the probability density function of the corresponding linearized system near the equilibrium point are obtained. Finally, numerical simulations are applied to verify the theoretical results.

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