The survival analysis of a stochastic Lotka-Volterra competition model with a coexistence equilibrium

Junjing Xiong; Xiong Li; Hao Wang; Junjing Xiong; Xiong Li; Hao Wang

doi:10.3934/mbe.2019135

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 2717-2737. doi: 10.3934/mbe.2019135

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The survival analysis of a stochastic Lotka-Volterra competition model with a coexistence equilibrium

1.
School of Mathematics Sciences, Beijing Normal University, Beijing 100875, P.R. China
2.
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

Received: 01 January 2019 Accepted: 17 March 2019 Published: 27 March 2019

In this paper, we propose and analyze a two-species Lotka-Volterra competition model with random perturbations that relate to the inter-specific competition rates and the coexistence equilibrium of the corresponding deterministic system. The stochasticity in inter-specific competition (between species) is more important than that in intra-specific competition (within species). We pose two assumptions and then obtain su cient conditions for coexistence and for competitive exclusion respectively, and find that small random perturbations will not destroy the dynamic behaviors of the corresponding deterministic system. Moreover, if one species goes extinct, the convergence rate to zero is obtained by investigating the Lyapunov exponent. Finally, we provide several numerical examples to illustrate our mathematical results.
- A Lotka-Volterra competition model,
- coexistence equilibrium,
- stochastic coexistence,
- competitive exclusion,
- Lyapunov exponent
Citation: Junjing Xiong, Xiong Li, Hao Wang. The survival analysis of a stochastic Lotka-Volterra competition model with a coexistence equilibrium[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2717-2737. doi: 10.3934/mbe.2019135

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Abstract

In this paper, we propose and analyze a two-species Lotka-Volterra competition model with random perturbations that relate to the inter-specific competition rates and the coexistence equilibrium of the corresponding deterministic system. The stochasticity in inter-specific competition (between species) is more important than that in intra-specific competition (within species). We pose two assumptions and then obtain su cient conditions for coexistence and for competitive exclusion respectively, and find that small random perturbations will not destroy the dynamic behaviors of the corresponding deterministic system. Moreover, if one species goes extinct, the convergence rate to zero is obtained by investigating the Lyapunov exponent. Finally, we provide several numerical examples to illustrate our mathematical results.

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