Extinction and stationary distribution of stochastic predator-prey model with group defense behavior

Yansong Pei; Bing Liu; Haokun Qi; Yansong Pei; Bing Liu; Haokun Qi

doi:10.3934/mbe.2022610

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 12: 13062-13078. doi: 10.3934/mbe.2022610

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Research article

Extinction and stationary distribution of stochastic predator-prey model with group defense behavior

1.
Department of Mathematics, Liaoning Normal University, Dalian 116029, China
2.
College of Mathematics and Information Science, Anshan Normal University, Anshan 114007, China
3.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Academic Editor: Hermann J. Eberl

Received: 10 June 2022 Revised: 17 August 2022 Accepted: 22 August 2022 Published: 06 September 2022

Considering that many prey populations in nature have group defense behavior, and the relationship between predator and prey is usually affected by environmental noise, a stochastic predator-prey model with group defense behavior is established in this paper. Some dynamical properties of the model, including the existence and uniqueness of global positive solution, sufficient conditions for extinction and unique ergodic stationary distribution, are investigated by using qualitative theory of stochastic differential equations, Lyapunov function analysis method, Itô formula, etc. Furthermore, the effects of group defense behavior and environmental noise on population stability are also discussed. Finally, numerical simulations are carried out to illustrate that the effects of environmental noise on both populations are negative, the appropriate group defense level of prey can maintain the stability of the relationship between two populations, and the survival threshold is strongly influenced by the intrinsic growth rate of prey population and the intensity of environmental noise.
- stochastic predator-prey model,
- group defense,
- extinction,
- stationary distribution,
- Itô formula
Citation: Yansong Pei, Bing Liu, Haokun Qi. Extinction and stationary distribution of stochastic predator-prey model with group defense behavior[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 13062-13078. doi: 10.3934/mbe.2022610

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Abstract

Considering that many prey populations in nature have group defense behavior, and the relationship between predator and prey is usually affected by environmental noise, a stochastic predator-prey model with group defense behavior is established in this paper. Some dynamical properties of the model, including the existence and uniqueness of global positive solution, sufficient conditions for extinction and unique ergodic stationary distribution, are investigated by using qualitative theory of stochastic differential equations, Lyapunov function analysis method, Itô formula, etc. Furthermore, the effects of group defense behavior and environmental noise on population stability are also discussed. Finally, numerical simulations are carried out to illustrate that the effects of environmental noise on both populations are negative, the appropriate group defense level of prey can maintain the stability of the relationship between two populations, and the survival threshold is strongly influenced by the intrinsic growth rate of prey population and the intensity of environmental noise.

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