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Stationary distribution of a stochastic three species predator–prey model with anti-predator behavior

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Abstract

A stochastic three species predator–prey model with intermediate predator and anti-predator behavior is established and studied in this paper. By constructing suitable Lyapunov functions and combining knowledge of stochastic differential equations, the behavior of global positive solutions to the model are investigated. Firstly, we prove that there is unique global positive solution to the model, and establish the stochastic boundedness of the positive solution as well. Secondly, the sufficient condition for the existence of a unique ergodic stationary distribution is provided. Thirdly, the persistence and extinction of the populations are discussed. Finally, some numerical simulations demonstrate that the results obtained in this paper are true, moreover, the influence of white noise on the populations is revealed.

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Acknowledgements

We are very grateful to the editor and reviewers for their careful reading and valuable comments, which greatly improved the expressiveness of our paper. This work was supported by the National Natural Science Foundation of China (Grant No. 12001502).

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Authors and Affiliations

  1. School of Science, China University of Geosciences (Beijing), Xueyuan Road, Beijing, 10083, China

    Ming Kang, Xiang Zhang, Fengjie Geng & Zhaohai Ma

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  1. Ming Kang
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  2. Xiang Zhang
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  3. Fengjie Geng
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  4. Zhaohai Ma
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Correspondence to Fengjie Geng.

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Kang, M., Zhang, X., Geng, F. et al. Stationary distribution of a stochastic three species predator–prey model with anti-predator behavior. J. Appl. Math. Comput. 70, 1365–1393 (2024). https://doi.org/10.1007/s12190-024-02007-5

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  • DOI: https://doi.org/10.1007/s12190-024-02007-5

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