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Influence of Fear Effect on Bifurcation Dynamics of Predator-prey System in a Predator-poisoned Environment

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Abstract

In this paper, we propose a predator-prey system with fear effect and predator death due to toxins based on the phenomenon that African predators death due to eating toxin-carrying carcasses. First, the existence and stability of equilibria are proved. Second, the conditions for the occurrence of different types of bifurcations are established by varying parameters. Our results indicate that (1) when the cost of minimum fear is taken as the bifurcation parameter, it can determine the stability and direction of periodic solution of Hopf bifurcation; (2) when the predation rate and the cost of minimum fear are taken as the bifurcation parameters, the system undergoes Bogdanov-Takens (BT) bifurcation of codimension two, and the BT bifurcation will disappear when the system is not affected by toxins. At last, some numerical simulations verify the validity of theoretical results, and we obtain that toxins not only inhibit the survival of predator populations but also affect the density of prey populations. Moreover, only an appropriate cost of minimum fear maintains long-term coexistence of prey and predator populations.

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Acknowledgements

We would like to thank Professor Huaiping Zhu for their helpful comments and suggestions which really helped us to improve the manuscript. We also thank the reviewer for very helpful comments and suggestions which greatly improved the presentation of the paper. This work was supported by Shandong Provincial Natural Science Foundation of China (ZR2019MA003), the Research Fund for the Taishan Scholar Project of Shandong Province of China, the SDUST Research Fund (2014TDJH102), and the SDUST Innovation Fund for Graduate Students (YC20210215).

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Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Haokun Qi & Xinzhu Meng

  2. Nonlinear Analysis and Applied Mathematics(NAAM)-Research Group, Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Xinzhu Meng & Aatef Hobiny

  3. Department of Mathematics, Quaid-i-Azam University 45320, Isamabad, 44000, Pakistan

    Tasawar Hayat

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  1. Haokun Qi
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  2. Xinzhu Meng
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Correspondence to Xinzhu Meng.

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Qi, H., Meng, X., Hayat, T. et al. Influence of Fear Effect on Bifurcation Dynamics of Predator-prey System in a Predator-poisoned Environment. Qual. Theory Dyn. Syst. 21, 27 (2022). https://doi.org/10.1007/s12346-021-00555-w

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