Bifurcation and optimal harvesting analysis of a discrete-time predator–prey model with fear and prey refuge effects

Jie Liu; Qinglong Wang; Xuyang Cao; Ting Yu; Jie Liu; Qinglong Wang; Xuyang Cao; Ting Yu

doi:10.3934/math.20241281

AIMS Mathematics

2024, Volume 9, Issue 10: 26283-26306. doi: 10.3934/math.20241281

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Research article

Bifurcation and optimal harvesting analysis of a discrete-time predator–prey model with fear and prey refuge effects

School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, China

Received: 09 July 2024 Revised: 26 August 2024 Accepted: 04 September 2024 Published: 11 September 2024
MSC : 39A28, 39A30

In this contribution, the complicated dynamical behaviors and optimal harvesting policy of a discrete-time predator–prey model with fear and refuge effects are formulated. Both the fear and prey refuge effects refer to an interaction between predator and prey. In the first place, the existence and local stability of three fixed points of proposed model are investigated by virtue of our methodology, that is, the eigenvalues of the Jacobian matrix. One step further, it is worth mentioning that the model undergoes flip bifurcation (i.e., period–doubling bifurcation) and Neimark–Sacker bifurcation at the interior fixed point by the utilization of bifurcation theory and center manifold theory. Also, optimal harvesting strategy is investigated, and the expressions of optimal harvesting efforts are determined. Two examples, in the end, are put forward to prove that they are consistent with the previous theoretical results.
- discrete predator–prey system,
- fear effect,
- prey refuge,
- flip bifurcation,
- Neimark–Sacker bifurcation,
- optimal harvesting
Citation: Jie Liu, Qinglong Wang, Xuyang Cao, Ting Yu. Bifurcation and optimal harvesting analysis of a discrete-time predator–prey model with fear and prey refuge effects[J]. AIMS Mathematics, 2024, 9(10): 26283-26306. doi: 10.3934/math.20241281

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Abstract

In this contribution, the complicated dynamical behaviors and optimal harvesting policy of a discrete-time predator–prey model with fear and refuge effects are formulated. Both the fear and prey refuge effects refer to an interaction between predator and prey. In the first place, the existence and local stability of three fixed points of proposed model are investigated by virtue of our methodology, that is, the eigenvalues of the Jacobian matrix. One step further, it is worth mentioning that the model undergoes flip bifurcation (i.e., period–doubling bifurcation) and Neimark–Sacker bifurcation at the interior fixed point by the utilization of bifurcation theory and center manifold theory. Also, optimal harvesting strategy is investigated, and the expressions of optimal harvesting efforts are determined. Two examples, in the end, are put forward to prove that they are consistent with the previous theoretical results.

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