Hopf bifurcation analysis of a multiple delays stage-structure predator-prey model with refuge and cooperation

San-Xing Wu; Xin-You Meng; San-Xing Wu; Xin-You Meng

doi:10.3934/era.2025045

Electronic Research Archive

2025, Volume 33, Issue 2: 995-1036. doi: 10.3934/era.2025045

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Research article

Hopf bifurcation analysis of a multiple delays stage-structure predator-prey model with refuge and cooperation

San-Xing Wu ^1,2,
Xin-You Meng ^{1
,
,}

1.
School of Science, Lanzhou University of Technology, Lanzhou 730050, Gansu, China
2.
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Gansu, China

Received: 22 November 2024 Revised: 18 January 2025 Accepted: 12 February 2025 Published: 24 February 2025

In this paper, a multiple delays stage-structure predator-prey model with refuge and cooperation is established. First, the local asymptotic stability of the trivial equilibrium and the predator extinction equilibrium are discussed by analyzing the characteristic equations of the system. Second, taking time delays as the bifurcation parameters, the existence of Hopf bifurcation at the positive equilibrium is given. Next, the direction of Hopf bifurcation and the stability of the periodic solutions are analyzed based on the center manifold theorem and normal form theory. Moreover, the optimal harvesting policy of the system is showed by using Pontryagin's maximum principle. Finally, we give the global sensitivity analysis of some parameters by calculating the partial rank correlation coefficients, and some numerical simulations are performed to verify the correctness and feasibility of the theoretical results by using the MATLAB software.
- stage-structure,
- prey refuge,
- cooperation,
- time delay,
- Hopf bifurcation
Citation: San-Xing Wu, Xin-You Meng. Hopf bifurcation analysis of a multiple delays stage-structure predator-prey model with refuge and cooperation[J]. Electronic Research Archive, 2025, 33(2): 995-1036. doi: 10.3934/era.2025045

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Abstract

In this paper, a multiple delays stage-structure predator-prey model with refuge and cooperation is established. First, the local asymptotic stability of the trivial equilibrium and the predator extinction equilibrium are discussed by analyzing the characteristic equations of the system. Second, taking time delays as the bifurcation parameters, the existence of Hopf bifurcation at the positive equilibrium is given. Next, the direction of Hopf bifurcation and the stability of the periodic solutions are analyzed based on the center manifold theorem and normal form theory. Moreover, the optimal harvesting policy of the system is showed by using Pontryagin's maximum principle. Finally, we give the global sensitivity analysis of some parameters by calculating the partial rank correlation coefficients, and some numerical simulations are performed to verify the correctness and feasibility of the theoretical results by using the MATLAB software.

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