Hopf bifurcation in a predator-prey model under fuzzy parameters involving prey refuge and fear effects

Xuyang Cao; Qinglong Wang; Jie Liu; Xuyang Cao; Qinglong Wang; Jie Liu

doi:10.3934/math.20241164

AIMS Mathematics

2024, Volume 9, Issue 9: 23945-23970. doi: 10.3934/math.20241164

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Research article

Hopf bifurcation in a predator-prey model under fuzzy parameters involving prey refuge and fear effects

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

Received: 01 July 2024 Revised: 29 July 2024 Accepted: 06 August 2024 Published: 12 August 2024
MSC : 34C23, 34D20

In ecology, the most significant aspect is that the interactions between predators and prey are extremely complicated. Numerous experiments have shown that both direct predation and the fear induced in prey by the presence of predators lead to a reduction in prey density in predator-prey interactions. In addition, a suitable shelter can effectively stop predators from attacking as well as support the persistence of prey population. There has been less exploration of the effects of not only fear but also refuge factors on the dynamics of predator prey interactions. In this paper, we unveil several conclusions about a predator-prey system with fuzzy parameters, considering the cost of fear in two prey species and the effect of shelter on two prey species and one predator. As the first step of the investigation, the boundedness and non-negativity of the solutions to the system are put forward. Using the Jocabian matrix and Lyapunov function methods, we further analyze the existence and stability of the available equilibria and also the existence of Hopf bifurcation, considering the fear parameter as the bifurcation parameter that has been observed by applying the normal form theory. Finally, numerical simulations help us better understand the dynamics of the model, in which some interesting chaotic phenomena are also exhibited.
- fear effect,
- refuge,
- Hopf bifurcation,
- fuzzy parameter,
- chaotic phenomena
Citation: Xuyang Cao, Qinglong Wang, Jie Liu. Hopf bifurcation in a predator-prey model under fuzzy parameters involving prey refuge and fear effects[J]. AIMS Mathematics, 2024, 9(9): 23945-23970. doi: 10.3934/math.20241164

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Abstract

In ecology, the most significant aspect is that the interactions between predators and prey are extremely complicated. Numerous experiments have shown that both direct predation and the fear induced in prey by the presence of predators lead to a reduction in prey density in predator-prey interactions. In addition, a suitable shelter can effectively stop predators from attacking as well as support the persistence of prey population. There has been less exploration of the effects of not only fear but also refuge factors on the dynamics of predator prey interactions. In this paper, we unveil several conclusions about a predator-prey system with fuzzy parameters, considering the cost of fear in two prey species and the effect of shelter on two prey species and one predator. As the first step of the investigation, the boundedness and non-negativity of the solutions to the system are put forward. Using the Jocabian matrix and Lyapunov function methods, we further analyze the existence and stability of the available equilibria and also the existence of Hopf bifurcation, considering the fear parameter as the bifurcation parameter that has been observed by applying the normal form theory. Finally, numerical simulations help us better understand the dynamics of the model, in which some interesting chaotic phenomena are also exhibited.

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