Hopf bifurcation induced by fear: A Leslie-Gower reaction-diffusion predator-prey model

Jiani Jin; Haokun Qi; Bing Liu; Jiani Jin; Haokun Qi; Bing Liu

doi:10.3934/era.2024304

Electronic Research Archive

2024, Volume 32, Issue 12: 6503-6534. doi: 10.3934/era.2024304

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Hopf bifurcation induced by fear: A Leslie-Gower reaction-diffusion predator-prey model

1.
School of Mathematics, Anshan Normal University, Anshan Liaoning 114007, China
2.
School of Mathematics, Liaoning Normal University, Dalian Liaoning 116029, China

Received: 17 October 2024 Revised: 15 November 2024 Accepted: 22 November 2024 Published: 02 December 2024

The aim of this paper was to explore the impact of fear on the dynamics of prey and predator species. Specifically, we investigated a reaction-diffusion predator-prey model in which the prey was subjected to Beddington-DeAngelis type and the predator was subjected to modified Leslie-Gower type. First, we analyzed the existence and stability of equilibria of the nonspatial model, and further investigated the global stability and Hopf bifurcation at the unique positive equilibrium point. For the spatial model, we studied the local and global stability of the unique constant positive steady state solution and captured the existence of Turing instability, which depended on the diffusion rate ratio between the two species. Then, we demonstrated the existence of Hopf bifurcations and discussed the direction and stability of spatially homogeneous and inhomogeneous periodic solutions. Finally, the impact of fear and spatial diffusion on the dynamics of populations were probed by numerical simulations. Results revealed that spatial diffusion and fear both broaden the dynamical properties of this model, facilitating the emergence of periodic solutions and the formation of biodiversity.
- predator-prey model,
- reaction-diffusion,
- fear effect,
- Turing instability,
- Hopf bifurcation
Citation: Jiani Jin, Haokun Qi, Bing Liu. Hopf bifurcation induced by fear: A Leslie-Gower reaction-diffusion predator-prey model[J]. Electronic Research Archive, 2024, 32(12): 6503-6534. doi: 10.3934/era.2024304

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Abstract

The aim of this paper was to explore the impact of fear on the dynamics of prey and predator species. Specifically, we investigated a reaction-diffusion predator-prey model in which the prey was subjected to Beddington-DeAngelis type and the predator was subjected to modified Leslie-Gower type. First, we analyzed the existence and stability of equilibria of the nonspatial model, and further investigated the global stability and Hopf bifurcation at the unique positive equilibrium point. For the spatial model, we studied the local and global stability of the unique constant positive steady state solution and captured the existence of Turing instability, which depended on the diffusion rate ratio between the two species. Then, we demonstrated the existence of Hopf bifurcations and discussed the direction and stability of spatially homogeneous and inhomogeneous periodic solutions. Finally, the impact of fear and spatial diffusion on the dynamics of populations were probed by numerical simulations. Results revealed that spatial diffusion and fear both broaden the dynamical properties of this model, facilitating the emergence of periodic solutions and the formation of biodiversity.

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