Turing instability and bifurcation in a predator-prey model with nonlocal fear and nonlinear cross-diffusion

Meiling Zhao; Sen Wang; Biao Liu; Meiling Zhao; Sen Wang; Biao Liu

doi:10.3934/era.2025327

Electronic Research Archive

2025, Volume 33, Issue 12: 7428-7441. doi: 10.3934/era.2025327

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Research article

Turing instability and bifurcation in a predator-prey model with nonlocal fear and nonlinear cross-diffusion

1.
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
2.
Operations Research and Data Science Laboratory, Anhui Jianzhu University, Hefei 230601, China

Received: 27 September 2025 Revised: 19 November 2025 Accepted: 25 November 2025 Published: 09 December 2025

This paper has studied a predator-prey model that incorporates a nonlocal fear effect and nonlinear cross-diffusion under homogeneous Neumann boundary conditions. We have derived necessary and sufficient conditions for Turing instability in the presence of both nonlocal fear and nonlinear cross-diffusion by means of linear stability analysis. Moreover, we have investigated steady-state bifurcations induced by the nonlocal fear effect using the Lyapunov-Schmidt reduction.
- predator-prey model,
- nonlinear cross-diffusion,
- nonlocal fear effect,
- Turing instability,
- steady-state bifurcation
Citation: Meiling Zhao, Sen Wang, Biao Liu. Turing instability and bifurcation in a predator-prey model with nonlocal fear and nonlinear cross-diffusion[J]. Electronic Research Archive, 2025, 33(12): 7428-7441. doi: 10.3934/era.2025327

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Abstract

This paper has studied a predator-prey model that incorporates a nonlocal fear effect and nonlinear cross-diffusion under homogeneous Neumann boundary conditions. We have derived necessary and sufficient conditions for Turing instability in the presence of both nonlocal fear and nonlinear cross-diffusion by means of linear stability analysis. Moreover, we have investigated steady-state bifurcations induced by the nonlocal fear effect using the Lyapunov-Schmidt reduction.

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