Global well-posedness and pattern formation in a predator–prey reaction–diffusion system with prey-taxis

Mohamed Hafdane; Nossaiba Baba; Saida Khiyar; Mohamed Hafdane; Nossaiba Baba; Saida Khiyar

doi:10.3934/mbe.2026044

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 5: 1203-1250. doi: 10.3934/mbe.2026044

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Research article

Global well-posedness and pattern formation in a predator–prey reaction–diffusion system with prey-taxis

Analysis, Modeling and Simulation Laboratory, Hassan Ⅱ University, Morocco

Received: 24 January 2026 Revised: 13 March 2026 Accepted: 20 March 2026 Published: 31 March 2026

This paper investigates a spatiotemporal predator–prey model that incorporates the Allee effect, the fear effect, prey-taxis, and harvesting within a Beddington–DeAngelis functional framework. The model captures the combined influence of biological interactions, behavioral responses, and harvesting activities on population dynamics in a spatially heterogeneous environment. The global existence, positivity, and boundedness of classical solutions are first established under appropriate parameter conditions. The existence and local stability of homogeneous steady states are then analyzed, and the conditions for diffusion-driven instability are derived to characterize the onset of spatial patterns. Using weakly nonlinear analysis, amplitude equations are developed to describe the modulation of spatial modes near the bifurcation threshold. Numerical investigations are conducted to complement the theoretical analysis: bifurcation diagrams are employed to examine the effects of biological parameters such as the Allee threshold ($ \beta $), fear intensity ($ \gamma $), and conversion efficiency ($ \varepsilon $), while spatiotemporal simulations are performed to visualize different scenarios and demonstrate the impact of prey-taxis on pattern formation and population organization.
- predator-prey model,
- spatiotemporal dynamics,
- prey-taxis,
- turing instability,
- amplitude equation,
- pattern formation
Citation: Mohamed Hafdane, Nossaiba Baba, Saida Khiyar. Global well-posedness and pattern formation in a predator–prey reaction–diffusion system with prey-taxis[J]. Mathematical Biosciences and Engineering, 2026, 23(5): 1203-1250. doi: 10.3934/mbe.2026044

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Abstract

This paper investigates a spatiotemporal predator–prey model that incorporates the Allee effect, the fear effect, prey-taxis, and harvesting within a Beddington–DeAngelis functional framework. The model captures the combined influence of biological interactions, behavioral responses, and harvesting activities on population dynamics in a spatially heterogeneous environment. The global existence, positivity, and boundedness of classical solutions are first established under appropriate parameter conditions. The existence and local stability of homogeneous steady states are then analyzed, and the conditions for diffusion-driven instability are derived to characterize the onset of spatial patterns. Using weakly nonlinear analysis, amplitude equations are developed to describe the modulation of spatial modes near the bifurcation threshold. Numerical investigations are conducted to complement the theoretical analysis: bifurcation diagrams are employed to examine the effects of biological parameters such as the Allee threshold ($ \beta $), fear intensity ($ \gamma $), and conversion efficiency ($ \varepsilon $), while spatiotemporal simulations are performed to visualize different scenarios and demonstrate the impact of prey-taxis on pattern formation and population organization.

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