Memory-based prey-taxis and environmental stress shape spatiotemporal predator-prey dynamics

Dazhuo Liu; Xuewen Tan; Dazhuo Liu; Xuewen Tan

doi:10.3934/math.2026729

AIMS Mathematics

2026, Volume 11, Issue 6: 17880-17916. doi: 10.3934/math.2026729

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Research article

Memory-based prey-taxis and environmental stress shape spatiotemporal predator-prey dynamics

Dazhuo Liu ^{1
,
,},
Xuewen Tan ²

1.
College of Education and Psychology Science, Minnan Normal University, Zhangzhou Fujian 363000, China
2.
School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650504, China

Received: 19 March 2026 Revised: 10 May 2026 Accepted: 27 May 2026 Published: 18 June 2026
MSC : 92D25, 35B36

This paper investigates a diffusive predator–prey system incorporating memory-based prey-taxis, the Allee effect, and environmental stressors. After establishing global well-posedness and existence conditions for equilibria, we use prey-taxis sensitivity and memory delay as parameters to identify Turing and Hopf bifurcation thresholds. Our results show that strong memory-based prey-taxis suppresses diffusion-driven Turing patterns and restores spatial homogeneity. However, this spatial stabilization does not necessarily imply greater delay tolerance: The critical Hopf delay may decrease with taxis sensitivity, revealing a trade-off between taxis-induced spatial stabilization and delay-induced temporal oscillations. Additionally, we analyze environmental stress, revealing a predator release effect where moderate stress disproportionately suppresses predators, which indirectly leads to an increase in the prey's density. We also identify a mode-jumping phenomenon in the critical delay threshold during stress-induced transitions from ordinary differential equation (ODE) to Turing instability. Finally, the numerical simulations provide a two-parameter stability map delineating four dynamic regions: Stable homogeneous states, stationary Turing patterns, spatially nonhomogeneous periodic solutions, and complex spatiotemporal dynamics from interacting instabilities.
- predator–prey model,
- memory-based prey-taxis,
- environmental stress,
- spatiotemporal patterns,
- mathematical modeling ability
Citation: Dazhuo Liu, Xuewen Tan. Memory-based prey-taxis and environmental stress shape spatiotemporal predator-prey dynamics[J]. AIMS Mathematics, 2026, 11(6): 17880-17916. doi: 10.3934/math.2026729

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Abstract

This paper investigates a diffusive predator–prey system incorporating memory-based prey-taxis, the Allee effect, and environmental stressors. After establishing global well-posedness and existence conditions for equilibria, we use prey-taxis sensitivity and memory delay as parameters to identify Turing and Hopf bifurcation thresholds. Our results show that strong memory-based prey-taxis suppresses diffusion-driven Turing patterns and restores spatial homogeneity. However, this spatial stabilization does not necessarily imply greater delay tolerance: The critical Hopf delay may decrease with taxis sensitivity, revealing a trade-off between taxis-induced spatial stabilization and delay-induced temporal oscillations. Additionally, we analyze environmental stress, revealing a predator release effect where moderate stress disproportionately suppresses predators, which indirectly leads to an increase in the prey's density. We also identify a mode-jumping phenomenon in the critical delay threshold during stress-induced transitions from ordinary differential equation (ODE) to Turing instability. Finally, the numerical simulations provide a two-parameter stability map delineating four dynamic regions: Stable homogeneous states, stationary Turing patterns, spatially nonhomogeneous periodic solutions, and complex spatiotemporal dynamics from interacting instabilities.

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