Hopf bifurcation in a memory-based diffusion predator-prey model with spatial heterogeneity

• Published in: Journal of Differential Equations Vol. 397; pp. 377 - 403
• Main Authors: Liu, Di, Jiang, Weihua
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.07.2024
• Subjects: Bifurcation; Memory-based diffusion; Non-constant steady-state; Non-homogeneous periodic solution; Spatial heterogeneity; Stability; Memory-based diffusion; Bifurcation; Spatial heterogeneity; Stability; Non-constant steady-state; Non-homogeneous periodic solution
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2024.04.015

In this paper, we present a memory-based diffusion predator-prey model that incorporates spatial heterogeneity and is subject to homogeneous Dirichlet boundary conditions. Prey species lack memory or cognitive abilities, exhibiting only random diffusion. In contrast, predators utilize memory-based self-diffusion. For the proposed model, we establish the existence and explicit expression of a spatially non-constant positive steady-state. Furthermore, we demonstrate that memory-based diffusion and the averaged memory period can lead to richer dynamics. Specifically, when the memory-based diffusion coefficient is not dominant, the averaged memory period has no impact on the non-constant steady-state. However, when the memory-based diffusion coefficient takes precedence, the averaged memory period can destabilize the non-constant steady-state, resulting in Hopf bifurcation and the emergence of spatially non-homogeneous periodic solutions.