Dynamics Analysis for Diffusive Resource-Consumer Model With Nonlocal Discrete Memory

In this paper, based on the importance of consumer memory on spatial resource distribution, we propose a novel consumer-resource model that incorporates nonlocal discrete memory. By conducting thorough bifurcation and stability analysis, we determine the conditions for the occurrence of Hopf and Turing bifurcations and reveal a unique dynamic phenomenon termed Turing–Hopf bifurcation, which is uncommon in models without nonlocal discrete memory. We also show that as the memory delay increases, both the spatially nonhomogeneous periodic and steady-state solutions may vanish, and the unstable positive homogeneous steady state may regain stability. Furthermore, leveraging the theory of normal forms, we derive a new effective algorithm to determine the direction and stability of Hopf bifurcation in a model where the diffusion component incorporates an integral term with delay. In addition, we perform numerical simulations to validate our theoretical findings, particularly to assess the direction and stability of the delay-induced mode-1 Hopf bifurcation. Our new method is used for this purpose, and the results have been confirmed by rigorous numerical analysis.