Bifurcation Analysis of a Resource–Consumer System With Explicit Spatiotemporal Memory

ABSTRACT In ecological systems, animal movement is often influenced by memory and spatial cognition, especially in advanced species. This paper investigates the dynamics of a diffusive resource–consumer model incorporating explicit spatiotemporal distributed memory, where memory effects are modeled as distributed delays in both time and space. The memory kernel functions represent processes of knowledge accumulation and decay. By using the mean memory delay and the memory‐based diffusion coefficient as bifurcation parameters, we analyze the occurrence of Hopf and Turing bifurcations under both weak and strong kernel cases. Notably, in the strong kernel case, rare and complex dynamical behaviors such as double Hopf, Turing–Hopf, and Turing–Turing bifurcations are observed—phenomena that are typically absent in single‐species models with distributed delays. The normal form theory is applied to determine the direction and stability of Hopf bifurcations, and numerical simulations validate the theoretical results. We find that when consumers are repelled by resources, distributed memory has minimal effect on system stability. However, when consumers are attracted to resources, memory can induce oscillatory and patterned behaviors. Fast memory decay leads to homogeneous periodic oscillations, while slow decay results in more diverse spatial–temporal patterns. This study provides new insights into the role of distributed memory in shaping animal movement and ecological pattern formation.