Spatiotemporal Patterns Induced by Memory Delay and Nonlocal Perception

In this paper, we are concerned with the spatiotemporal dynamics of the memory-based diffusion with nonlocal perception. The influence of memory-induced diffusion coefficient, perception strength, memory delay and perception scale on the stability of the positive constant steady-state is thoroughly investigated. Our research uncovers a wide array of dynamical behaviors that arise from the combined effects of nonlocal perception and memory. Specifically, under the conditions where memory-induced diffusion prevails, the constant steady-state remains unstable and is unaffected by the strength of perception, the extent of the perceptual scale, and the memory delay. Conversely, when the magnitude of memory-induced diffusion falls below a critical threshold, meaning that nonlocal perception assumes dominance, a greater perception scale can maintain the constant steady-state to be locally asymptotically stable. However, a reduced perception scale, in conjunction with the memory delay, can trigger a variety of complex dynamics, such as Turing, Hopf, Turing–Hopf, and double Hopf bifurcations.