Bifurcations induced by nonlocal spatial memory versus nonlocal perception

Spatial memory and perception are two key mechanisms driving animal movement's decisions. In this paper, we formulate a reaction-diffusion model incorporating nonlocal spatial memory and nonlocal perception with both kernels characterized by a top-hat function. To understand the impact of species' memory and instantaneous perception on their movement, we investigate how memory-induced diffusion coefficient, perceptual strength, memory delay, and perceptual scale affect the stability and spatiotemporal dynamics of positive steady states. For spatial memory versus perception, we sketch bifurcation curves in the planes of memory delay and perception scale. When memory and perception are weak, the positive constant steady state remains locally asymptotically stable, indicating minimal impact on stability. A larger perception scale preserves stability, whereas a smaller one can induce instability through bifurcations. Specifically, when both the memory-induced diffusion coefficient and perceptual strength are large and share the same sign (or differ in sign), the system undergoes Turing bifurcation to produce spatially nonhomogeneous steady states (or spatially nonhomogeneous periodic solutions via Hopf bifurcation with increased memory delay). When one of these two parameters is large and the other is small, the stability boundary of the positive constant steady state may be governed by Turing bifurcation or a combination of Turing and Hopf bifurcations, potentially leading to higher codimension bifurcations such as Turing-Hopf and Hopf-Hopf bifurcations.