Diffusive Spatial Movement with Memory - Shi, Junping; Wang, Chuncheng; Wang, Hao; Yan, Xiangping (Journal of dynamics and differential equations, 2020)
Animal movements and their underlying mechanisms are an extremely important research area in biology and have been extensively studied for centuries. However, spatial memory and cognition, which is the most significant difference between animal movements and chemical movements, has been ignored in the modeling of animal movements. To incorporate cognition and memory of “clever” animals in the simplest and self-contained way, we propose a delayed diffusion model via a modified Fick’s law, whereas in literature the standard diffusion for chemical movements was applied to describe “drunk” animal movements. Our mathematical model is expressed by a reaction–diffusion equation with an additional delayed diffusion term, which makes rigorous analysis intriguing and challenging. We show the wellposedness and analyze the asymptotic stability of steady state in the spatial memory model. It is shown that for the three possible reaction schemes, the stability of a spatially homogeneous steady state fully depends on the relationship between the two diffusion coefficients but is independent of the time delay. Finally, we numerically illustrate possible spatialtemporal patterns when the system is divergent.