Spatial Movement with Distributed Memory and Maturation Delay

Distributed memory reflects the phenomenon that memory can decay over time. In this paper, we are concerned with the joint effect of distributed memory and maturation delay on spatiotemporal dynamics of the model that illustrates spatial movement of animals. Without maturation delay, distributed memory with weak temporal kernel gives rise to Turing and double Turing bifurcations and there is no Hopf bifurcation. When maturation delay exists, it has been shown that the joint effect of distributed memory and maturation delay can lead to rich dynamics due to the occurrence of double Hopf, Turing–Hopf, codimension 3 and codimension 4 bifurcations. The Wright–Hutchinson equation and its modification are employed to illustrate the theoretical results and maturation delay-induced stability switches, spatially inhomogeneous steady states/periodic solutions and quasi-periodic solution are found.