Spatial Movement With Explicit Memory and Nonlocal Pregnancy Delay

In depicting animal movement, besides considering spatial memory, the pregnancy period of the animals themselves should also be taken into account. This article proposes a novel single-population model with discrete memory delay and nonlocal spatiotemporal gestation delay. Assuming the positive equilibrium is locally stable without spatiotemporal delay, the study investigates two types of directional movements using memory delay and gestation delay as control parameters. The research shows that for the positive memory-based diffusion coefficient, the system destabilizes through homogeneous/nonhomogeneous Hopf bifurcations, but for the negative memory diffusion coefficient, the system not only undergo Hopf bifurcations but also becomes unstable through Turing bifurcations, and may even exhibit Turing–Hopf bifurcation. Finally, we use a diffusive logistic model with predation behavior as an example to numerically simulate the theoretical results and also discover stability switch phenomena.