Diffusive spatial movement with memory and maturation delays

A single species spatial population model that incorporates Fickian diffusion, memory-based diffusion, and reaction with maturation delay is formulated. The stability of a positive equilibrium and the crossing curves in the two-delay parameter plane on which the characteristic equation has purely imaginary roots are studied. With Neumann boundary condition, the crossing curve that separates the stable and unstable regions of the equilibrium may consist of two components, where spatially homogeneous and inhomogeneous periodic solutions are generated through Hopf bifurcation respectively. This phenomenon rarely emerges from standard partial functional differential equations with Neumann boundary condition, which indicates that the memory-based diffusion can induce more complicated spatiotemporal dynamics.