Memory-based movement with spatiotemporal distributed delays in diffusion and reaction.

• Propose a general single species animal movement model with distributed delays in diffusion and reaction. • Investigate the impact of spatiotemporal delays on stability of memory-based model. • Obtain conditions for the occurrence of Hopf and steady state bifurcations. • Explore dynamics due to the interaction of Hopf and steady state bifurcations. In this paper, we investigate the spatiotemporal dynamics of a single-species model with spatiotemporal delays characterizing spatial memory and maturation. Through stability and bifurcation analysis, we find that the spatial memory-based diffusion coefficient, the spatiotemporal diffusive delay and spatiotemporal reaction delay have important effects on the dynamics of the model and their combined impact can cause the destabilization of the positive constant steady state and give rise to steady state and Hopf bifurcations. Taking the coefficient of spatial memory diffusion as the bifurcation parameter, the critical values of steady state and Hopf bifurcations are rigorously determined. Furthermore, we apply the theoretical results to a modified diffusive logistic model with predation and obtain spatially inhomogeneous steady states and spatially homogeneous and inhomogeneous periodic solutions via numerical simulations. [ABSTRACT FROM AUTHOR]