Spatiotemporal patterns of a diffusive prey-predator model with spatial memory and pregnancy period in an intimidatory environment.

Spatial memory and predator-induced fear have recently been considered in modeling population dynamics of animals independently. This paper is the first to integrate both aspects in a prey-predator model with pregnancy cycle to investigate the direct and indirect effects of predation on the spatial distribution of prey. We extensively study Turing instability and Hopf bifurcation. When the prey population has slow memory-based diffusion, the model is easier to generate Turing patterns. While when the prey population has fast memory-based diffusion, the model can exhibit rich dynamics. Specifically, (1) for the model with spatial memory delay only, the prey population with long term memory shows a spatially nonhomogeneous periodic distribution; (2) for the model with pregnancy delay only, the prey population with long pregnancy cycles shows a spatially homogeneous (or nonhomogeneous) periodic distribution, and (3) for the model with both the two time delays, more interesting spatiotemporal dynamics can be observed for long memory delay and (or) long pregnancy cycles. Our findings indicate that both spatial memory and pregnancy cycle play significant roles in the pattern formation of prey-predator interactions.