Rich dynamics in a spatial predator-prey model with delay

In this paper, we study the spatiotemporal dynamics of a diffusive Holling-Tanner predator-prey model with discrete time delay. Via analytically and numerically analysis, we unveil six types of patterns with and without time delay. Among them, of particular novel is the observation of linear pattern (consisting of a series of parallel lines), whose formation is closely related with the temporal Hopf bifurcation threshold. Moreover, we also find that larger time delay or diffusion of predator may induce the extinction of both prey and predator. Theoretical analysis and numerical simulations validate the well-known conclusion: diffusion is usually beneficial for stabilizing pattern formation, yet discrete time delay plays a destabilizing role in the generation of pattern.