Pattern dynamics of a predator-prey reaction-diffusion model with spatiotemporal delay

Using the tool of Turing instability for partial differential equations, we investigate the spatiotemporal distributions for solutions of a predator-prey-type reaction-diffusion model with spatiotemporal delay. The linear stability conditions of Turing instability, which induce bifurcation patterns in this model, are obtained. Moreover, according to these conditions, we numerically calculate the bifurcation diagrams by using time delay and the predator rate as parameters. The effects of two parameters in the different bifurcation diagrams are also demonstrated through numerical computations and lead to some spatiotemporal patterns of this model, which enrich the pattern formation of predator-prey models.