Spatiotemporal dynamics in a delayed diffusive predator model

In this paper, we investigate the spatiotemporal dynamics of a delayed reaction–diffusion Leslie–Gower model. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: Pure Turing instability gives birth to spots, spots–stripes-mixture, stripes, stripes–holes-mixture and holes patterns, pure Hopf instability to spiral wave pattern, and Hopf–Turing instability to chaotic wave pattern. Our results well extend the findings of spatiotemporal dynamics in the delayed reaction–diffusion predator–prey model, and indicate that time delay play an important roles in pattern formation.