Delay-driven pattern formation in a reaction–diffusion predator–prey model incorporating a prey refuge

Abstract

In this paper, we consider a diffusive predation model with a delay effect, which is based on a modified version of the Leslie–Gower scheme incorporating a prey refuge. We mainly investigate the effects of time delay on the stability of the homogeneous state point and the formation of spatial patterns. We give the conditions for diffusion-driven and delay–diffusion-driven instability in detail. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the model dynamics exhibits a delay and diffusion controlled formation growth not only of spots, stripes and holes, but also of self-replicating spiral patterns. The results indicate that the delay plays an important role in the pattern selection. This may enrich the pattern dynamics for a delay diffusive model.