Simulation of Pattern in a Delay and Cross-Diffusive Predation Model with the Allee Effect

In this paper, we consider the effect of time delay and cross diffusion on the dynamics of a predator-prey model with the Allee effect. We mainly investigate the stability of the homogeneous state points and give the conditions of time delay and cross diffusion driven instability in details. 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 stripes and holes, but also spots pattern. The results indicate that time delay and cross diffusion play important roles in pattern selection.