Pattern dynamics of a diffusive predator-prey model with delay effect

We study the spatiotemporal dynamics in a diffusive predator-prey system with time delay. By investigating the dynamical behavior of the system in the presence of Turing-Hopf bifurcations, we present a classification of the pattern dynamics based on the dispersion relation for the two unstable modes. More specifically, we researched the existence of the Turing pattern when control parameters lie in the Turing space. Particularly, when parameter values are taken in Turing-Hopf domain, we numerically investigate the formation of all the possible patterns, including time-dependent wave pattern, persistent short-term competing dynamics and stationary Turing pattern. Furthermore, the effect of time delay on the formation of spatial pattern has also been analyzed from the aspects of theory and numerical simulation. We speculate that the interaction of spatial and temporal instabilities in the reaction-diffusion system might bring some insight to the finding of patterns in spatial predator-prey models.