Analysis of a spatial predator-prey model with delay

In this paper, we present a Holling-Tanner model combined with diffusion and time delay. We found that, when time delay is small, there is no the synchronization of prey and the predator. However, when it is larger, there is antiphase synchronization. Furthermore, a transition from anti-phase synchronization to in-phase synchronization emerges as time delay further increases. Since synchronization of populations could lead them to be extinct, the results obtained may indicate that time delay plays an important role on the persistence of the populations.