Self-organized wave pattern in a predator-prey model

In this paper, pattern formation of a predator-prey model with spatial effect is investigated. We obtain the conditions for Hopf bifurcation and Turing bifurcation by mathematical analysis. When the values of the parameters can ensure a stable limit cycle of the no-spatial model, our study shows that the spatially extended models have spiral waves dynamics. Moreover, the stability of the spiral wave is given by the theory of essential spectrum. Furthermore, although the environment is heterogeneous, the system still exhibit spiral waves. The obtained results confirm that diffusion can form the population in the stable motion, which well enrich the finding of spatiotemporal dynamics in the predator-prey interactions and may well explain the field observed in some areas.