Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment

We investigate the effect of dispersal and spatial heterogeneity of the environment on the dynamics of a predator-prey model. In contrast with the homogeneous environment, the dynamics of the model in spatially heterogeneous environment is more complex. For instance, for certain ranges of death and dispersal rates of the predator, the semi-trivial steady state of the model in the heterogeneous case could change its stability multiple times as the dispersal rate of the prey varies from small to large, whereas the stability of the semi-trivial steady state is unaffected by the dispersal rates of the predator and prey in the homogeneous case.