Multiple positive steady states of a diffusive predator-prey model in spatially heterogeneous environments

In this paper, we discuss a diffusive predator-prey model with predator cannibalism in spatially heterogeneous environments. In contrast with spatially homogeneous environments, we find the dynamics of the model in spatially heterogenous environments is more complicated. For the spatially heterogeneous case, we could classify death rate of the predator into four different regions and demonstrate that for every region, this model with Neumann boundary conditions has at least one positive steady state for some ranges of dispersal rates of the predator and prey via Leray-Schauder degree theory. In particular, for some scopes of death rate of the predator, this model has multiple positive steady states for certain ranges of dispersal rates of the predator and prey.