The effects of dispersal and spatial heterogeneity on the dynamics of a predator–prey model

We investigate a diffusive predator–prey model in spatially heterogeneous environments. When the intrinsic growth rate of the prey is constant and the intrinsic growth rate of the predator is non-constant, we completely study how the semi-trivial steady states step-wisely change their stability as the dispersal rates of the prey and predator vary. Moreover, we can obtain multiple positive steady states of this model and determine their stability. In particular, if the dispersal rate of the prey is considered as bifurcation parameter, then the local bifurcation results can be generalized to a global one. We also investigate the stability of the semi-trivial steady states when both the intrinsic growth rate of the prey and the intrinsic growth rate of the predator are non-constant. Finally, when the dispersal rates of the prey and predator are simultaneously regarded as bifurcation parameters, we can deduce positive steady state of this model and derive its stability.