We consider a diffusive predator-prey model in a spatially heterogeneous environment. In contrast to existing models that operate in spatially homogeneous environments, our model can describe natural environments that are basically heterogeneous. We explain how the linearly stability of semi-trivial steady state of our model changes from stable to unstable step-wise as the death rate of the predator decreases. Based on the results of stability of the semi-trivial steady state, we regard the dispersal rates of the predator and prey as bifurcation parameters, and deduce corresponding bifurcation conclusions. In particular, considering the dispersal rate of the predator as a bifurcation parameter, the bifurcation result can be extended to the global bifurcation case.

B. Wang was supported by the National Science Foundation of China (No. 11801436) and Natural Science Basic Research Plan in Shaanxi Province of China (No. 2019JQ-346). Z. C. Zhang was supported by the National Science Foundation of China (No. 11371286).