Qualitative analysis for a diffusive predator-prey model with a transmissible disease in the prey population

In this paper, we study a diffusive predator-prey model with a transmissible disease in the prey population. We offer a complete discussion of the dynamical properties under the homogeneous Neumann boundary condition. We analyze local and global stabilities of nonnegative constant steady states and long time behaviors of the positive solutions. Moreover, we study the existence, nonexistence and bifurcation of nonconstant positive stationary solutions. Finally, some numerical simulations are presented to support and strengthen our analytical analysis.