Stability and bifurcation of a predator–prey model with disease in the prey and temporal–spatial nonlocal effect

In this paper, we consider the dynamics of a predator–prey model with disease in the prey and ratio-dependent Michaelis–Menten functional response. The model is a reaction–diffusion system with a nonlocal term representing the temporal–spatial weighted average for the prey density. The limiting case of the system reduces to the Lotka–Volterra diffusive system with logistic growth of the prey. We study the linear stability of the two non-trivial steady states either with or without nonlocal term. The bifurcations to three types of periodic solutions occurring from the coexistence steady state are investigated for two particular kernels, which reveal the important significance of temporal–spatial nonlocal effects.