Permanence and stability of a diffusive predator–prey model with disease in the prey

We investigate a 3-dimensional system of the predator–prey dynamics with disease for the prey. This system is a reaction–diffusion model with ratio-dependent Michaelis–Menten functional response and diffusion in a bounded domain. By applying the comparison principle, Liapunov function and linearized method, we are able to achieve the sufficient conditions of the permanence, the global stability of the boundary equilibria and the positive equilibrium, and the local stability of the positive equilibrium. Numerical simulations are also given to illustrate the applications.