Spatiotemporal pattern formation of a ratio-dependent invasion-diffusion predator-prey system with disease in the predator

This paper is purported to investigate a ratio-dependent invasion-diffusion predator-prey epidemic system in a bounded domain with no flux boundary condition. The local and global stabilities of positive equilibrium are investigated to this system without invasion-diffusion. The sufficient conditions to nonexistence and existence of non-constant positive solution are found for this invasion-diffusion epidemic system, which implies the existence of spatiotemporal pattern formation. The results show that in wide space for the predator and the prey to diffuse by the self pressure and with a little tendency of predator to catch prey, they can't coexist and there isn't endemic disease extensively, but they can coexist and endemic disease exists extensively when the coefficient of invasion-diffusion for predator is big enough with other random diffusion coefficients being fixed and satisfying curtain conditions.