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This paper is concerned with a nonlocal reaction–diffusion prey–predator model with free boundary. The spatial–temporal risk index R 0 p (t) which is threshold value, is introduced for this model. The global existence of positive solutions is proved and the long-time behavior of positive solutions is investigated. The sufficient conditions are found for vanishing of predators and the establishment of themselves in new environment. Furthermore, the spreading–vanishing criteria are established. The results show that the predators will vanish finally and not establish themselves if the sizes of habitant and spreading coefficient are relatively small or the initial value of predators is also relatively small with R 0 p (0) < 1 , and the predators will achieve if initial values of predators and preys are relatively big or R 0 p (t 0) > 1 for t 0 > 0. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]