Spatial dynamics of a nonlocal dispersal Leslie-Gower predator-prey model with some shifting habitats

In this paper, we will consider the spatial dynamical behavior of a Leslie-Gower predator-prey model with nonlocal diffusion under shifting environments. Compared with the classical Laplace diffusion, we introduce the integral operator $ \int_{\mathbb{R}}J(x-y)u(t,y)dy-u(t,x) $ reflected the relationship between the position $ x $ and all other positions $ y $. By comparing the different spreading speed of species with the speed of environmental worsening, we get the different conditions for three situations of extinction of both species, existence of only one specie and coexistence of two species.