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In this paper, a Lévy-diffusion Leslie-Gower predator-prey model with a nonmonotonic functional response is studied. We show the existence, uniqueness and attractiveness of the globally positive solution to this model. Moreover, to its corresponding steady-state model, we obtain the stability of the semi-trivial solutions, the existence and nonexistence of coexistence states by the method of topological degree, the uniqueness and stability of coexistence state, and the multiplicity and stability of coexistence states by Grandall-Rabinowitz bifurcation theorem. In addition, to get these results, we study the property of the Lévy diffusion operator, and give out the comparison principle of the generalized parabolic Lévy-diffusion differential equation, as well as the existence and stability of the solution for the steady-state Logistic equation with Lévy diffusion. Furthermore, we obtain the comparison principle of the steady-state Lévy-diffusion equation. As far as we know, these results are new in the ecological model.
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