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Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion

In this paper we consider a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion under zero Dirichlet boundary condition. By using topological degree theory, bifurcation theory, energy estimates and asymptotic behavior analysis, we prove the existence, uniqueness and multiplicity of positive steady states solutions under certain conditions on the parameters.