Dynamics of a diffusive Leslie-Gower predator-prey system with ratio-dependent Holling III functional response

This paper is devoted to investigating the dynamics of a diffusive Leslie-Gower predator-prey system with ratio-dependent Holling III functional response. We first establish the stability of positive constant equilibrium, and show the condition under which system undergoes a Hopf bifurcation with the explicit computational formulas for determining the bifurcating properties. Especially, when the positive constant equilibrium loses its stability, a supercritical Hopf bifurcation with spatial homogeneous and stable bifurcating periodic solution occurs. Finally, we discuss the existence and nonexistence of nonconstant positive solutions with the help of Leray-Schauder degree theory and the implicit function theorem.