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.