Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie–Gower predator–prey model

We consider a diffusive Leslie–Gower predator–prey model subject to the homogeneous Neumann boundary condition. Treating the diffusion coefficient dd as a parameter, the Hopf bifurcation and steady-state bifurcation from the positive constant solution branch are investigated. Moreover, the global structure of the steady-state bifurcations from simple eigenvalues is established by bifurcation theory. In particular, the local structure of the steady-state bifurcations from double eigenvalues is also obtained by the techniques of space decomposition and implicit function theorem.