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 $d$ 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.