Hopf bifurcation and steady-state bifurcation for an autocatalysis reaction–diffusion model

This paper is concerned with an autocatalysis model subject to no-flux boundary conditions. The existence of Hopf bifurcation are firstly obtained. Then by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions are established. On the other hand, the steady-state bifurcations from simple and double eigenvalues are intensively studied. The techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalues. Finally, some numerical simulations are shown to verify the analytical results.