Dynamics Analysis in a Gierer-Meinhardt Reaction-Diffusion Model with Homogeneous Neumann Boundary Condition

A reaction-diffusion Gierer-Meinhardt system with homogeneous Neumann boundary condition on one-dimensional bounded spatial domain is considered in the present article. Local asymptotic stability, Turing instability and existence of Hopf bifurcation of the constant positive equilibrium are explored by analyzing in detail the associated eigenvalue problem. Moreover, properties of spatially homogeneous Hopf bifurcation are carried out by employing the normal form method and the center manifold technique for reaction-diffusion equations. Finally, numerical simulations are also provided in order to check the obtained theoretical conclusions.