Analysis of the Hopf bifurcation in a Diffusive Gierer-Meinhardt Model

In this work, we consider an activator-inhibitor system, known as the Gierer-Meinhardt model. Using the linear stability analysis at the unique positive equilibrium, we derive the conditions of the Hopf bifurcation. We compute the normal form of this bifurcation up to the third degree and obtain the direction of the Hopf bifurcation. Finally, we provide numerical simulations to illustrate the theoretical results of this paper. In this study, we will use the technique of normal form and center manifold theorem.