Hopf Bifurcation Analysis in a Gierer–Meinhardt Activator–Inhibitor Model

Abstract

In this paper, we study the Hopf bifurcation in a Gierer–Meinhardt model of the activator–inhibitor type with different sources. In the absence of diffusion, we determine the dynamics of the corresponding kinetic equations. Then, we investigate the impact of the diffusion rates on the stability of the homogeneous steady state. By choosing a proper bifurcation parameter, we prove that, under some suitable conditions of the parameters, a Hopf bifurcation occurs in the nonhomogeneous system. We compute the normal form of this bifurcation up to the third order. Next, we specify the direction of the Hopf bifurcation by the normal form theory. Moreover, we provide numerical simulations to illustrate our analytical results.