Stability and Hopf Bifurcation Analysis of a Reduced Gierer–Meinhardt Model

Abstract

In this paper, we consider a reduction of the Gierer–Meinhardt Activator–Inhibitor model. In the absence of diffusion, we determine the global dynamics of the homogeneous system. Then, we study the effect of the diffusion constants on the stability of a homogeneous steady state. By choosing a proper bifurcation parameter, we prove that, under some suitable conditions on the parameters, a generalized Hopf bifurcation occurs in the inhomogeneos model. We compute the normal form of this bifurcation up to the fifth order. Furthermore, the direction of the Hopf bifurcation is obtained by the normal form theory. Finally, we provide some numerical simulations to justify our theoretical results.