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

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.

Automated summary

In this paper, we study the Hopf bifurcation in a Gierer-Meinhardt model with different sources and analyze the impact of diffusion rates on system stability. The normal form of this bifurcation is computed up to the third order, and the direction of Hopf bifurcation is determined using normal form theory. Numerical simulations are also provided to illustrate the analytical results.