Bifurcation analysis of the Gierer-Meinhardt system with a saturation in the activator production

The reaction-diffusion Gierer-Meinhardt system with a saturation in the activator production is considered. Stability of the unique positive constant steady state solution is analysed, and associated Hopf bifurcations and steady state bifurcations are obtained. A global bifurcation diagram of non-trivial periodic orbits and steady state solutions with respect to key system parameters is obtained, which improves the understanding of dynamics of Gierer-Meinhardt system with a saturation in different parameter regimes.