Stripe and spot patterns for the Gierer-Meinhardt model with saturated activator production

The Gierer-Meinhardt model of morphogenesis with saturated activator production is considered. For the unique positive equilibrium of the kinetic equations, the precise parameter conditions of stability, instability and Hopf bifurcation are obtained. It is shown that the equilibrium can either undergo supercritical or subcritical Hopf bifurcation under certain parameter range. Furthermore, it is proved that there exists at least one stable limit cycle besides the periodic solution bifurcating from Hopf bifurcation. In addition, Turing instability conditions on the parameters and diffusion coefficients for the positive equilibrium and the periodic solution bifurcating from Hopf bifurcation are given. The dynamics of the model are illustrated by numerical simulations which exhibit that Turing patterns are of either stripe or spot type.