Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources

We study the pattern generating mechanism of a generalized Gierer-Meinhardt model with diffusions. We show the existence and stability of the Hopf bifurcation for the corresponding kinetic system under certain conditions. With spatial uneven diffusions, the obtained stable Hopf periodic solution may become unstable, which results in Turing instability. We derive conditions for the existence of Turing instability. Numerical simulations reveal that the Turing patterns are of stripe and spot shapes. In the analysis, we use bifurcation analysis, center manifold reduction for ordinary differential equations and partial differential equations. Though the Gierer-Meinhardt system is classical, our system with more general settings has yet to be analyzed in the literature.