Pattern formation in morphogenesis. Analytical treatment of the Gierer-Meinhardt model on a sphere

We first treat the Gierer-Meinhardt equations by linear stability analysis to determine the critical parameter, at which the homogeneous distributions of activator and inhibitor concentrations become unstable. We find two types of instabilities: one leading to spatial pattern formation and another one leading to temporal oscillations. We consider the case where two instabilities are present. Using the method of generalized Ginzburg-Landau equations introduced earlier we then analyze the nonlinear equations. As we are mainly interested in spatial pattern formation on a sphere we consider the problem under an appropriate constraint. Combining the two occurring solutions we find patterns well-known in biology, such as a gradient system and temporal oscillations.