Numerical investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system

This work is concerned with the numerical simulations on the Gierer-Meinhardt, activator-inhibitor models. We consider the case when the inhibitor time constant τ is non-zero. In this case, oscillations and pulse splitting are observed numerically. Numerical experiments are carried out to investigate the dynamical behaviors and instabilities of the spike patterns. The numerical schemes used are based upon an efficient moving mesh finite element method which distributes more grid points near the localized spike regions.