Stability of the static spike autosolitons in the Gray-Scott model

We performed an asymptotic linear stability analysis of the static spike autosolitons (ASs) self-sustained solitary inhomogeneous states in the Gray Scott model of an autocatalytic chemical reaction. We found that in one dimension these ASs destabilize with respect to pulsations or the onset of traveling motion when the inhibitor is slow enough. In higher dimensions, the one-dimensional static spike ASs are always unstable with respect to corrugation and wriggling. The higher-dimensional radially symmetric static spike ASs may destabilize with respect to the radially nonsymmetric fluctuations leading to their splitting when the inhibitor is fast or with respect to pulsations when the inhibitor is slow.