Existence and stability analysis of spiky solutions for the Gierer–Meinhardt system with large reaction rates

We study the Gierer–Meinhardt system in one dimension in the limit of large reaction rates. First we construct three types of solution: (i) an interior spike; (ii) a boundary spike and (iii) two boundary spikes. Second we prove results on their stability. It is found that an interior spike is always unstable; a boundary spike is always stable. The two boundary spike configuration can be either stable or unstable, depending on the parameters. We fully classify the stability in this case. We characterize the destabilizing eigenfunctions in all cases. Numerical simulations are shown which are in full agreement with the analytical results.