Pulse-splitting for some reaction-diffusion systems in one-space dimension

Pulse-splitting, or self-replication, behavior is studied for some two-component singularly perturbed reaction-diffusion systems on a one-dimensional spatial domain. For the Gierer-Meinhardt model in the weak interaction regime, characterized by asymptotically small activator and inhibitor diffusivities, a numerical approach is used to verify the key bifurcation and spectral conditions of Ei et al. [Japan. J. Indust. Appl. Math., 18, (2001)] that are believed to be essential for the occurrence of pulse-splitting in a reaction-diffusion system. The pulse- splitting that is observed here is edge-splitting, where only the spikes that are closest to the boundary are able to replicate. For the Gray-Scott model, it is shown numerically that there are two types of pulse- splitting behavior depending on the parameter regime: edge-splitting in the weak interaction regime, and a simultaneous splitting in the semi-strong interaction regime. For the semi-strong spike interaction regime, where only one of the solution components is localized, we construct several model reaction-diffusion systems where all of the pulse- splitting conditions of Ei et al. can be verified analytically, yet no pulse- splitting is observed. These examples suggest that an extra condition, referred to here as the multi-bump transition condition, is also required for pulse- splitting behavior. This condition is in fact satisfied by the Gierer-Meinhardt and Gray-Scott systems in their pulse- splitting parameter regimes.