Pulse‐Splitting for Some Reaction‐Diffusion Systems in One‐Space Dimension

Abstract

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.