Spike Patterns in the Gray-Scott Model: Equilibria, Stability, and Pulse-Splitting Behavior
by
Michael Ward
University of British Columbia
Coauthors: Theo Kolokolnikov (UBC), Juncheng Wei (CUHK)

The existence, stability, and pulse-splitting behavior of spike patterns in the one-dimensional Gray-Scott model on a finite domain is analyzed in the semi-strong spike-interaction regime. This regime is characterized by a localization of one of the components of the reaction near certain spike locations, while the other component exhibits a more global variation across the domain. Asymptotic analysis is used to construct k-spike equilibria in terms of a certain core problem. This core problem is studied numerically and qualitatively. For each positive integer k, it is shown that there are two branches of k-spike equilibria that meet at a saddle-node bifurcation value. For small values of the diffusivity D of the second component, these saddle node bfurcation points occur at approximately the same value. A combination of asymptotic and numerical methods is used to analyze the stability of these branches of k-spike equilibria with respect to both drift instabilities associated with the small eigenvalues and oscillatory instabilities of the spike profile. In this way, the key bifurcation and spectral conditions of Ei, Nishiura, Ueda [Japan J. Indus. Appl. Math. 18, (2001)] believed to be essential to pulse-splitting behavior in a reaction-diffusion system are verified. A simple analytical criterion for the occurrence of pulse-splitting is formulated and verified with full numerical simulations of the Gray-Scott model.

Date received: October 9, 2003