Slow translational instabilities of symmetric k-spike equilibria for the one-dimensional singularly perturbed two-component Gray–Scott (GS) model are analyzed. These symmetric spike patterns are characterized by a common value of the spike amplitude. The GS model is studied on a finite interval in the semi-strong spike-interaction regime, where the diffusion coefficient of only one of the two chemical species is asymptotically small. Two distinguished limits for the GS model are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime it is shown analytically that k−1 small eigenvalues, governing the translational stability of a symmetric k-spike pattern, simultaneously cross through zero at precisely the same parameter value at which k−1 different asymmetric k-spike equilibria bifurcate off of the symmetric k-spike equilibrium branch. These asymmetric equilibria have the general form SBB...BS (neglecting the positioning of the B and S spikes in the overall spike sequence). For a one-spike equilibrium solution in the intermediate regime it is shown that a translational, or drift, instability can occur from a Hopf bifurcation in the spike-layer location when a reaction-time parameter τ is asymptotically large as ε→0. Locally, this instability leads to small-scale oscillations in the spike-layer location. For a certain parameter range within the intermediate regime such a drift instability for the GS model is shown to be the dominant instability mechanism. Numerical experiments are performed to validate the asymptotic theory.
Theodore Kolokolnikov
1
;
Michael J. Ward
1
;
Juncheng Wei
1
1
University of British Columbia, Vancouver, Canada
Theodore Kolokolnikov; Michael J. Ward; Juncheng Wei. Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model. Interfaces and free boundaries, Tome 8 (2006) no. 2, pp. 185-222. doi: 10.4171/ifb/140
@article{10_4171_ifb_140,
author = {Theodore Kolokolnikov and Michael J. Ward and Juncheng Wei},
title = {Slow translational instabilities of spike patterns in the one-dimensional {Gray-Scott} model},
journal = {Interfaces and free boundaries},
pages = {185--222},
year = {2006},
volume = {8},
number = {2},
doi = {10.4171/ifb/140},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/140/}
}
TY - JOUR
AU - Theodore Kolokolnikov
AU - Michael J. Ward
AU - Juncheng Wei
TI - Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model
JO - Interfaces and free boundaries
PY - 2006
SP - 185
EP - 222
VL - 8
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/140/
DO - 10.4171/ifb/140
ID - 10_4171_ifb_140
ER -
%0 Journal Article
%A Theodore Kolokolnikov
%A Michael J. Ward
%A Juncheng Wei
%T Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model
%J Interfaces and free boundaries
%D 2006
%P 185-222
%V 8
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4171/ifb/140/
%R 10.4171/ifb/140
%F 10_4171_ifb_140
