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Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model

Published online by Cambridge University Press:  01 April 2009

WAN CHEN  and
MICHAEL J. WARD
WAN CHEN
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, CanadaV6T 1Z2 email: ward@math.ubc.ca
MICHAEL J. WARD
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, CanadaV6T 1Z2 email: ward@math.ubc.ca
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Abstract

The dynamics and oscillatory instabilities of multi-spike solutions to the one-dimensional Gray-Scott reaction–diffusion system on a finite domain are studied in a particular parameter regime. In this parameter regime, a formal singular perturbation method is used to derive a novel ODE–PDE Stefan problem, which determines the dynamics of a collection of spikes for a multi-spike pattern. This Stefan problem has moving Dirac source terms concentrated at the spike locations. For a certain subrange of the parameters, this Stefan problem is quasi-steady and an explicit set of differential-algebraic equations characterizing the spike dynamics is derived. By analysing a nonlocal eigenvalue problem, it is found that this multi-spike quasi-equilibrium solution can undergo a Hopf bifurcation leading to oscillations in the spike amplitudes on an O(1) time scale. In another subrange of the parameters, the spike motion is not quasi-steady and the full Stefan problem is solved numerically by using an appropriate discretization of the Dirac source terms. These numerical computations, together with a linearization of the Stefan problem, show that the spike layers can undergo a drift instability arising from a Hopf bifurcation. This instability leads to a time-dependent oscillatory behaviour in the spike locations.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 20 , Issue 2 , April 2009 , pp. 187 - 214
Copyright
Copyright © Cambridge University Press 2008

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