Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots
type: Item
instance of: scholarly article
title: Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots
zbMATH Open document ID: 1065.35133
author: Tasso J. Kaper
author: David S. Morgan
published in: Physica D
publication date: 2004-11-22T00:00:00Z
review text: The work addresses instability and nonlinear evolution of axially symmetric (ring-shaped) patterns in the Gray-Scott model of the reaction-diffusion type: \[ \begin{aligned} U_t & =D_U\nabla^2U - UV^2 +A(1-U),\\ V_t & =D_V\nabla^2V + UV^2 -BV,\end{aligned} \] for real fields \(U\) and \(V\) with real control parameters \(A\) and \(V\), which is known to demonstrate quite complex behavior (spontaneous spot multiplication, etc.) in two-dimensional simulations. In this work, solutions for ring-shaped patterns in the monostable sector of the model are constructed by means of an asymptotic method (starting from the one-dimensional solution in the radial direction as the zero-order approximation), and their stability is investigated against infinitesimal azimuthal perturbations, which tend to destroy the axial symmetry and split each ring into a set of localized spots. The stability problem is solved by reducing the one for the two coupled equations to an effective problem for a single nonlocal equation, which is achieved by eliminating the field \(U\). As a result, a type of the unstable eigenmode (the most unstable particular azimuthal harmonic) is found, thus making it possible to predict the number of spots into which the ring(s) are split. The predictions are confirmed by direct numerical simulations (which show the establishment of stationary patterns consisting of several layers of necklace-shaped arrays of spots). The analysis is also performed for the same model in the bistable section, where it reduces to the consideration of an instability of the Turing type.
reviewed by: B. A. Malomed
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 92C15
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B32
zbMATH DE Number: 2117592
zbMATH Keywords: nonlocal eigenvalue problem
zbMATH Keywords: Turing instability
zbMATH Keywords: pattern formation
describes a project that uses: Mathematica
describes a project that uses: CWRESU
describes a project that uses: CWRESX
describes a project that uses: Algorithm 731
describes a project that uses: VLUGR2
MaRDI profile type: Publication
full work available at URL: j.physd.2003.12.012
OpenAlex ID: W2115651779
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DOI: 10.1016/J.PHYSD.2003.12.012
description: scientific article; zbMATH DE number 2117592