Mode-doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model

EJ Crampin, Eamonn Gaffney, PK Maini

Research output: Contribution to journalArticle

56 Citations (Scopus)


Reaction-diffusion equations are ubiquitous as models of biological pattern formation. In a recent paper we have shown that incorporation of domain growth in a reaction-diffusion model generates a sequence of quasi-steady patterns and can provide a mechanism for increased reliability of pattern selection. In this paper we analyse the model to examine the transitions between patterns in the sequence. Introducing a piecewise linear approximation we find closed form approximate solutions for steady-state patterns by exploiting a small parameter, the ratio of diffusivities, in a singular perturbation expansion. We consider the existence of these steady-state solutions as a parameter related to the domain length is varied and predict the point at which the solution ceases to exist, which we identify with the onset of transition between patterns for the sequence generated on the growing domain. Applying these results to the model in one spatial dimension we are able to predict the mechanism and timing of transitions between quasi-steady patterns in the sequence. We also highlight a novel sequence behaviour, mode-tripling, which is a consequence of a symmetry in the reaction term of the reaction-diffusion system.
Original languageEnglish
Pages (from-to)107-128
Number of pages22
JournalJournal of Mathematical Biology
Issue number2
Publication statusPublished - 1 Feb 2002


  • growing domain
  • frequency-doubling
  • reaction-diffusion
  • Turing system
  • pattern formation
  • matched asymptotic expansion


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