Skip to main content
Log in

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

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract.

 Reaction-diffusion equations are ubiquitous as models of biological pattern formation. In a recent paper [4] 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Author information

Authors and Affiliations

Authors

Additional information

Received: 19 December 2000 / Revised version: 24 May 2001 / Published online: 7 December 2001

Rights and permissions

Reprints and permissions

About this article

Cite this article

Crampin, E., Gaffney, E. & Maini, P. Mode-doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model. J Math Biol 44, 107–128 (2002). https://doi.org/10.1007/s002850100112

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s002850100112

Keywords

Navigation