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Journal of Mathematical Biology

, Volume 61, Issue 1, pp 133–164 | Cite as

Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains

  • Anotida MadzvamuseEmail author
  • Eamonn A. Gaffney
  • Philip K. Maini
Article

Abstract

By using asymptotic theory, we generalise the Turing diffusively-driven instability conditions for reaction-diffusion systems with slow, isotropic domain growth. There are two fundamental biological differences between the Turing conditions on fixed and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions and (ii) the restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Our theoretical findings are confirmed and reinforced by numerical simulations for the special cases of isotropic linear, exponential and logistic growth profiles. In particular we illustrate an example of a reaction-diffusion system which cannot exhibit a diffusively-driven instability on a fixed domain but is unstable in the presence of slow growth.

Keywords

Convection-reaction-diffusion systems Turing diffusively-driven instability Pattern formation Growing domains asymptotic theory Domain-induced diffusively-driven instability 

Mathematics Subject Classification (2000)

34C23 34D05 34D35 35K55 35K57 35B36 37C60 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Anotida Madzvamuse
    • 1
    Email author
  • Eamonn A. Gaffney
    • 2
    • 3
  • Philip K. Maini
    • 2
    • 3
    • 4
  1. 1.Department of MathematicsUniversity of SussexBrightonUK
  2. 2.Centre for Mathematical Biology, Mathematical InstituteUniversity of OxfordOxfordUK
  3. 3.Oxford Centre for Collaborative Applied MathematicsUniversity of OxfordOxfordUK
  4. 4.Department of BiochemistryOxford Centre for Integrative Systems BiologyOxfordUK

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