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

Type
Article

Authors
Madzvamuse, Anotida
Gaffney, Eamonn A.
Maini, Philip K.

KAUST Grant Number
KUK-C1-013-04

Online Publication Date
2009-08-29

Print Publication Date
2010-07

Date
2009-08-29

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. © Springer-Verlag 2009.

Citation
Madzvamuse A, Gaffney EA, Maini PK (2009) Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains. Journal of Mathematical Biology 61: 133–164. Available: http://dx.doi.org/10.1007/s00285-009-0293-4.

Acknowledgements
AM would like to acknowledge Professors Georg Hetzer and Wenxian Shen (Auburn University, USA) for fruitful discussions. EAG: This publication is based on work supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). PKM was partially supported by a Royal Society Wolfson Merit Award.

Publisher
Springer Nature

Journal
Journal of Mathematical Biology

DOI
10.1007/s00285-009-0293-4

PubMed ID
19727733

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