Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains
- 432 Downloads
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.
KeywordsConvection-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
Unable to display preview. Download preview PDF.
- Gjorgjieva J, Jacobsen J (2007) Turing patterns on growing spheres: the exponential case. Dynamical systems and differential equations. In: Proceedings of the 6th AIMS international conference. Discrete continuous dynamical systems supplement, USA, pp 436–445Google Scholar
- Golub GH, Van Loan CF (1996) Matrix computations. JHU Press ISBN 0801854148Google Scholar
- Maini PK, Crampin EJ, Madzvamuse A, Wathen AJ, Thomas RDK (2002) Implications of domain growth in morphogenesis. In: Capaso V (ed) Mathematical modelling and computing in biology and medicine. Proceedings of the 5th European conference for mathematics and theoretical biology: conference, Milan, vol 153, pp 67–73Google Scholar
- Murray JD (2002) Mathematical biology I and II, 3rd edn. Springer, BerlinGoogle Scholar