title: Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains.
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.
authors.affiliation: Department of Mathematics, University of Sussex, Mantell Building, Brighton, BN1 9RF, UK. a.madzvamuse@sussex.ac.uk
authors.forename: Anotida
authors.initials: A
authors.lastname: Madzvamuse
authors.forename: Eamonn A
authors.initials: EA
authors.lastname: Gaffney
authors.forename: Philip K
authors.initials: PK
authors.lastname: Maini
mesh_terms: D004058:Diffusion; D006128:Growth; D007700:Kinetics; D008954:Models, Biological; D008956:Models, Chemical; D015233:Models, Statistical