The Mathematical Institute, University of Oxford, Eprints Archive

Turing instabilities in general systems

Satnoianu, R. A. and Menzinger, M. and Maini, P. K. (2000) Turing instabilities in general systems. Journal of Mathematical Biology, 41 (6). pp. 493-512.

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Abstract

We present necessary and sufficient conditions on the stability matrix of a general n(S2)-dimensional reaction-diffusion system which guarantee that its uniform steady state can undergo a Turing bifurcation. The necessary (kinetic) condition, requiring that the system be composed of an unstable (or activator) and a stable (or inhibitor) subsystem, and the sufficient condition of sufficiently rapid inhibitor diffusion relative to the activator subsystem are established in three theorems which form the core of our results. Given the possibility that the unstable (activator) subsystem involves several species (dimensions), we present a classification of the analytically deduced Turing bifurcations into p (1 h p h (n m 1)) different classes. For n = 3 dimensions we illustrate numerically that two types of steady Turing pattern arise in one spatial dimension in a generic reaction-diffusion system. The results confirm the validity of an earlier conjecture [12] and they also characterise the class of so-called strongly stable matrices for which only necessary conditions have been known before [23, 24]. One of the main consequences of the present work is that biological morphogens, which have so far been expected to be single chemical species [1-9], may instead be composed of two or more interacting species forming an unstable subsystem.

Item Type:Article
Uncontrolled Keywords:Turing systems - Pattern formation - Generalised morphogen - Activator-inhibitor system - Differential-flow instability (DIFI)
Subjects:A - C > Biology and other natural sciences
Research Groups:Centre for Mathematical Biology
ID Code:410
Deposited By:Philip Maini
Deposited On:22 Nov 2006
Last Modified:20 Jul 2009 14:21

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