Satnoianu, R. A. and Menzinger, M. and Maini, P. K. (2000) Turing instabilities in general systems. Journal of Mathematical Biology, 41 (6). pp. 493512.

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Abstract
We present necessary and sufficient conditions on the stability matrix of a general n(S2)dimensional reactiondiffusion 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 reactiondiffusion system. The results confirm the validity of an earlier conjecture [12] and they also characterise the class of socalled 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 [19], 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  Activatorinhibitor system  Differentialflow 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:  29 May 2015 18:21 
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