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Turing instabilities in general systems

By R. A. Satnoianu, M. Menzinger and P. K. Maini


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

Topics: Biology and other natural sciences
Year: 2000
DOI identifier: 10.1007/s002850000056
OAI identifier:

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  8. (1994). H.L.: Experimental observation of self-replicating spots in a reaction-diffusion system.
  9. (1991). H.L.: Transition to chemical turbulence.
  10. (1997). H.N.P.: Spatial pattern formation in chemical and biological media.
  11. (1971). I.: Thermodynamic theory of structure, stability and fluctuations.
  12. (1993). I.R.: Turing structures in simple chemical reactions.
  13. (1969). Interactions of reaction and diffusion in open systems.
  14. (1938). Mathematical Biophysics.
  15. (1982). Models of biological pattern formation, doi
  16. (2000). Physical mechanism for segmentation in early development, submitted doi
  17. (1965). Qualitative economics and the stability of equilibrium.
  18. (1986). S.K.: A new model for oscillatory behaviour in closed systems: the autocatalator.
  19. (1998). S.K.: Interaction between Hopf and convective instabilities in a flow reactor with cubic autocatalator kinetics.
  20. (1993). Self-organization induced by the differential flow of activator and inhibitor.
  21. (1998). Spatio-temporal structures in differential-flow reactors. doi
  22. (1980). Synchronized and differentiated modes of cellular dynamics.
  23. (1952). The chemical basis of morphogenesis.
  24. (1993). The Origins of Order: Self-Organization and Selection in Evolution. doi
  25. (1978). Three types of matrix stability. Linear algebra and its applications 20,
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