Pattern formation in generalized Turing systems

1994 ◽  
Vol 32 (4) ◽  
pp. 345-393 ◽  
Author(s):  
R. Dillon ◽  
P. K. Maini ◽  
H. G. Othmer
Keyword(s):  
Pattern Formation ◽  
Turing Systems

Friction-Induced Pattern Formation and Turing Systems

Langmuir ◽  
2011 ◽  
Vol 27 (8) ◽  
pp. 4772-4779 ◽  
Author(s):  
Vahid Mortazavi ◽  
Michael Nosonovsky
Keyword(s):  
Pattern Formation ◽  
Turing Systems

scholarly journals Pattern formation from spatially heterogeneous reaction–diffusion systems

2021 ◽  
Vol 379 (2213) ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Pattern Formation ◽  
Spatial Heterogeneity ◽  
Heterogeneous Systems ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Instability Criterion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatially Homogeneous ◽  
Turing Systems

First proposed by Turing in 1952, the eponymous Turing instability and Turing pattern remain key tools for the modern study of diffusion-driven pattern formation. In spatially homogeneous Turing systems, one or a few linear Turing modes dominate, resulting in organized patterns (peaks in one dimension; spots, stripes, labyrinths in two dimensions) which repeats in space. For a variety of reasons, there has been increasing interest in understanding irregular patterns, with spatial heterogeneity in the underlying reaction–diffusion system identified as one route to obtaining irregular patterns. We study pattern formation from reaction–diffusion systems which involve spatial heterogeneity, by way of both analytical and numerical techniques. We first extend the classical Turing instability analysis to track the evolution of linear Turing modes and the nascent pattern, resulting in a more general instability criterion which can be applied to spatially heterogeneous systems. We also calculate nonlinear mode coefficients, employing these to understand how each spatial mode influences the long-time evolution of a pattern. Unlike for the standard spatially homogeneous Turing systems, spatially heterogeneous systems may involve many Turing modes of different wavelengths interacting simultaneously, with resulting patterns exhibiting a high degree of variation over space. We provide a number of examples of spatial heterogeneity in reaction–diffusion systems, both mathematical (space-varying diffusion parameters and reaction kinetics, mixed boundary conditions, space-varying base states) and physical (curved anisotropic domains, apical growth of space domains, chemicalsimmersed within a flow or a thermal gradient), providing a qualitative understanding of how spatial heterogeneity can be used to modify classical Turing patterns. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

scholarly journals Turing systems as models of complex pattern formation

2004 ◽  
Vol 34 (2a) ◽  
Author(s):  
Teemu Leppänen ◽  
Mikko Karttunen ◽  
R.A. Barrio ◽  
Kimmo Kaski
Keyword(s):  
Pattern Formation ◽  
Complex Pattern ◽  
Turing Systems

Pattern formation in the splay Freedericks transition of a nematic side-group polysiloxane

1993 ◽  
Vol 3 (6) ◽  
pp. 865-889 ◽  
Author(s):  
Norbert Schwenk ◽  
Hans Wolfgang Spiess
Keyword(s):  
Pattern Formation ◽  
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