Turing instability of the periodic solutions for reaction-diffusion systems with cross-diffusion and the patch model with cross-diffusion-like coupling

2021 ◽  
Vol 281 ◽  
pp. 379-410
Author(s):  
Fengqi Yi
Keyword(s):  
Periodic Solutions ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Patch Model ◽  
Reaction Diffusion Systems ◽  
Cross Diffusion ◽  
Diffusion Systems

Turing space in reaction-diffusion systems with density-dependent cross diffusion

2013 ◽  
Vol 87 (3) ◽  
Author(s):  
E. P. Zemskov ◽  
K. Kassner ◽  
M. J. B. Hauser ◽  
W. Horsthemke
Keyword(s):  
Reaction Diffusion ◽  
Density Dependent ◽  
Reaction Diffusion Systems ◽  
Cross Diffusion ◽  
Diffusion Systems

Instability induced by cross-diffusion in reaction–diffusion systems

2010 ◽  
Vol 11 (2) ◽  
pp. 1036-1045 ◽  
Author(s):  
Canrong Tian ◽  
Zhigui Lin ◽  
Michael Pedersen
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Cross Diffusion ◽  
Diffusion Systems

scholarly journals CROSS-DIFFUSION INDUCED INSTABILITY AND STABILITY IN REACTION-DIFFUSION SYSTEMS

2011 ◽  
Vol 1 (1) ◽  
pp. 95-119 ◽  
Author(s):  
Junping Shi ◽  
◽  
Zhifu Xie ◽  
Kristina Little ◽  
◽  
...  
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Cross Diffusion ◽  
Diffusion 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’.

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