A theory of pattern formation for reaction–diffusion systems on temporal networks

Gorder, Robert A. Van

doi:10.1098/rspa.2020.0753

Cited by 21 publications

(18 citation statements)

References 68 publications

(76 reference statements)

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“…Furthermore, in addition to static networks, it is possible to study temporal networks, which have topology or structure that evolves in time. A theory of pattern formation on undirected (or, purely diffusive) temporal networks was recently presented [74], and it may be interesting to attempt a theory for spatio-temporal pattern formation on directed temporal networks by combining some of the techniques we describe in the present paper with those of [74].…”

Section: Discussionmentioning

confidence: 99%

Pattern formation from spatially heterogeneous reaction–diffusion systems

Gorder

2021

Phil. Trans. R. Soc. A.

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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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Section: Discussionmentioning

confidence: 99%

Pattern formation from spatially heterogeneous reaction–diffusion systems

Gorder

2021

Phil. Trans. R. Soc. A.

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“…Following from these ideas, Van Gorder [ 149 ] further developed the approach presented in [ 138 ], generalizing both Turing and Benjamin-Feir-type instabilities to a large class of non-autonomous reaction–diffusion systems incorporating time-dependent kinetics, diffusion coefficients and base states. This analysis was recently applied to time-varying networks [ 150 ].…”

Section: Heterogeneitymentioning

confidence: 99%

Modern perspectives on near-equilibrium analysis of Turing systems

Krause

Gaffney

Maini

et al. 2021

Phil. Trans. R. Soc. A.

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In the nearly seven decades since the publication of Alan Turing’s work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction–diffusion theory. Some of these developments were nascent in Turing’s paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction–diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of ‘trivial’ base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

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“…Following from these ideas, Van Gorder [189] further developed the approach presented in [193], generalizing both Turing and Benjamin-Feir-type instabilities to a large class of non-autonomous reaction-diffusion systems incorporating time-dependent kinetics, diffusion coefficients, and base states. This analysis was recently applied to time-varying networks [191].…”

Section: (B) Evolving Domains and Non-autonomous Forcingmentioning

confidence: 99%

Modern Perspectives on Near-Equilibrium Analysis of Turing Systems

Krause,

Gaffney,

Maini

et al. 2021

Preprint

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In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations, and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of 'trivial' base states. We emphasise important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality.

show abstract

A theory of pattern formation for reaction–diffusion systems on temporal networks

Cited by 21 publications

References 68 publications

Pattern formation from spatially heterogeneous reaction–diffusion systems

Pattern formation from spatially heterogeneous reaction–diffusion systems

Modern perspectives on near-equilibrium analysis of Turing systems

Modern Perspectives on Near-Equilibrium Analysis of Turing Systems

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