ReviewPhilosophical transactions. Series A, Mathematical, physical, and engineering sciences2021
Modern perspectives on near-equilibrium analysis of Turing systems.
Review in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences, 2021. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 25 papers.
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Who cites it
25 citing papers in PubMed.
- Morphogenetic mechanical metamaterials: Emerging tensor properties from self-organized structures.Nature communications · 2026Article
- Development of whole-limb skeletal patterning through the coordination of growth and self-organization models.PLoS computational biology · 2026Article
- Solution landscape of reaction-diffusion systems reveals a nonlinear mechanism and spatial robustness of pattern formation.Fundamental research · 2026Article
- Virtual twins and the future of human developmental biology.NPJ systems biology and applications · 2026Review
- Numerical bifurcation analysis of turing and symmetry broken patterns of a PDE model for vegetation dynamics.Journal of mathematical biology · 2026Article
- Revisiting Turing's Chemical Basis of Morphogenesis.Bulletin of mathematical biology · 2026Review
- Persistent Homology Classifies Parameter Dependence of Patterns in Turing Systems.Bulletin of mathematical biology · 2025Article
- Actin network heterogeneity tunes activator-inhibitor dynamics at the cell cortex.Proceedings of the National Academy of Sciences of the United States of America · 2025Article
- Transient Instability and Patterns of Reactivity in Diffusive-Chemotaxis Soil Carbon Dynamics.Bulletin of mathematical biology · 2025Article
- Designing reaction-cross-diffusion systems with Turing and wave instabilities.Journal of mathematical biology · 2025Article
- Pattern Formation as a Resilience Mechanism in Cancer Immunotherapy.Bulletin of mathematical biology · 2025Article
- Article
- Widespread biochemical reaction networks enable Turing patterns without imposed feedback.Nature communications · 2024Article
- Interspecific dispersal constraints suppress pattern formation in metacommunities.Philosophical transactions of the Royal Society of London. Series B, Biological sciences · 2024Article
- Review
- Turing Instabilities are Not Enough to Ensure Pattern Formation.Bulletin of mathematical biology · 2024Article
- Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry.Bulletin of mathematical biology · 2024Article
- Physical interactions in non-ideal fluids promote Turing patterns.Journal of the Royal Society, Interface · 2023Article
- Disappearance, division, and route change of excitable reaction-diffusion waves in deformable membranes.Scientific reports · 2023Article
- Concentration-Dependent Domain Evolution in Reaction-Diffusion Systems.Bulletin of mathematical biology · 2023Article
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Authors and funding
4 authors.
Funding
No grant is acknowledged in the PubMed record.
Abstract
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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