History
A short history of reaction–diffusion
How a single idea—that reacting and diffusing substances can give rise to patterns—grew into a field spanning biology, chemistry, and mathematics, and where the field stands today.
Reaction–diffusion is the mathematics of things that both react and spread. Picture chemicals that turn into one another while diffusing through a tissue, or animals that breed while wandering across a landscape: the same family of partial differential equations (PDEs) describes both. Out of this simple combination comes a surprising amount of order.
In 1952 Alan Turing showed that reaction and diffusion together can make pattern out of uniformity — the spots, stripes and spirals now called Turing patterns. This helped launch pattern formation and mathematical biology as research fields and tied them to chemistry, physics and, more recently, to image analysis.
A harder cousin of the theory is cross-diffusion, where the movement of one species or substance depends on the others. Such systems are notoriously difficult to analyze, and a key modern tool for them is the entropy method. This page traces the story from Fisher and Turing to today’s cross-diffusion and entropy-based analysis.
Milestones
- 1937
Fisher–KPP: foundations of the classical theory
In June 1937 Ronald Fisher published a nonlinear equation for the spatial spread of an advantageous gene. In the same year Kolmogorov, Petrovsky and Piskunov independently obtained the classic Fisher–KPP results on traveling waves, including the minimal wave speed. Together these works laid the foundations of the classical Fisher–KPP theory.
- 1952
Turing’s theory of morphogenesis
In his 1952 paper The Chemical Basis of Morphogenesis, Alan Turing showed mathematically that two chemicals (‘morphogens’) that react and diffuse at different speeds can, on their own, turn a featureless state into spots and stripes. The mechanism is now widely known as diffusion-driven instability — a later term, not Turing’s own — and underlies Turing patterns.
- 1950s–1964
The Belousov–Zhabotinsky reaction
In the 1950s Boris Belousov discovered a chemical reaction that oscillates in time, changing color back and forth. His early reports met skepticism and did not gain wide recognition for years. Anatol Zhabotinsky later refined and explained the system, giving science its textbook chemical oscillator and, when left unstirred, its traveling reaction-diffusion waves.
- 1961–1962
The FitzHugh–Nagumo model of excitable nerves
In 1961 Richard FitzHugh reduced the complicated Hodgkin–Huxley description of a nerve signal to a simple two-variable model of excitable dynamics. In 1962 Nagumo and colleagues implemented an equivalent active transmission line. Its spatially extended form became a standard reaction-diffusion model of excitable media — systems that fire and reset, like nerves and heart tissue.
- 1968–1977
Prigogine’s dissipative structures and the Brusselator
In 1968 Ilya Prigogine and René Lefever introduced the ‘Brusselator’, a theoretical reaction scheme that exhibits oscillations and, with diffusion, Turing-type patterns. Prigogine received the 1977 Nobel Prize in Chemistry for his contributions to nonequilibrium thermodynamics and dissipative structures.
- 1972
The Gierer–Meinhardt activator–inhibitor model
Alfred Gierer and Hans Meinhardt proposed a now-classic account of biological pattern formation: short-range self-enhancing activation balanced by longer-range inhibition. Tuning the ingredients yields spots, stripes and nets, and their model was closely tied to regeneration and pattern formation in Hydra.
- 1977; 1979
Fife–McLeod and Fife’s 1979 book
In 1977 Paul Fife and J. B. McLeod proved key results showing that solutions of nonlinear diffusion equations can settle into traveling fronts. Fife’s 1979 book Mathematical Aspects of Reacting and Diffusing Systems then organized the subject, helping establish reaction-diffusion as a rigorous branch of PDE analysis.
- 1979
The Shigesada–Kawasaki–Teramoto cross-diffusion model
In their 1979 paper Spatial segregation of interacting species, Nanako Shigesada, Kohkichi Kawasaki and Ei Teramoto let each population’s movement depend on the crowding of the other — ‘cross-diffusion’. This small change created a system so mathematically stubborn that fully understanding it took decades.
- 1980
Cross-diffusion and spatial segregation
Masayasu Mimura and Kohkichi Kawasaki showed that population-pressure cross-diffusion can induce spatial segregation and stabilize the coexistence of two competing species. It was an early sign that cross-diffusion is a genuine pattern-forming mechanism, not just a technical add-on.
- 1981–1989
Murray’s animal coat patterns and ‘Mathematical Biology’
In 1981 James Murray showed how a reaction-diffusion mechanism acting on domains of different size and shape could explain why a leopard has spots while its tail has stripes. His 1988 popular article How the Leopard Gets Its Spots and 1989 textbook Mathematical Biology brought these ideas to a wide audience and trained a generation of researchers.
- 1990
Turing patterns seen in the lab
Nearly forty years after Turing’s prediction, Patrick De Kepper’s group in Bordeaux produced a widely accepted experimental realization of stationary Turing patterns — hexagonal spots and stripes — in a carefully designed gel-based chemical reactor (the CIMA reaction). It gave the theory strong experimental support.
- 2009
Turing patterns in living animals
Shigeru Kondo’s team in Osaka showed pigment-cell dynamics and interactions in zebrafish that are consistent with a Turing-type reaction-diffusion mechanism. The experiments did not establish a classical chemical reaction-diffusion origin in the strict sense, but gave striking evidence that Turing-type dynamics can operate in living biology.
- 2015
The boundedness-by-entropy method
Ansgar Jüngel at TU Wien developed a method for a broad class of cross-diffusion systems with entropy or gradient-flow structure. By rewriting the equations in ‘entropy variables’, he proved global existence of bounded weak solutions under the corresponding structural assumptions — without the usual maximum-principle tricks.
- 2017/2018
Cross-diffusion for many species
Building on the entropy approach, Xiuqing Chen, Esther Daus and Ansgar Jüngel proved global existence for reaction–cross-diffusion systems with any number of competing species under detailed-balance or weak-cross-diffusion assumptions. Published online in 2017 and in print in 2018, the result showed how entropy methods extend to broad multi-species models.
Scientists who shaped the field
Ronald A. Fisher
University College London / Cambridge · 1890–1962
Independently of the KPP group, he helped launch reaction-diffusion theory in 1937. His Fisher equation models an advantageous gene spreading through a population as a traveling wave; he was also a founder of modern statistics.
Andrey Kolmogorov
Moscow State University · 1903–1987
With Ivan Petrovsky and Nikolai Piskunov (the ‘KPP’ paper of 1937), he gave the first rigorous analysis of a reaction-diffusion equation, including the minimal speed of its traveling wave.
Alan Turing
Manchester / Cambridge · 1912–1954
In 1952 he showed that differing diffusion rates can destabilize a uniform state and create stationary patterns. This mechanism, later called diffusion-driven instability, underlies Turing patterns and much of modern pattern formation.
Ilya Prigogine
Université Libre de Bruxelles / University of Texas · 1917–2003
Leader of the Brussels school of nonequilibrium thermodynamics. He developed the theory of dissipative structures and co-introduced the Brusselator model; in 1977 he received the Nobel Prize in Chemistry for contributions to nonequilibrium thermodynamics and dissipative structures.
Boris Belousov
Institute of Biophysics, Moscow · 1893–1970
A chemist who discovered a color-changing oscillating reaction in the 1950s. His early reports met skepticism and did not gain wide recognition for years; refined by Zhabotinsky, the system became the Belousov–Zhabotinsky reaction, a classic experimental reaction-diffusion system.
Hans Meinhardt
Max Planck Institute, Tübingen · 1938–2016
With Alfred Gierer he created the activator–inhibitor model of biological pattern formation (1972) and spent his career at the Max Planck Institute in Tübingen explaining natural patterns, from Hydra to seashells.
James D. Murray
Oxford / University of Washington · b. 1931
A founder of modern mathematical biology. He applied reaction-diffusion theory to animal coat patterns in 1981, brought the idea to a broad audience in 1988 with How the Leopard Gets Its Spots, and wrote the influential 1989 textbook Mathematical Biology.
Masayasu Mimura
Hiroshima / Meiji University, Japan · 1941–2021
A leading Japanese researcher of reaction-diffusion pattern formation. With Kawasaki he showed how cross-diffusion segregates competing species, and he founded the MIMS institute at Meiji University.
Paul C. Fife
University of Arizona / University of Utah · 1930–2014
He helped turn reaction-diffusion into a rigorous part of PDE analysis, proving key results on traveling fronts and writing the 1979 standard reference Mathematical Aspects of Reacting and Diffusing Systems.
Nanako Shigesada
Mathematical biologist and theoretical ecologist · co-author of the SKT model (1979)
With Kawasaki and Teramoto she introduced the cross-diffusion population model (SKT, 1979), in which each species’ movement responds to the crowding of the others — a foundational model for the mathematical theory of cross-diffusion.
Ansgar Jüngel
TU Wien (Vienna), Austria · professor since 2006
A leading figure in the analysis of cross-diffusion systems. His boundedness-by-entropy method uses ‘entropy variables’ to obtain global bounded weak solutions for broad classes with the required entropy structure; he extended it to systems with many species and was awarded a 2021 ERC Advanced Grant.
Universities and research centers
TU Wien
Vienna, Austria
Its Institute for Analysis and Scientific Computing is home to Ansgar Jüngel’s rigorous PDE work on cross-diffusion and the boundedness-by-entropy method. He was awarded a 2021 ERC Advanced Grant.
University of Oxford
Oxford, United Kingdom
Its Wolfson Centre for Mathematical Biology was founded in 1983 under James Murray; Philip Maini has led it since 1992. It is a major home for reaction-diffusion pattern formation and biomedical modeling.
University of Cambridge
Cambridge, United Kingdom
Alan Turing was a fellow of King’s College here; its DAMTP department remains active in reaction-diffusion and Turing-instability research.
Meiji University (MIMS)
Tokyo, Japan
Home of the Meiji Institute for Advanced Study of Mathematical Sciences (MIMS), created in 2007. Masayasu Mimura was its founder and first director; the institute studies reaction-diffusion and self-organized pattern formation.
Université de Bordeaux / CNRS
Bordeaux, France
At its Centre de Recherche Paul Pascal, Patrick De Kepper’s group produced a widely accepted experimental realization of stationary Turing patterns in the CIMA chemical reaction.
Université Libre de Bruxelles
Brussels, Belgium
Home of Prigogine’s ‘Brussels school’ — birthplace of the Brusselator and the theory of dissipative structures (Nobel Prize in Chemistry, 1977).
Osaka University
Osaka, Japan
Shigeru Kondo’s work at the Graduate School of Frontier Biosciences investigated cellular mechanisms of zebrafish stripes, with dynamics consistent with a Turing-type reaction-diffusion mechanism.
Heidelberg University
Heidelberg, Germany
Anna Marciniak-Czochra studies reaction-diffusion–ODE systems, diffusion-driven instability, and mathematical models of stem-cell dynamics at Heidelberg University.
How this connects to our work
Our group works in the living continuation of this story. The cross-diffusion systems pioneered by Shigesada, Kawasaki and Teramoto, and the entropy methods brought to maturity by Ansgar Jüngel, are exactly the tools we build on — strongly coupled PDEs with a gradient-flow (entropy) structure, and estimates that keep solutions physically meaningful.
We carry the same circle of ideas beyond biology: into Dirichlet-type uncertainty models, variational regularization, and medical image segmentation, where diffusion, energy and entropy again help turn noisy data into clear structure. Seen this way, the history of reaction-diffusion, cross-diffusion and pattern formation is not a closed chapter but the foundation of our current research.