
Reaction–diffusion systems are mathematical models that correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: local
chemical reaction
A chemical reaction is a process that leads to the chemistry, chemical transformation of one set of chemical substances to another. When chemical reactions occur, the atoms are rearranged and the reaction is accompanied by an Gibbs free energy, ...
s in which the substances are transformed into each other, and
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
which causes the substances to spread out over a surface in space.
Reaction–diffusion systems are naturally applied in
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in
biology
Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
,
geology
Geology (). is a branch of natural science concerned with the Earth and other astronomical objects, the rocks of which they are composed, and the processes by which they change over time. Modern geology significantly overlaps all other Earth ...
and
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
(neutron diffusion theory) and
ecology
Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ...
. Mathematically, reaction–diffusion systems take the form of semi-linear
parabolic partial differential equation
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena in, for example, engineering science, quantum mechanics and financial ma ...
s. They can be represented in the general form
:
where represents the unknown vector function, is a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagon ...
of
diffusion coefficient
Diffusivity, mass diffusivity or diffusion coefficient is usually written as the proportionality constant between the molar flux due to molecular diffusion and the negative value of the gradient in the concentration of the species. More accurate ...
s, and accounts for all local reactions. The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation of
travelling waves and wave-like phenomena as well as other
self-organized patterns like stripes, hexagons or more intricate structure like
dissipative solitons. Such patterns have been dubbed "
Turing patterns". Each function, for which a reaction diffusion differential equation holds, represents in fact a ''concentration variable''.
One-component reaction–diffusion equations
The simplest reaction–diffusion equation is in one spatial dimension in plane geometry,
:
is also referred to as the
Kolmogorov–Petrovsky–Piskunov equation. If the reaction term vanishes, then the equation represents a pure diffusion process. The corresponding equation is
Fick's second law. The choice yields
Fisher's equation that was originally used to describe the spreading of biological
population
Population is a set of humans or other organisms in a given region or area. Governments conduct a census to quantify the resident population size within a given jurisdiction. The term is also applied to non-human animals, microorganisms, and pl ...
s, the Newell–Whitehead-Segel equation with to describe
Rayleigh–Bénard convection, the more general
Zeldovich–Frank-Kamenetskii equation with and (
Zeldovich number) that arises in
combustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combustion ...
theory, and its particular degenerate case with that is sometimes referred to as the Zeldovich equation as well.
The dynamics of one-component systems is subject to certain restrictions as the evolution equation can also be written in the variational form
:
and therefore describes a permanent decrease of the "free energy"
given by the functional
:
with a potential such that
In systems with more than one stationary homogeneous solution, a typical solution is given by travelling fronts connecting the homogeneous states. These solutions move with constant speed without changing their shape and are of the form with , where is the speed of the travelling wave. Note that while travelling waves are generically stable structures, all non-monotonous stationary solutions (e.g. localized domains composed of a front-antifront pair) are unstable. For , there is a simple proof for this statement:
[P. C. Fife]
Mathematical Aspects of Reacting and Diffusing Systems
Springer (1979) if is a stationary solution and is an infinitesimally perturbed solution, linear stability analysis yields the equation
:
With the ansatz we arrive at the eigenvalue problem
:
of
Schrödinger type where negative eigenvalues result in the instability of the solution. Due to translational invariance is a neutral
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
with the
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
, and all other eigenfunctions can be sorted according to an increasing number of nodes with the magnitude of the corresponding real eigenvalue increases monotonically with the number of zeros. The eigenfunction should have at least one zero, and for a non-monotonic stationary solution the corresponding eigenvalue cannot be the lowest one, thereby implying instability.
To determine the velocity of a moving front, one may go to a moving coordinate system and look at stationary solutions:
:
This equation has a nice mechanical analogue as the motion of a mass with position in the course of the "time" under the force with the damping coefficient c which allows for a rather illustrative access to the construction of different types of solutions and the determination of .
When going from one to more space dimensions, a number of statements from one-dimensional systems can still be applied. Planar or curved wave fronts are typical structures, and a new effect arises as the local velocity of a curved front becomes dependent on the local
radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius ...
(this can be seen by going to
polar coordinates
In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are
*the point's distance from a reference ...
). This phenomenon leads to the so-called curvature-driven instability.
[A. S. Mikhailov, Foundations of Synergetics I. Distributed Active Systems, Springer (1990)]
Two-component reaction–diffusion equations
Two-component systems allow for a much larger range of possible phenomena than their one-component counterparts. An important idea that was first proposed by
Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer ...
is that a state that is stable in the local system can become unstable in the presence of
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
.
A linear stability analysis however shows that when linearizing the general two-component system
:
a
plane wave
In physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
perturbation
:
of the stationary homogeneous solution will satisfy
:
Turing's idea can only be realized in four
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of systems characterized by the signs of the
Jacobian of the reaction function. In particular, if a finite wave vector is supposed to be the most unstable one, the Jacobian must have the signs
:
This class of systems is named ''activator-inhibitor system'' after its first representative: close to the ground state, one component stimulates the production of both components while the other one inhibits their growth. Its most prominent representative is the
FitzHugh–Nagumo equation
:
with which describes how an
action potential
An action potential (also known as a nerve impulse or "spike" when in a neuron) is a series of quick changes in voltage across a cell membrane. An action potential occurs when the membrane potential of a specific Cell (biology), cell rapidly ri ...
travels through a nerve.
Here, and are positive constants.
When an activator-inhibitor system undergoes a change of parameters, one may pass from conditions under which a homogeneous ground state is stable to conditions under which it is linearly unstable. The corresponding
bifurcation
Bifurcation or bifurcated may refer to:
Science and technology
* Bifurcation theory, the study of sudden changes in dynamical systems
** Bifurcation, of an incompressible flow, modeled by squeeze mapping the fluid flow
* River bifurcation, the for ...
may be either a
Hopf bifurcation to a globally oscillating homogeneous state with a dominant wave number or a ''Turing bifurcation'' to a globally patterned state with a dominant finite wave number. The latter in two spatial dimensions typically leads to stripe or hexagonal patterns.
Image:Turing_bifurcation_1.gif, Noisy initial conditions at ''t'' = 0.
Image:Turing_bifurcation_2.gif, State of the system at ''t'' = 10.
Image:Turing_bifurcation_3.gif, Almost converged state at ''t'' = 100.
For the Fitzhugh–Nagumo example, the neutral stability curves marking the boundary of the linearly stable region for the Turing and Hopf bifurcation are given by
:
If the bifurcation is subcritical, often localized structures (
dissipative solitons) can be observed in the
hysteretic region where the pattern coexists with the ground state. Other frequently encountered structures comprise pulse trains (also known as
periodic travelling waves), spiral waves and target patterns. These three solution types are also generic features of two- (or more-) component reaction–diffusion equations in which the local dynamics have a stable limit cycle
Image:reaction_diffusion_spiral.gif, Rotating spiral.
Image:reaction_diffusion_target.gif, Target pattern.
Image:reaction_diffusion_stationary_ds.gif, Stationary localized pulse (dissipative soliton).
Three- and more-component reaction–diffusion equations
For a variety of systems, reaction–diffusion equations with more than two components have been proposed, e.g. the
Belousov–Zhabotinsky reaction, for
blood clotting
Coagulation, also known as clotting, is the process by which blood changes from a liquid to a gel, forming a thrombus, blood clot. It results in hemostasis, the cessation of blood loss from a damaged vessel, followed by repair. The process of co ...
, fission waves or planar
gas discharge systems.
It is known that systems with more components allow for a variety of phenomena not possible in systems with one or two components (e.g. stable running pulses in more than one spatial dimension without global feedback). An introduction and systematic overview of the possible phenomena in dependence on the properties of the underlying system is given in.
Applications and universality
In recent times, reaction–diffusion systems have attracted much interest as a prototype model for
pattern formation
The science of pattern formation deals with the visible, (statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature.
In developmental biology, pattern formation refers to the generation of c ...
.
The above-mentioned patterns (fronts, spirals, targets, hexagons, stripes and dissipative solitons) can be found in various types of reaction–diffusion systems in spite of large discrepancies e.g. in the local reaction terms. It has also been argued that reaction–diffusion processes are an essential basis for processes connected to
morphogenesis
Morphogenesis (from the Greek ''morphê'' shape and ''genesis'' creation, literally "the generation of form") is the biological process that causes a cell, tissue or organism to develop its shape. It is one of three fundamental aspects of deve ...
in biology and may even be related to animal coats and skin pigmentation.
Other applications of reaction–diffusion equations include ecological invasions, spread of epidemics, tumour growth, dynamics of fission waves,
wound healing and visual hallucinations. Another reason for the interest in reaction–diffusion systems is that although they are nonlinear partial differential equations, there are often possibilities for an analytical treatment.
Experiments
Well-controllable experiments in chemical reaction–diffusion systems have up to now been realized in three ways. First, gel reactors or filled capillary tubes may be used. Second,
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
pulses on
catalytic surfaces have been investigated. Third, the propagation of running nerve pulses is modelled using reaction–diffusion systems.
Aside from these generic examples, it has turned out that under appropriate circumstances electric transport systems like plasmas or semiconductors can be described in a reaction–diffusion approach. For these systems various experiments on pattern formation have been carried out.
Numerical treatments
A reaction–diffusion system can be solved by using methods of
numerical mathematics. There exist several numerical treatments in research literature.
Numerical solution methods for complex
geometries
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.
Types, methodologies, and terminologies of geometry. ...
are also proposed. Reaction-diffusion systems are described to the highest degree of detail with particle based simulation tools like SRSim or ReaDDy which employ among others reversible interacting-particle reaction dynamics.
[Fröhner, Christoph, and Frank Noé. "Reversible interacting-particle reaction dynamics." The Journal of Physical Chemistry B 122.49 (2018): 11240-11250.]
See also
*
Autowave
*
Diffusion-controlled reaction
Diffusion-controlled (or diffusion-limited) chemical reaction, reactions are reactions in which the reaction rate is equal to the rate of transport of the reactants through the reaction medium (usually a solution). The process of chemical reactio ...
*
Chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is different from chemical thermodynamics, which deals with the direction in which a ...
*
Phase space method
*
Autocatalytic reactions and order creation
*
Pattern formation
The science of pattern formation deals with the visible, (statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature.
In developmental biology, pattern formation refers to the generation of c ...
*
Patterns in nature
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, wave ...
*
Periodic travelling wave
*
Self-similar solutions
*
Diffusion equation
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
*
Stochastic geometry
In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which exten ...
*
MClone
*
The Chemical Basis of Morphogenesis
"The Chemical Basis of Morphogenesis" is an article that the English mathematician Alan Turing wrote in 1952. It describes how patterns in nature, such as stripes and spirals, can arise naturally from a homogeneous, uniform state. The theory, w ...
*
Turing pattern
*
Multi-state modeling of biomolecules
Examples
*
Fisher's equation
*
Zeldovich–Frank-Kamenetskii equation
*
FitzHugh–Nagumo model
The FitzHugh–Nagumo model (FHN) describes a prototype of an excitable system (e.g., a neuron
A neuron (American English), neurone (British English), or nerve cell, is an membrane potential#Cell excitability, excitable cell (biology), cell t ...
*Wrinkle paint
References
External links
Reaction–Diffusion by the Gray–Scott Model: Pearson's parameterizationa visual map of the parameter space of Gray–Scott reaction diffusion.
A thesis on reaction–diffusion patterns with an overview of the field
{{DEFAULTSORT:Reaction-diffusion system
Mathematical modeling
Parabolic partial differential equations
Reaction mechanisms
Functions of space and time