### Reaction-diffusion equation

**Reaction–diffusion systems** are mathematical models which explain how the concentration of one or more substances distributed in space changes under the influence of two processes: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space.

Reaction–diffusion systems are naturally applied in chemistry. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in biology, geology and physics and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form

- $$

\partial_t \boldsymbol{q} = \underline{\underline{\boldsymbol{D}}} \,\nabla^2 \boldsymbol{q} + \boldsymbol{R}(\boldsymbol{q}),

where each component of the vector **q**(**x**,*t*) represents the concentration of one substance, * D* is a diagonal matrix of diffusion coefficients, and

**R**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.

## Contents

## One-component reaction–diffusion equations

The most simple reaction–diffusion equation concerning the concentration *u* of a single substance in one spatial dimension,

- $$

\partial_t u = D \partial^2_x u + R(u),

is also referred to as the KPP (Kolmogorov-Petrovsky-Piskounov) equation.^{[1]} If the reaction term vanishes, then the equation represents a pure diffusion process. The corresponding equation is Fick's second law. The choice *R*(*u*) = *u*(1-*u*) yields Fisher's equation that was originally used to describe the spreading of biological populations,^{[2]} the Newell-Whitehead-Segel equation with *R*(*u*) = *u*(1 − *u*^{2}) to describe Rayleigh-Benard convection,^{[3]}^{[4]} the more general Zeldovich equation with *R*(*u*) = *u*(1 − *u*)(*u* − *α*) and 0 < *α* < 1 that arises in combustion theory,^{[5]} and its particular degenerate case with *R*(*u*) = *u*^{2} − *u*^{3} that is sometimes referred to as the Zeldovich equation as well.^{[6]}

The dynamics of one-component systems is subject to certain restrictions as the evolution equation can also be written in the variational form

- $$

\partial_t u=-\frac{\delta\mathfrak L}{\delta u}

and therefore describes a permanent decrease of the "free energy" $\backslash mathfrak\; L$ given by the functional

- $\backslash mathfrak\; L=\backslash int\backslash limits\_\{-\backslash infty\}^\backslash infty\backslash left[\backslash frac$

D2(\partial_xu)^2-V(u)\right]\text{d}x

with a potential *V*(*u*) such that *R*(*u*)=d*V*(*u*)/d*u*.

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 *u*(*x*, *t*) = û(*ξ*) with *ξ* = *x* − *ct*, where *c* 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 *c* = 0, there is a
simple proof for this statement:^{[7]} if *u _{0}*(

*x*) is a stationary solution and

*u*=

*u*

_{0}(

*x*) +

*ũ*(

*x*,

*t*) is an infinitesimally perturbed solution, linear stability analysis yields the equation

- $$

\partial_t \tilde{u}=D\partial_x^2 \tilde{u}-U(x)\tilde{u},\quad U(x) = -R^{\prime}(u)|_{u=u_0(x)}.

With the ansatz *ũ* = *ψ*(*x*)exp(−*λt*) we arrive at the eigenvalue problem

- $\backslash hat\; H\backslash psi=\backslash lambda\backslash psi,\; \backslash qquad$

\hat H=-D\partial_x^2+U(x),

of Schrödinger type where negative eigenvalues result in the instability of the solution. Due to translational invariance *ψ* = ∂_{x}*u*_{0}(*x*) is a neutral eigenfunction with the eigenvalue λ = 0, and all other eigenfunctions can be sorted according to an increasing number of knots with the magnitude of the corresponding real eigenvalue increases monotonically with the number of zeros. The eigenfunction *ψ* = ∂_{x} *u*_{0}(*x*) should have at least one zero, and for a non-monotonic stationary solution the corresponding eigenvalue *λ* = 0 cannot be the
lowest one, thereby implying instability.

To determine the velocity *c* of a moving front, one may go to a moving coordinate system and look at stationary solutions:

- $$

D \partial^2_{\xi}\hat{u}(\xi)+ c\partial_{\xi} \hat{u}(\xi)+R(\hat{u}(\xi))=0.

This equation has a nice mechanical analogue as the motion of a
mass *D* with position *û* in the course of the "time" *ξ* under
the force *R* with the damping coefficient c which allows for a
rather illustrative access to the construction of different
types of solutions and the determination of *c*.

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 (this can be
seen by going to polar coordinates). This phenomenon leads
to the so-called curvature-driven instability.^{[8]}

## 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 is that a state
that is stable in the local system should become unstable in
the presence of diffusion.^{[9]}

A linear stability analysis however shows that when linearizing the general two-component system

- $\backslash left(\; \backslash begin\{array\}\{c\}$

\partial_t u\\ \partial_t v \end{array} \right) = \left(\begin{array}{cc} D_u &0\\0&D_v \end{array}\right) \left( \begin{array}{c} \partial_{xx} u\\ \partial_{xx} v \end{array}\right) + \left(\begin{array}{c} F(u,v)\\G(u,v) \end{array}\right)

a plane wave perturbation

- $$

\tilde{\boldsymbol{q}}_{\boldsymbol{k}}(\boldsymbol{x},t) = \left(\begin{array}{c} \tilde{u}(t)\\\tilde{v}(t)\end{array}\right) e^{i \boldsymbol{k} \cdot \boldsymbol{x}}

of the stationary homogeneous solution will satisfy

- $$

\left( \begin{array}{c} \partial_t \tilde{u}_{\boldsymbol{k}}(t)\\ \partial_t \tilde{v}_{\boldsymbol{k}}(t) \end{array}

\right) = -k^2\left(

\begin{array}{c} D_u \tilde{u}_{\boldsymbol{k}}(t)\\ D_v\tilde{v}_{\boldsymbol{k}}(t) \end{array}

\right) + \boldsymbol{R}^{\prime} \left(

\begin{array}{c} \tilde{u}_{\boldsymbol{k}}(t)\\ \tilde{v}_{\boldsymbol{k}}(t) \end{array}

\right).

Turing's idea can only be realized in four
equivalence classes of systems characterized
by the signs of the Jacobian
**R'** of the reaction function. In particular, if a finite
wave vector **k** is supposed to be the most unstable one,
the Jacobian must have the signs

- $\backslash left(\backslash begin\{array\}\{cc\}\; +\&-\backslash \backslash +\&-\backslash end\{array\}\backslash right),$

\quad \left(\begin{array}{cc} +&+\\-&-\end{array}\right), \quad \left(\begin{array}{cc} -&+\\-&+\end{array}\right), \quad \left(\begin{array}{cc} -&-\\+&+\end{array}\right).

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

- $$

\begin{align} \partial_t u &= d_u^2 \,\nabla^2 u + f(u) - \sigma v, \\ \tau \partial_t v &= d_v^2 \,\nabla^2 v + u - v \end{align}

with *ƒ*(*u*) = *λu* − *u*^{3} − *κ* which describes how an action potential travels
through a nerve.^{[10]}^{[11]} Here, *d _{u}*,

*d*,

_{v}*τ*,

*σ*and

*λ*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 may be either
a Hopf bifurcation to a globally oscillating homogeneous
state with a dominant wave number *k* = 0 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.

- Turing bifurcation 1.gif
Noisy initial conditions at

*t*= 0. - Turing bifurcation 2.gif
State of the system at

*t*= 10. - 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

- $$

\begin{align} q_{\text{n}}^H(k): &{}\quad \frac{1}{\tau} + (d_u^2 + \frac{1}{\tau} d_v^2)k^2 & =f^{\prime}(u_{h}),\\[6pt] q_{\text{n}}^T(k): &{}\quad \frac{\kappa}{1 + d_v^2 k^2}+ d_u^2 k^2 & = f^{\prime}(u_{h}). \end{align}

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^{[12]}

- Reaction diffusion spiral.gif
Rotating spiral.

- Reaction diffusion target.gif
Target pattern.

- 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. as models
for the Belousov-Zhabotinsky reaction,
,^{[13]} for blood clotting^{[14]} or planar gas discharge systems.
^{[15]}

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),.^{[16]} An introduction and systematic
overview of the possible phenomena in dependence on the properties
of the underlying system is given in.^{[17]}

## Applications and universality

In recent times, reaction–diffusion systems have attracted much interest as a prototype model for pattern formation. 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 in biology^{[18]} and may even be related to animal coats and skin pigmentation.^{[19]}^{[20]} **
Other applications of reaction-diffusion equations include ecological invasions,**^{[21]} spread of epidemics,^{[22]} tumour growth^{[23]}^{[24]}^{[25]} and wound healing.^{[26]} 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.^{[7]}^{[8]}^{[27]}^{[28]}^{[29]}

## Experiments

Well-controllable experiments in chemical reaction-diffusion systems have up to now
been realized in three ways. First, gel reactors^{[30]} or filled capillary tubes^{[31]} may be used. Second, temperature pulses on catalytic surfaces
have been investigated.^{[32]}^{[33]}
Third, the propagation of running nerve pulses is modelled
using reaction-diffusion systems.^{[10]}^{[34]}

Aside from these generic examples, it has turned out that under appropriate
circumstances electric transport systems like plasmas^{[35]} or semiconductors^{[36]} can be
described in a reaction-diffusion approach. For these systems various experiments
on pattern formation have been carried out.

## See also

- Autowave
- Diffusion-controlled reaction
- Chemical kinetics
- Phase space method
- Autocatalytic reactions and order creation
- Pattern formation
- Patterns in nature
- Periodic travelling wave
- Stochastic geometry
- MClone

## References

## External links

- Java applet showing a reaction–diffusion simulation
- Another applet showing Gray-Scott reaction-diffusion.
- Java applet Uses reaction-diffusion to simulate pattern formation in several snake species.
- Gallery of reaction-diffusion images and movies.
- TexRD software random texture generator based on reaction-diffusion for graphists and scientific use
- Reaction-Diffusion by the Gray-Scott Model: Pearson's parameterization a visual map of the parameter space of Gray-Scott reaction diffusion.
- A Thesis on reaction-diffusion patterns with an overview of the field
- ReDiLab - Reaction Diffusion Laboratory Flash & GPU based application simulating Belousov-Zhabotinsky, Gray Scott, Willamowski–Rössler and FitzHugh-Nagumo with full source code.