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: localchemical reactions in which the substances are transformed into each other, anddiffusion which causes the substances to spread out over a surface in space.

Reaction–diffusion systems are naturally applied inchemistry. However, the system can also describe dynamical processes of non-chemical nature. Examples are found inbiology,geology andphysics (neutron diffusion theory) andecology. Mathematically, reaction–diffusion systems take the form of semi-linearparabolic partial differential equations. They can be represented in the general form

${\displaystyle {\mathrm{\partial }}_t\mathit{q}=\underset{\_}{\underset{\_}{\mathit{D}}}{\mathrm{\nabla }}^2\mathit{q}+\mathit{R}(\mathit{q}),}$

whereq(x,t) represents the unknown vector function,D is adiagonal matrix ofdiffusion coefficients, andR accounts for all local reactions. The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation oftravelling waves and wave-like phenomena as well as otherself-organizedpatterns like stripes, hexagons or more intricate structure likedissipative solitons. Such patterns have been dubbed "Turing patterns".^[1] Each function, for which a reaction diffusion differential equation holds, represents in fact aconcentration variable.

The simplest reaction–diffusion equation is in one spatial dimension in plane geometry,

${\displaystyle {\mathrm{\partial }}_{t}u=D{\mathrm{\partial }}_x^{2}u+R(u),}$

is also referred to as theKolmogorov–Petrovsky–Piskunov equation.^[2] If the reaction term vanishes, then the equation represents a purediffusion process. The corresponding equation isFick's second law. The choiceR(u) =u(1 −u) yieldsFisher's equation that was originally used to describe the spreading of biologicalpopulations,^[3] the Newell–Whitehead-Segel equation withR(u) =u(1 −u²) to describeRayleigh–Bénard convection,^[4][5] the more generalZeldovich–Frank-Kamenetskii equation withR(u) =u(1 −u)e^-β(1-u) and0 <β <∞ (Zeldovich number) that arises incombustion theory,^[6] and its particular degenerate case withR(u) =u² −u³ that is sometimes referred to as the Zeldovich equation as well.^[7]

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

${\displaystyle {\mathrm{\partial }}_{t}u=-\frac{\delta \mathfrak{L}}{\delta u}}$

and therefore describes a permanent decrease of the "free energy"${\displaystyle \mathfrak{L}}$ given by the functional

${\displaystyle \mathfrak{L}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[\frac{D}{2}{\left({\mathrm{\partial }}_{x}u\right)}^2-V(u)\right]\text{d}x}$

with a potentialV(u) such thatR(u) =⁠dV(u)/du⁠.

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 formu(x,t) =û(ξ) withξ =x −ct, wherec 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. Forc = 0, there is a simple proof for this statement:^[8] ifu₀(x) is a stationary solution andu =u₀(x) +ũ(x,t) is an infinitesimally perturbed solution,linear stability analysis yields the equation

${\displaystyle {\mathrm{\partial }}_t\tilde{u}=D{\mathrm{\partial }}_x^2\tilde{u}-U(x)\tilde{u},U(x)=-R^{\mathrm{\prime }}(u){|}_{u=u_0(x)}.}$

With the ansatzũ =ψ(x)exp(−λt) we arrive at the eigenvalue problem

${\displaystyle \hat{H}\psi =\lambda \psi ,\hat{H}=-D{\mathrm{\partial }}_x^2+U(x),}$

ofSchrödinger type where negative eigenvalues result in the instability of the solution. Due to translational invarianceψ = ∂_x u₀(x) is a neutraleigenfunction with theeigenvalueλ = 0, 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 eigenfunctionψ = ∂_x u₀(x) should have at least one zero, and for a non-monotonic stationary solution the corresponding eigenvalueλ = 0 cannot be the lowest one, thereby implying instability.

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

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

This equation has a nice mechanical analogue as the motion of a massD with positionû in the course of the "time"ξ under the forceR with the damping coefficient c which allows for a rather illustrative access to the construction of different types of solutions and the determination ofc.

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 localradius of curvature (this can be seen by going topolar coordinates). This phenomenon leads to the so-called curvature-driven instability.^[9]

Two-component systems allow for a much larger range of possible phenomena than their one-component counterparts. An important idea that was first proposed byAlan Turing is that a state that is stable in the local system can become unstable in the presence ofdiffusion.^[10]

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

aplane wave perturbation

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

of the stationary homogeneous solution will satisfy

${\displaystyle \left(\begin{array}{c}{\mathrm{\partial }}_t{\tilde{u}}_{\mathit{k}}(t)\\ {\mathrm{\partial }}_t{\tilde{v}}_{\mathit{k}}(t)\end{array}\right)=-k^2\left(\begin{array}{c}{D}_u{\tilde{u}}_{\mathit{k}}(t)\\ D_v{\tilde{v}}_{\mathit{k}}(t)\end{array}\right)+{\mathit{R}}^{\mathrm{\prime }}\left(\begin{array}{c}{\tilde{u}}_{\mathit{k}}(t)\\ {\tilde{v}}_{\mathit{k}}(t)\end{array}\right).}$

Turing's idea can only be realized in fourequivalence classes of systems characterized by the signs of theJacobianR′ of the reaction function. In particular, if a finite wave vectork is supposed to be the most unstable one, the Jacobian must have the signs

This class of systems is namedactivator-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 theFitzHugh–Nagumo equation

with f (u) =λu −u³ −κ which describes how anaction potential travels through a nerve.^[11][12] Here,d_u,d_v,τ,σ andλ 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 correspondingbifurcation may be either aHopf bifurcation to a globally oscillating homogeneous state with a dominant wave numberk = 0 or aTuring 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.

• Subcritical Turing bifurcation: formation of a hexagonal pattern from noisy initial conditions in the above two-component reaction–diffusion system of Fitzhugh–Nagumo type.
• 
  Noisy initial conditions att = 0.
• 
  State of the system att = 10.
• 
  Almost converged state att = 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 thehysteretic region where the pattern coexists with the ground state. Other frequently encountered structures comprise pulse trains (also known asperiodic 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^[13]

• Other patterns found in the above two-component reaction–diffusion system of Fitzhugh–Nagumo type.
• 
  Rotating spiral.
• 
  Target pattern.
• 
  Stationary localized pulse (dissipative soliton).

For a variety of systems, reaction–diffusion equations with more than two components have been proposed, e.g. theBelousov–Zhabotinsky reaction,^[14] forblood clotting,^[15] fission waves^[16] or planargas discharge systems.^[17]

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).^[18] An introduction and systematic overview of the possible phenomena in dependence on the properties of the underlying system is given in.^[19]

In recent times, reaction–diffusion systems have attracted much interest as a prototype model forpattern formation.^[20] 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 tomorphogenesis in biology^[21][22] and may even be related to animal coats and skin pigmentation.^[23][24] Other applications of reaction–diffusion equations include ecological invasions,^[25] spread of epidemics,^[26] tumour growth,^[27][28][29] dynamics of fission waves,^[30] wound healing^[31] and visual hallucinations.^[32] 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.^{[8][9][33][34][35][20]}

Well-controllable experiments in chemical reaction–diffusion systems have up to now been realized in three ways. First, gel reactors^[36] or filled capillary tubes^[37] may be used. Second,temperature pulses oncatalytic surfaces have been investigated.^[38][39] Third, the propagation of running nerve pulses is modelled using reaction–diffusion systems.^[11][40]

Aside from these generic examples, it has turned out that under appropriate circumstances electric transport systems like plasmas^[41] or semiconductors^[42] can be described in a reaction–diffusion approach. For these systems various experiments on pattern formation have been carried out.

A reaction–diffusion system can be solved by using methods ofnumerical mathematics. There exist several numerical treatments in research literature.^[43][20][44] Numerical solution methods for complexgeometries are also proposed.^[45][46] Reaction-diffusion systems are described to the highest degree of detail with particle based simulation tools like SRSim or ReaDDy^[47] which employ among others reversible interacting-particle reaction dynamics.^[48]

• Autowave – Self-supporting non-linear waves in active media
• Diffusion-controlled reaction – Reaction rate equals rate of transport
• Chemical kinetics – Study of the rates of chemical reactions
• Phase space method
• Autocatalytic reactions and order creation – Chemical reaction whose product is also its catalystPages displaying short descriptions of redirect targets
• Pattern formation – Study of how patterns form by self-organization in nature
• Patterns in nature – Visible regularity of form found in the natural world
• Periodic travelling wave – Constant speed wavetrain
• Self-similar solutions – Concept in partial differential equations
• Diffusion equation – Equation that describes density changes of a material that is diffusing in a medium
• Stochastic geometry – Study of random spatial patterns
• MClone
• The Chemical Basis of Morphogenesis – 1952 scholarly article by Alan Turing
• Turing pattern – Concept from evolutionary biology
• Multi-state modeling of biomolecules

• Fisher's equation
• Zeldovich–Frank-Kamenetskii equation
• FitzHugh–Nagumo model
• Wrinkle paint

• 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
• RD Tool: an interactive web application for reaction-diffusion simulation

• Mathematical modeling
• Parabolic partial differential equations
• Reaction mechanisms
• Functions of space and time

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