View source for Reaction–diffusion system

{{Use mdy dates|date = March 2019}}
{{Short description|Type of mathematical model}}
{{Use American English|date = March 2019}}
[[File:Reaction-Diffusion.gif|thumb|A simulation of two virtual chemicals reacting and diffusing on a Torus using the [https://groups.csail.mit.edu/mac/projects/amorphous/GrayScott/ Gray–Scott model] ]]
'''Reaction–diffusion systems''' are mathematical models which 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]]s 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]] (neutron diffusion theory) and [[ecology]]. Mathematically, reaction–diffusion systems take the form of semi-linear [[parabolic partial differential equation]]s. They can be represented in the general form

:<math>\partial_t \boldsymbol{q} = \underline{\underline{\boldsymbol{D}}} \,\nabla^2 \boldsymbol{q} + \boldsymbol{R}(\boldsymbol{q}), </math>

where {{math|'''''q'''''('''''x''''', ''t'')}} represents the unknown vector function, {{math|{{uuline|'''''D'''''}}}} is a [[diagonal matrix]] of [[diffusion coefficient]]s, and {{math|'''''R'''''}} accounts for all local reactions. The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation of [[travelling wave]]s and wave-like phenomena as well as other [[Self-organization|self-organized]] [[pattern formation|patterns]] like stripes, hexagons or more intricate structure like [[dissipative soliton]]s. Such patterns have been dubbed "[[Turing pattern]]s".<ref>Wooley, T. E., Baker, R. E. [[Philip Maini|Maini, P. K.]], Chapter 34, ''Turing's theory of morphogenesis''. In {{cite book |author-link1 = Jack Copeland|last1 = Copeland | first1=B. Jack | authorlink2=Jonathan Bowen | last2=Bowen | first2=Jonathan P. | authorlink3=Robin Wilson (mathematician)| last3=Wilson | first3=Robin | last4=Sprevak | first4=Mark | title=The Turing Guide | publisher=[[Oxford University Press]] | year=2017 | isbn=978-0198747826 |title-link = The Turing Guide }}</ref> 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,

:<math>\partial_t u = D \partial^2_x u + R(u),</math>

is also referred to as the [[Kolmogorov–Petrovsky–Piskunov equation]].<ref>Kolmogorov, A., Petrovskii, I. and Piskunov, N. (1937) Study of a Diffusion Equation That Is Related to the Growth of a Quality of Matter and Its Application to a Biological Problem. Moscow University Mathematics Bulletin, 1, 1-26.</ref> If the reaction term vanishes, then the equation represents a pure diffusion process. The corresponding equation is [[Fick's law|Fick's second law]]. The choice {{math|''R''(''u'') {{=}} ''u''(1 − ''u'')}} yields [[Fisher's equation]] that was originally used to describe the spreading of biological [[population]]s,<ref>R. A. Fisher, Ann. Eug. 7 (1937): 355</ref> the Newell–Whitehead-Segel equation with {{math|''R''(''u'') {{=}} ''u''(1 − ''u''<sup>2</sup>)}} to describe [[Rayleigh–Bénard convection]],<ref>{{cite journal | last=Newell | first=Alan C. | last2=Whitehead | first2=J. A. | title=Finite bandwidth, finite amplitude convection | journal=Journal of Fluid Mechanics | publisher=Cambridge University Press (CUP) | volume=38 | issue=2 | date=1969-09-03 | issn=0022-1120 | doi=10.1017/s0022112069000176 | pages=279–303}}</ref><ref>{{cite journal | last=Segel | first=Lee A. | title=Distant side-walls cause slow amplitude modulation of cellular convection | journal=Journal of Fluid Mechanics | publisher=Cambridge University Press (CUP) | volume=38 | issue=1 | date=1969-08-14 | issn=0022-1120 | doi=10.1017/s0022112069000127 | pages=203–224}}</ref> the more general [[Zeldovich]] equation with {{math|''R''(''u'') {{=}} ''u''(1 − ''u'')(''u'' − ''α'')}} and {{math|0 < ''α'' < 1}} that arises in [[combustion]] theory,<ref>Y. B. Zeldovich and D. A. Frank-Kamenetsky, Acta Physicochim. 9 (1938): 341</ref> and its particular degenerate case with {{math|''R''(''u'') {{=}} ''u''<sup>2</sup> − ''u''<sup>3</sup>}} that is sometimes referred to as the Zeldovich equation as well.<ref>B. H. Gilding and R. Kersner, [https://books.google.com/books?id=tdIACAAAQBAJ&printsec=frontcover#v=onepage&q&f=false Travelling Waves in Nonlinear Diffusion Convection Reaction], Birkhäuser (2004)</ref>

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

:<math>\partial_t u=-\frac{\delta\mathfrak L}{\delta u}</math>

and therefore describes a permanent decrease of the "free energy" <math>\mathfrak L</math> given by the functional

:<math> \mathfrak L=\int_{-\infty}^\infty \left[\tfrac{D}{2} \left (\partial_xu \right )^2-V(u)\right] \, \text{d}x</math>

with a potential {{math|''V''(''u'')}} such that {{math|''R''(''u'') {{=}} {{sfrac|d''V''(''u'')|d''u''}}.}}

[[Image:Travelling wave for Fisher equation.svg|thumb|right|A travelling wave front solution for Fisher's equation.]]  
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 {{math|''u''(''x'', ''t'') {{=}} ''û''(''ξ'')}} with {{math|''ξ'' {{=}} ''x'' − ''ct''}}, where {{mvar|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 {{math|''c'' {{=}} 0}}, there is a simple proof for this statement:<ref name="fife">P. C. Fife, [https://books.google.com/books?id=FJXsCAAAQBAJ&printsec=frontcover#v=onepage&q&f=false Mathematical Aspects of Reacting and Diffusing Systems], Springer (1979)</ref> if {{math|''u''<sub>0</sub>(''x'')}} is a stationary solution and {{math|''u'' {{=}} ''u''<sub>0</sub>(''x'') + ''ũ''(''x'', ''t'')}} is an infinitesimally perturbed solution, linear stability analysis yields the equation

:<math> \partial_t \tilde{u}=D\partial_x^2 \tilde{u}-U(x)\tilde{u},\qquad U(x) = -R^{\prime}(u)\Big|_{u=u_0(x)}.</math>

With the ansatz {{math|''ũ'' {{=}} ''ψ''(''x'')exp(−''λt'')}} we arrive at the eigenvalue problem

:<math> \hat H\psi=\lambda\psi, \qquad \hat H=-D\partial_x^2+U(x),</math>

of [[Schrödinger equation|Schrödinger type]] where negative eigenvalues result in the instability of the solution. Due to translational invariance {{math|''ψ'' {{=}} ∂<sub>''x''</sub>&thinsp;''u''<sub>0</sub>(''x'')}} is a neutral [[eigenfunction]] with the [[eigenvalue]] {{math|''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 {{math|''ψ'' {{=}} ∂<sub>''x''</sub>&thinsp;''u''<sub>0</sub>(''x'')}} should have at least one zero, and for a non-monotonic stationary solution the corresponding eigenvalue {{math|''λ'' {{=}} 0}} cannot be the lowest one, thereby implying instability.

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

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

This equation has a nice mechanical analogue as the motion of a mass {{mvar|D}} with position {{math|''û''}} in the course of the "time" {{mvar|ξ}} under the force {{mvar|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 {{mvar|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 [[curvature|radius of curvature]] (this can be seen by going to [[polar coordinates]]). This phenomenon leads to the so-called curvature-driven instability.<ref name="mikhailov">A. S. Mikhailov, Foundations of Synergetics I. Distributed Active Systems, Springer (1990)</ref>

==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 can become unstable in the presence of [[diffusion]].<ref>{{cite journal |last=Turing|first=A. M.|author-link=Alan Turing| title=The chemical basis of morphogenesis | journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences | publisher=The Royal Society | volume=237 | issue=641 | date=1952-08-14 | issn=2054-0280 | doi=10.1098/rstb.1952.0012 | pages=37–72}}</ref>

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

:<math> \begin{pmatrix} \partial_t u \\ \partial_t v \end{pmatrix} = \begin{pmatrix} D_u &0 \\0&D_v \end{pmatrix} \begin{pmatrix} \partial_{xx} u\\ \partial_{xx} v \end{pmatrix} + \begin{pmatrix} F(u,v)\\G(u,v)\end{pmatrix}</math>

a [[plane wave]] perturbation

:<math> \tilde{\boldsymbol{q}}_{\boldsymbol{k}}(\boldsymbol{x},t) = \begin{pmatrix} \tilde{u}(t)\\\tilde{v}(t) \end{pmatrix} e^{i \boldsymbol{k} \cdot \boldsymbol{x}} </math>

of the stationary homogeneous solution will satisfy

:<math>\begin{pmatrix} \partial_t \tilde{u}_{\boldsymbol{k}}(t)\\ \partial_t \tilde{v}_{\boldsymbol{k}}(t) \end{pmatrix} = -k^2\begin{pmatrix} D_u \tilde{u}_{\boldsymbol{k}}(t)\\ D_v\tilde{v}_{\boldsymbol{k}}(t) \end{pmatrix} + \boldsymbol{R}^{\prime} \begin{pmatrix}\tilde{u}_{\boldsymbol{k}}(t) \\ \tilde{v}_{\boldsymbol{k}}(t) \end{pmatrix}.</math>

Turing's idea can only be realized in four [[equivalence class]]es of systems characterized by the signs of the [[Jacobian matrix and determinant|Jacobian]] {{math|'''''R'''''′}} of the reaction function. In particular, if a finite wave vector {{math|'''''k'''''}} is supposed to be the most unstable one, the Jacobian must have the signs

:<math> \begin{pmatrix} +&-\\+&-\end{pmatrix}, \quad \begin{pmatrix} +&+\\-&-\end{pmatrix}, \quad \begin{pmatrix} -&+\\-&+\end{pmatrix}, \quad
\begin{pmatrix} -&-\\+&+\end{pmatrix}. </math>

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]]

:<math>\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}</math>

with {{math|&thinsp;''f''&thinsp;(''u'') {{=}} ''λu'' − ''u''<sup>3</sup> − ''κ''}} which describes how an [[action potential]] travels through a nerve.<ref name="fitzhugh">{{cite journal | last=FitzHugh | first=Richard | title=Impulses and Physiological States in Theoretical Models of Nerve Membrane | journal=Biophysical Journal | publisher=Elsevier BV | volume=1 | issue=6 | year=1961 | issn=0006-3495 | doi=10.1016/s0006-3495(61)86902-6 | pages=445–466|doi-access=free}}</ref><ref>J. Nagumo et al., Proc. Inst. Radio Engin. Electr. 50 (1962): 2061</ref> Here, {{math|''d<sub>u</sub>'', ''d<sub>v</sub>'', ''τ'', ''σ''}} and {{math|''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 theory|bifurcation]] may be either a [[Hopf bifurcation]] to a globally oscillating homogeneous state with a dominant wave number {{math|''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.

<gallery caption="Subcritical Turing bifurcation: formation of a hexagonal pattern from noisy initial conditions in the above two-component reaction–diffusion system of Fitzhugh–Nagumo type." widths="257" heights="235" perrow="3">
Image:Turing_bifurcation_1.gif| Noisy initial conditions at ''t''&nbsp;=&nbsp;0. 
Image:Turing_bifurcation_2.gif| State of the system at ''t''&nbsp;=&nbsp;10. 
Image:Turing_bifurcation_3.gif| Almost converged state at ''t''&nbsp;=&nbsp;100.
</gallery>

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

:<math>\begin{align}
q_{\text{n}}^H(k): &{}\quad \frac{1}{\tau} + \left  (d_u^2 + \frac{1}{\tau} d_v^2 \right )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}</math>

If the bifurcation is subcritical, often localized structures ([[dissipative solitons]]) can be observed in the [[Hysteresis|hysteretic]] region where the pattern coexists with the ground state. Other frequently encountered structures comprise pulse trains (also known as [[periodic travelling wave]]s), 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<ref>{{cite journal | last=Kopell | first=N. | last2=Howard | first2=L. N. | title=Plane Wave Solutions to Reaction-Diffusion Equations | journal=Studies in Applied Mathematics | publisher=Wiley | volume=52 | issue=4 | year=1973 | issn=0022-2526 | doi=10.1002/sapm1973524291 | pages=291–328}}</ref>

<gallery caption="Other patterns found in the above two-component reaction–diffusion system of Fitzhugh–Nagumo type." widths="265" heights="235" perrow="3">
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).
</gallery>

==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 regulation of [[lymphangiogenesis]] by [[Vascular endothelial growth factor C|VEGFC]], [[MMP2]], and [[Type I collagen|collagen I]];<ref name="Roose 1–37">{{Cite journal|last=Roose|first=Tiina|last2=Wertheim|first2=Kenneth Y.|date=2019-01-03|title=Can VEGFC Form Turing Patterns in the Zebrafish Embryo?|journal=Bulletin of Mathematical Biology|volume=81|issue=4|pages=1201–1237|doi=10.1007/s11538-018-00560-2|pmid=30607882|pmc=6397306|issn=1522-9602}}</ref> the [[Belousov–Zhabotinsky reaction]],<ref>{{cite journal | last=Vanag | first=Vladimir K. | last2=Epstein | first2=Irving R. | title=Stationary and Oscillatory Localized Patterns, and Subcritical Bifurcations | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=92 | issue=12 | date=2004-03-24 | issn=0031-9007 | doi=10.1103/physrevlett.92.128301 | page=128301}}</ref> for [[blood clotting]]<ref>{{cite journal | last=Lobanova | first=E. S. | last2=Ataullakhanov | first2=F. I. | title=Running Pulses of Complex Shape in a Reaction-Diffusion Model | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=93 | issue=9 | date=2004-08-26 | issn=0031-9007 | doi=10.1103/physrevlett.93.098303 | page=098303}}</ref> or planar [[gas discharge]] systems.<ref>H.-G. Purwins et al. in: Dissipative Solitons, Lectures Notes in Physics, Ed. N. Akhmediev and A. Ankiewicz, Springer (2005)</ref>

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),.<ref>{{cite journal | last=Schenk | first=C. P. | last2=Or-Guil | first2=M. | last3=Bode | first3=M. | last4=Purwins | first4=H.-G. | title=Interacting Pulses in Three-Component Reaction-Diffusion Systems on Two-Dimensional Domains | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=78 | issue=19 | date=1997-05-12 | issn=0031-9007 | doi=10.1103/physrevlett.78.3781 | pages=3781–3784}}</ref> An introduction and systematic overview of the possible phenomena in dependence on the properties of the underlying system is given in.<ref>A. W. Liehr: ''Dissipative Solitons in Reaction Diffusion Systems. Mechanism, Dynamics, Interaction.'' Volume 70 of Springer Series in Synergetics, Springer, Berlin Heidelberg 2013, {{isbn|978-3-642-31250-2}}</ref>

The challenges posed by multi-component systems are rooted in their analytically intractable nature; one solution is to explore the parametric space of such a model one point at a time and then solve the model numerically, as was done in a theoretical study about [[lymphangiogenesis]].<ref name="Roose 1–37"/>

== Applications and universality==
In recent times, reaction–diffusion systems have attracted much interest as a prototype model for [[pattern formation]].<ref name=":0">{{Cite journal|last=Gupta|first=Ankur|last2=Chakraborty|first2=Saikat|date=January 2009|title=Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors|journal=Chemical Engineering Journal|volume=145|issue=3|pages=399–411|doi=10.1016/j.cej.2008.08.025|issn=1385-8947}}</ref> 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<ref>L.G. Harrison, Kinetic Theory of Living Pattern, Cambridge University Press (1993)</ref> and may even be related to animal coats and skin pigmentation.<ref>H. Meinhardt, Models of Biological Pattern Formation, Academic Press (1982)</ref><ref name="Murray2013">{{cite book|last=Murray|first=James D.|title=Mathematical Biology|url=https://books.google.com/books?id=K3LmCAAAQBAJ&pg=PA436|date=9 March 2013|publisher=Springer Science & Business Media|isbn=978-3-662-08539-4|pages=436–450}}</ref> Other applications of reaction–diffusion equations include ecological invasions,<ref>{{cite journal | last=Holmes | first=E. E. | last2=Lewis | first2=M. A. | last3=Banks | first3=J. E. | last4=Veit | first4=R. R. | title=Partial Differential Equations in Ecology: Spatial Interactions and Population Dynamics | journal=Ecology | publisher=Wiley | volume=75 | issue=1 | year=1994 | issn=0012-9658 | doi=10.2307/1939378 | pages=17–29}}</ref> spread of epidemics,<ref>{{cite journal |last=Murray|first=James D.|last2=Stanley|first2=E. A.|last3=Brown|first3=D. L.| title=On the spatial spread of rabies among foxes | journal=Proceedings of the Royal Society of London. Series B. Biological Sciences | publisher=The Royal Society | volume=229 | issue=1255 | date=1986-11-22 | issn=2053-9193 | doi=10.1098/rspb.1986.0078 | pages=111–150}}</ref> tumour growth<ref>{{cite journal | last=CHAPLAIN | first=M. A. J. | title=REACTION–DIFFUSION PREPATTERNING AND ITS POTENTIAL ROLE IN TUMOUR INVASION | journal=Journal of Biological Systems | publisher=World Scientific Pub Co Pte Lt | volume=03 | issue=04 | year=1995 | issn=0218-3390 | doi=10.1142/s0218339095000824 | pages=929–936}}</ref><ref>{{cite journal |last=Sherratt|first=J. A.|last2=Nowak|first2=M. A.| title=Oncogenes, anti-oncogenes and the immune response to cancer : a mathematical model | journal=Proceedings of the Royal Society of London. Series B: Biological Sciences | publisher=The Royal Society | volume=248 | issue=1323 | date=1992-06-22 | issn=0962-8452 | doi=10.1098/rspb.1992.0071 | pages=261–271}}</ref><ref>R.A. Gatenby and E.T. Gawlinski, Cancer Res. 56 (1996): 5745</ref> and wound healing.<ref>{{cite journal |last=Sherratt|first=J. A.|last2=Murray|first2=J. D.| title=Models of epidermal wound healing | journal=Proceedings of the Royal Society of London. Series B: Biological Sciences | publisher=The Royal Society | volume=241 | issue=1300 | date=1990-07-23 | issn=0962-8452 | doi=10.1098/rspb.1990.0061 | pages=29–36}}</ref> 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.<ref name="fife" /><ref name="mikhailov" /><ref>P. Grindrod, Patterns and Waves: The Theory and Applications of Reaction-Diffusion Equations, Clarendon Press (1991)</ref><ref>J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer (1994)</ref><ref>B. S. Kerner and V. V. Osipov, Autosolitons. A New Approach to Problems of Self-Organization and Turbulence, Kluwer Academic Publishers (1994)</ref><ref name=":0" />

== Experiments ==
Well-controllable experiments in chemical reaction–diffusion systems have up to now been realized in three ways. First, gel reactors<ref>{{cite journal | last=Lee | first=Kyoung-Jin | last2=McCormick | first2=William D. | last3=Pearson | first3=John E. | last4=Swinney | first4=Harry L. | title=Experimental observation of self-replicating spots in a reaction–diffusion system | journal=Nature | publisher=Springer Nature | volume=369 | issue=6477 | year=1994 | issn=0028-0836 | doi=10.1038/369215a0 | pages=215–218}}</ref> or filled capillary tubes<ref>{{cite journal | last=Hamik | first=Chad T | last2=Steinbock | first2=Oliver | title=Excitation waves in reaction-diffusion media with non-monotonic dispersion relations | journal=New Journal of Physics | publisher=IOP Publishing | volume=5 | date=2003-06-06 | issn=1367-2630 | doi=10.1088/1367-2630/5/1/358 | pages=58–58|doi-access=free}}</ref> may be used. Second, [[temperature]] pulses on [[Catalytic converter|catalytic surfaces]] have been investigated.<ref>{{cite journal | last=Rotermund | first=H. H. | last2=Jakubith | first2=S. | last3=von Oertzen | first3=A. | last4=Ertl | first4=G. | title=Solitons in a surface reaction | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=66 | issue=23 | date=1991-06-10 | issn=0031-9007 | doi=10.1103/physrevlett.66.3083 | pages=3083–3086}}</ref><ref>{{cite journal | last=Graham | first=Michael D. | last2=Lane | first2=Samuel L. | last3=Luss | first3=Dan | title=Temperature pulse dynamics on a catalytic ring | journal=The Journal of Physical Chemistry | publisher=American Chemical Society (ACS) | volume=97 | issue=29 | year=1993 | issn=0022-3654 | doi=10.1021/j100131a028 | pages=7564–7571}}</ref> Third, the propagation of running nerve pulses is modelled using reaction–diffusion systems.<ref name="fitzhugh" /><ref>{{cite journal | last=Hodgkin | first=A. L. | last2=Huxley | first2=A. F. | title=A quantitative description of membrane current and its application to conduction and excitation in nerve | journal=The Journal of Physiology | publisher=Wiley | volume=117 | issue=4 | date=1952-08-28 | issn=0022-3751 | doi=10.1113/jphysiol.1952.sp004764 | pages=500–544|doi-access=free}}</ref>

Aside from these generic examples, it has turned out that under appropriate circumstances electric transport systems like plasmas<ref>{{cite journal | last=Bode | first=M. | last2=Purwins | first2=H.-G. | title=Pattern formation in reaction-diffusion systems - dissipative solitons in physical systems | journal=Physica D: Nonlinear Phenomena | publisher=Elsevier BV | volume=86 | issue=1-2 | year=1995 | issn=0167-2789 | doi=10.1016/0167-2789(95)00087-k | pages=53–63}}</ref> or semiconductors<ref>E. Schöll, Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors, Cambridge University Press (2001)</ref> 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 are existing several numerical treatments in research literature.<ref>S.Tang et al., J.Austral.Math.Soc. Ser.B 35(1993): 223–243</ref><ref name=":0" /><ref>Tim Hutton, Robert Munafo, Andrew Trevorrow, Tom Rokicki, Dan Wills. "Ready, a cross-platform implementation of various reaction-diffusion systems." https://github.com/GollyGang/ready</ref> Also for complex [[geometry|geometries]] numerical solution methods are proposed.<ref>{{cite journal | last1 = Isaacson | first1 = Samuel A. | last2 = Peskin | first2 = Charles S.| title = Incorporating Diffusion in Complex Geometries into Stochastic Chemical Kinetics Simulations | journal = SIAM J. Sci. Comput.| year = 2006| volume = 28| issue = 1| pages = 47–74| doi = 10.1137/040605060| citeseerx = 10.1.1.105.2369}}</ref><ref>{{cite journal | last = Linker | first = Patrick | title = Numerical methods for solving the reactive diffusion equation in complex geometries | journal = The Winnower| year = 2016| url = https://www.thewinnower.com/papers/4159-numerical-methods-for-solving-the-reactive-diffusion-equation-in-complex-geometries}}</ref>

==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]]
*[[The Chemical Basis of Morphogenesis]]
*[[Turing pattern]]

==Examples==
*[[Fisher's equation]]
*[[Fisher–Kolmogorov equation]]
*[[FitzHugh–Nagumo model]]

{{Div col end}}

==References==
{{reflist|28em}}

==External links==
* [http://mrob.com/pub/comp/xmorphia/ Reaction–Diffusion by the Gray–Scott Model: Pearson's parameterization] a visual map of the parameter space of Gray–Scott reaction diffusion.
* [http://hantz.web.elte.hu/cikkfile/hantzth.pdf A thesis on reaction–diffusion patterns with an overview of the field]

{{DEFAULTSORT:Reaction-diffusion system}}
[[Category:Mathematical modeling]]
[[Category:Parabolic partial differential equations]]