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Reaction–diffusion system

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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 <a href="Chemical_reaction" title="Chemical reaction">chemical reactions</a> in which the substances are transformed into each other, and <a href="Diffusion" title="Diffusion">diffusion</a> which causes the substances to spread out over a surface in space.

Reaction–diffusion systems are naturally applied in <a href="Chemistry" title="Chemistry">chemistry</a>. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in <a href="Biology" title="Biology">biology</a>, <a href="Geology" title="Geology">geology</a> and <a href="Physics" title="Physics">physics</a> (neutron diffusion theory) and <a href="Ecology" title="Ecology">ecology</a>. Mathematically, reaction–diffusion systems take the form of semi-linear <a href="Parabolic_partial_differential_equation" title="Parabolic partial differential equation">parabolic partial differential equations</a>. They can be represented in the general form

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<annotation encoding="application/x-tex">{\displaystyle \partial _{t}{\boldsymbol {q}}={\underline {\underline {\boldsymbol {D}}}}\,\nabla ^{2}{\boldsymbol {q}}+{\boldsymbol {R}}({\boldsymbol {q}}),}</annotation>

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</math><img src="./a0781eca8a47e8a52e6836f80b7e29d86cf92a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.02ex; margin-bottom: -0.818ex; width:21.9ex; height:4.176ex;" alt="{\displaystyle \partial _{t}{\boldsymbol {q}}={\underline {\underline {\boldsymbol {D}}}}\,\nabla ^{2}{\boldsymbol {q}}+{\boldsymbol {R}}({\boldsymbol {q}}),}" loading="lazy"></dd></dl>

where q(x, t) represents the unknown vector function, D is a <a href="Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> of <a href="Diffusion_coefficient" class="mw-redirect" title="Diffusion coefficient">diffusion coefficients</a>, and R accounts for all local reactions. The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation of <a href="Travelling_wave" class="mw-redirect" title="Travelling wave">travelling waves</a> and wave-like phenomena as well as other <a href="Self-organization" title="Self-organization">self-organized</a> <a href="Pattern_formation" title="Pattern formation">patterns</a> like stripes, hexagons or more intricate structure like <a href="Dissipative_soliton" title="Dissipative soliton">dissipative solitons</a>. Such patterns have been dubbed "<a href="Turing_pattern" title="Turing pattern">Turing patterns</a>".<a href="#cite_note-1">[1]</a> Each function, for which a reaction diffusion differential equation holds, represents in fact a concentration variable.

<div class="mw-heading mw-heading2"><h2 id="One-component_reaction–diffusion_equations">One-component reaction–diffusion equations</h2></div>

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

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<annotation encoding="application/x-tex">{\displaystyle \partial _{t}u=D\partial _{x}^{2}u+R(u),}</annotation>

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</math><img src="./0443e49841dc2a787c73679991a9abc43cbe710e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.54ex; height:3.009ex;" alt="{\displaystyle \partial _{t}u=D\partial _{x}^{2}u+R(u),}" loading="lazy"></dd></dl>

is also referred to as the <a href="Kolmogorov%E2%80%93Petrovsky%E2%80%93Piskunov_equation" class="mw-redirect" title="Kolmogorov–Petrovsky–Piskunov equation">Kolmogorov–Petrovsky–Piskunov equation</a>.<a href="#cite_note-2">[2]</a> If the reaction term vanishes, then the equation represents a pure diffusion process. The corresponding equation is <a href="Fick's_law" class="mw-redirect" title="Fick's law">Fick's second law</a>. The choice R(u) = u(1 − u) yields <a href="Fisher's_equation" class="mw-redirect" title="Fisher's equation">Fisher's equation</a> that was originally used to describe the spreading of biological <a href="Population" title="Population">populations</a>,<a href="#cite_note-3">[3]</a> the Newell–Whitehead-Segel equation with R(u) = u(1 − u2) to describe <a href="Rayleigh%E2%80%93B%C3%A9nard_convection" title="Rayleigh–Bénard convection">Rayleigh–Bénard convection</a>,<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> the more general <a href="ZFK_equation" title="ZFK equation">Zeldovich–Frank-Kamenetskii equation</a> with R(u) = u(1 − u)e-β(1-u) and 0 < β < ∞ (<a href="Zeldovich_number" title="Zeldovich number">Zeldovich number</a>) that arises in <a href="Combustion" title="Combustion">combustion</a> theory,<a href="#cite_note-6">[6]</a> and its particular degenerate case with R(u) = u2 − u3 that is sometimes referred to as the Zeldovich equation as well.<a href="#cite_note-7">[7]</a>

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

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<annotation encoding="application/x-tex">{\displaystyle \partial _{t}u=-{\frac {\delta {\mathfrak {L}}}{\delta u}}}</annotation>

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</math><img src="./3f8df5439e1a29a2155a8e257e705e3c9714ef78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.729ex; height:5.509ex;" alt="{\displaystyle \partial _{t}u=-{\frac {\delta {\mathfrak {L}}}{\delta u}}}" loading="lazy"></dd></dl>

and therefore describes a permanent decrease of the "free energy" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {L}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {L}}}</annotation>

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</math><img src="./f1cc2d02222bcba1e741979a145f0317df3cda81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.548ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {L}}}" loading="lazy"> given by the functional

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {L}}=\int _{-\infty }^{\infty }\left[{\tfrac {D}{2}}\left(\partial _{x}u\right)^{2}-V(u)\right]\,{\text{d}}x}</annotation>

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</math><img src="./ae3c05e847595f155e3a7cb073798c6d01220d3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.635ex; height:6.009ex;" alt="{\displaystyle {\mathfrak {L}}=\int _{-\infty }^{\infty }\left[{\tfrac {D}{2}}\left(\partial _{x}u\right)^{2}-V(u)\right]\,{\text{d}}x}" loading="lazy"></dd></dl>

with a potential V(u) such that R(u) = <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠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 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:<a href="#cite_note-fife-8">[8]</a> if u0(x) is a stationary solution and u = u0(x) + ũ(x, t) is an infinitesimally perturbed solution, linear stability analysis yields the equation

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<annotation encoding="application/x-tex">{\displaystyle \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)}.}</annotation>

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</math><img src="./fd91e155f2d1905ab95486b5897a53b3d9b2877b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.072ex; height:4.676ex;" alt="{\displaystyle \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)}.}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle {\hat {H}}\psi =\lambda \psi ,\qquad {\hat {H}}=-D\partial _{x}^{2}+U(x),}</annotation>

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</math><img src="./222949a844db3ed895349992e2b84a79419595af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.933ex; height:3.343ex;" alt="{\displaystyle {\hat {H}}\psi =\lambda \psi ,\qquad {\hat {H}}=-D\partial _{x}^{2}+U(x),}" loading="lazy"></dd></dl>

of <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger type</a> where negative eigenvalues result in the instability of the solution. Due to translational invariance ψ = ∂x u0(x) is a neutral <a href="Eigenfunction" title="Eigenfunction">eigenfunction</a> with the <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> λ = 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 eigenfunction ψ = ∂x u0(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:

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<annotation encoding="application/x-tex">{\displaystyle D\partial _{\xi }^{2}{\hat {u}}(\xi )+c\partial _{\xi }{\hat {u}}(\xi )+R({\hat {u}}(\xi ))=0.}</annotation>

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</math><img src="./7045a405413bd0e33f63a2a62f0e730c02d1a1c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:34.193ex; height:3.509ex;" alt="{\displaystyle D\partial _{\xi }^{2}{\hat {u}}(\xi )+c\partial _{\xi }{\hat {u}}(\xi )+R({\hat {u}}(\xi ))=0.}" loading="lazy"></dd></dl>

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 <a href="Curvature" title="Curvature">radius of curvature</a> (this can be seen by going to <a href="Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a>). This phenomenon leads to the so-called curvature-driven instability.<a href="#cite_note-mikhailov-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="Two-component_reaction–diffusion_equations">Two-component reaction–diffusion equations</h2></div>

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 <a href="Alan_Turing" title="Alan Turing">Alan Turing</a> is that a state that is stable in the local system can become unstable in the presence of <a href="Diffusion" title="Diffusion">diffusion</a>.<a href="#cite_note-10">[10]</a>

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

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<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

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</math><img src="./ca343c7178a61986b2c711b4d566c6c7d651de5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.197ex; height:6.176ex;" alt="{\displaystyle {\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}}}" loading="lazy"></dd></dl>

a <a href="Plane_wave" title="Plane wave">plane wave</a> perturbation

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<annotation encoding="application/x-tex">{\displaystyle {\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}}}}</annotation>

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</math><img src="./db4123001a865ddf902aa598c12bf9cb2e36f5fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.561ex; height:6.176ex;" alt="{\displaystyle {\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}}}}" loading="lazy"></dd></dl>

of the stationary homogeneous solution will satisfy

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<annotation encoding="application/x-tex">{\displaystyle {\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}}.}</annotation>

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</math><img src="./1f27fd6a250d8a56514c190220b5efa1dd06b001.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.684ex; height:6.176ex;" alt="{\displaystyle {\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}}.}" loading="lazy"></dd></dl>

Turing's idea can only be realized in four <a href="Equivalence_class" title="Equivalence class">equivalence classes</a> of systems characterized by the signs of the <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> 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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}+&-\\+&-\end{pmatrix}},\quad {\begin{pmatrix}+&+\\-&-\end{pmatrix}},\quad {\begin{pmatrix}-&+\\-&+\end{pmatrix}},\quad {\begin{pmatrix}-&-\\+&+\end{pmatrix}}.}</annotation>

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</math><img src="./01e7bcb262f9e772ba0974035133e4ae1e51b555.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.162ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}+&-\\+&-\end{pmatrix}},\quad {\begin{pmatrix}+&+\\-&-\end{pmatrix}},\quad {\begin{pmatrix}-&+\\-&+\end{pmatrix}},\quad {\begin{pmatrix}-&-\\+&+\end{pmatrix}}.}" loading="lazy"></dd></dl>

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 <a href="FitzHugh%E2%80%93Nagumo_equation" class="mw-redirect" title="FitzHugh–Nagumo equation">FitzHugh–Nagumo equation</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation>

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</math><img src="./2d25a5c106bc991dea276879d69b3359bafa72de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.531ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\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{aligned}}}" loading="lazy"></dd></dl>

with f (u) = λu − u3 − κ which describes how an <a href="Action_potential" title="Action potential">action potential</a> travels through a nerve.<a href="#cite_note-fitzhugh-11">[11]</a><a href="#cite_note-12">[12]</a> Here, du, dv, 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 <a href="Bifurcation_theory" title="Bifurcation theory">bifurcation</a> may be either a <a href="Hopf_bifurcation" title="Hopf bifurcation">Hopf bifurcation</a> 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.

<li class="gallerycaption">Subcritical Turing bifurcation: formation of a hexagonal pattern from noisy initial conditions in the above two-component reaction–diffusion system of Fitzhugh–Nagumo type.</li>

<div class="gallerytext"> Noisy initial conditions at t = 0. </div>

</li>

<div class="gallerytext"> State of the system at t = 10. </div>

</li>

<div class="gallerytext"> Almost converged state at t = 100.</div>

</li>

</ul>

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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation>

</semantics>

</math><img src="./44e1fc5b936101f66caea7f346e2c29a3b879ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:44.71ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}" loading="lazy"></dd></dl>

If the bifurcation is subcritical, often localized structures (<a href="Dissipative_solitons" class="mw-redirect" title="Dissipative solitons">dissipative solitons</a>) can be observed in the <a href="Hysteresis" title="Hysteresis">hysteretic</a> region where the pattern coexists with the ground state. Other frequently encountered structures comprise pulse trains (also known as <a href="Periodic_travelling_wave" title="Periodic travelling wave">periodic travelling waves</a>), 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<a href="#cite_note-13">[13]</a>

<li class="gallerycaption">Other patterns found in the above two-component reaction–diffusion system of Fitzhugh–Nagumo type.</li>

<div class="gallerytext"> Rotating spiral. </div>

</li>

<div class="gallerytext"> Target pattern. </div>

</li>

<div class="gallerytext"> Stationary localized pulse (dissipative soliton).</div>

</li>

</ul>

<div class="mw-heading mw-heading2"><h2 id="Three-_and_more-component_reaction–diffusion_equations">Three- and more-component reaction–diffusion equations</h2></div>

For a variety of systems, reaction–diffusion equations with more than two components have been proposed, e.g. the <a href="Belousov%E2%80%93Zhabotinsky_reaction" title="Belousov–Zhabotinsky reaction">Belousov–Zhabotinsky reaction</a>,<a href="#cite_note-14">[14]</a> for <a href="Blood_clotting" class="mw-redirect" title="Blood clotting">blood clotting</a>,<a href="#cite_note-15">[15]</a> fission waves<a href="#cite_note-16">[16]</a> or planar <a href="Gas_discharge" class="mw-redirect" title="Gas discharge">gas discharge</a> systems.<a href="#cite_note-17">[17]</a>

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).<a href="#cite_note-18">[18]</a> An introduction and systematic overview of the possible phenomena in dependence on the properties of the underlying system is given in.<a href="#cite_note-19">[19]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications_and_universality">Applications and universality</h2></div>

In recent times, reaction–diffusion systems have attracted much interest as a prototype model for <a href="Pattern_formation" title="Pattern formation">pattern formation</a>.<a href="#cite_note-:0-20">[20]</a> 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 <a href="Morphogenesis" title="Morphogenesis">morphogenesis</a> in biology<a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a> and may even be related to animal coats and skin pigmentation.<a href="#cite_note-23">[23]</a><a href="#cite_note-Murray2013-24">[24]</a> Other applications of reaction–diffusion equations include ecological invasions,<a href="#cite_note-25">[25]</a> spread of epidemics,<a href="#cite_note-26">[26]</a> tumour growth,<a href="#cite_note-27">[27]</a><a href="#cite_note-28">[28]</a><a href="#cite_note-29">[29]</a> dynamics of fission waves,<a href="#cite_note-Osborne_100588-30">[30]</a> wound healing<a href="#cite_note-31">[31]</a> and visual hallucinations.<a href="#cite_note-32">[32]</a> 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.<a href="#cite_note-fife-8">[8]</a><a href="#cite_note-mikhailov-9">[9]</a><a href="#cite_note-33">[33]</a><a href="#cite_note-34">[34]</a><a href="#cite_note-35">[35]</a><a href="#cite_note-:0-20">[20]</a>

<div class="mw-heading mw-heading2"><h2 id="Experiments">Experiments</h2></div>

Well-controllable experiments in chemical reaction–diffusion systems have up to now been realized in three ways. First, gel reactors<a href="#cite_note-36">[36]</a> or filled capillary tubes<a href="#cite_note-37">[37]</a> may be used. Second, <a href="Temperature" title="Temperature">temperature</a> pulses on <a href="Catalytic_converter" title="Catalytic converter">catalytic surfaces</a> have been investigated.<a href="#cite_note-38">[38]</a><a href="#cite_note-39">[39]</a> Third, the propagation of running nerve pulses is modelled using reaction–diffusion systems.<a href="#cite_note-fitzhugh-11">[11]</a><a href="#cite_note-40">[40]</a>

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

<div class="mw-heading mw-heading2"><h2 id="Numerical_treatments">Numerical treatments</h2></div>

A reaction–diffusion system can be solved by using methods of <a href="Numerical_mathematics" class="mw-redirect" title="Numerical mathematics">numerical mathematics</a>. There exist several numerical treatments in research literature.<a href="#cite_note-43">[43]</a><a href="#cite_note-:0-20">[20]</a><a href="#cite_note-44">[44]</a> Numerical solution methods for complex <a href="Geometry" title="Geometry">geometries</a> are also proposed.<a href="#cite_note-45">[45]</a><a href="#cite_note-46">[46]</a> Reaction-diffusion systems are described to the highest degree of detail with particle based simulation tools like SRSim or ReaDDy<a href="#cite_note-47">[47]</a> which employ among others reversible interacting-particle reaction dynamics.<a href="#cite_note-48">[48]</a>

<ul><li><a href="Autowave" title="Autowave">Autowave</a></li>

<li><a href="Diffusion-controlled_reaction" title="Diffusion-controlled reaction">Diffusion-controlled reaction</a></li>

<li><a href="Chemical_kinetics" title="Chemical kinetics">Chemical kinetics</a></li>

<li><a href="Phase_space_method" title="Phase space method">Phase space method</a></li>

<li><a href="Autocatalytic_reactions_and_order_creation" class="mw-redirect" title="Autocatalytic reactions and order creation">Autocatalytic reactions and order creation</a></li>

<li><a href="Pattern_formation" title="Pattern formation">Pattern formation</a></li>

<li><a href="Patterns_in_nature" title="Patterns in nature">Patterns in nature</a></li>

<li><a href="Periodic_travelling_wave" title="Periodic travelling wave">Periodic travelling wave</a></li>

<li><a href="Self-similar_solution" title="Self-similar solution">Self-similar solutions</a></li>

<li><a href="Diffusion_equation" title="Diffusion equation">Diffusion equation</a></li>

<li><a href="Stochastic_geometry" title="Stochastic geometry">Stochastic geometry</a></li>

<li><a href="MClone" title="MClone">MClone</a></li>

<li><a href="The_Chemical_Basis_of_Morphogenesis" title="The Chemical Basis of Morphogenesis">The Chemical Basis of Morphogenesis</a></li>

<li><a href="Turing_pattern" title="Turing pattern">Turing pattern</a></li>

<li><a href="Multi-state_modeling_of_biomolecules" title="Multi-state modeling of biomolecules">Multi-state modeling of biomolecules</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li><a href="Fisher's_equation" class="mw-redirect" title="Fisher's equation">Fisher's equation</a></li>

<li><a href="Zeldovich%E2%80%93Frank-Kamenetskii_equation" class="mw-redirect" title="Zeldovich–Frank-Kamenetskii equation">Zeldovich–Frank-Kamenetskii equation</a></li>

<li><a href="FitzHugh%E2%80%93Nagumo_model" title="FitzHugh–Nagumo model">FitzHugh–Nagumo model</a></li>

<li>Wrinkle paint</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> R. A. Fisher, Ann. Eug. 7 (1937): 355

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFNewellWhitehead1969" class="citation journal cs1">Newell, Alan C.; Whitehead, J. A. (September 3, 1969). "Finite bandwidth, finite amplitude convection". Journal of Fluid Mechanics. 38 (2). Cambridge University Press (CUP): 279–303. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1969JFM....38..279N">1969JFM....38..279N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs0022112069000176">10.1017/s0022112069000176</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-1120">0022-1120</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:73620481">73620481</a>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> Y. B. Zeldovich and D. A. Frank-Kamenetsky, Acta Physicochim. 9 (1938): 341

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFTuring1952" class="citation journal cs1"><a href="Alan_Turing" title="Alan Turing">Turing, A. M.</a> (August 14, 1952). <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstb.1952.0012">"The chemical basis of morphogenesis"</a>. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences. 237 (641). The Royal Society: 37–72. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1952RSPTB.237...37T">1952RSPTB.237...37T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstb.1952.0012">10.1098/rstb.1952.0012</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2054-0280">2054-0280</a>.</cite>

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<li id="cite_note-45"><a href="#cite_ref-45">^</a> <cite id="CITEREFIsaacsonPeskin2006" class="citation journal cs1">Isaacson, Samuel A.; Peskin, Charles S. (2006). "Incorporating Diffusion in Complex Geometries into Stochastic Chemical Kinetics Simulations". SIAM J. Sci. Comput. 28 (1): 47–74. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006SJSC...28...47I">2006SJSC...28...47I</a>. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.105.2369">10.1.1.105.2369</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F040605060">10.1137/040605060</a>.</cite>

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<li id="cite_note-47"><a href="#cite_ref-47">^</a> <cite id="CITEREFSchönebergUllrichNoé2014" class="citation journal cs1">Schöneberg, Johannes; Ullrich, Alexander; Noé, Frank (October 24, 2014). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4347613">"Simulation tools for particle-based reaction-diffusion dynamics in continuous space"</a>. BMC Biophysics. 7 (1): 11. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1186%2Fs13628-014-0011-5">10.1186/s13628-014-0011-5</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2046-1682">2046-1682</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4347613">4347613</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25737778">25737778</a>.</cite>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://mrob.com/pub/comp/xmorphia/">Reaction–Diffusion by the Gray–Scott Model: Pearson's parameterization</a> a visual map of the parameter space of Gray–Scott reaction diffusion.</li>

<li><a rel="nofollow" class="external text" href="http://hantz.web.elte.hu/cikkfile/hantzth.pdf">A thesis on reaction–diffusion patterns with an overview of the field</a></li>

<li><a rel="nofollow" class="external text" href="http://www.karlsims.com/rdtool.html">RD Tool: an interactive web application for reaction-diffusion simulation</a></li></ul>

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