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| '''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 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.
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| 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 equation]]s. They can be represented in the general form
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| :<math>
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| \partial_t \boldsymbol{q} = \underline{\underline{\boldsymbol{D}}} \,\nabla^2 \boldsymbol{q}
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| + \boldsymbol{R}(\boldsymbol{q}), </math>
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| where each component of the vector '''q'''('''x''',''t'') represents the concentration of one substance, {{uuline|'''''D'''''}} is a [[diagonal matrix]] of [[diffusion coefficient]]s, and '''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
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| and wave-like phenomena as well as other [[Self-organization|self-organized]] [[pattern formation|patterns]] like stripes, hexagons or more intricate structure like [[dissipative solitons]].
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| == One-component reaction–diffusion equations ==
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| The simplest reaction–diffusion equation concerning the concentration ''u'' of a single substance in one spatial dimension,
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| :<math>
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| \partial_t u = D \partial^2_x u + R(u),
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| </math>
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| is also referred to as the KPP (Kolmogorov-Petrovsky-Piskounov) equation.<ref>A. Kolmogorov
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| et al., Moscow Univ. Bull. Math. A 1 (1937): 1</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 ''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 ''R''(''u'') = ''u''(1 − ''u''<sup>2</sup>) to describe [[Bénard cells|Rayleigh-Benard convection]],<ref>A. C. Newell and J. A. Whitehead, J. Fluid Mech. 38 (1969): 279</ref><ref>L. A. Segel,
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| J. Fluid Mech. 38 (1969): 203</ref> the more general [[Zeldovich]] equation with ''R''(''u'') = ''u''(1 − ''u'')(''u'' − ''α'') and 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 ''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, Travelling Waves in Nonlinear Diffusion Convection Reaction, Birkhäuser (2004)</ref>
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| 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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| :<math>
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| \partial_t u=-\frac{\delta\mathfrak L}{\delta u}
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| </math>
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| and therefore describes a permanent decrease of the "free energy" <math>\mathfrak L</math> given by the functional
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| :<math> \mathfrak L=\int\limits_{-\infty}^\infty\left[\frac
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| D2(\partial_xu)^2-V(u)\right]\text{d}x
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| </math>
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| with a potential ''V''(''u'') such that ''R''(''u'')=d''V''(''u'')/d''u''.
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| [[Image:Travelling wave for Fisher equation.svg|thumb|right|A travelling wave front solution for Fisher's equation.]]
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| 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
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| simple proof for this statement:<ref name="fife">P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer (1979)</ref> if ''u<sub>0</sub>''(''x'') is a stationary solution and ''u''=''u''<sub>0</sub>(''x'') + ''ũ''(''x'', ''t'') is an infinitesimally perturbed solution, linear stability analysis yields the equation
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| :<math>
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| \partial_t \tilde{u}=D\partial_x^2
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| \tilde{u}-U(x)\tilde{u},\quad U(x) =
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| -R^{\prime}(u)|_{u=u_0(x)}.</math>
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| With the ansatz ''ũ'' = ''ψ''(''x'')exp(−''λt'') we arrive at the eigenvalue problem
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| :<math> \hat H\psi=\lambda\psi, \qquad
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| \hat H=-D\partial_x^2+U(x),
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| </math>
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| of [[Schrödinger equation|Schrödinger type]] where negative eigenvalues result in the instability of the solution. Due to translational invariance ''ψ'' = ∂<sub>''x''</sub>''u''<sub>0</sub>(''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'' = ∂<sub>''x''</sub> ''u''<sub>0</sub>(''x'') should have at least one zero, and for a non-monotonic stationary solution the corresponding eigenvalue ''λ'' = 0 cannot be the
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| lowest one, thereby implying instability.
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| 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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| :<math>
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| D \partial^2_{\xi}\hat{u}(\xi)+ c\partial_{\xi} \hat{u}(\xi)+R(\hat{u}(\xi))=0.
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| </math>
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| This equation has a nice mechanical analogue as the motion of a
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| mass ''D'' with position ''û'' in the course of the "time" ''ξ'' under
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| the force ''R'' with the damping coefficient c which allows for a
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| rather illustrative access to the construction of different
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| types of solutions and the determination of ''c''.
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| When going from one to more space dimensions, a number of
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| statements from one-dimensional systems can still be applied.
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| Planar or curved wave fronts are typical structures, and a new
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| effect arises as the local velocity of a curved front becomes
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| dependent on the local [[curvature|radius of curvature]] (this can be
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| seen by going to [[polar coordinates]]). This phenomenon leads
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| to the so-called curvature-driven instability.<ref name="mikhailov">
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| A. S. Mikhailov, Foundations of Synergetics I.
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| Distributed Active Systems, Springer (1990)</ref>
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| == Two-component reaction–diffusion equations ==
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| Two-component systems allow for a much larger range of possible
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| phenomena than their one-component counterparts. An important
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| idea that was first proposed by [[Alan Turing]] is that a state
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| that is stable in the local system should become unstable in
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| the presence of [[diffusion]].<ref>A. M. Turing, Phil.
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| Transact. Royal Soc. B 237 (1952): 37</ref>
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| A linear stability analysis however shows that when linearizing
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| the general two-component system
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| :<math> \left( \begin{array}{c}
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| \partial_t u\\ \partial_t v
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| \end{array} \right) =
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| \left(\begin{array}{cc} D_u &0\\0&D_v
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| \end{array}\right)
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| \left( \begin{array}{c} \partial_{xx} u\\ \partial_{xx} v
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| \end{array}\right) + \left(\begin{array}{c} F(u,v)\\G(u,v)
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| \end{array}\right)
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| </math>
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| a [[plane wave]] perturbation
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| :<math>
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| \tilde{\boldsymbol{q}}_{\boldsymbol{k}}(\boldsymbol{x},t) =
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| \left(\begin{array}{c}
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| \tilde{u}(t)\\\tilde{v}(t)\end{array}\right) e^{i
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| \boldsymbol{k} \cdot \boldsymbol{x}} </math>
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| of the stationary homogeneous solution will satisfy
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| :<math>
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| \left(
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| \begin{array}{c}
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| \partial_t \tilde{u}_{\boldsymbol{k}}(t)\\
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| \partial_t \tilde{v}_{\boldsymbol{k}}(t)
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| \end{array}
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| \right) = -k^2\left(
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| \begin{array}{c}
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| D_u \tilde{u}_{\boldsymbol{k}}(t)\\
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| D_v\tilde{v}_{\boldsymbol{k}}(t)
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| \end{array}
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| \right) + \boldsymbol{R}^{\prime} \left(
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| \begin{array}{c}
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| \tilde{u}_{\boldsymbol{k}}(t)\\
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| \tilde{v}_{\boldsymbol{k}}(t)
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| \end{array}
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| \right). </math>
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| Turing's idea can only be realized in four
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| [[equivalence class]]es of systems characterized
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| by the signs of the [[Jacobian]]
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| '''R'''' of the reaction function. In particular, if a finite
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| wave vector '''k''' is supposed to be the most unstable one,
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| the Jacobian must have the signs
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| :<math> \left(\begin{array}{cc} +&-\\+&-\end{array}\right),
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| \quad \left(\begin{array}{cc} +&+\\-&-\end{array}\right), \quad
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| \left(\begin{array}{cc} -&+\\-&+\end{array}\right), \quad
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| \left(\begin{array}{cc} -&-\\+&+\end{array}\right). </math>
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| This class of systems is named ''activator-inhibitor system''
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| after its first representative: close to the ground state, one
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| component stimulates the production of both components while
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| the other one inhibits their growth. Its most prominent
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| representative is the [[FitzHugh–Nagumo equation]]
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| :<math>
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| \begin{align}
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| \partial_t u &= d_u^2 \,\nabla^2 u + f(u) - \sigma v, \\
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| \tau \partial_t v &= d_v^2 \,\nabla^2 v + u - v
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| \end{align}
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| </math>
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| with ''ƒ''(''u'') = ''λu'' − ''u''<sup>3</sup> − ''κ'' which describes how an [[action potential]] travels
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| through a nerve.<ref name="fitzhugh">R. FitzHugh, Biophys. J. 1 (1961):
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| 445</ref><ref>J. Nagumo et al., Proc. Inst. Radio Engin.
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| Electr. 50 (1962): 2061</ref> Here, ''d<sub>u</sub>'',
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| ''d<sub>v</sub>'', ''τ'', ''σ'' and ''λ'' are
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| positive constants.
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| When an activator-inhibitor system undergoes a change of parameters, one may pass
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| from conditions under which a homogeneous ground state is
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| stable to conditions under which it is linearly unstable. The
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| corresponding [[Bifurcation theory|bifurcation]] may be either
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| a [[Hopf bifurcation]] to a globally oscillating homogeneous
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| state with a dominant wave number ''k'' = 0 or a
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| ''Turing bifurcation'' to a globally patterned state with
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| a dominant finite wave number. The latter in two
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| spatial dimensions typically leads to stripe or hexagonal
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| patterns.
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| <gallery caption="Subcritical Turing bifurcation: formation of
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| a hexagonal pattern from noisy initial conditions in the above
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| two-component reaction-diffusion system of Fitzhugh-Nagumo
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| type. " widths="257" heights="235" perrow="3">
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| Image:Turing_bifurcation_1.gif| Noisy initial conditions at ''t'' = 0.
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| Image:Turing_bifurcation_2.gif| State of the system at ''t'' = 10.
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| Image:Turing_bifurcation_3.gif| Almost converged state at ''t'' = 100.
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| </gallery>
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| For the Fitzhugh-Nagumo example, the neutral stability curves marking the
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| boundary of the linearly stable region for the Turing and Hopf
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| bifurcation are given by
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| :<math>
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| \begin{align}
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| 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]
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| q_{\text{n}}^T(k): &{}\quad \frac{\kappa}{1 + d_v^2 k^2}+ d_u^2 k^2 & = f^{\prime}(u_{h}).
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| \end{align}
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| </math>
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| If the bifurcation is subcritical, often localized structures
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| ([[dissipative solitons]]) can be observed in the
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| [[Hysteresis|hysteretic]] region where the pattern coexists
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| with the ground state. Other frequently encountered structures
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| comprise pulse trains (also known as [[periodic travelling waves]]),
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| 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>N. Kopell and L.N. Howard, Stud. Appl. Math. 52 (1973): 291</ref>
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| <gallery caption="Other patterns found in the above
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| two-component reaction-diffusion system of Fitzhugh-Nagumo
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| type. " widths="265" heights="235" perrow="3">
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| Image:reaction_diffusion_spiral.gif| Rotating spiral.
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| Image:reaction_diffusion_target.gif| Target pattern.
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| Image:reaction_diffusion_stationary_ds.gif| Stationary localized pulse (dissipative soliton).
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| </gallery>
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| == Three- and more-component reaction–diffusion equations ==
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| For a variety of systems, reaction-diffusion equations with
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| more than two components have been proposed, e.g. as models
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| for the [[Belousov-Zhabotinsky reaction]],
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| ,<ref>V. K. Vanag and I. R. Epstein, Phys. Rev. Lett.
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| 92 (2004): 128301</ref> for [[blood clotting]]<ref>E. S. Lobanova
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| and F. I. Ataullakhanov, Phys. Rev. Lett.
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| 93 (2004): 098303</ref> or planar [[gas discharge]] systems.
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| <ref>H.-G. Purwins et al. in: Dissipative Solitons,
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| Lectures Notes in Physics, Ed. N. Akhmediev and A. Ankiewicz,
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| Springer (2005)</ref>
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| It is known that systems with more components allow for
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| a variety of phenomena not possible in systems with one or two
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| components (e.g. stable running pulses in more than one spatial
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| dimension without global feedback),.<ref>C. P. Schenk et al.,
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| Phys. Rev. Lett. 78 (1997): 3781</ref> An introduction and systematic
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| overview of the possible phenomena in dependence on the properties
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| 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, [http://www.springer.com/physics/complexity/book/978-3-642-31250-2 ISBN 978-3-642-31250-2]</ref>
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| == Applications and universality==
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| 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<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>J. D. Murray, Mathematical Biology, Springer (1993)</ref> {{see also|The chemical basis of morphogenesis}}
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| Other applications of reaction-diffusion equations include ecological invasions,<ref>E.E. Holmes et al, Ecology 75 (1994): 17</ref> spread of epidemics,<ref>J.D. Murray et al, Proc. R. Soc. Lond. B 229 (1986: 111</ref> tumour growth<ref>M.A.J. Chaplain J. Bio. Systems 3 (1995): 929</ref><ref>J.A. Sherratt and M.A. Nowak, Proc. R. Soc. Lond. B 248 (1992): 261</ref><ref>R.A. Gatenby and E.T. Gawlinski, Cancer Res. 56 (1996): 5745</ref> and wound healing.<ref>J.A. Sherratt and J.D. Murray, Proc. R. Soc. Lond. B 241 (1990): 29</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>
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| == Experiments ==
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| Well-controllable experiments in chemical reaction-diffusion systems have up to now
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| been realized in three ways. First, gel reactors<ref>K.-J. Lee et al.,
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| Nature 369 (1994): 215</ref> or filled capillary tubes<ref>C. T. Hamik and O. Steinbock,
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| New J. Phys. 5 (2003): 58</ref> may be used. Second, [[temperature]] pulses on [[Catalytic converter|catalytic surfaces]]
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| have been investigated.<ref>H. H. Rotermund et al.,
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| Phys. Rev. Lett. 66 (1991): 3083</ref><ref>M. D. Graham et al.,
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| J. Phys. Chem. 97 (1993): 7564</ref>
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| Third, the propagation of running nerve pulses is modelled
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| using reaction-diffusion systems.<ref name="fitzhugh" /><ref>A. L. Hodgkin and A. F. Huxley,
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| J. Physiol. 117 (1952): 500</ref>
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| Aside from these generic examples, it has turned out that under appropriate
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| circumstances electric transport systems like plasmas<ref>M. Bode and H.-G. Purwins,
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| Physica D 86 (1995): 53</ref> or semiconductors<ref>E. Schöll,
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| Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors,
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| Cambridge University Press (2001)</ref> can be
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| described in a reaction-diffusion approach. For these systems various experiments
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| on pattern formation have been carried out.
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| == See also ==
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| {{Div col}}
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| *[[Autowave]]
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| *[[Diffusion-controlled reaction]]
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| *[[Chemical kinetics]]
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| *[[Phase space method]]
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| *[[Autocatalytic reactions and order creation]]
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| *[[Pattern formation]]
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| *[[Patterns in nature]]
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| *[[Periodic travelling wave]]
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| *[[Stochastic geometry]]
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| *[[MClone]]
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| {{Div col end}}
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| == References ==
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| {{reflist}}
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| == External links ==
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| * [http://texturegarden.com/java/rd/ Java applet] showing a reaction–diffusion simulation
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| * [http://www.joakimlinde.se/java/ReactionDiffusion/index.php Another applet] showing Gray-Scott reaction-diffusion.
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| * [http://cornguide.com/rd.php Java applet] Uses reaction-diffusion to simulate pattern formation in several snake species.
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| * [http://softology.com.au/gallery/galleryrd.htm Gallery] of reaction-diffusion images and movies.
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| * [http://www.texrd.com TexRD software] random texture generator based on reaction-diffusion for graphists and scientific use
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| * [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.
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| * [http://hantz.web.elte.hu/cikkfile/hantzth.pdf A Thesis] on reaction-diffusion patterns with an overview of the field
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| * [http://flexmonkey.blogspot.co.uk/search/label/ReDiLab ReDiLab - Reaction Diffusion Laboratory] Flash & GPU based application simulating Belousov-Zhabotinsky, Gray Scott, Willamowski–Rössler and FitzHugh-Nagumo with full source code.
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| {{DEFAULTSORT:Reaction-diffusion system}}
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| [[Category:Mathematical modeling]]
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| [[Category:Parabolic partial differential equations]]
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