Reaction–diffusion system

From Wikipedia, the free encyclopedia
  (Redirected from Reaction-diffusion system)
Jump to: navigation, search



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

where each component of the vector q(x,t) represents the concentration of one substance, D is a diagonal matrix of diffusion coefficients, and R accounts for all local reactions. The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation of travelling waves and wave-like phenomena as well as other self-organized patterns like stripes, hexagons or more intricate structure like dissipative solitons.

One-component reaction–diffusion equations[edit]

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

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

is also referred to as the KPP (Kolmogorov-Petrovsky-Piskounov) equation.[1] If the reaction term vanishes, then the equation represents a pure diffusion process. The corresponding equation is Fick's second law. The choice R(u) = u(1-u) yields Fisher's equation that was originally used to describe the spreading of biological populations,[2] the Newell-Whitehead-Segel equation with R(u) = u(1 − u2) to describe Rayleigh-Benard convection,[3][4] the more general Zeldovich equation with R(u) = u(1 − u)(u − α) and 0 < α < 1 that arises in combustion theory,[5] and its particular degenerate case with R(u) = u2 − u3 that is sometimes referred to as the Zeldovich equation as well.[6]


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

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

 \mathfrak L=\int\limits_{-\infty}^\infty\left[\frac


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 u(xt) = û(ξ) with ξ = x − ct, where c is the speed of the travelling wave. Note that while travelling waves are generically stable structures, all non-monotonous stationary solutions (e.g. localized domains composed of a front-antifront pair) are unstable. For c = 0, there is a simple proof for this statement:[7] if u0(x) is a stationary solution and u=u0(x) + ũ(xt) is an infinitesimally perturbed solution, linear stability analysis yields the equation

\partial_t \tilde{u}=D\partial_x^2
\tilde{u}-U(x)\tilde{u},\quad U(x) =


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



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



Two-component reaction–diffusion equations[edit]

Two-component systems allow for a much larger range of possible phenomena than their one-component counterparts. An important idea that was first proposed by Alan Turing is that a state that is stable in the local system should become unstable in the presence of diffusion.[9]


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

a plane wave perturbation

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

of the stationary homogeneous solution will satisfy

    \partial_t \tilde{u}_{\boldsymbol{k}}(t)\\
    \partial_t \tilde{v}_{\boldsymbol{k}}(t)
\right) = -k^2\left(
\right) + \boldsymbol{R}^{\prime} \left(

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

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


\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

with ƒ(u) = λu − u3 − κ which describes how an action potential travels through a nerve.[10][11] Here, du, dv, τ, σ and λ are positive constants.



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



For a variety of systems, reaction-diffusion equations with more than two components have been proposed, e.g. as models for the Belousov-Zhabotinsky reaction, ,[13] for blood clotting[14] or planar gas discharge systems. [15]


Applications and universality[edit]






See also[edit]


  20. ^ J. D. Murray, Mathematical Biology, Springer (1993)

External links[edit]