Turing patterns
A Turing pattern is a spatial pattern that can arise when diffusion destabilises a homogeneous equilibrium that is stable without spatial movement. The idea was introduced by Alan Turing in 1952 as a mathematical mechanism for biological pattern formation.
Reaction–diffusion model
For two interacting quantities \(u(x,t)\) and \(v(x,t)\),
\[\frac{\partial u}{\partial t}=D_u\nabla^2u+f(u,v),\] \[\frac{\partial v}{\partial t}=D_v\nabla^2v+g(u,v).\]| Term | Meaning |
|---|---|
| \(f(u,v),g(u,v)\) | local reaction or interaction dynamics |
| \(D_u,D_v\) | diffusion coefficients |
| \(\nabla^2\) | Laplacian describing spatial spreading |
A homogeneous equilibrium \((u^*,v^*)\) satisfies \(f(u^*,v^*)=g(u^*,v^*)=0\).
Stable without diffusion
The reaction Jacobian at the equilibrium is
\[J=\begin{pmatrix}f_u&f_v\\g_u&g_v\end{pmatrix}.\]For the non-spatial two-variable system, local stability requires
\[\operatorname{tr}(J)=f_u+g_v<0,\qquad \det(J)=f_ug_v-f_vg_u>0.\]So a small uniform disturbance decays when diffusion is absent.
Spatial perturbations
A small spatial perturbation can be decomposed into modes such as
\[e^{\lambda t}\cos(kx),\]where \(k\) is the wavenumber and \(\lambda\) is its growth rate. The wavelength is
\[L=\frac{2\pi}{k}.\]After including diffusion, the matrix governing a mode of wavenumber \(k\) is
\[J_k=J-k^2\begin{pmatrix}D_u&0\\0&D_v\end{pmatrix}.\]The growth rates are the eigenvalues of \(J_k\). A Turing instability occurs if at least one spatial mode has a positive largest eigenvalue.
A numerical Turing example
Consider the linearised reaction matrix and diffusion coefficients
\[J=\begin{pmatrix}1&-2\\2&-3\end{pmatrix},\qquad D_u=0.05,\qquad D_v=1.\]Without diffusion, \(\operatorname{tr}(J)=-2<0\) and \(\det(J)=1>0\), so the homogeneous equilibrium is stable. For a spatial mode,
\[J_k=\begin{pmatrix}1-0.05k^2&-2\\2&-3-k^2\end{pmatrix}.\]The two growth rates are calculated from the characteristic equation
\[\lambda^2-\operatorname{tr}(J_k)\lambda+\det(J_k)=0.\]Notice that \(k=0\), the non-spatial mode, has negative growth rate. A band of non-zero wavenumbers has positive growth rate. This is precisely the diffusion-driven instability.
What the growing mode looks like
If a mode with wavenumber \(k\) grows, its spatial shape can be represented by
\[u(x)=u^*+A\cos(kx).\]For illustration, take \(u^*=1\), \(A=0.18\) and \(k=2\):
As the unstable mode grows, peaks and troughs become more pronounced. Nonlinear reaction terms eventually become important and can limit the growth, producing a persistent spatial pattern.
Classical two-component conditions
For a stable reaction equilibrium to become unstable through diffusion, a common set of conditions is
\[f_u+g_v<0,\] \[f_ug_v-f_vg_u>0,\] \[D_vf_u+D_ug_v>0,\] \[(D_vf_u+D_ug_v)^2>4D_uD_v(f_ug_v-f_vg_u).\]The first two give stability without diffusion. The last two allow diffusion to destabilise a range of spatial modes.
Why unequal diffusion matters
If the two components diffuse at very different rates, local reaction feedback and spatial spreading act on different scales. In the classical activator–inhibitor interpretation, a slowly spreading activator can reinforce a local increase while a faster spreading inhibitor suppresses nearby regions.
From instability to biological pattern
The linear analysis predicts which small perturbations initially grow. It does not by itself determine the final nonlinear pattern. The final structure also depends on the nonlinear reaction terms, domain size, geometry, boundary conditions and initial perturbations.
Turing mechanisms have been investigated in developmental biology, pigmentation, tissue organisation and other systems in which interacting substances or signals can generate spatial structure.