Turing pattern

A Turing pattern is a spatially structured state produced by interactions between local reactions and diffusion. A Turing instability explains linear pattern onset when a homogeneous equilibrium is stable without diffusion but unstable to a nonzero spatial mode.

Table of contents
- Turing instability Turing pattern
  - Fast-inhibitor cubic activator growth rate Turing instability
  - Two-species diffusion-driven instability criterion Turing instability
    - Impossibility of a two-species Turing instability at equal diffusivities Two-species diffusion-driven instability criterion
    - Near-unity diffusivity ratio for a Turing instability Two-species diffusion-driven instability criterion

Turing instability

 0  0

A Turing, or diffusion-driven, instability occurs when a spatially homogeneous equilibrium is stable to homogeneous perturbations but unstable to a nonzero spatial Fourier mode after unequal diffusion is introduced.

Fast-inhibitor cubic activator growth rate ( $σ (k) = - r - D k^{2} + ρ k^{2} / (1 + k^{2})$ )

 0  0

For

u_{t} = D u_{xx} - u (u - r) (u - 1) - ρ (v - u)

and

0 = v_{xx} - v + u

, a Fourier mode perturbation about zero has

v = u / (1 + k^{2})

and growth rate

σ (k) = - r - D k^{2} + ρ k^{2} / (1 + k^{2})

. If

ρ > D

, its maximum is

- r + (ρ - D)^{2}

k^{2} = ρ / D - 1

; if

ρ \leq D

, its maximum is

- r

at zero. Replacing

(u, v, r)

(1 - u, 1 - v, 1 - r)

gives the corresponding criterion at the other homogeneous equilibrium.

Two-species diffusion-driven instability criterion

 0  0

Let a stable two-species reaction equilibrium have Jacobian

J = (a c b d), tr J < 0, det J > 0,

(1)

and diffusion matrix

diag (D_{u}, D_{v})

. A mode with

q = k^{2}

has determinant

det (J - q D) = det J - (D_{v} a + D_{u} d) q + D_{u} D_{v} q^{2} .

(2)

Since its trace decreases with

q

, diffusion-driven instability occurs exactly when

D_{v} a + D_{u} d > 0, (D_{v} a + D_{u} d)^{2} > 4 D_{u} D_{v} det J .

(3)

Impossibility of a two-species Turing instability at equal diffusivities

 0  0

When

D_{u} = D_{v} = D

, every spatial Fourier mode replaces the reaction Jacobian

J

J - D k^{2} I

. Each eigenvalue is shifted left by

D k^{2}

, so diffusion cannot destabilize a stable homogeneous equilibrium.

Near-unity diffusivity ratio for a Turing instability

 0  0

For reaction Jacobian

(- 1 1 + δ - 1 1 - δ),

(1)

the critical diffusivity ratio is

d_{c} = (\frac{2 δ + 1 + δ}{1 - δ})^{2} = 1 + 2 2 δ + O (δ) .

(2)

It approaches one as the stable reaction system approaches marginality.

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A Turing pattern refers to a mathematical model that describes how complex patterns can emerge in biological systems through the interaction of two or more substances that diffuse and react with each other. This concept was introduced by the British mathematician and logician Alan Turing in his 1952 paper titled "The Chemical Basis of Morphogenesis.

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