---
title: "Turing patterns Interactive Lab | Sandbox Physics"
description: "Why can diffusion help a uniform mixture grow a pattern? Two reacting substances can amplify small spatial differences when they spread at different rates."
canonical_url: "https://sandboxphysics.com/labs/turing-patterns/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/turing-patterns/"
last_modified: "2026-10-28"
---

FR21 · Reaction and diffusion

# Turing patterns

**Why can diffusion help a uniform mixture grow a pattern?** Two reacting substances can amplify small spatial differences when they spread at different rates. The growing wavelengths come from the reaction rates, diffusion and the size of the domain.

Watch small concentration differences become spots. Change diffusion, paint a disturbance and compare measured growth with a predicted spectrum.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Two dimensionless Schnakenberg species react on a periodic square. Color shows activator concentration on one fixed scale. The growth-spectrum record uses separate tiny-perturbation runs; no biological rates are fitted.

Evolution time $30$

Mean concentration $1$

Spatial contrast $0.1$

Peak concentration $2$

SIMULATION RECORD AND ANALYTIC REFERENCE

## Which spatial scales grow?

Cyan: continuum growth rate. Dashed gold: the 32-cell finite-difference prediction. White points: slopes fitted to nine saved Fourier-amplitude samples from separate tiny-perturbation simulations. Those runs use the same current reaction parameters as the field.

The field uses one fixed color scale: dark blue at zero, cyan near one and amber at 2.5 and above. Opposite edges touch.

[Read the derivation and worked examples](https://sandboxphysics.com/stories/morphogenesis/)

Physics tutorial

## How diffusion selects a spacing

**Background** A mixture can return to equilibrium after a uniform disturbance and still amplify a striped disturbance. Different diffusion rates change the response to each spatial scale.

**Why it matters** This gives a precise way to ask how a pattern starts: calculate which small ripples grow, then measure their growth before the peaks become large.

Start with the essentials

- **Focus question**: Which sizes of disturbance grow in this mixture?
- **One-sentence intuition**: Diffusion changes the two coupled responses by different amounts, and an intermediate range of spatial scales can become unstable.

### Core mathematical model

#### Reaction and transport

$$
\partial_t u=a-u+u^2v+D_u\nabla^2u,\qquad \partial_t v=b-u^2v+D_v\nabla^2v
$$

The activator consumes substrate to make more activator. Both species diffuse, and the surrounding reservoir supplies each at a fixed rate. Every quantity here is dimensionless.

#### A small spatial ripple

$$
\partial_t\begin{pmatrix}\delta u_k\\\delta v_k\end{pmatrix}=\left[J-k^2\begin{pmatrix}D_u&0\\0&D_v\end{pmatrix}\right]\begin{pmatrix}\delta u_k\\\delta v_k\end{pmatrix}
$$

The matrix tells how a sinusoidal perturbation changes. The largest real part of its eigenvalues is the exponential growth rate. Positive values predict early growth.

### Common difficulties

#### Fast transport can destabilize a mixture

**Typical misconception** Diffusion always smooths things out, so it cannot make a pattern.

**Better mental model** Each species does smooth out. But when substrate moves much faster, its redistribution changes the local reaction feedback. The coupled system can amplify some disturbances.

#### A mature spot is outside the linear approximation

**Typical misconception** The predicted growth rate should describe every frame.

**Better mental model** The spectrum applies close to the homogeneous equilibrium. Large spots deplete their surroundings and interact nonlinearly. Their late growth is governed by the full reaction equations.

### Run the experiment

1. 01

   #### Follow a growing band

   Start with Spots emerge. Find the white points above the zero line in the spectrum.

   **What to observe:** Those modes grow from tiny perturbations. Their finite-grid reference differs slightly from the smooth continuum prediction.

2. 02

   #### Make diffusion equal

   Select Diffusion smooths, then drag across the field to add activator.

   **What to observe:** The disturbance spreads and fades. Equal diffusion subtracts the same damping from both coupled modes.

3. 03

   #### Separate oscillation from patterning

   Select Mixture oscillates and watch the field change in time.

   **What to observe:** The uniform mode is now unstable too. A changing color across the mixture is a temporal oscillation; a stationary pattern grows at nonzero spatial wave number.

Physical conclusion

A pattern begins when a spatial mode grows faster than the homogeneous mixture relaxes.

## Related experiments and articles

Morphogenesis and reaction–diffusion · Reading path

### Reaction, diffusion, growth and active stress

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How diffusion can help a pattern grow**](https://sandboxphysics.com/stories/morphogenesis/reaction-and-diffusion/)
2. [Article **Calculate which ripples will grow**](https://sandboxphysics.com/stories/morphogenesis/turing-instability/)
3. [Article **Measure a growth rate from a simulation record**](https://sandboxphysics.com/stories/morphogenesis/measuring-growth/)
4. [Article **Recalculate Turing’s two-cell example**](https://sandboxphysics.com/stories/morphogenesis/turing-paper-guide/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)