Watch spots, stripes and coral grow themselves out of nothing but two chemicals spreading and reacting. This is the Gray-Scott model — the classic demonstration of Turing patterns, where a fast diffuser and a slow one break their own symmetry into structure. Like our Game of Life, it's rich global order emerging from nothing but simple local rules. Tune the feed and kill rates, or paint your own reaction fronts onto the canvas.
Click or drag on the canvas to paint reaction fronts
PatternCoral
Feed rate f0.0545
Kill rate k0.0620
RegimeCoral fronts
Steps0
Where the current f and k sit in Gray-Scott parameter space. Move the sliders or pick a preset and the ring jumps to the matching zone — a tiny step across a boundary can switch the pattern entirely.
In 1952 Alan Turing showed that two reacting, diffusing chemicals can break their own symmetry into spots and stripes. His paper on morphogenesis proposed that a slow-spreading activator and a fast-spreading inhibitor are enough to turn a featureless start into an ordered pattern — a mechanism now widely invoked for leopard spots, zebra stripes and seashell markings. The Gray-Scott model shown here is one of the simplest systems that reproduces them: change nothing but the feed and kill rates and the same two equations grow coral, dividing cells, mazes or moving waves.
Reading the simulation
What the colours, the readouts and the ring on the f–k map are telling you.
1
Colour tracks the second chemical
The canvas shows how much of chemical V sits in each cell: dark where there is almost none, brightening through magenta and hot pink to near-white at the peaks, while chemical U fills the dark background it is consumed from. Watch the two spread at different rates — the fast, smooth halo is U diffusing, the sharp self-multiplying fronts are V. Click or drag to inject a fresh blob of V and a new front nucleates on the spot.
2
Feed and kill place you on the map
The Feed rate f and Kill rate k readouts echo the two sliders, and the ring on the f–k parameter map shows exactly where that pair lands among the labelled zones. The Regime readout names the nearest family — coral fronts, dividing cells, isolated spots, a labyrinth or travelling waves — so you can see the same two equations settle into completely different behaviour as the ring crosses a boundary.
3
Local rules, a global pattern
Every cell follows the identical rule using only its neighbours, yet the Steps counter climbs and evenly spaced marks fill the whole grid — a Turing pattern no single cell planned. Pick a preset to jump to a known-good setting and re-seed, or nudge the sliders and watch the spacing and shape drift as the pattern re-organises around the new balance.
How It Works
Two chemicals, a reaction, and two speeds of spreading — nothing more.
1
React and diffuse
Chemical U is fed in everywhere; the reaction U + 2V → 3V turns it into V; and V is steadily removed. Both chemicals also diffuse, but U spreads about twice as fast as V. Each cell updates from just its neighbours — a local rule applied everywhere at once.
2
The feed/kill balance picks the pattern
Two dials decide everything: the feed rate f (how fast fresh U arrives) and the kill rate k (how fast V is removed). A low kill lets V spread into mazes and coral; a higher kill pinches it into discrete spots; special balances make cells that keep dividing or fronts that travel as waves.
3
Local rules become a global Turing pattern
No cell knows the shape it belongs to, yet the fast inhibitor and slow activator settle into stable, evenly spaced marks — a Turing pattern. Paint a fresh blob of V and watch a new front nucleate and organise, the same way structure is thought to emerge on a growing animal's skin.
What is reaction-diffusion?
Reaction-diffusion describes what happens when substances both spread out (diffuse) and transform into one another (react) at the same time. When two chemicals diffuse at different speeds and feed back on each other, the smooth mixture can spontaneously organise into stable spots, stripes and labyrinths instead of blurring into uniform grey. It is a standard model for pattern formation in chemistry, biology and physics.
What is the Gray-Scott model?
The Gray-Scott model is a simple two-chemical reaction-diffusion system. A chemical U is steadily fed in, a reaction U + 2V → 3V converts it into V, and V is removed at a set kill rate. Each chemical also diffuses, with U spreading roughly twice as fast as V. Despite only two parameters — the feed rate f and the kill rate k — it reproduces a remarkable zoo of patterns: dividing cells, coral, moving spots, mazes and travelling waves.
What are Turing patterns, and how do they relate to animal coats?
Turing patterns are the spots and stripes that emerge when a slow-diffusing activator and a fast-diffusing inhibitor react together — a mechanism Alan Turing proposed in 1952 to explain how a featureless embryo can develop structure. The same short-range-activation, long-range-inhibition idea is widely used to explain leopard spots, zebra and tiger stripes, the ridges on seashells and the markings on tropical fish, where the balance of the two signals sets the size and spacing of the marks. The Gray-Scott system running here has the same ingredient — the self-multiplying chemical V diffuses at half the rate of the U it consumes — which is why nudging the feed and kill rates shifts the pattern between spots, stripes and mazes instead of simply making it brighter.
What do the feed and kill rates do?
The feed rate f sets how fast fresh U is supplied, and the kill rate k sets how fast V is removed. Together they decide which pattern the system settles into. Low kill relative to feed lets V spread into mazes and coral; higher kill isolates it into discrete spots; particular balances produce cells that keep dividing (mitosis) or fronts that travel as waves. Small changes to f and k can switch the pattern entirely, which is why they are the two dials that matter most.
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Classroom activity
A ready-to-assign lab using the simulation above. Free to use — nothing to install or sign up for, and no student data leaves the browser.
High school, College10–20 minInteractive simulation
Learning objectives
Explain how patterns emerge from reaction and diffusion.
Describe the activator–inhibitor mechanism.
Connect the patterns to nature.
Variables to change
Feed rate
Kill rate
Diffusion rates
Procedure
Start the simulation and watch spots or stripes emerge from near-uniform starting conditions.
Adjust the feed and kill rates and observe how the pattern changes.
Compare patterns that form spots versus mazes/stripes.
Observations
Describe the patterns that form and how parameter changes shift them between spots, stripes, and mazes.
Questions
What two processes create the patterns?
What is an activator–inhibitor system?
Do the patterns require a pre-drawn template?
Who first proposed this mechanism for biological patterns?
Give a natural example of such patterns.
Explanation
Reaction–diffusion systems pair a self-activating chemical with a faster-diffusing inhibitor. From near-uniform starts they self-organize into spots, stripes, and mazes — Turing patterns that model animal markings and other natural patterning.
Answer key (for instructors)
1. Local chemical reaction and diffusion (spreading) of the reacting substances.
2. One substance promotes its own production (activator) while another suppresses it (inhibitor) and diffuses faster, producing spaced patterns.
3. No — they self-organize from small fluctuations; the pattern is emergent.
4. Alan Turing — these are called Turing patterns.
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