Diffusion has a reputation to maintain.

Drop ink in water and the boundary softens. Heat a cold room and the gradients flatten. Put a Laplacian into a differential equation and it punishes high frequency variation. Diffusion is the mathematical scent of smoothing.

So the reaction-diffusion story is a little mischievous:

mixing can help make a pattern

Not by itself. Pure diffusion still smooths. The trick is that diffusion is coupled to local reactions. One chemical may catalyze the production of another. One species may spread faster than the other. A spatially uniform state can be stable if every point is forced to stay identical, yet unstable to a spatial perturbation once neighboring points can exchange material.

That is the door Alan Turing opened in 1952. In “The Chemical Basis of Morphogenesis,” he argued that morphogens reacting and diffusing through tissue could account for the spontaneous emergence of spatial structure from an initially homogeneous state.1 The modern phrase is diffusion-driven instability: a homogeneous steady state is safe against well-mixed perturbations but unsafe against perturbations with certain wavelengths.

The idea is bigger than any one stripe, spot, or shell. It is a mechanism. Local rules, spatial transport, and nonlinear feedback can turn sameness into geometry.

A Toy That Pushes Back

The lab below uses the Gray-Scott reaction-diffusion system, a two-species model popular because it produces a zoo of patterns with only a few parameters. Pearson’s 1993 paper showed that this simple model can generate regular hexagons, irregular steady patterns, dividing spots, and chaotic spatiotemporal behavior from finite-amplitude perturbations.2

The dimensionless equations are:

\[\frac{\partial u}{\partial t} = D_u \nabla^2 u - u v^2 + F(1-u)\] \[\frac{\partial v}{\partial t} = D_v \nabla^2 v + u v^2 - (F+k)v.\]

There are two chemicals, \(u\) and \(v\). The term \(u v^2\) is autocatalytic: where enough \(v\) already exists, it helps make more \(v\) by consuming \(u\). The feed term replenishes \(u\). The kill term removes \(v\). Diffusion moves both species across the grid, but at different rates.

The browser simulation uses periodic boundaries and an explicit finite difference update. It is intentionally small enough to inspect. The point is not to build the fastest PDE solver. The point is to watch the qualitative regimes: quiet, spots, crowded texture, and decay.

V concentration Textured parameter cells Active area trace V histogram

Deterministic explicit simulation with periodic boundaries. The local parameter map reruns nearby feed/kill settings at lower resolution to show how narrow the patterning region can be.

Try it like a little bench experiment.

First, lower rollout steps to watch the pattern before it has had time to organize. The trace starts with a finite perturbation and records how contrast and active area evolve. In many settings the field needs a while before it reveals what regime it is in.

Second, move feed and kill by only a few ticks. Some changes look ridiculously small in the equations and large in the field. That is the parameter-map panel’s point: pattern formation often lives on thin regions of parameter space. Nearby feed/kill pairs can mean spots, worms, or decay.

Third, raise V diffusion. If the active species spreads too quickly relative to the reaction, local peaks can wash out. If it spreads too slowly, patches may overcrowd and compete. The pattern is not stored in either chemical alone. It is stored in the loop between reaction and transport.

The metrics are intentionally physical rather than aesthetic. V mass says how much product remains. Contrast measures spatial variation. Active area counts cells above a concentration threshold. Edge energy responds to fine boundaries. Histogram spread tells you whether the grid is mostly one state or split across many concentration levels.

The Seed Is Not the Artist

The initial perturbation matters. Pearson emphasized that many Gray-Scott patterns arise in response to finite-amplitude perturbations, not simply from infinitesimal linear instability.2 That does not make the pattern random in the casual sense.

Randomness seeds the field. The equations select, amplify, and prune.

This distinction shows up everywhere in simulation work. A random seed is often less interesting than the attractor landscape it samples. If many seeds end in the same texture, the system has a robust regime. If tiny parameter changes move the simulation from spots to extinction, the model is telling you that the regime boundary matters more than the seed.

Reaction-diffusion systems are therefore a good antidote to a lazy phrase:

emergence means it came from nowhere

No. Emergence here means the visible scale is not written explicitly into the local rule. The local rule says diffuse, feed, kill, and react. Spot size, spacing, splitting, and decay are consequences of iterating that rule across space.

Do Not Mistake the Grid for Biology

It is tempting to look at the image and say “nature solved stripes this way.” Sometimes reaction-diffusion is the right mechanistic language. Sometimes it is only a useful analogy. Real developmental systems include gene regulation, growth, mechanics, geometry, cell movement, signaling networks, and boundary conditions. A two-chemical PDE on a square grid is not a universal explanation.

Turing’s contribution was more precise and more durable: he showed that known physical processes could, in principle, break spatial symmetry without a hidden prepattern. Later work has refined, criticized, extended, and experimentally tested that idea in many settings. Modern PDE datasets even use Gray-Scott systems as challenging benchmarks for surrogate models because different feed/kill parameters produce qualitatively different long-time dynamics.3

That computational angle is worth noticing. A reaction-diffusion simulator is a tiny world where the prediction problem is not just “next pixel.” It is long-horizon stability, regime change, and sensitivity to parameters. A model trained only on spots may not understand worms. A one-step predictor may look good until repeated rollout slowly erases the pattern.

This is why toy physics can be useful for machine learning even when the biology is not the target. The toy forces us to ask whether a learned simulator has captured the rule or merely interpolated the pictures.

The Smoother Becomes a Pen

The good mental picture is not that diffusion creates pattern by betraying smoothing. It is that diffusion couples local nonlinear reactors.

Every cell is doing simple chemistry. Diffusion lets neighbors perturb one another. The feedback loop decides whether those perturbations die, spread, split, or stabilize. Under the right conditions, the smoothing operator becomes part of a pattern-writing machine.

That is the beautiful discomfort of the subject.

You add diffusion to remove differences. The system uses it to decide which differences survive.

  1. Alan M. Turing, “The Chemical Basis of Morphogenesis”, Philosophical Transactions of the Royal Society of London B, 1952. ↩

  2. John E. Pearson, “Complex Patterns in a Simple System”, Science, 1993. A PDF copy is available from the University of Notre Dame: pearson.pdf. ↩ ↩2

  3. The Well, “Pattern formation in the Gray-Scott reaction-diffusion equations”, accessed June 14, 2026. ↩