Brusselator

Parameter A 1.0

Parameter B 3.0

Diffusion X 0.16

Diffusion Y 0.08

Actions

X (activator)

Y (inhibitor)

step: 0 regime: — Click canvas to perturb

The Reaction Scheme

The Brusselator is defined by four steps involving external species A and B (held constant) and intermediates X and Y:

A → X
B + X → Y + D
2X + Y → 3X
X → E

The third step is autocatalytic: X catalyzes its own production, creating a positive feedback loop that drives oscillations.

PDEs with Diffusion

Adding spatial diffusion gives the reaction-diffusion system:

∂X/∂t = A − (B+1)X + X²Y + Dₓ∇²X
∂Y/∂t = BX − X²Y + D_y∇²Y

The steady state is (A, B/A). It is unstable when B > 1 + A², leading to limit-cycle oscillations. With unequal diffusion, Turing instability can produce stationary spatial patterns.

Turing Instability

Alan Turing showed (1952) that a homogeneous steady state can be destabilized by diffusion if the activator diffuses slower than the inhibitor. Here X is the activator, Y the inhibitor.

Try: set B ≈ 4.5, Dₓ ≈ 0.05, D_y ≈ 0.5 and reset — stationary spots or stripes emerge, resembling animal coat patterns.

Exploring Regimes

Oscillating (B > 1+A²): the whole field oscillates in color — a spatially uniform limit cycle.

Traveling waves: with moderate diffusion ratio, spiral waves propagate through the medium, resembling the BZ reaction.

Stationary Turing patterns: slow activator + fast inhibitor → frozen spatial structure. Click the canvas to nucleate patterns from perturbations.