Λ — 005Substrate

Gray-Scott

Reaction-diffusion morphogenesis

Pearson (1993) · Gray & Scott (1984)

Two partial differential equations. Eight classifications. One pattern-forming substrate.

The Gray-Scott reaction-diffusion system models a catalytic reaction where substrate u feeds at rate F, and catalyst v is removed at rate k. Pearson (1993) showed that small changes in these two parameters produce radically different steady-state morphologies — mitosis, solitons, coral, worms. Each a phase of the same substrate.

Pearson, J. E. (1993). Science 261(5118), 189–192.

200² grid · mitosis · step 0 · click or drag on the field to seed

Pearson classification

Self-replicating spots

parameters

feed rate · F0.0367

kill rate · k0.0649

integration speed8×

color mode

playback

derived, not chosen

D_u · u-diffusion
0.2097

D_v · v-diffusion
0.1050

ratio D_u / D_v
Turing condition requires > 1
2.00

Gray-Scott equations

∂u/∂t = D_u ∇²u − uv² + F(1−u)
∂v/∂t = D_v ∇²v + uv² − (F+k)v

F controls the feed rate of the u substrate.
k controls the removal rate of the v catalyst.

on reading the patterns

The eight preset classes are not a complete taxonomy — they are landmarks in a continuous two-dimensional parameter space. Drag F and k through small increments and the field morphology glides between them; drag past certain thresholds and the pattern destabilises into a new class entirely. The boundaries between classes are themselves a phase diagram.

This is the shape of the argument the paper makes about homeostatic minds: architectures in a continuous configuration space, with regions of stable pattern formation separated by destabilising boundaries. Finding the right region is not a matter of solving an optimisation problem — it is a matter of landing somewhere the field can sustain itself.