Ant Colony Double-Bridge: Pheromone Phase-Space Map

Bridge Setup

Long / short path length ratio 2.0×

Pheromone evaporation rate 0.12 /s

Choice nonlinearity n 2.0

Colony size 50 ants

Live State

short 50%long 50%

Pheromone (short)

0.0

Pheromone (long)

0.0

Round trips completed

How it works

Same mechanism as the 3D double-bridge diorama (Deneubourg, Aron, Goss & Pasteels, 1990), rendered as a phase-space map instead of a scene of ants on bridges. The colony's entire collective state at any instant is just two numbers — the pheromone level on the short branch (φ_S) and on the long branch (φ_L) — so this view plots that pair directly as a point moving in the (φ_S, φ_L) plane, leaving a fading trail behind it. The colored background is the exact choice-probability field every ant samples from at the fork:

P(short | φ_S, φ_L) = (c + φ_S)^n
                      ───────────────────────────
                      (c + φ_S)^n + (c + φ_L)^n

φ evaporates:  φ(t+dt) = φ(t) · e^(−ρ·dt)
φ on arrival:  φ ← φ + Δ  (one ant's deposit)

Green = the field favors the short branch at that (φ_S, φ_L) pair, blue = it favors the long branch. Early on, with little pheromone laid, the state sits near the origin in the neutral gray region; as trips accumulate, positive feedback pushes the trajectory away from the diagonal φ_S = φ_L and out toward one axis — visually, the point walks into ever-greener or ever-bluer territory and stays there.

Length ratio — how much longer the long branch is. Ants on the short branch complete round trips faster, so they deposit more often per unit time, biasing the trajectory's drift toward the short axis before any preference even exists.
Evaporation rate ρ — higher ρ pulls the trajectory back toward the origin faster between deposits, keeping the colony flexible; ρ → 0 lets any early random lead lock in permanently, freezing the trajectory far from the diagonal.
Nonlinearity n — n = 1 gives a shallow, gradual color gradient (linear probability matching); n > 1 sharpens the field into a near step function across the diagonal, so a tiny pheromone lead snowballs into a near-vertical or near-horizontal trajectory almost immediately.
Colony size — more ants sample the fork more often per second, so the trajectory moves through phase space faster relative to evaporation.

This is the identical stigmergic reinforcement rule Marco Dorigo formalised in 1992 as Ant Colony Optimization — only visualized here as the state-space trajectory of a 2-variable dynamical system rather than a diorama of individual foragers.

Trips: 0

Field favors short branch Field favors long branch Live (φ_short, φ_long) trajectory

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