Two branches join the nest to a food source. Each ant walking from one end chooses a branch with the Deneubourg choice function (Deneubourg, Aron, Goss & Pasteels 1990, fitted to Argentine-ant data):
P_A = (k + A)^n / [ (k + A)^n + (k + B)^n ], P_B = 1 - P_A
A, B = pheromone on each branch (in units of one ant crossing), k about 20, n about 2. Every ant that completes a crossing adds delta to the branch it used; the trail evaporates with time constant tau. Ants on the short branch finish earlier, so they re-mark sooner: a pure positive-feedback decision (Goss et al. 1989: with a long/short ratio r = 2 most colonies end on the short branch; equal branches give a random but strongly one-sided choice).
Exact model: each of the N ants is simulated individually (random choice, travel time T and r·T, pause at the ends). Mean field:
dA/dt = delta·Phi·P_A - A/tau, dB/dt = delta·Phi·(1-P_A) - B/tau
Phi = N / ( P_A·T_A + P_B·T_B + pause ) (crossings per second)
With equal branches the symmetric state loses stability when delta·Phi·tau > 2k/(n-1) (40 for k=20, n=2); then two one-sided states appear. The ODE starting at A = B = 0 stays exactly at 0.5 because it contains no noise: the spontaneous choice is a fluctuation effect, which the ant-by-ant model and the Monte-Carlo ensemble show.
Late short branch. If the short branch appears after the long one is saturated, P_A is almost 0 (A = 0, B large) and with no evaporation the colony stays on the long branch. Evaporation weakens the old trail and a few explorers (fraction epsilon who ignore the trail) can discover the new one; the hysteresis (bistability) then allows switches.
Simplifications: a branch is one number (trail amount), ants never turn back, deposition happens at the end of a crossing, ants do not interact otherwise, the choice parameters are those of one species, and times are in simulated seconds. Real colonies add recruitment variants, memory and individual differences.