Male fiddler crabs court by waving one enlarged claw. Neighbours entrain to each other's rhythm the same way fireflies or pacemaker cells do — each crab is a phase oscillator, and its wave phase θᵢ is pulled toward the phase of every visible neighbour. This is the Kuramoto coupled-oscillator model:
dθᵢ/dt = ωᵢ + (K / nᵢ) · Σⱼ sin(θⱼ − θᵢ)
ωᵢ is each crab's own natural wave rate (small random spread per individual), K is the "Signal coupling" slider, and the sum runs over the nᵢ neighbours within visual range. When K is high, phases lock and the colony wave-synchronizes; when K is low, crabs wave independently. The global synchrony readout is the Kuramoto order parameter:
r·e^(iψ) = (1/N) Σⱼ e^(iθⱼ)
r runs from 0 (phases scattered — no sync) to 1 (every visible crab waves in lock-step). Rising tide floods burrows from the shoreline inward; a submerged crab retreats and drops out of both the oscillator network and the synchrony count until the water recedes, exactly as real fiddler crabs plug their burrows at high tide and only signal while the flat is exposed.
- Claw — swings up when sin(θ) > 0; the flash brightens near the peak of the wave.
- Water — the blue band rising from the bottom edge; crabs under it are inactive.
- Sync arrow — top-left compass needle; its length is r, its heading is the mean phase ψ.