Arapaima gigas is an obligate air-breather: its swim bladder is modified into a vascularized lung, so it must surface every few minutes even when gills alone would suffice — a trait that lets it survive the hypoxic, oxygen-poor water of Amazon floodplains where gill-only fish suffocate. Each fish here carries an unbounded "air-debt" phase θᵢ that grows at its own natural rate ωᵢ and is nudged by every shoal-mate's phase through a Kuramoto coupling term, integrated with 4th-order Runge–Kutta at a fixed sub-step:
θᵢ' = ωᵢ + (K/N)·Σⱼ sin(θⱼ − θᵢ)
ωᵢ = ω₀·(1 + deficit) + Δᵢ (Δᵢ fixed per fish, individual metabolism)
Every time θᵢ crosses a multiple of 2π the fish gulps air at the surface — that is the "breathing event" you see as a ripple. Raising the oxygen deficit raises ω₀ for everyone (hotter, less-oxygenated water forces faster gill-driven debt build-up), which is why surfacing gets more frequent but not, on its own, more synchronized. Raising social coupling K pushes the population past the Kuramoto critical coupling: below it phases drift independently (r stays low), above it they lock into a shared rhythm (r → 1) — a real phase transition, not an animation trick, and it is exactly the mechanism proposed for why shoaling air-breathers surface together (shared vigilance dilutes each individual's predation exposure at the surface).
- River pane — top-down view; each fish's fill brightens as its air-debt nears the surfacing threshold, and flashes white with a ripple at the moment it breathes.
- Phase circle — every θᵢ plotted mod 2π on the unit circle; the arrow is the Kuramoto order parameter r·e^(iψ), the population's mean phase vector.
- Synchrony trace — r(t) over time; watch it climb past the critical coupling as you raise K.