Presets
Playback
Swarm
Number of cicadas, mean natural flashing rate and the spread of natural rates (rad/s).
Coupling
Pacemaker: a common external cue (temperature or sound) at rate Om and strength F. Mutual: each cicada hears the swarm with strength K.
Layers
Model checks
The science
Phase oscillators
Each cicada is a phase oscillator dθ/dt = ωi + K r sin(ψ − θi) + F sin(Ωt − θi). A flash happens when θ passes a multiple of 2π. The first term is the natural rate, the second is the pull toward the swarm's mean phase ψ, the third is the pull toward a common pacemaker (a temperature wave or a loud neighbour) at rate Ω.
Order parameter
The complex order parameter r eiψ = (1/N) ∑ eiθj measures synchrony: r = 1 is perfect lock, r ≈ 1/√N is random. For natural rates drawn from a Lorentzian of half-width γ and no pacemaker, theory gives r = √(1 − 2γ/K) above the critical coupling Kc = 2γ, and r = 0 below it.
Pacemaker locking
In the frame of the pacemaker the phase difference ψ obeys dψ/dt = Δω − F sin ψ (the Adler equation): it locks if |Δω| ≤ F, which gives the lock range. With Δω = 0 the solution is tan(ψ/2) = tan(ψ0/2) e−Ft, used as an independent check.
Honest simplifications
Real periodical cicadas (Magicicada) emerge when the soil reaches about 17 °C after 13 or 17 years underground; this lab models only the short-term chorus synchrony, not emergence timing. The 3D scene uses the same pacemaker coupling with identical rates; the Lorentzian spread and mutual coupling are added here to show the Kuramoto transition. Natural rates use deterministic quantiles of the Lorentzian, so finite-N results differ slightly from the infinite-N formula.
What to try
- Start with the default preset: all cicadas are identical, so the pacemaker pulls every dot to the yellow ring and r goes to 1.
- Pick Just below Kc, then raise K slowly past 2 gamma: r jumps from about 0.05 toward the blue theory curve.
- Pick Detuned pacemaker: the offset exceeds F, so no lock occurs and the dots keep circling.
- Widen the spread gamma with the pacemaker on: cicadas outside the lock range (beyond the yellow lines on the last graph) drift away.