Cricket Chorus Synchronisation

What is modelled

A cricket chirps at a rate set by body temperature. Each singer has a slightly different natural rate, and each adjusts the timing of its next chirp by what it hears from the others. This is the classic setting of the Kuramoto model: coupled oscillators with spread natural frequencies lock together once the coupling is strong enough.

Equations

• Dolbear's law N60 = 4 (TF − 40) chirps per minute, with T_F in degrees Fahrenheit.
• Kuramoto phases dθi/dt = ωi + (K/N) Σ sin(θj − θi) in the frame of the mean chirp rate; the order parameter r eiψ = (1/N) Σ eiθj.
• Threshold for a Gaussian spread of width σ Kc = 2 σ √(2π) / π; above it r = K r ∫ cos²θ g(K r sinθ) dθ.
• Speed of sound c = 331.3 √(1 + T/273.15) m/s.

Checks run in the page

• Dolbear's law inverted and against the 20 C textbook case.
• The simulated order parameter against the self-consistency equation.
• Incoherence at 0.6 K_c and locking at 1.6 K_c.
• The mean phase advance is unchanged by the coupling.

Honest simplifications

Real crickets do not interact through a smooth sine coupling: they respond to the sound of a neighbour at a certain phase, and many species do not synchronise at all. Dolbear's law is an empirical rule of thumb that works best for one species in a limited temperature range. The page models phase only, with no sound propagation, distance or noise.

• Choose Independent singers: the loudness history is flat noise and r stays near zero.
• Choose Near threshold and raise K slowly through the red K_c line in the second plot.
• Choose Cold night: the tempo drops with the temperature but the lock is unchanged.