Each firefly is a phase oscillator θi' = ωi + (K/ni) Σj in N(i) sin(θj − θi), with natural angular frequencies drawn from a Gaussian of mean 4.2 rad/s (flash period about 1.5 s) and spread σ. A flash is drawn each time a phase wraps through 2π. The neighbours N(i) are all fireflies within the interaction radius.
The collective state is the order parameter r eiψ = (1/N) Σ eiθj. In the all-to-all (mean-field) Kuramoto model, partial synchrony appears at Kc = 2/(π g(0)) = 2σ√(2π)/π ≈ 1.6σ, and above it r solves r = K r ∫−π/2π/2 cos²θ g(K r sinθ) dθ. The numerical check compares an N=1500 run with this equation.
Honest simplifications: real fireflies (e.g. Pteroptyx) use pulse-coupled phase resetting rather than smooth sinusoidal coupling; positions are static; Kc shown is the global-coupling value and is not exact for a short interaction radius.