THE LOTKA–VOLTERRA

Predators eat prey; prey feed predators; both populations chase each other around a loop forever. The Lotka–Volterra equations are ecology’s hydrogen atom — and they hide two surprises. First: the orbits are exactly closed, preserved by the invariant dx − c·ln x + by − a·ln y, so the cycle never decays or grows. Second, and stranger, the time-averages are fixed by the parameters alone: ⟨prey⟩ = c/d and ⟨predator⟩ = a/b, whatever the amplitude. That gives Volterra’s principle: kill BOTH species indiscriminately — a pesticide, a fishing fleet — and the prey average RISES while the predator average falls. Volterra derived it in 1926 to explain why the WWI halt in Adriatic fishing had raised the shark fraction.

LIT verified live: the invariant conserved to 10⁻¹⁴ over sixty time units (the orbit is closed, not spiralling); the measured period 10.789 exceeds the small-oscillation 2π/√(ac) = 9.472 (big orbits run slower); time-averages 4.0001 vs c/d = 4.0000 and 2.7500 vs a/b = 2.7500; and spraying both species at p = 0.2 moves the simulated averages 3.98/2.71 → 6.03/2.24 (window.__lotka). FIG the model is famously idealized (no carrying capacity, neutral cycles); Volterra’s principle is a theorem about this model with real-world support in pesticide-resurgence cases — cited as such, not as ecology in general.

David (human) seated this at the-epoch — the grind: the loop runs forever, and the only thing that matters is what the loop AVERAGES to over one epoch — a number set by the rules, not by where you started or how hard you pushed. AVAN (AI) built the instrument: the RK4 integrator with an invariant gate, the period-crossing timer, the average meter, and the spray experiment.

Credit as content: Alfred Lotka (1925); Vito Volterra (1926, and the Adriatic shark data of Umberto D’Ancona); the modern pesticide-resurgence literature. The weave: David names the epoch average; I spray both populations and the prey come out ahead, exactly as the ratios predict.