Counted by brute force over 6000 years, compared with closed forms gcd(P,q)/q and lcm(P1,P2).
Toy model of the prime-number hypothesis for periodical cicadas (magicicadas emerge every 13 or 17 years in North America). Predators and competitors with cycle length q peak every q years, all in phase at year 0. A brood emerging every P years meets a peak when q divides P·m for its m-th emergence.
gcd(P,q)/q. It is the smallest, 1/q, exactly when P and q share no factor, which is why a prime P is hit at the minimum rate by every shorter cycle.lcm(P1,P2) years (221 for 13 and 17).s = ∏(1 − k) over coinciding predators. Long-run growth per year: r = (ln R + ⟨ln s⟩)/P, averaged over the repeating pattern.xP(t) ∝ xP(0)·erP t, normalised each step.Honest simplifications. Whether primes win depends on the numbers: with weak predation or plentiful offspring the shortest period wins (see the presets). The real cause of 13 and 17 year cycles is debated (predator avoidance, hybridisation between broods, Allee effects, glacial-era climate). This model only shows the predator-overlap argument. Predator peaks are perfectly periodic and aligned at year 0 (worst case), the loss k and the offspring number R are illustrative numbers, there is no density dependence and no mutation. The cicada periods 13 and 17 and the 221-year overlap are well-known facts; everything else here is for exploring, not a field measurement. Longer periods always pay a cost in this model because a strain reproduces only every P years, so the optimum depends on R and k: try lowering R.