The Magical Maths of Magicicadas
Autumn leaves drift down, silently ushering the chilly advent of fall. Gone are the noisy days of summer, synonymous with the incessant, and insistent, chirp of the cicadas made by rapid vibration of abdominal tymbals and orchestrated by a frenzied mass of mating males. Did you know that the chirp of a cicada clocks in at 120 decibels, enough to cause permanent hearing loss in humans?!
A Plague of Primes: Periodical cicadas, of the North American genus Magicada, have a bizarre life cycle, spending 13 or 17 years underground as immature nymphs, emerging briefly to live, love and die as adults.
Seventeen years of peaceful dreaming,
Followed by a week of screaming*.
Their coordinated emergence, triggered when the soil warms to precisely 64 degrees F, guarantees a plague of biblical proportions: the densest broods can number 1,000 cicadas per square meter! Is there a mathematical basis for 13 or 17 year life cycles? You may have noticed that both are prime numbers: divisible only by themselves and the number 1.
Mathemagics: In the computer simulation graphed below, notice that 13 and 17 year periods produce the most survivors. The cicadas only defense against predators is their sheer number, and their survival strategy is simple- predator satiation. The prime numbers work better because they decrease the chance that the life cycle of the cicada matches that of its predators. A 12 year life cycle, in contrast, is a particularly bad choice: predators that reproduce every 2, 4, or 6 years (all divisors of 12) would feast on the hapless cicadas. Hiding underground for long periods helps survival, but reproduction cycles that are too long may result in being out competed by other species. Shorter prime numbered cycles may be weeded out if co-emergence of different broods results in hybridization and altered life cycles in the offspring.
Allee Effect: Biologists refer to the penalty of small population size on individual fitness as Allee Effect, named after W.C. Allee who showed, in 1932, that goldfish survived better in larger populations. The Allee effect means that there is a critical population size, below which the population becomes extinct. If the Allee effect is applied to simulations of cicada populations, successful cycles are in the order 17> 13 >> 19 year cycles, all others become extinct. Without the Allee effect, all brood cycles survive (see Fig. 1 of Tanaka et al., cited below).
So the next time you hear the chirp of the cicada, take a moment to appreciate the simple maths hidden within their lives!
*“A Cicada’s Life” by Alan Rubin
Graph and blog: http://arachnoid.com/is_math_a_science/
#OpenAccess Ref: Allee effect in the selection for prime-numbered cycles in periodical cicadas. Tanaka et al., 2009 PNAS.