The Cicada’s Love Affair With Prime Numbers

As far back as the seventeen-hundreds, fur trappers for the Hudson’s Bay Company noted that while in some years they would collect an enormous number of Canadian lynx pelts, in the following years hardly any of the wild snow cats could be found—until, some years later, when the trappers found themselves again deluged with an abundance of lynx. Later research revealed that the rise and fall (and subsequent rise and fall) of the lynx population correlated with the rise and fall of the lynx’s favorite food: the snowshoe hare. A bountiful year for the hares meant a plentiful year for lynxes, while dismal hare years were often followed by bad lynx years. The hare booms and busts followed, on average, a ten-year cycle.

That still left an unanswered question: What was behind the rise and fall of the hare populations? A recent hypothesis is that the population of hares rises and falls due to a mixture of population pressure and predation: when hares overpopulate their environment, the population becomes stressed—the fact that the food supply is gobbled up certainly doesn’t help—which can lead to decreased reproduction, resulting in a drop in next year’s hare count. Meanwhile, predators like lynxes and raptors celebrate the hare bubble by gorging themselves and reproducing like mad. The subsequent decline in hares can lead to a drop in the swollen predator population; fewer predators can then result in more hares surviving to reproduce; and the cycle begins again.

Now, imagine an animal that emerges every twelve years, like a cicada. According to the paleontologist Stephen J. Gould, in his essay “Of Bamboo, Cicadas, and the Economy of Adam Smith,” these kind of boom-and-bust population cycles can be devastating to creatures with a long development phase. Since most predators have a two-to-ten-year population cycle, the twelve-year cicadas would be a feast for any predator with a two-, three-, four-, or six-year cycle. By this reasoning, any cicada with a development span that is easily divisible by the smaller numbers of a predator’s population cycle is vulnerable.

Prime numbers, however, can only be divided by themselves and one; they cannot be evenly divided into smaller integers. Cicadas that emerge at prime-numbered year intervals, like the seventeen-year Brood II set to swarm the East Coast, would find themselves relatively immune to predator population cycles, since it is mathematically unlikely for a short-cycled predator to exist on the same cycle. In Gould’s example, a cicada that emerges every seventeen years and has a predator with a five-year life cycle will only face a peak predator population once every eighty-five (5 x 17) years, giving it an enormous advantage over less well-adapted cicadas.Video From The New YorkerPerfecting a Roast Chicken, Without Having to Roast

To test this hypothesis, researchers from Brazil’s Universidade Estadual de Campinas used a computer simulation, very similar to John Conway’s Game of Life, in which simulated cicadas and predators battled it out in a hundred-by-hundred-cell matrix. They found exactly what Gould had suggested: cicadas with a prime-numbered life cycle had the most successful evolutionary strategy. If we discount those cicadas with life cycles of ten years or fewer (as being too close to predator life cycles), we find that the most successful emergence rates for cyber cicadas are thirteen and seventeen years—precisely what we find in the wild.

So how do the cicadas know how to calculate prime numbers? They don’t. They’re cicadas. The pattern probably emerged as a result of Darwinian natural selection: cicadas that naturally matured in easily divisible years were gobbled up by predators, and simply didn’t live long enough to produce as many offspring. Those who, by chance, had long, prime-numbered life spans fared best, survived longest, and left the most offspring, becoming the dominant variation of the species. (There are now at least fifteen distinct populations of periodical cicadas.) As things stand now, cicada emergences are so tightly timed, with the bulk of the insects emerging within a span of a few weeks, that any cicada that tries to break the pattern is simply taking her offspring’s life into her own hands.

Not everyone buys into this hypothesis. Lou Sorkin, the American Museum of Natural History’s cicada expert, pointed out that cicadas evolved during the Pleistocene epoch, about 1.8 million years ago, when the earth was much cooler. Since cicadas don’t survive well in the cold, he explained, it is possible that the cicadas that were naturally adapted to stay underground longer were less likely to face an unexpectedly cold spring.

But perhaps Adam Smith had it right, as Gould points out: the “invisible hand” of millions of mindless drones working for their own self-interest can create highly organized, mathematically precise behavior that benefits the group. At least for cicadas.

Ali Kaya


Ali Kaya

This is Ali. Bespectacled and mustachioed father, math blogger, and soccer player. I also do consult for global math and science startups.