Nature’s Prime Number Generators

The cicada timing riddle 

By David Rothenberg

It’s the slowest sonic beat in the animal world. It’s a sound that can be used to mark the phases of a human life. It’s a mathematical conundrum, an unearthly wonder of animal sound. The cloud of insect music you can barely recall. When you last heard it, you were just settling down. The time before that, you were a teenager. Before that it was the year you were born. The next time you will hear it you might be a grandfather.

This time the song arrives, you are smack in the middle of your journey through life.

“You are a Cicada Boy,” my friend John P. O’Grady insists. Once an English professor, he is now a photographer, flaneur, and part-time astrologer. “Believing that the stars affect us is a useful fiction,” he smiles. “One might also believe the appearance of cicadas every seventeen years touches us in a similar way.” He looks up the year 1962. It was one of those years that the great thrumming insects arrived en masse in the trees of late spring in the Northeast. One month later I was born in July. The cicadas were back when I graduated high school, and again when I moved from the city to the country. That last time back in 1996, O’Grady took me out to the Mohonk Mountain House to experience the great emergence.

In the years since then I have been making music together with all kinds of animals. It has humbled me a bit and taught me to appreciate many more kinds of sounds. Now I am even more in awe of this longest animal rhythm, a great beat that emerges out of silence only once every seventeen years. It is only a North American thing, nowhere else has this kind of wave of cicada appearances, following these strange great cycles of prime-numbered years.

Most cicadas all over the world come out in large enough numbers every year to enthrall people with their volume, energy, and roughness of sound, and annoy us as well, especially those who would wish that nature was closer to silence.

When did people first notice the prime-numbered cycles of cicadas, something that only happens in the New World of North America? Certainly the early Pilgrims experienced an emergence a few years after Plymouth Rock. They were not always sure just what to call the insects they were astonished by. William Bradford wrote in 1633: “all the month of May, there was such a quantity of a great sort of flyes like for bigness to wasps or bumblebees, which came out of holes in the ground and replenished all the woods, and ate the green things, and made such a constant yelling noise as made all the woods ring of them, and ready to deaf the hearers.” Music indeed. He goes on: “They have not by the English been heard or seen before. But the Indians told them that sickness would follow, and so it did in June, July, August, and the chief heat of summer.”

Scientists attempting to explain why periodical cicadas return every thirteen or seventeen years often seem to be doing little more than tracking coincidences, tossing out possible explanations, and hoping for the most interesting thing that can be said.

In fact, if you read the popular science literature, sometimes it seems that this issue is completely settled. It even appears as one of the first examples in The Math Book, a beautiful, coffee-table volume on what makes mathematics cool. The book proceeds by date, from the earliest to the latest, and the third entry, timed at 1 million BC, is the description of periodical cicadas as nature’s prime number generators. There are, it seems, relatively few prime numbers in the world of natural cycles, so the example of the cicadas having cycles of both thirteen and seventeen years is most notable and unusual. A simple explanation is often given, and it is one suggested by biologists for decades, and made popular by Stephen Jay Gould in Ever Since Darwin, one of his popular collections of science essays. The idea is simple, and that is to suppose that the cicadas appear in prime number cycles simply because no predators appear in such cycles: Most animals that have cyclical rises and falls of population do so in more regular numbers of seasons like every two, four, six, or eight years. And since prime numbers are defined as numbers that cannot be divided into other numbers, such other cycles would hardly ever line up with the cicadas’ emergences. Voilà! A perfectly ingenious and reasonable explanation, one that is repeated in many biology textbooks and all over the media.

Only problem is, there is absolutely no evidence for it whatsoever. No one has ever identified even a single predator of the periodical cicada whose population follows such a cycle. Of course such a predator cycle still might exist, but we haven’t found it, or even spent much time looking for it. This explanation might fall into that category of “just so” story that science puts forward because it sounds like it should work, and the public is so intrigued by the idea that hardly anyone seems to notice that there is little evidence for the hypothesis in the first place.

Although the predator/prey story sounds pretty cool, there is no predator that follows a periodic cycle of a more even number of years. Besides, there are so many cicadas that emerge during the brood years that the sheer number of the insects overwhelms any possible predators; it is a much simpler strategy that doesn’t need prime numbers for it to work. It’s a simple situation of the enemy being overrun by the horde!

Much of the work on trying to decode the prime number periodicity has been a kind of bioinformatic mathematical modeling based on theory and running computer models, rather than collecting any evidence. Mathematical biologists have tried to see if there is something inevitable about the prime number cycles that appear simply by running the numbers through hypo-thetical situations of competition. Although such models are basically simulations that depend entirely on their assumptions, it seems that a likely hypothesis is starting to emerge. Whether or not there are actual predators who are foiled by prime number cycles, it seems that when the math is done the predation model alone is not enough to generate prime number cycles. One has to also add the variable that different populations of cicadas inherently contain a mechanism to avoid hybridization; that is, they want to ensure that as a species they are kept separate. This is also coupled with a tendency that a small number of individuals in the seventeen-year broods have of emerging four years early. So it is believed this tendency might have originally led to the establishment of thirteen-year species at the start.

It also appears that the establishment of prime number cycles may have something to do with a population that is nearly decimated, such as the few remnants of a species remaining after a catastrophic ecological event, like the advance of glaciers during an ice age compromising cicada habitat except for a few isolated pockets. It turns out that in such situations of very small populations, the reproduction rate increases with high population density, as opposed to large populations where if things get too dense, there is too much competition for resources and the population is no longer able to increase. This might explain why many animals congregate in leks, or concentrated groups for the purpose of mating, and that this may have evolved especially with very small populations who have to work hard to find each other when their habitat has been stressed.

This situation was identified by ecologist Warder Clyde Allee in the 1920s and is today called the Allee Effect. In a mathematical model devised by Yumi Tanaka and Jin Yoshimura in Japan together with Chris Simon in Connecticut, prime number cycles were observed to develop over just one thousand hypothetical years, only if the Allee Effect is considered. Though still a purely theoretical exercise, it does give some clues for how these cycles might have evolved in a situation where the cicadas became nearly extinct and then had to congregate in order to find enough of each other to survive. In a second paper the authors suggest that “Our results indicate that prime number selection is a very rare event, occurring at the verge of extinction. This is probably why the evolution of prime-numbered periodicity was likely only in what is now the Central and Eastern United States, where glacial advances created many refuges during the uneven Pleistocene glaciation.” The model is based on the idea that a small population of insects, sequestered in the few available habitats that remain during glaciation, congregate in order to best survive. The even-year gestation periods tend to disappear because the insects hybridize with each other when they emerge, losing the species’ distinctiveness. After even 150 hypothetical years only prime number years remain, 13, 17, and 19. Perhaps 19 is just too many years of development for an instinct. No matter—keep running the model, and after 500 years only the 13- and 17-year cycles remain.

The latest mathematical thinking on the mysterious prime-numbered cicada cycles thus suggests that the predation idea is not enough to lead to such neat prime results. We also have to have competition within the species to avoid hybridization, and then also the Allee Effect of the small, concentrated populations. The specific glaciation situation might help to explain why this phenomenon is only found in Eastern North America, with its peculiar glacial history. Then there is also the strange fact that a certain number of seventeen-year cicadas do emerge four years early, more than two, three, five, six, or any number of years early. Thus the thirteen-year species could have evolved from the seventeen-year species. Mysteries, mysteries.

Philosopher and jazz musician David Rothenberg, a professor of philosophy and music at the New Jersey Institute of Technology, is the author of Survival of the Beautiful, Why Birds Sing, and Thousand Mile Song. He lives in the Hudson Valley, New York.

This essay is excerpted from the book Bug Music by David Rothenberg. Copyright © 2013 by David Rothenberg. Reprinted with permission of St. Martin’s Press.

Photo by David Rothenberg

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