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Deciphering the Strange Mathematics of Cicadas [Video]


“Periodical cicadas have the longest life cycles known for insects. They are called ‘periodical’ because in any one population all but a trivially small fraction are exactly the same age. The nymphs suck juices from the roots of forest trees and finally emerge from the ground, become adults, mate, lay their eggs, and die, all within the same few weeks of every 17th (or in the South, every 13th) year. Not one species does this, but three, and they always do it together.”

—Monte Lloyd and Henry S. Dybas, 1966

There is safety in numbers or, at least, there is survival in numbers. That is the maxim that periodical cicadas live by.

Periodical cicadas—insects of the genus Magicicada—are remarkable creatures. They develop extremely slowly, underground, before surfacing en masse at either 13- or 17-year intervals, when the ground temperature reaches 64 degrees Fahrenheit. As described in the epigraph above, they quickly mate, lay eggs and die, disappearing from view until their offspring crawl out of the ground more than a dozen years later.

Much has been made of this year’s cicada emergence, but in fact periodical cicadas rise to reproduce in most years: there are 15 different geographic “broods” of periodical cicadas, each on its own synchronized life cycle. This year, brood II—which stretches from North Carolina to New York and Connecticut—is emerging for the first time since 1996. Meanwhile the 14 other broods are maturing underground, awaiting their turn in the limelight.

Staying Alive

Periodical cicadas are only found in the forests of the Eastern U.S. (Other, more numerous species of nonperiodical cicadas appear more often and in more locations.) The genus Magicicada includes seven species: three 17-year cicadas in the northern U.S. and four 13-year cicadas in the south. Those species are more broadly divided into three groups: decula, cassini and decim. For each group there exists a 13- and a 17-year species—for instance, the 13-year Magicicada tredecula and the 17-year Magicicada septendecula—which, other than their lifespans and their geographic range, are almost indistinguishable. Amazingly, a single brood often contains multiple species, which grow alongside one another as nymphs and emerge from the ground in synchrony but do not interbreed. The video above shows the subtle visual differences between the species and the much more dramatic differences between their respective mating calls.

Synchronized life cycles deliver Magicicada a major benefit: when all the insects emerge at once, their predators, which can only eat so much, become sated before consuming the entire population. It is a brute-force survival tactic—akin to storming a fortress, unarmed, in huge numbers—but it works.

The long life cycles of periodical cicadas have fascinated entomologists for decades. As Richard D. Alexander and Thomas E. Moore of the University of Michigan at Ann Arbor put it in 1962:

Their incredible ability to merge by the million as noisy, flying, gregarious, photo-positive adults within a matter of hours after having spent 13 or 17 years underground as silent, burrowing, solitary, sedentary juveniles is without parallel in the animal kingdom.

And for years researchers have sought to explain how the Magicicada life cycles developed, why they are so long, and why they are both prime numbers.

Why So Slow?

The pronounced elongation of the Magicicada life span may trace to about 20,000 years ago, during the last glacial period. The colder conditions then may simply have slowed the growth and development of cicadas—as suggested in 1997 by Jin Yoshimura, now of Shizuoka University in Japan—thereby extending what had been a somewhat shorter lifespan toward the long life cycles that exist today. (Warmer ground temperatures in the south as compared to the north allow cicadas to develop more rapidly, which may account for their shorter 13-year life cycles.)

Alternately, the insects may have adapted to life in the glacial period by extending their life cycle so as to limit the chances of emerging in an unusually cold year that would prevent mating. As proposed in 1988 by Randel Tom Cox, now at the University of Memphis, and C. E. Carlton, now at Louisiana State University, the lifespan of Magicicada ancestors may have “increased progressively to lengths similar to those observed today as an adaptive strategy during glacial stades in which maximum annual temperatures may occasionally not have reached the critical level for flight and copulation…. The longer the nymphal life cycle, the smaller the chance of emerging during a cold summer.” Such an effect would be strongest in colder climates, which would also explain the longer 17-year lifespans of northern Magicicada species.

Primed for Success

If a cold climate forced cicadas to develop long lifespans, the insects may have emerged from the last glacial period with a spectrum of life cycles, perhaps ranging from 12 to 20 years. Eventually two of those life cycles, 13 and 17 years, won out.

The fact that the surviving periodical cicadas have life cycles built on prime numbers may have conferred key survival advantages. A prime-numbered lifespan means that predators cannot match their own shorter life cycles to the availability of cicada prey. For instance, if the cicadas had even-numbered lifespans, a predator with a two-year life cycle could expect a cicada feast, and a subsequent population bump, every few generations, because all even numbers are divisible by two. As explained in 2001 by a trio of researchers from the University of Chile and the Max Planck Institute of Molecular Physiology in Germany, “a prey with a 12-year cycle will meet—every time it appears—properly synchronized predators appearing every 1, 2, 3, 4, 6 or 12 years, whereas a mutant with a 13-year period has the advantage of being subject to fewer predators.” Prime numbers are still divisible by themselves and by 1, of course, but they have no other divisors.

On the other hand, prime lifespans may relate to periodic overlaps between different cicada species, rather than overlaps between cicadas and their predators. The two prime-numbered life cycles of Magicicada ensure that asynchronous broods rarely interact where their geographic ranges overlap—a 13-year cycle and a 17-year cycle match up only once every 221 years. Those rare meetups may confer the advantage of preventing the two groups from mating and producing hybrid offspring. As Cox and Carlton wrote in 2003, “cicadas that are hybrids of two populations with different life cycle lengths will suffer greater predation losses, as many may emerge on years before or after the main population. Cicadas with prime-numbered cycles (13 years and 17 years) will hybridize significantly less frequently than cicadas with non-prime (composite) cycles and thus will have larger emergences and a greater advantage of predator satiation.”

More recently, in a study in the Proceedings of the National Academy of Sciences (PNAS), researchers in Japan and the U.S. have deployed genetic evidence in support of a different model: that periodical cicadas simply jumped from one life cycle to the other. “They are time travelers,” says study co-author Chris Simon of the University of Connecticut. “They undergo these four-year accelerations or decelerations in their life cycles. If you go to a site where 17-year cicadas emerge, you’ll find a lot of them coming out four years early.” Some stragglers will also emerge four years late.

Given a large enough population of Magicicada, the insects emerging four years off-schedule could form a group numerous enough to survive predation, thus spinning off their own population on a new life cycle. Four years is a key offset—populations separated by less than that seem not to be able to coexist in the same location, perhaps because of fierce competition for resources between nymphs growing underground.

Once a 13-year brood had successfully spun off from a 17-year brood, or vice versa, that new population would act as a “nurse brood,” Simon and her colleagues argue, protecting invading cicada populations—provided that the invaders adapted to the new life cycle as well. “Natural selection would have promoted synchronization of invading populations to resident populations because invaders would gain protection from predation and, consequently, avoid Allee effects (failure to reproduce due to low population density),” the researchers propose. In that case, the four-year gap between the two life cycles may be more important than the fact that both are prime numbers. Says Simon: “It’s hard to say whether 13 and 17 is an accident or whether it has an advantage.”

As brood II makes its long-awaited 21st-century debut, Simon and her colleagues are out in the field studying these unique insects and working to uncover the basis for their extreme behaviors. She notes that anyone in the vicinity of a cicada brood can aid in this ongoing investigation by reporting cicada sightings to, a Web site run by her University of Connecticut colleague and PNAS study co-author John Cooley.

And now, for your viewing pleasure, here are the expected emergences of periodical cicadas in the next few hundred years:

The views expressed are those of the author and are not necessarily those of Scientific American.

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