PrimeTime Cicadas
Ivars Peterson
Cicadas
are flying, planteating insects. Most cicada species have life cycles
that span 2 to 8 years. They spend most of their lives underground
before emerging as adults. In a few species, almost all the individuals
in a given location come out of hiding at the same time. These are
known as periodical cicadas, and they generally belong to the genus Magicicada.
Periodical cicadas usually have 13 or 17year life cycles.
Their development is so synchronized that practically no adults are
present in the 12 or 16 years between emergences. When these cicadas do
come out of their underground homes, they appear in huge numbers and
create a cacophonous, throbbing din during their brief period of mating
frenzy in the open air.
Curiously, 13 and 17 are both prime numbers, evenly divisible
only by themselves and 1. The fact that periodical cicadas emerge after
a prime number of years could be just a coincidence. Or it might
reflect some sort of evolutionary pressure that leads to primenumber
cycles.
For example, prime cycles might occur so that periodical
cicadas can more readily evade shorterlived predators or parasites. If
periodical cicadas had 12year life cycles, all predators with 2, 3,
4, or 6year cycles would get a chance to eat them, potentially wiping
out an entire population. With primenumber cycles, the chances of
predator and prey coinciding would be much less.
A few years ago, Mario Markus of the Max Planck Institute for
Molecular Physiology in Dortmund, Germany, and his coworkers decided to
see whether such primenumber cycles could come out of a simple
evolutionary mathematical model of interactions between predator and
prey.
In such a mathematical model, predator and prey have randomly
assigned lifecycle durations. If cicadas appear when many predators
are waiting, their population drops. If cicadas come out when few
predators are around, they flourish. In the meantime, random
"mutations" change the lifecycle durations of succeeding generations,
subject to the requirement that the predator's life cycle stays shorter
than that of the prey.
The researchers observed that, in their simulations, a sequence
of mutations would eventually lock the cicadas (prey) into a stable
primenumber cycle. The fact that a simple predatorprey mathematical model leads
to primenumber cycles, however, doesn't really explain why periodical
cicadas have 13 or 17year cycles. For one thing, no one has yet
identified predators or parasites that would fit the bill biologically.
Moreover, the model says nothing about why many species have cycles
that are not prime numbers.
Interestingly, the mathematical model developed by Markus and
his colleagues can serve as a machine for generating prime numbers.
Starting with a cycle of any length, the steps of their procedure
inevitably lead to a prime number.
It's not a particularly efficient way to generate a prime number, but it certainly does the job.
"The remarkable feature of the present work, however, is the
biological rationale underlying the primegenerating algorithm," Markus
and his coworkers reported in a paper describing their work. "Our
algorithm displays the merging of two seemingly unrelated subjects:
number theory and population biology."
References: Goles, E., O. Schulz, and M. Markus. 2001. Prime number selection of cycles in a predatorprey model. Complexity 6(No. 4):3338. Abstract available at http://www3.interscience.wiley.com/cgibin/abstract/84502365/START.
______. 2000. A biological generator of prime numbers. Nonlinear Phenomena in Complex Systems 3(No. 2):208213. Available at http://alpha01.dm.unito.it/personalpages/cerruti/primality/biologicalprimes.pdf.
Klarreich, E. 2001. Cicadas appear in their prime. Nature Science Update (July 23). Available at http://www.nature.com/nsu/010726/0107263.html.
Markus, M., and E. Goles. 2002. Cicadas showing up after a prime number of years. Mathematical Intelligencer 24(No. 2):3032.
Milius, S. 2000. Cicada subtleties. Science News 157(June 24):408410. Available at http://www.sciencenews.org/20000624/bob8.asp.
Mario Markus has a web page on population dynamics and his cicada models at http://www.mpidortmund.mpg.de/departments/swo/markus/hp9.php3.
Information about periodical cicadas can be found at http://ummz.lsa.umich.edu/magicicada/Periodical/Index.html.
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A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.
Comments are welcome. Please send messages
to Ivars Peterson at ip@sciencenews.org.
Ivars Peterson is the mathematics/computer writer and online editor
at Science News and
Science News for Kids.
He is the author of The Mathematical Tourist, Islands of Truth,
Newton's Clock, Fatal Defect, The Jungles of Randomness,
and Fragments of Infinity. He also writes for the children's
magazine Muse.
He is coauthor of the children's books Math Trek: Adventures in
the MathZone and Math Trek 2: A Mathematical Space Odyssey.

