Prime-Time Cicadas
Ivars Peterson
Cicadas are flying, plant-eating 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 Magicada.
Periodical cicadas usually have 13- or 17-year 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 prime-number cycles.
For example, prime cycles might occur so that periodical cicadas can more
readily evade shorter-lived predators or parasites. If periodical cicadas had
12-year life cycles, all predators with 2-, 3-, 4-, or 6-year cycles would get a
chance to eat them, potentially wiping out an entire population. With
prime-number 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
prime-number 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
life-cycle 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 life-cycle 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 prime-number cycle.
The fact that a simple predator-prey mathematical model leads to prime-number
cycles, however, doesn't really explain why periodical cicadas have 13- or
17-year 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 prime-generating 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 predator-prey model. Complexity 6(No. 4):33-38.
Abstract available at http://www3.interscience.wiley.com/cgi-bin/abstract/84502365/START.
______. 2000. A biological generator of prime numbers.
Nonlinear Phenomena in Complex Systems 3(No. 2):208-213. Available at http://alpha01.dm.unito.it/personalpages/cerruti/primality/biological-primes.pdf.
Klarreich, E. 2001. Cicadas appear in their prime. Nature
Science Update (July 23). Available at http://www.nature.com/nsu/010726/010726-3.html.
Markus, M., and E. Goles. 2002. Cicadas showing up after a
prime number of years. Mathematical Intelligencer 24(No. 2):30-32.
Milius, S. 2000. Cicada subtleties. Science News
157(June 24):408-410. 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.mpi-dortmund.mpg.de/departments/swo/markus/hp9.php3.
Information about periodical cicadas can be found at http://ummz.lsa.umich.edu/magicicada/Periodical/Index.html.
**********
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 writer and online editor at
Science News. 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 (see MatheMUSEments at http://home.att.net/~mathtrek/). The
Mathematical Association of America has published a collection of his online
MathTrek
articles.
|
| 
