ScienceAsia 28 (2002) : 199-203 |doi: 10.2306/scienceasia1513-1874.2002.28.199

Prey population cycles are stable in an
evolutionary model if and only if their periods are prime

Mario Markus^a,*, Oliver Schulz^a and Eric Goles^b

ABSTRACT: We present an evolutionary model that marks an encounter of two seemingly unrelated disciplines: population dynamics and number theory. Assuming mutations and selection of predators and prey, we show that prey cycles with non-prime lengths are unstable, while cycles with prime lenghts are stable. Allowing arbitrarily long cycles, this model is a number-theoretical tool for the calculation of large prime numbers. An extension of this purely temporal process to an evolutionary game on a spatial array leads to homogeneity, or to travelling or spiral waves having a predominance of prime prey cycle lengths. These results may be related to the appearance of cicadas (genus Magicicadae) every 13 or 17 years.

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a Max-Planck-Institut fuer molekulare Physiologie Postfach 500247, D-44202 Dortmund, Germany.
b Center for Mathematical Modelling, UMR 2071, CNRS- Universidad de Chile, Casilla 170-3, Santiago, Chile.


*Corresponding author, E-mail: Markus@mpi-dortmund.mpg.de

Received 25 May 2001, Accepted 5 Feb 2002