  Real-world examples of periods of periodical organisms range from cicadas
whose life-cycles are larger prime numbers, like 13 or 17, to bamboos whose
periods are large multiples of small primes, like 40 or even 120. The
periodicity is caused by interaction of species, be it a predator-prey
relationship, symbiosis, commensialism, or competition exclusion principle. We
propose a simple mathematical model which explains and models all those
principles, including listed extremel cases. This, rather universal,
qualitative model is based on the concept of a local fitness function, where a
randomly chosen new period is selected if the value of the global fitness
function of the species increases. Arithmetically speaking, the different
observed interactions are related to only four principles: given a couple of
integer periods either (1) their greatest common divisor is one, (2) one of the
periods is prime, (3) both periods are equal, or (4) one period is an integer
multiple of the other.
