In certain parts of eastern North America, it has been noted for centuries that some summers seem to bring a massive swarm of cicadas. Observant naturalists such as Pehr Kalm noted in the mid-1700’s that this mass emergence of adult cicadas happened every 17 years. Since then, a similar pattern has been observed with many different broods of cicadas, with precisely 17 or 13 years between emergences of mature cicadas.
What could possibly explain such a specific, long gap between these spikes?
This phenomenon has been well-researched and the species of cicadas (Magicicada) are known as periodical cicadas. They can be distinguished by their striking black bodies and red eyes. Like most cicadas, periodical cicadas start their lives as nymphs living underground, feeding on tree roots. They take 13 or 17 years (depending on the genus) until they emerge all at once in the summer as mature adults – far longer than the 1-9 years seen in other cicadas. After such a long period of growth, they emerge for a few glorious weeks in the sun to mate, before laying eggs and disappearing.
The astute reader would notice that both 13 and 17 are prime numbers (a number divisible only by itself or 1). Is this a sheer coincidence or a beautiful example of mathematics in nature?
This curious, specifically long period of maturation has been a great point of interest for scientists. The phenomenon of mass, synchronised maturation is a well-documented survival strategy known as predator satiation. Essentially, if the entire population emerges at the same time, predators feast on the large numbers, get full and stop hunting as much. The surviving proportion (still a great number), carry on to reproduce and the species survives.
One theory holds that the prime numbers are so that predators cannot synchronise their population booms with the cicadas. If the cicadas all emerged every 4 years, a predator who matures every 4 or 2 years could exploit this by having a reliable source of food in a cyclical pattern. 13 and 17 are large enough prime numbers that it would be very difficult for a predator to synchronise its maturation cycles with.
Another possible theory is that it is a remnant of a survival strategy from the Ice Age. Mathematical models have shown that staying as a nymph for a longer period increased the chances of adults emerging during a warm summer, rather than when it is too cold for reproduction. This resulted in broods of varying, lengthy cycles, but this created another problem: hybridisation. When broods of different cycle lengths intermingled, hybridisation could occur and disrupt the precise timing of maturation cycles, decreasing the brood’s survival rate. Prime number cycles such as 13 or 17 years have a much less chance of hybridisation, increasing the survival rate.
As Galileo Galilei said, mathematics is the language in which the universe is written. It is fascinating to see examples of how maths can influence natural phenomena, even the life cycles of insects.