Prime number cicadas — really!

By Frederic Friedel

You are not going to believe this one, and I won’t blame you for your skepticism. I could not believe it myself, when it happened.

Many years ago I was in Watchung, New Jersey, visiting my friend Ken Thompson (the Unix man) and his family. I have done it literally dozens of times, because they are wonderful people, and because it is always interesting. On that visit Ken introduced me to the potato gun, a home made pipe-based device that uses pressure, generated by the combustion of a flammable gas to fire a potato plug hundreds of feet into the air.

This is a potato gun being loaded — full instructions of how to build and operate one are to be found here.

It was exhilarating: we built one in Watchung, and then a number back home in Germany, to the delight of a population that had not seen anything like it before. But that is not the subject of today’s article.

On that visit, one night there was a horrible din, and in the morning I was able to see what was causing it: thousands and thousands of very beautiful insects crawling up the sides of trees, on fences, house walls, everywhere. I do not have photos from the time, so I am using a historical drawing.

Above is a cicada just like the ones I saw in New Jersey, painted by American entomologist and artist Robert Evans Snodgrass in 1930.

The insects were quite tame and to my delight would crawl onto my hand without fear of harm. They don’t bite or sting, just make a deafening racket. It’s only the males that do it. They have a noisemaker located on each side of their abdomen, which is largely hollow, as I discovered by opening dead insects (I had to find out how such a small creature could make such a loud noise). The abdomen acts as a sound box for a very rapid succession of clicks which are used to attract females.

Some days later, these beautiful, noisy creatures, to which I had grown quite attached, lay below the trees in their thousands — in their millions per acre, actually, as Setonian Online, from whom I have the picture below, tells us.

Now comes the interesting part: the cicadas spend all but a few weeks of their lives underground, as nymphs, feeding on the sap of tree roots. They moult and shed their skin until they are ready to emerge as the winged red-eyed insects you have seen above. But when does this happen?

Most cicadas have a life cycle that lasts between two to five years, but some remain underground for longer periods. The ones I encountered are 17-year cicadas — further south there is a 13-year variety. Amazingly the juveniles stay underground for 13 and 17 years, until it is time to emerge and mate, and then they do it all at once. Millions of cicadas crawling to the surface, but never will one emerge in a different year, or at a non-synchronized time.

There is a reason for this: insects like cicadas have to survive predators who specialize in eating them. One strategy is to procreate in such large numbers that at least a few individuals will survive (and then lay very large numbers of eggs). But there is a downside: when there are ample feeding resources the predators become more abundant (it is known as predator satiation). By leaving out years the victims make it more difficult for the predators to flourish. In years when no cicadas are there, the birds do not have the satiation banquet and their numbers drop.

But what’s with the 13 and 17 year cycles? They are weirdly long and, as the mathematically inclined will have noticed: both are prime numbers! Coincidence? Not at all. There is a fascinating evolutionary reason for it. Ken explained it to me, and I did not believe him until he showed me that paleontologist Stephen J. Gould confirmed it in his writings.

If the cicadas come out very often, the birds develop their own cycles and are ready for the cicadas, every second year, or every third. So longer cycles are advantageous — the insects are making sure that fewer specialized predators will be waiting for them.

But why 13 and 17? Why the prime numbers?

Most cicada predators have a population cycle of between two and ten years. Now imagine twelve-year cicadas: they would be a feast for predators with two, three, four, and six year cycles. That would not be good. Any number of years that is divisible by smaller numbers corresponding with a predator’s population cycle leaves the victims vulnerable.

As Gould explained it, by cycling at a large prime number cicadas minimize the number of times their periods coincide with those of the predators. For instance, three-year predators can expect to feed on 17-year cicadas once every 3×17 = 51 years; five-year predators can expect to feed on them once every 5×17 = 85 years. The 13 and the 17 year cicadas even avoid two-year predators every second coming.

Evolution figured all this out, with no formal mathematical training. And stand in awe: these cicadas can count to seventeen! Try it out yourself, if you are inclined: lock yourself in a dark cellar and come out exactly seventeen years later, on a specific day when millions of other cellar dwellers do the same. Or take my word for it — it’s hard.

Incidentally the two species of long-term cicadas will occasionally emerge together, because their cycles coincide. This happens once every 221 years! The next time you will see 13 and 17 year cicadas together on trees and balconies is in the year 2115. Better have ear plugs ready for that.