The clock of life ticks at different speeds for every species, yet the hand marking the peak of reproduction always stops at the same mark: 37% of the way. In beech trees, cod, albatrosses, and humans, the maximum effort to continue the line comes when a little more than a third of the maximum lifespan has been lived. This is not a coincidence but a mathematical law as reliable as gravity.
An organism that reproduces too early has not yet grown and stored enough resources. One that waits too long risks dying without offspring. The optimal moment is when enough strength has been accumulated, yet the chance of surviving to the next season is still high. The math sharpens this: that moment arrives when 1/e of all the opportunities life can offer have passed. For a steady stream of chances, that is exactly 37% of maximum lifespan.
A check across 2197 species showed that in 96% of them maturity arrives no later than this mark. In 967 fish species, the age when growth slows falls at 0.367 of maximum age. Fifteen detailed studies, from trees to humans, put the reproductive peak in the window 0.32–0.42. Even survival to maximum age is predicted by the same model: about 1% of individuals make it to the end, which matches observations.
This pattern ties together growth physiology, demography, and probability theory. From the age of the reproductive peak, one can estimate a species' maximum lifespan, and vice versa. For rare or hard-to-reach animals, this is a simple way to obtain vital parameters without long observations.
The clock of life, ticking off 37% before the main decision, runs for all species. The mathematics of optimal stopping explains why the hand stops exactly here: it is the best compromise between preparation and risk. Nature does not do mental arithmetic, but evolution spent millions of years selecting those who happened to land on this point. Now we know what number lies behind that choice.
The number 1/e (about 0.37) is known as the "37% rule" in the fussy suitor problem: to maximize your chance of picking the best partner, you should reject the first 37% of candidates, then choose the first one better than all previous ones. In biology, the same number governs the timing of the reproductive peak.
In Stanisław Lem's novel Solaris, the planet's ocean displays behavior reminiscent of optimal stopping on the scale of geological epochs, but there is no direct analogy to the 1/e law in well-known science fiction.
Not published on arXiv. Here are its page, the PDF and the formalization: you can check it yourself, not only read our account.