Every species faces the question of when during its life it should pour maximum effort into reproduction. Reproducing too early forfeits the benefits of further growth and accumulated parental experience, while reproducing too late raises the risk of dying without leaving offspring. Classic work linked the age of maturation and the age of peak reproduction to growth, mortality and fecundity curves, but there was no simple cross-species numerical benchmark for the age of peak effort relative to maximum lifespan. This work offers such a benchmark and connects it to a well-known optimum from optimal stopping theory. Around this paper, our archive holds studies on evolution, horizontal gene transfer, phylogenetic networks and cell biology, but none of them derived a universal temporal fraction of the life cycle from a mathematical theorem.
The authors applied Bruss's 1/e law to biological reproduction. In the model, opportunities to reproduce (partners, suitable conditions) arrive over a lifetime as independent random variables with a common distribution; you can only choose the current opportunity, and you cannot go back to a missed one. Bruss's theorem states that the strategy "wait until time t_w, then choose the first opportunity better than all previous ones" guarantees a success probability of at least 1/e for any number of opportunities, provided t_w satisfies the integral condition F(t_w)=1/e. For an approximately uniform arrival of opportunities, this yields t_w = t_max/e. To test it, the researchers assembled data on the age of first maturation in 2197 species (fish, invertebrates, birds, mammals, trees), on the age of peak reproductive effort from 15 detailed studies (from beech to human), and on the ratio of the inflection age of the growth curve to maximum age in 967 fish species. They also compared the predicted survival to maximum age (about 1%) with empirical estimates for 202 species.
The peak of reproductive effort in every species examined falls at 1/e of maximum lifespan. In 96% of the 2197 species, maturation occurs no later than 1/e of the lifespan. For 967 fish species, the median ratio of the inflection age of the growth curve to maximum age was 0.367 (interquartile range 0.25–0.48). Fifteen detailed "age–reproductive effort" series, from trees to humans, show a peak in the window 0.32–0.42 of maximum age. The predicted survival to maximum age of about 1.1% agrees with empirical data (median 1.5%, range 0.9–2.2%). The shape of the reproductive effort distribution is universal: a steep rise after maturation, a peak near 1/e, then an exponential decline that tracks population survival.
The work puts the timing of the reproductive peak on a solid mathematical footing: the generation turns out to be a moment of optimal stopping. Unlike neighboring studies that describe individual mechanisms or special cases, this one is the first to derive a numerical fraction of the life cycle, universal across all taxa, from Bruss's theorem. It is not an empirical pattern but a first-principles prediction confirmed on a broad dataset. The result links growth physiology, demography and probability theory and offers a simple way to estimate maximum lifespan from the age of the reproductive peak, and vice versa.
The topic could develop toward more precise testing of the theorem's conditions in specific populations, especially where opportunities to reproduce arrive very unevenly. It would be interesting to study species with menopause and a post-reproductive period, as well as cases where the number of opportunities can be estimated, which would allow stronger algorithms such as the odds-algorithm to be applied. How exactly organisms implement the stopping strategy at the physiological level remains an open question.
The result is applicable in demography, conservation biology and the management of exploited populations: from the age of the reproductive peak one can estimate maximum lifespan, and vice versa. This provides a tool for quickly assessing vital parameters of species for which direct data are hard to obtain.
The next step is to test the prediction on a broader range of species with reliable data on the age of the reproductive peak and maximum lifespan, especially among invertebrates and plants. It is also worth investigating how deviations from a uniform arrival of opportunities affect the position of the peak.
The work directly links an unsolved problem in evolutionary biology, the optimization of the life cycle, to a mathematical theorem. It gives a quantitative answer to why the age of the reproductive peak is so stable relative to lifespan, and proposes a mechanism that can be tested in further empirical studies.
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.