Evolutionary Entropy Shapes Reproductive Lifespan in Age-Structured Populations
Evolutionary entropy measures the temporal organization of reproductive contributions along the life cycle of an age-structured population. We develop a mathematical and empirical framework showing that, in iteroparous animal populations represented by Leslie-type demographic matrices, reproductive windows are frequently organized near the age classes selected by entropy maximization. Evolutionary entropy complements the classical net reproductive number and asymptotic growth rate: whereas these measure lifetime replacement and growth, entropy measures the temporal dispersion of the growth-adjusted reproductive distribution. Our central result is a reduction principle: under Euler--Lotka normalization, evolutionary entropy and generation time are invariant under multiplicative rescaling of survivorship and fertility on the reproductive interval. The relevant entropy is determined not by absolute survivorship, fertility, or juvenile mortality, but by the normalized post-maturity reproductive distribution. We derive explicit entropy functionals for finite and open-group Leslie models, including geometric reproductive tails. For the geometric regime, governed by we prove a sharp critical threshold separating populations with a unique finite entropy-maximizing endpoint from those whose entropy increases toward an asymptotic value in terms solely of the age at first reproduction. The theory is tested on 130 animal species. Entropy-derived predictions, computed from the demographic matrices alone, are compared with independent life-history variables. Predicted and observed reproductive medians coincide exactly for a majority of species, over 90% are predicted within three reproductive classes, and associations remain strong after phylogenetic correction. These results identify a quantitative regularity across taxa, with geometric reproductive distributions playing a central role.