Stochastic neutral fractions and the effective population size
Raphaël Forien, Emmanuel Schertzer, Zsófia Talyigás, Julie Tourniaire
TL;DR
The paper develops a forward‑time framework for structured populations using an ∞‑decomposable SDE, introducing neutral fractions that evolve on a fast ecological scale while the fractions themselves undergo slow evolutionary drift. Under small demographic noise, the neutral fractions converge to a standard Wright–Fisher diffusion on the (K−1)‑simplex when time is accelerated by the factor $\Sigma^2$, with $\Sigma^2$ defined via Perron–Frobenius eigenvectors and local covariances; the effective population size is $N_e = N/\Sigma^2$. The work provides two rigorous proof strategies (Katzenberger’s method and a Perron–Frobenius projection approach) and illustrates the framework across diverse examples including asexual and sexual populations, gut microbiome models, and FKPP fronts with Allee effects, while also discussing non‑uniqueness of neutral fractions and the ancestral/coalescent interpretation. By linking forward‑time neutral fractions to Kingman coalescence under time scaling, the results offer a tractable, forward‑time pathway to understanding genealogies in structured populations without recourse to backward‑in‑time constructions. The findings have broad implications for estimating $N_e$ in complex populations and for understanding how microscale structure shapes macro‑scale genetic drift and genealogies.
Abstract
The dynamics of a general structured population is modelled using a general stochastic differential equation (SDE) with an infinite decomposability property. This property allows the population to be divided into an arbitrary number of allelic components, also known as stochastic neutral fractions. When demographic noise is small, a fast-slow principle provides a general formula for the effective population size in structured populations. To illustrate this approach, we revisit several examples from the literature, including expansion fronts.