From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics

TL;DR

This paper unifies discrete evolutionary models into continuous equations, showing how different scalings reveal genetic drift, natural selection, or both, with rigorous proofs and fixation probability analysis.

Contribution

It introduces a unifying framework connecting drift-diffusion and replicator dynamics through different scalings of the Moran process.

Findings

Continuous models depend on scalings, highlighting genetic drift or natural selection.

Singular diffusion equations describe genetic drift effects.

Hyperbolic equations embed replicator dynamics with conservation laws.

Abstract

We study the large population limit of the Moran process, assuming weak-selection, and for different scalings. Depending on the particular choice of scalings, we obtain a continuous model that may highlight the genetic-drift (neutral evolution) or natural selection; for one precise scaling, both effects are present. For the scalings that take the genetic-drift into account, the continuous model is given by a singular diffusion equation, together with two conservation laws that are already present at the discrete level. For scalings that take into account only natural selection, we obtain a hyperbolic singular equation that embeds the Replicator Dynamics and satisfies only one conservation law. The derivation is made in two steps: a formal one, where the candidate limit model is obtained, and a rigorous one, where convergence of the probability density is proved. Additional results on the fixation probabilities are also presented.