Dynamical compatibility for finite and infinite population models used in genetics

TL;DR

This paper explores the relationship between finite population Moran processes and infinite population replicator equations in genetics, characterizing when their dynamics align and how to model one with the other.

Contribution

It provides a complete characterization of when finite and infinite population models exhibit similar dynamics and introduces methods to derive compatible Moran processes from replicator dynamics.

Findings

Asymptotic fixation probability is a convex combination of simpler cases.

Models with inner metastable states may require complex d-player game theory.

Complete characterization of when Moran and replicator models exhibit similar attractors.

Abstract

Finite and infinite population models are frequently used in population dynamics. However, their interrelationship is rarely discussed. In this work, we examine the limits of large populations of the Moran process (a finite-population birth-death process) and the replicator equation (an ordinary differential equation) as paradigmatic examples of finite and infinite population models, respectively, both of which are extensively used in population genetics. Except for certain degenerate cases, we completely characterize when these models exhibit similar dynamics, i.e., when there is a one-to-one relation between the stable attractors of the replicator equations and the metastable states of the Moran process. To achieve this goal, we first show that the asymptotic expression for the fixation probability in the Moran process, when the population size is large and individual interaction is almost arbitrary (including cases modeled through $d$-player game theory), is a convex combination of the asymptotic approximations obtained in the constant fitness case or 2-player game theory. We discuss several examples and the inverse problem, i.e., how to derive a Moran process that is compatible with a given replicator dynamics. In particular, we prove that modeling a Moran process with an inner metastable state may require the use of $d$-player game theory with possibly large $d$ values, depending on the precise location of the inner equilibrium.