Fitness potentials and qualitative properties of the Wright-Fisher dynamics

TL;DR

This paper introduces a formalism based on fitness potentials to analyze Wright-Fisher dynamics, offering new insights into fixation processes, stability, and long-term behavior in evolutionary models.

Contribution

It develops a graphical, mechanics-inspired analysis of Wright-Fisher models using fitness potentials, including a new definition of stable states in finite populations.

Findings

Provides a graphical tool for short and long-term dynamics analysis.

Defines a new concept of evolutionary stability for finite populations.

Shows the theory's consistency with multi-type and mutation-including scenarios.

Abstract

We present a mechanistic formalism for the study of evolutionary dynamics models based on the diffusion approximation described by the Kimura Equation. In this formalism, the central component is the fitness potential, from which we obtain an expression for the amount of work necessary for a given type to reach fixation. In particular, within this interpretation, we develop a graphical analysis --- similar to the one used in classical mechanics --- providing the basic tool for a simple heuristic that describes both the short and long term dynamics. As a by-product, we provide a new definition of an evolutionary stable state in finite populations that includes the case of mixed populations. We finish by showing that our theory -- rigorous for two types evolution without mutations-- is also consistent with the multi-type case, and with the inclusion of rare mutations.