# Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective

## TL;DR

This paper unifies classical biological evolution models by reformulating the Moran process, Kimura Equation, and Replicator Equation as gradient flows, revealing their geometric relationships and convergence properties.

## Contribution

It introduces a gradient flow framework for these models, connecting discrete and continuous evolutionary dynamics through geometric and variational principles.

## Key findings

- Reformulation of Moran process and Kimura Equation as gradient flows.
- Conditions for convergence of gradient structures from Moran to Kimura, and Kimura to Replicator.
- Geometric characterization of evolutionary processes as energy minimization problems.

## Abstract

We consider three classical models of biological evolution: (i) the Moran process, an example of a reducible Markov Chain; (ii) the Kimura Equation, a particular case of a degenerated Fokker-Planck Diffusion; (iii) the Replicator Equation, a paradigm in Evolutionary Game Theory. While these approaches are not completely equivalent, they are intimately connected, since (ii) is the diffusion approximation of (i), and (iii) is obtained from (ii) in an appropriate limit. It is well known that the Replicator Dynamics for two strategies is a gradient flow with respect to the celebrated Shahshahani distance. We reformulate the Moran process and the Kimura Equation as gradient flows and in the sequel we discuss conditions such that the associated gradient structures converge: (i) to (ii) and (ii) to (iii). This provides a geometric characterisation of these evolutionary processes and provides a reformulation of the above examples as time minimization of free energy functionals.