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

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.

Analysis of Multilevel Replicator Dynamics for General Two-Strategy Social Dilemma

“…The result in this paper on Hölder exponents serves a similar purpose to the idea suggested by Dawson, but characterizing the equivalent approach with moments may provide an alternate strategy for proving convergence to steady state and bears resemblence to the strategy used to analyze the long-time behavior for Becker-D 'oring models of aggregation-fragmentation processes [47][48][49]. In related models possesing individual level selection and replicator-mutator or replicator-diffusion dynamics, formulation of these systems as gradient flows has provided a strategy for proving convergence of the dynamics to a steady-state solution [50,51], and others models with diffusion have used arguments regarding the principle eigenvalue of the diffusion operator [52,53] or decay of an energy-like function [54,55] to prove convergence to steady-state. In addition to analytical attempts to charactertize the long-time behavior, effort should be placed into developing numerical methods for solving Equations 2.5 and 2.13 and comparing the numerical solutions to analytical predictions.…”

An Optimal Mass Transport Method for Random Genetic Drift

“…In this work, we will take advantage of the fact that the following free energy admits (1.1) as a gradient flow with respect to a variation of optimal transport distances (see e.g. [36,6,4,10])…”

“…More recently, the gradient flow structure (1.7) has been discussed in [10]. The analysis of the model is built upon the classical steepest descent variational schemes for nonlinear Fokker-Planck equations which are introduced in [20] and generalized in [1,2].…”