Gradient Flow Formulations of Discrete and Continuous Evolutionary Models: A Unifying Perspective

Fabio A. C. C. Chalub, Léonard Monsaingeon, Ana Margarida Ribeiro, Max O. Souza

Research output: Contribution to journalArticlepeer-review

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 minimisation of free energy functionals.

Original languageEnglish
Article number24
JournalActa Applicandae Mathematicae
Volume171
Issue number1
DOIs
Publication statusPublished - Feb 2021

Keywords

  • Gradient flow structure
  • Kimura equation
  • Optimal transport
  • Reducible Markov chains
  • Replicator dynamics
  • Shahshahani distance

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