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 minimisation of free energy functionals.

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Acknowledgements

FACCC and AMR were partially supported by FCT/Portugal Strategic Project UID/MAT/00297/2019 (Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa). FACCC also benefited from an “Investigador FCT” grant. LM was supported by FCT/Portugal projects PTDC/MAT-STA/0975/2014 and PTDC/MAT-STA/28812/2017. MOS was partially supported by CNPq under grants # 309079/2015-2, 310293/2018-9 and by CAPES — Finance Code 001. We also thank an anonymous reviewer for many comments that helped us to improve the paper.

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Chalub, F.A.C.C., Monsaingeon, L., Ribeiro, A.M. et al. Gradient Flow Formulations of Discrete and Continuous Evolutionary Models: A Unifying Perspective. Acta Appl Math 171, 24 (2021). https://doi.org/10.1007/s10440-021-00391-9

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Keywords

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

Mathematics Subject Classification (2010)

  • 35Q92
  • 60J25
  • 92D15
  • 58E30