TY - JOUR

T1 - Gradient Flow Formulations of Discrete and Continuous Evolutionary Models

T2 - A Unifying Perspective

AU - Chalub, Fabio A. C. C.

AU - Monsaingeon, Léonard

AU - Ribeiro, Ana Margarida

AU - Souza, Max O.

N1 - info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UID%2FMAT%2F00297%2F2019/PT#
info:eu-repo/grantAgreement/FCT/3599-PPCDT/PTDC%2FMAT-STA%2F0975%2F2014/PT#
info:eu-repo/grantAgreement/FCT/3599-PPCDT/PTDC%2FMAT-STA%2F28812%2F2017/PT#
FACCC also benefited from an "Investigador FCT" grant. LM was supported by MOS was partially supported by CNPq under grants # 309079/2015-2, 310293/2018-9 and by CAPES - Finance Code 001.

PY - 2021/2

Y1 - 2021/2

N2 - 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.

AB - 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.

KW - Gradient flow structure

KW - Kimura equation

KW - Optimal transport

KW - Reducible Markov chains

KW - Replicator dynamics

KW - Shahshahani distance

UR - http://www.scopus.com/inward/record.url?scp=85100563213&partnerID=8YFLogxK

U2 - 10.1007/s10440-021-00391-9

DO - 10.1007/s10440-021-00391-9

M3 - Article

AN - SCOPUS:85100563213

SN - 0167-8019

VL - 171

JO - Acta Applicandae Mathematicae

JF - Acta Applicandae Mathematicae

IS - 1

M1 - 24

ER -