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Probabilistic analysis of replicator–mutator equations

Part of: Stochastic processes Mathematical biology in general Stochastic analysis

Published online by Cambridge University Press:  23 March 2022

Lijun Bo  and
Huafu Liao
Lijun Bo*
Affiliation:
Xidian University and University of Science and Technology of China
Huafu Liao*
Affiliation:
National University of Singapore
*
*Postal address: School of Mathematics and Statistics, Xidian University, Xi’an 710071, China; School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China. Email address: lijunbo@ustc.edu.cn
**Postal address: Department of Mathematics, National University of Singapore, Singapore 119076, Singapore. Email address: mathuaf@nus.edu.sg
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Abstract

This paper discusses a general class of replicator–mutator equations on a multidimensional fitness space. We establish a novel probabilistic representation of weak solutions of the equation by using the theory of Fokker–Planck–Kolmogorov (FPK) equations and a martingale extraction approach. We provide examples with closed-form probabilistic solutions for different fitness functions considered in the existing literature. We also construct a particle system and prove a general convergence result to the unique solution of the FPK equation associated with the extended replicator–mutator equation with respect to a Wasserstein-like metric adapted to our probabilistic framework.

Keywords

Replicator–mutator equationsprobabilistic representationFPK equationsmartingale extractionparticle system

MSC classification

Primary: 92B05: General biology and biomathematics
Secondary: 60H10: Stochastic ordinary differential equations 60G46: Martingales and classical analysis
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 1 , March 2022 , pp. 167 - 201
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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