Derivation of replicator–mutator equations from a model in population genetics

Joe Yuichiro Wakano; Tadahisa Funaki; Satoshi Yokoyama

doi:10.1007/s13160-017-0249-9

Derivation of replicator–mutator equations from a model in population genetics

Joe Yuichiro Wakano^*, Tadahisa Funaki, Satoshi Yokoyama

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

We introduce a Markov chain model to study evolution of a continuous trait based on population genetics. It corresponds to individual-based model which includes frequency dependent selection caused by m-player game interactions and stochastic fluctuations due to random genetic drift and mutation. We prove that under a proper scaling limit as the population size increases the system converges to the solution of replicator–mutator equations. Our result establishes an affirmative mathematical base to the adaptive dynamics formulation employed in the theory of the mathematical biology.

Original language	English
Pages (from-to)	473-488
Number of pages	16
Journal	Japan Journal of Industrial and Applied Mathematics
Volume	34
Issue number	2
DOIs	https://doi.org/10.1007/s13160-017-0249-9
Publication status	Published - 2017 Aug 1

Keywords

Adaptive dynamics
Population genetic model
Scaling limits
replicator–mutator equation

ASJC Scopus subject areas

Engineering(all)
Applied Mathematics

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10.1007/s13160-017-0249-9

Cite this

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title = "Derivation of replicator–mutator equations from a model in population genetics",

abstract = "We introduce a Markov chain model to study evolution of a continuous trait based on population genetics. It corresponds to individual-based model which includes frequency dependent selection caused by m-player game interactions and stochastic fluctuations due to random genetic drift and mutation. We prove that under a proper scaling limit as the population size increases the system converges to the solution of replicator–mutator equations. Our result establishes an affirmative mathematical base to the adaptive dynamics formulation employed in the theory of the mathematical biology.",

keywords = "Adaptive dynamics, Population genetic model, Scaling limits, replicator–mutator equation",

author = "Wakano, {Joe Yuichiro} and Tadahisa Funaki and Satoshi Yokoyama",

note = "Funding Information: T. Funaki is supported in part by the JSPS KAKENHI Grant Numbers (B) 26287014 and 26610019. Funding Information: J. Y. Wakano is supported in part by the JSPF KAKENHI Grant Number (C) 16K05283. Funding Information: S. Yokoyama is supported in part by the JSPS KAKENHI Grant Number (S) 24224004. Publisher Copyright: {\textcopyright} 2017, The JJIAM Publishing Committee and Springer Japan KK.",

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doi = "10.1007/s13160-017-0249-9",

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T1 - Derivation of replicator–mutator equations from a model in population genetics

AU - Wakano, Joe Yuichiro

AU - Funaki, Tadahisa

AU - Yokoyama, Satoshi

N1 - Funding Information: T. Funaki is supported in part by the JSPS KAKENHI Grant Numbers (B) 26287014 and 26610019. Funding Information: J. Y. Wakano is supported in part by the JSPF KAKENHI Grant Number (C) 16K05283. Funding Information: S. Yokoyama is supported in part by the JSPS KAKENHI Grant Number (S) 24224004. Publisher Copyright: © 2017, The JJIAM Publishing Committee and Springer Japan KK.

PY - 2017/8/1

Y1 - 2017/8/1

N2 - We introduce a Markov chain model to study evolution of a continuous trait based on population genetics. It corresponds to individual-based model which includes frequency dependent selection caused by m-player game interactions and stochastic fluctuations due to random genetic drift and mutation. We prove that under a proper scaling limit as the population size increases the system converges to the solution of replicator–mutator equations. Our result establishes an affirmative mathematical base to the adaptive dynamics formulation employed in the theory of the mathematical biology.

AB - We introduce a Markov chain model to study evolution of a continuous trait based on population genetics. It corresponds to individual-based model which includes frequency dependent selection caused by m-player game interactions and stochastic fluctuations due to random genetic drift and mutation. We prove that under a proper scaling limit as the population size increases the system converges to the solution of replicator–mutator equations. Our result establishes an affirmative mathematical base to the adaptive dynamics formulation employed in the theory of the mathematical biology.

KW - Adaptive dynamics

KW - Population genetic model

KW - Scaling limits

KW - replicator–mutator equation

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JF - Japan Journal of Industrial and Applied Mathematics

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ER -

Derivation of replicator–mutator equations from a model in population genetics

Abstract

Keywords

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