Solving the migration-recombination equation from a genealogical point of view.

F Alberti, E Baake, I Letter, S Martínez

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Article in Journal of mathematical biology, 2021. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Not yet cited in PubMed.

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4 authors.

F AlbertiFaculty of Mathematics, Bielefeld University, Postbox 100131, 33501, Bielefeld, Germany.

E BaakeFaculty of Mathematics, Bielefeld University, Postbox 100131, 33501, Bielefeld, Germany. ebaake@techfak.uni-bielefeld.de.ORCID 0000-0001-8970-2102

I LetterStatistics Department, University of Oxford, 24-29 St Giles, Oxford, OX1 3LB, UK.

S MartínezDepartment of Mathematical Engineering and Center of Mathematical Modeling, UMI 2807 UCHILE-CNRS, Universidad de Chile, Santiago, Chile.

Funding

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6 · The paper itself

Abstract

We consider the discrete-time migration-recombination equation, a deterministic, nonlinear dynamical system that describes the evolution of the genetic type distribution of a population evolving under migration and recombination in a law of large numbers setting. We relate this dynamics (forward in time) to a Markov chain, namely a labelled partitioning process, backward in time. This way, we obtain a stochastic representation of the solution of the migration-recombination equation. As a consequence, one obtains an explicit solution of the nonlinear dynamics, simply in terms of powers of the transition matrix of the Markov chain. The limiting and quasi-limiting behaviour of the Markov chain are investigated, which gives immediate access to the asymptotic behaviour of the dynamical system. We finally sketch the analogous situation in continuous time.

Indexed as

Models, GeneticAnimalsBiological EvolutionGenealogy and HeraldryHumansMarkov ChainsNonlinear DynamicsPopulation DynamicsRecombination, GeneticAncestral recombination graphDualityHaldane linearisationLabelled partitioning processMigration–recombination equationQuasi-stationarity

Identifiers

PMID33774735

DOI10.1007/s00285-021-01584-4

PMCPMC8004498

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