Solving the migrationrecombination equation from a genealogical point of view
Abstract
We consider the discretetime migrationrecombination 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 migrationrecombination 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. Finally, we investigate the limiting and quasilimiting behaviour of the Markov chain, which gives immediate access to the asymptotic behaviour of the dynamical system. We finally sketch the analogous situation in continuous time.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.08958
 Bibcode:
 2020arXiv200408958A
 Keywords:

 Mathematics  Probability;
 Mathematics  Dynamical Systems;
 Quantitative Biology  Populations and Evolution;
 92D15;
 60C05;
 05C80;
 37N25
 EPrint:
 J. Math. Biol. 82 (2021) 127