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

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. Finally, we investigate the limiting and quasi-limiting 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.

References in corpus (4)

• Closed-form asymptotic sampling distributions under the coalescent with recombination for an arbitrary number of loci
• The general recombination equation in continuous time and its solution
• Single--crossover recombination and ancestral recombination trees
• Chromosome Painting: how recombination mixes ancestral colors