an:07331662
Zbl 7331662
Alberti, F.; Baake, E.; Letter, I.; MartÃnez, S.
Solving the migration-recombination equation from a genealogical point of view
EN
J. Math. Biol. 82, No. 5, Paper No. 41, 27 p. (2021)
0303-6812 1432-1416
2021
j
92D15 60J20
migration-recombination equation; ancestral recombination graph; duality; labelled partitioning process; quasi-stationarity; Haldane linearisation
Summary: 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.