  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.
