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We consider the discrete-time migration-recombination equation, adeterministic, nonlinear dynamical system that describes the evolution of thegenetic type distribution of a population evolving under migration andrecombination 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 thesolution of the migration-recombination equation. As a consequence, one obtainsan explicit solution of the nonlinear dynamics, simply in terms of powers ofthe transition matrix of the Markov chain. Finally, we investigate the limitingand quasi-limiting behaviour of the Markov chain, which gives immediate accessto the asymptotic behaviour of the dynamical system. We finally sketch theanalogous situation in continuous time.

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