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Solving the migration-recombination equation from a genealogical point of view. (English) Zbl 07331662
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.
92D15 Problems related to evolution
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
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