# American Institute of Mathematical Sciences

January  2016, 36(1): 63-95. doi: 10.3934/dcds.2016.36.63

## The general recombination equation in continuous time and its solution

 1 Technische Fakultät, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany, Germany 2 Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld

Received  August 2014 Revised  March 2015 Published  June 2015

The process of recombination in population genetics, in its deterministic limit, leads to a nonlinear ODE in the Banach space of finite measures on a locally compact product space. It has an embedding into a larger family of nonlinear ODEs that permits a systematic analysis with lattice-theoretic methods for general partitions of finite sets. We discuss this type of system, reduce it to an equivalent finite-dimensional nonlinear problem, and establish a connection with an ancestral partitioning process, backward in time. We solve the finite-dimensional problem recursively for generic sets of parameters and briefly discuss the singular cases, and how to extend the solution to this situation.
Citation: Ellen Baake, Michael Baake, Majid Salamat. The general recombination equation in continuous time and its solution. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 63-95. doi: 10.3934/dcds.2016.36.63
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##### References:
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