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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

The general recombination equation in continuous time and its solution

Pages: 63 - 95, Volume 36, Issue 1, January 2016      doi:10.3934/dcds.2016.36.63

 
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Ellen Baake - Technische Fakultät, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany (email)
Michael Baake - Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany (email)
Majid Salamat - Technische Fakultät, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany (email)

Abstract: 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.

Keywords:  Nonlinear recombination equation, population genetics, measure-valued ODE, lattice of partitions, recursive solution.
Mathematics Subject Classification:  34G20, 06B23, 92D10.

Received: August 2014;      Revised: March 2015;      Available Online: June 2015.

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