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

Haldane linearisation done right: Solving the nonlinear recombination equation the easy way

Pages: 6645 - 6656, Volume 36, Issue 12, December 2016      doi:10.3934/dcds.2016088

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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)

Abstract: The nonlinear recombination equation from population genetics has a long history and is notoriously difficult to solve, both in continuous and in discrete time. This is particularly so if one aims at full generality, thus also including degenerate parameter cases. Due to recent progress for the continuous time case via the identification of an underlying stochastic fragmentation process, it became clear that a direct general solution at the level of the corresponding ODE itself should also be possible. This paper shows how to do it, and how to extend the approach to the discrete-time case as well.

Keywords:  Nonlinear recombination equation, continuous and discrete time, population genetics, measure-valued equations, closed solution.
Mathematics Subject Classification:  34G20, 06B23, 39A12, 92D10.

Received: February 2016;      Revised: July 2016;      Available Online: October 2016.