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

Ellen  Baake; Michael Baake

doi:10.3934/dcds.2016088

Article Contents

2016, Volume 36, Issue 12: 6645-6656. Doi: 10.3934/dcds.2016088 open access

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

Ellen Baake^1, and
Michael Baake^2,

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

Received: January 31, 2016

Revised: June 30, 2016

Published: May 2017

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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.

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Mathematics Subject Classification: 34G20, 06B23, 39A12, 92D10.

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