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

TL;DR

This paper presents a novel method for solving the nonlinear recombination equation in population genetics, leveraging stochastic fragmentation processes to obtain a general solution for both continuous and discrete time cases.

Contribution

It introduces a direct, unified approach to solve the nonlinear recombination equation, extending previous stochastic methods to the discrete-time setting.

Findings

Provides a general solution to the nonlinear recombination equation

Extends stochastic fragmentation techniques to discrete time

Simplifies solving complex population genetics models

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.