The general recombination equation in continuous time and its solution

TL;DR

This paper analyzes the deterministic recombination process in population genetics using nonlinear ODEs, providing a systematic solution approach and connecting it with ancestral partitioning processes.

Contribution

It introduces a systematic method to solve the general recombination equation in continuous time, linking it with lattice theory and ancestral processes.

Findings

Reduced the infinite-dimensional problem to a finite-dimensional nonlinear ODE

Established a recursive solution method for generic parameters

Discussed extensions to singular cases and broader parameter sets

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.