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

@article{Baake2016HaldaneLD,
title={Haldane linearisation done right: Solving the nonlinear recombination equation the easy way},
author={Ellen Baake and Michael Baake},
journal={arXiv: Classical Analysis and ODEs},
year={2016}
}
• Published 16 June 2016
• Mathematics
• arXiv: Classical Analysis and ODEs
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…

## Figures from this paper

Solving the migration–recombination equation from a genealogical point of view
• Mathematics
Journal of mathematical biology
• 2021
The discrete-time migration–recombination equation is considered, a deterministic, nonlinear dynamical system that describes the evolution of the genetic type distribution of a population evolving under migration and recombination in a law of large numbers setting, and the limiting and quasi-limiting behaviour of the Markov chain are investigated.
The recombination equation for interval partitions
• Mathematics, Computer Science
• 2015
The general deterministic recombination equation in continuous time is analysed for various lattices, with special emphasis on the lattice of interval (or ordered) partitions and the corresponding solution for interval partitions is derived and analysed in detail.
Solving the selection-recombination equation: Ancestral lines under selection and recombination
• Biology
• 2020
This contribution uses a probabilistic, genealogical approach for the case of an \emph{arbitrary} number of neutral sites that are linked to one selected site to obtain a stochastic representation of the deterministic solution, along with the Markov semigroup in closed form.
Partitioning, duality, and linkage disequilibria in the Moran model with recombination
• Mathematics
Journal of mathematical biology
• 2016
It is proved that the partitioning process (backward in time) is dual to the Moran population process (forward in time), where the sampling function plays the role of the duality function.
A probabilistic analysis of a continuous-time evolution in recombination
• Mathematics
• 2018
We study the continuous-time evolution of the recombination equation of population genetics. This evolution is given by a differential equation that acts on a product probability space, and its
Genetic recombination as a generalised gradient flow
It is well known that the classical recombination equation for two parent individuals is equivalent to the law of mass action of a strongly reversible chemical reaction network, and can thus be
Ancestral lines under recombination
• Biology
Probabilistic Structures in Evolution
• 2020
With the help of an ancestral partitioning process, which is obtained by letting population size tend to infinity (without rescaling parameters or time) in an ancestral recombination graph, the solution to the recombination equation is obtained in a transparent form.
The general labelled partitioning process in action: recombination, selection, mutation, and more
This work aims to unify and generalise a recursive construction of the solution of the selectionrecombination equation by introducing a new labelled partitioning process with general Markovian labels.
Lines of descent in the deterministic mutation-selection model with pairwise interaction
• Mathematics
• 2018
With the help of the stratified ancestral selection graph, the mutation-selection differential equation with pairwise interaction is considered and results about the ancestral type distribution in the case of unidirectional mutation are obtained.
Selection, recombination, and the ancestral initiation graph.
• Biology
Theoretical population biology
• 2021

## References

SHOWING 1-10 OF 26 REFERENCES
The general recombination equation in continuous time and its solution
• Mathematics
• 2015
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
An Exactly Solved Model for Mutation, Recombination and Selection
• Mathematics
• 2003
Abstract It is well known that rather general mutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that
Single-crossover recombination in discrete time
• Mathematics
Journal of mathematical biology
• 2010
This work considers a particular case of recombination in discrete time, allowing only for single crossovers, and transforms the equations to a solvable system in a two-step procedure: linearisation followed by diagonalisation.
The recombination equation for interval partitions
• Mathematics, Computer Science
• 2015
The general deterministic recombination equation in continuous time is analysed for various lattices, with special emphasis on the lattice of interval (or ordered) partitions and the corresponding solution for interval partitions is derived and analysed in detail.
Deterministic and stochastic aspects of single-crossover recombination
A closed solution of the deterministic continuous-time system, for the important special case of single crossovers, is presented and an analogous deterministic discrete-time dynamics is provided, in terms of its generalised eigenvalues and a simple recursion for the corresponding coefficients.
Mathematical structures in population genetics
• Mathematics
• 1992
In the theory of population genetics, fundamental results on its dynamical processes and equilibrium laws have emerged during the last few decades. This monograph systematically reviews these
Recombination Semigroups on Measure Spaces
The dynamics of recombination in genetics leads to an interesting nonlinear differential equation, which has a natural generalization to a measure valued version which admits a closed formula for the semigroup of nonlinear positive operators that emerges from the forward flow.
Convergence of multilocus systems under weak epistasis or weak selection
• Economics
Journal of mathematical biology
• 1999
It is proved that if epistasis or selection is sufficiently weak (and satisfies a certain nondegeneracy assumption whose genericity the authors establish), then there is always convergence to some equilibrium point, and cycling cannot occur.