Solving the migration–recombination equation from a genealogical point of view

  title={Solving the migration–recombination equation from a genealogical point of view},
  author={F. Alberti and E. Baake and I. Letter and S. Mart{\'i}nez},
  journal={Journal of Mathematical Biology},
We consider the discrete-time migration–recombination equation, 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. We relate this dynamics (forward in time) to a Markov chain, namely a labelled partitioning process, backward in time. This way, we obtain a stochastic representation of the solution of the migration–recombination equation. As a… Expand

Figures from this paper


The general recombination equation in continuous time and its solution
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 Expand
The coalescent and the genealogical process in geographically structured population
  • M. Notohara
  • Geography, Medicine
  • Journal of mathematical biology
  • 1990
Kingman's coalescent is extended to the geographically structured population model with migration among colonies and a system of equations is derived for the spatial distribution of a common ancestor of sampled genes from colonies and the mean time to getting to one common ancestor. Expand
Single-crossover recombination and ancestral recombination trees
The ancestry of single individuals from the present population is traced back by a random tree, whose branching events correspond to the splitting of the sequence due to recombination, and the probabilities of the topologies of the ancestral trees are calculated. Expand
Single-crossover recombination in discrete time
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. Expand
Haldane linearisation done right: Solving the nonlinear recombination equation the easy way
This paper shows how to do a direct general solution at the level of the corresponding ODE itself to the nonlinear recombination equation from population genetics, and how to extend the approach to the discrete-time case as well. Expand
A probabilistic analysis of a discrete-time evolution in recombination
  • S. Martínez
  • Mathematics, Computer Science
  • Adv. Appl. Math.
  • 2017
This decomposition allows to define a Markov chain in a natural way and describes the geometric decay rate to the limit distribution, and the quasi-stationary behavior when conditioned to the event that the chain does not hit the limit distributions. Expand
Mathematical structures in population genetics
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 theseExpand
An Exactly Solved Model for Mutation, Recombination and Selection
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 thatExpand
Multilocus selection in subdivided populations I. Convergence properties for weak or strong migration
  • R. Bürger
  • Biology, Medicine
  • Journal of mathematical biology
  • 2009
It is proved that, in the absence of selection, all trajectories converge at a geometric rate to a manifold on which global linkage equilibrium holds and allele frequencies are identical across demes. Expand
Closed-Form Asymptotic Sampling Distributions under the Coalescent with Recombination for an Arbitrary Number of Loci
An arbitrary number of loci is considered and combinatorial approaches are employed to find closed-form expressions for the first couple of terms in an asymptotic expansion of the multi-locus sampling distribution. Expand