Cannings models, population size changes and multiple-merger coalescents

Fabian Freund

doi:10.1007/s00285-020-01470-5

Abstract

Multiple-merger coalescents, e.g. varLambda -n-coalescents, have been proposed as models of the genealogy of n sampled individuals for a range of populations whose genealogical structures are not captured well by Kingman’s n-coalescent. varLambda -n-coalescents can be seen as the limit process of the discrete genealogies of Cannings models with fixed population size, when time is rescaled and population size Nrightarrow infty . As established for Kingman’s n-coalescent, moderate population size fluctuations in the discrete population model should be reflected by a time-change of the limit coalescent. For varLambda -n-coalescents, this has been explicitly shown for only a limited subclass of varLambda -n-coalescents and exponentially growing populations. This article gives a more general construction of time-changed varLambda -n-coalescents as limits of specific Cannings models with rather arbitrary time changes.

Highlights

The genealogies of samples from populations with highly variant offspring numbers, for instance due to sweepstake reproduction or rapid selection, are not well modelled by Kingman’s n-coalescent
As for the fixed-size models, we consider the genealogy of a sample of n individuals, which is denoted by (Rr(N))r∈N0 The main results of the present paper show that the two allocation schemes allow one to construct Λ-n-coalescent limits of the genealogies of these modified Moran models if population sizes vary in the discrete models in ways described by Eq (4)
As for the Wright–Fisher model, genealogies of samples taken from modified Moran and other Cannings models can be approximated by a time-change of their limit coalescent process, when the population sizes of the discrete models are fluctuating, but are always of the same order of size

Summary

Introduction

The genealogies of samples from populations with highly variant offspring numbers, for instance due to sweepstake reproduction or rapid selection, are not well modelled by Kingman’s n-coalescent. For the Wright–Fisher model, which converges to Kingman’s n-coalescent if population size N is fixed for all generations, the same scaling c−N1 from discrete genealogy to limit is valid for population size changes which maintain a population size of order N at all times, see Griffiths and Tavare (1994) or Kaj and Krone (2003). While conditions for convergence of the discrete genealogies to a limit process are given in Möhle (2002), no explicit construction of haploid Cannings models leading to an analogous limit, a Λ-n-coalescent with changed time scale, is given. Other Λ-n-coalescents (or Cannings models which should converge to these) with changed time scale have been recently discussed and applied as models of genealogies, see Spence et al (2016), Kato et al (2017), Alter and Louzoun (2016) and Multiple-merger coalescents and population size changes. The focus in the present paper is slightly different though, the aim is to explicitly construct Cannings models that converge, after linear time scaling, to a time-changed Λ-n-coalescent, while Koskela and Wilke Berenguer (2019) concentrates on the convergence itself

Models and main results

Discussion

Interpretation of the Cannings models and allocation schemes used

Influence of the choice of Cannings model on the limit

Converging to a time-changed coalescent: sufficient conditions

Proofs of convergence to a time-changed coalescent: modified Moran models