Pareto genealogies arising from a Poisson branching evolution model with selection

TL;DR

This paper explores genealogical coalescent processes derived from Pareto-distributed sampling in evolving populations, identifying their limits and connections to models with selection effects.

Contribution

It introduces a new class of coalescent processes from Pareto-based sampling, analyzing their large population limits and relation to branching models with selection.

Findings

Large N limits lead to Poisson-Dirichlet, Beta, or Kingman coalescents depending on alpha.

The coalescent processes can be viewed as genealogies of evolving populations with selection.

The model incorporates fitness-dependent reproduction with power-law distributed intensities.

Abstract

We study a class of coalescents derived from a sampling procedure out of N i.i.d. Pareto(alpha) random variables, normalized by their sum, including beta-size-biasing on total length effects (beta < alpha). Depending on the range of alpha, we derive the large N limit coalescents structure, leading either to a discrete-time Poisson-Dirichlet(alpha,-beta) Xi-coalescent (alpha in [0,1)), or to a family of continuous-time Beta(2-alpha,alpha-beta) Lambda-coalescents (alpha in [1,2)), or to the Kingman coalescent (alpha geq 2). We indicate that this class of coalescent processes (and their scaling limits) may be viewed as the genealogical processes of some forward in time evolving branching population models including selection effects. In such constant-size population models, the reproduction step, which is based on a fitness-dependent Poisson Point Process with scaling power-law(alpha)intensity, is coupled to a selection step consisting of sorting out the N fittest individuals issued from the reproduction step.