The genealogy of branching Brownian motion with absorption

Abstract

We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately N particles. We show that the characteristic time scale for the evolution of this population is of order $(\log N)^3$, in the sense that when time is measured in these units, the scaled number of particles converges to a variant of Neveu's continuous-state branching process. Furthermore, the genealogy of the particles is then governed by a coalescent process known as the Bolthausen-Sznitman coalescent. This validates the nonrigorous predictions by Brunet, Derrida, Muller and Munier for a closely related model.Comment: Published in at http://dx.doi.org/10.1214/11-AOP728 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Genetic hitchhiking in spatially extended populations

“…In the discussion above, we have considered only the 'bulk' of the travelling wave and we have ignored the fluctuations in the position of the front. The analyses of Brunet and Derrida (2001); Brunet et al (2006); Berestycki et al (2012) of models which are believed to mirror the behaviour of solutions to (3) suggest that, at least for large η, for most of the time the deterministic approximation p cη (·) provides a good approximation to the shape of the wavefront, but at time intervals of order O((log η) 3 ) there are appreciable fluctuations. Roughly, these occur because an individual in the population manages to get significantly ahead of the bulk of the wave.…”

“…However, the work of Brunet et al (2006); Berestycki et al (2012) suggests that in fact it is coalescence due to fluctuations in the wavefront that dominate. More precisely, if we take a sample of lineages from the wavefront, the expected time until a pair of lineages coalesces is O((log η) 3 ), which determines the appropriate timescale on which to view coalescence.…”

“…(Here, and throughout, we use log to denote the natural logarithm.) This is sometimes referred to as Brunet-Derrida behaviour and has now been established rigorously for a whole raft of related models (see Berestycki et al (2012); Bérard and Gouéré (2011) for recent reviews) and it is reasonable to expect that the reduction of wavespeed in (2) compared to that in the limit as η → ∞ will (to leading order) also be proportional to 1/(log η) 2 . The wavespeed in (3) is just that in (4) multiplied by σ s/2.…”

“…However, the speed of the Fisher wave is determined by its behaviour in regions where the frequency of the favoured allele is extremely low and, consequently, it is very sensitive to stochastic fluctuations. In recent years there has been considerable progress in understanding the coupling between such fluctuations and the progress of the 'bulk' of the wave Derrida (1997, 2001);van Saarloos (2003); Brunet et al (2006); Hallatschek and Nelson (2008); Mueller et al (2011);Berestycki et al (2012)). Much of this work is concerned with the spread of a new species into an empty habitat, but the mathematical models apply equally to the spread of a selectively favoured allele through a stable population.…”

“…Three ramifications of the presence of small amounts of genetic drift have been explored: first, drift slows down the rate of advance of the wave by a factor proportional to the dimensionless quantity 1/(log(ρσ s/2)) 2 where ρ is population density, σ 2 is the rate of diffusion and s is the selection coefficient (Brunet et al, 2006;Mueller et al, 2011); second, the resulting cline shape (that is the shape of the wavefront) is well-approximated by a truncated Fisher wave (described in more detail in §2.1), Brunet and Derrida (1997); Mueller et al (2011); and third, the genealogy of a sample from the wavefront will be dominated by 'founder effects' (resulting from fluctuations in the wavefront) and in an appropriate timescale is approximated not by a Kingman coalescent, but instead by the so-called Bolthausen-Sznitman coalescent in which, in particular, multiple lineages merge in a single event (Brunet et al, 2006;Berestycki et al, 2012).…”