100 research outputs found
Rigorous results for a population model with selection II: genealogy of the population
We consider a model of a population of fixed size undergoing selection.
Each individual acquires beneficial mutations at rate , and each
beneficial mutation increases the individual's fitness by . Each
individual dies at rate one, and when a death occurs, an individual is chosen
with probability proportional to the individual's fitness to give birth. Under
certain conditions on the parameters and , we show that the
genealogy of the population can be described by the Bolthausen-Sznitman
coalescent. This result confirms predictions of Desai, Walczak, and Fisher
(2013), and Neher and Hallatschek (2013).Comment: 54 page
A coalescent model for the effect of advantageous mutations on the genealogy of a population
When an advantageous mutation occurs in a population, the favorable allele
may spread to the entire population in a short time, an event known as a
selective sweep. As a result, when we sample individuals from a population
and trace their ancestral lines backwards in time, many lineages may coalesce
almost instantaneously at the time of a selective sweep. We show that as the
population size goes to infinity, this process converges to a coalescent
process called a coalescent with multiple collisions. A better approximation
for finite populations can be obtained using a coalescent with simultaneous
multiple collisions. We also show how these coalescent approximations can be
used to get insight into how beneficial mutations affect the behavior of
statistics that have been used to detect departures from the usual Kingman's
coalescent
Critical branching Brownian motion with absorption: survival probability
We consider branching Brownian motion on the real line with absorption at
zero, in which particles move according to independent Brownian motions with
the critical drift of . Kesten (1978) showed that almost surely this
process eventually dies out. Here we obtain upper and lower bounds on the
probability that the process survives until some large time . These bounds
improve upon results of Kesten (1978), and partially confirm nonrigorous
predictions of Derrida and Simon (2007)
Beta-coalescents and continuous stable random trees
Coalescents with multiple collisions, also known as -coalescents,
were introduced by Pitman and Sagitov in 1999. These processes describe the
evolution of particles that undergo stochastic coagulation in such a way that
several blocks can merge at the same time to form a single block. In the case
that the measure is the
distribution, they are also known to describe the genealogies of large
populations where a single individual can produce a large number of offspring.
Here, we use a recent result of Birkner et al. to prove that Beta-coalescents
can be embedded in continuous stable random trees, about which much is known
due to the recent progress of Duquesne and Le Gall. Our proof is based on a
construction of the Donnelly--Kurtz lookdown process using continuous random
trees, which is of independent interest. This produces a number of results
concerning the small-time behavior of Beta-coalescents. Most notably, we
recover an almost sure limit theorem of the present authors for the number of
blocks at small times and give the multifractal spectrum corresponding to the
emergence of blocks with atypical size. Also, we are able to find exact
asymptotics for sampling formulae corresponding to the site frequency spectrum
and the allele frequency spectrum associated with mutations in the context of
population genetics.Comment: Published in at http://dx.doi.org/10.1214/009117906000001114 the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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