23,660 research outputs found
A construction of a -coalescent via the pruning of Binary Trees
Considering a random binary tree with labelled leaves, we use a pruning
procedure on this tree in order to construct a -coalescent
process. We also use the continuous analogue of this construction, i.e. a
pruning procedure on Aldous's continuum random tree, to construct a continuous
state space process that has the same structure as the -coalescent
process up to some time change. These two constructions unable us to obtain
results on the coalescent process such as the asymptotics on the number of
coalescent events or the law of the blocks involved in the last coalescent
event
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
Coalescent histories for lodgepole species trees
Coalescent histories are combinatorial structures that describe for a given
gene tree and species tree the possible lists of branches of the species tree
on which the gene tree coalescences take place. Properties of the number of
coalescent histories for gene trees and species trees affect a variety of
probabilistic calculations in mathematical phylogenetics. Exact and asymptotic
evaluations of the number of coalescent histories, however, are known only in a
limited number of cases. Here we introduce a particular family of species
trees, the \emph{lodgepole} species trees , in which
tree has taxa. We determine the number of coalescent
histories for the lodgepole species trees, in the case that the gene tree
matches the species tree, showing that this number grows with in the
number of taxa . This computation demonstrates the existence of tree
families in which the growth in the number of coalescent histories is faster
than exponential. Further, it provides a substantial improvement on the lower
bound for the ratio of the largest number of matching coalescent histories to
the smallest number of matching coalescent histories for trees with taxa,
increasing a previous bound of
to . We discuss the implications of our
enumerative results for phylogenetic computations
The peripatric coalescent
We consider a dynamic metapopulation involving one large population of size N
surrounded by colonies of size \varepsilon_NN, usually called peripheral
isolates in ecology, where N\to\infty and \varepsilon_N\to 0 in such a way that
\varepsilon_NN\to\infty. The main population periodically sends propagules to
found new colonies (emigration), and each colony eventually merges with the
main population (fusion). Our aim is to study the genealogical history of a
finite number of lineages sampled at stationarity in such a metapopulation. We
make assumptions on model parameters ensuring that the total outer population
has size of the order of N and that each colony has a lifetime of the same
order. We prove that under these assumptions, the scaling limit of the
genealogical process of a finite sample is a censored coalescent where each
lineage can be in one of two states: an inner lineage (belonging to the main
population) or an outer lineage (belonging to some peripheral isolate).
Lineages change state at constant rate and inner lineages (only) coalesce at
constant rate per pair. This two-state censored coalescent is also shown to
converge weakly, as the landscape dynamics accelerate, to a time-changed
Kingman coalescent.Comment: 17 pages,1 figur
A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks
Let be a finite measure on the unit interval. A
-Fleming-Viot process is a probability measure valued Markov process
which is dual to a coalescent with multiple collisions (-coalescent)
in analogy to the duality known for the classical Fleming Viot process and
Kingman's coalescent, where is the Dirac measure in 0.
We explicitly construct a dual process of the coalescent with simultaneous
multiple collisions (-coalescent) with mutation, the -Fleming-Viot
process with mutation, and provide a representation based on the empirical
measure of an exchangeable particle system along the lines of Donnelly and
Kurtz (1999). We establish pathwise convergence of the approximating systems to
the limiting -Fleming-Viot process with mutation. An alternative
construction of the semigroup based on the Hille-Yosida theorem is provided and
various types of duality of the processes are discussed.
In the last part of the paper a population is considered which undergoes
recurrent bottlenecks. In this scenario, non-trivial -Fleming-Viot
processes naturally arise as limiting models.Comment: 35 pages, 2 figure
On asymptotics of the beta-coalescents
We show that the total number of collisions in the exchangeable coalescent
process driven by the beta measure converges in distribution to a
1-stable law, as the initial number of particles goes to infinity. The stable
limit law is also shown for the total branch length of the coalescent tree.
These results were known previously for the instance , which corresponds
to the Bolthausen--Sznitman coalescent. The approach we take is based on
estimating the quality of a renewal approximation to the coalescent in terms of
a suitable Wasserstein distance. Application of the method to beta
-coalescents with leads to a simplified derivation of the known
-stable limit. We furthermore derive asymptotic expansions for the
moments of the number of collisions and of the total branch length for the beta
-coalescent by exploiting the method of sequential approximations.Comment: 25 pages, submitted for publicatio
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