183 research outputs found
The allelic partition for coalescent point processes
Assume that individuals alive at time in some population can be ranked in
such a way that the coalescence times between consecutive individuals are
i.i.d. The ranked sequence of these branches is called a coalescent point
process. We have shown in a previous work that splitting trees are important
instances of such populations. Here, individuals are given DNA sequences, and
for a sample of DNA sequences belonging to distinct individuals, we
consider the number of polymorphic sites (sites at which at least two
sequences differ), and the number of distinct haplotypes (sequences
differing at one site at least). It is standard to assume that mutations arrive
at constant rate (on germ lines), and never hit the same site on the DNA
sequence. We study the mutation pattern associated to coalescent point
processes under this assumption. Here, and grow linearly as
grows, with explicit rate. However, when the branch lengths have infinite
expectation, grows more rapidly, e.g. as for critical
birth--death processes. Then, we study the frequency spectrum of the sample,
that is, the numbers of polymorphic sites/haplotypes carried by individuals
in the sample. These numbers are shown to grow also linearly with sample size,
and we provide simple explicit formulae for mutation frequencies and haplotype
frequencies. For critical birth--death processes, mutation frequencies are
given by the harmonic series and haplotype frequencies by Fisher logarithmic
series
Random ultrametric trees and applications
Ultrametric trees are trees whose leaves lie at the same distance from the
root. They are used to model the genealogy of a population of particles
co-existing at the same point in time. We show how the boundary of an
ultrametric tree, like any compact ultrametric space, can be represented in a
simple way via the so-called comb metric. We display a variety of examples of
random combs and explain how they can be used in applications. In particular,
we review some old and recent results regarding the genetic structure of the
population when throwing neutral mutations on the skeleton of the tree.Comment: 20 pages, 7 figures, proceedings of MAS 2016, Grenoble, France
(Stochastic modeling and Statistics Conference, French Society for Applied
and Industrial Math, SMAI
Species abundance distributions in neutral models with immigration or mutation and general lifetimes
We consider a general, neutral, dynamical model of biodiversity. Individuals
have i.i.d. lifetime durations, which are not necessarily exponentially
distributed, and each individual gives birth independently at constant rate
\lambda. We assume that types are clonally inherited. We consider two classes
of speciation models in this setting. In the immigration model, new individuals
of an entirely new species singly enter the population at constant rate \mu
(e.g., from the mainland into the island). In the mutation model, each
individual independently experiences point mutations in its germ line, at
constant rate \theta. We are interested in the species abundance distribution,
i.e., in the numbers, denoted I_n(k) in the immigration model and A_n(k) in the
mutation model, of species represented by k individuals, k=1,2,...,n, when
there are n individuals in the total population. In the immigration model, we
prove that the numbers (I_t(k);k\ge 1) of species represented by k individuals
at time t, are independent Poisson variables with parameters as in Fisher's
log-series. When conditioning on the total size of the population to equal n,
this results in species abundance distributions given by Ewens' sampling
formula. In particular, I_n(k) converges as n\to\infty to a Poisson r.v. with
mean \gamma /k, where \gamma:=\mu/\lambda. In the mutation model, as
n\to\infty, we obtain the almost sure convergence of n^{-1}A_n(k) to a
nonrandom explicit constant. In the case of a critical, linear birth--death
process, this constant is given by Fisher's log-series, namely n^{-1}A_n(k)
converges to \alpha^{k}/k, where \alpha :=\lambda/(\lambda+\theta). In both
models, the abundances of the most abundant species are briefly discussed.Comment: 16 pages, 4 figures. To appear in Journal of Mathematical Biology.
The final publication is available at http://www.springerlink.co
The branching process with logistic growth
In order to model random density-dependence in population dynamics, we
construct the random analogue of the well-known logistic process in the
branching process' framework. This density-dependence corresponds to
intraspecific competition pressure, which is ubiquitous in ecology, and
translates mathematically into a quadratic death rate. The logistic branching
process, or LB-process, can thus be seen as (the mass of) a fragmentation
process (corresponding to the branching mechanism) combined with constant
coagulation rate (the death rate is proportional to the number of possible
coalescing pairs). In the continuous state-space setting, the LB-process is a
time-changed (in Lamperti's fashion) Ornstein-Uhlenbeck type process. We obtain
similar results for both constructions: when natural deaths do not occur, the
LB-process converges to a specified distribution; otherwise, it goes extinct
a.s. In the latter case, we provide the expectation and the Laplace transform
of the absorption time, as a functional of the solution of a Riccati
differential equation. We also show that the quadratic regulatory term allows
the LB-process to start at infinity, despite the fact that births occur
infinitely often as the initial state goes to \infty. This result can be viewed
as an extension of the pure-death process starting from infinity associated to
Kingman's coalescent, when some independent fragmentation is added.Comment: Published at http://dx.doi.org/10.1214/105051605000000098 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Splitting trees with neutral Poissonian mutations I: Small families
We consider a neutral dynamical model of biological diversity, where
individuals live and reproduce independently. They have i.i.d. lifetime
durations (which are not necessarily exponentially distributed) and give birth
(singly) at constant rate b. Such a genealogical tree is usually called a
splitting tree, and the population counting process (N_t;t\ge 0) is a
homogeneous, binary Crump--Mode--Jagers process. We assume that individuals
independently experience mutations at constant rate \theta during their
lifetimes, under the infinite-alleles assumption: each mutation instantaneously
confers a brand new type, called allele, to its carrier. We are interested in
the allele frequency spectrum at time t, i.e., the number A(t) of distinct
alleles represented in the population at time t, and more specifically, the
numbers A(k,t) of alleles represented by k individuals at time t,
k=1,2,...,N_t. We mainly use two classes of tools: coalescent point processes
and branching processes counted by random characteristics. We provide explicit
formulae for the expectation of A(k,t) in a coalescent point process
conditional on population size, which apply to the special case of splitting
trees. We separately derive the a.s. limits of A(k,t)/N_t and of A(t)/N_t
thanks to random characteristics. Last, we separately compute the expected
homozygosity by applying a method characterizing the dynamics of the tree
distribution as the origination time of the tree moves back in time, in the
spirit of backward Kolmogorov equations.Comment: 32 pages, 2 figures. Companion paper in preparation "Splitting trees
with neutral Poissonian mutations II: Large or old families
Evolution of discrete populations and the canonical diffusion of adaptive dynamics
The biological theory of adaptive dynamics proposes a description of the
long-term evolution of a structured asexual population. It is based on the
assumptions of large population, rare mutations and small mutation steps, that
lead to a deterministic ODE describing the evolution of the dominant type,
called the ``canonical equation of adaptive dynamics.'' Here, in order to
include the effect of stochasticity (genetic drift), we consider self-regulated
randomly fluctuating populations subject to mutation, so that the number of
coexisting types may fluctuate. We apply a limit of rare mutations to these
populations, while keeping the population size finite. This leads to a jump
process, the so-called ``trait substitution sequence,'' where evolution
proceeds by successive invasions and fixations of mutant types. Then we apply a
limit of small mutation steps (weak selection) to this jump process, that leads
to a diffusion process that we call the ``canonical diffusion of adaptive
dynamics,'' in which genetic drift is combined with directional selection
driven by the gradient of the fixation probability, also interpreted as an
invasion fitness. Finally, we study in detail the particular case of multitype
logistic branching populations and seek explicit formulae for the invasion
fitness of a mutant deviating slightly from the resident type. In particular,
second-order terms of the fixation probability are products of functions of the
initial mutant frequency, times functions of the initial total population size,
called the invasibility coefficients of the resident by increased fertility,
defence, aggressiveness, isolation or survival.Comment: Published at http://dx.doi.org/10.1214/105051606000000628 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Splitting trees stopped when the first clock rings and Vervaat's transformation
We consider a branching population where individuals have i.i.d.\ life
lengths (not necessarily exponential) and constant birth rate. We let
denote the population size at time . %(called homogeneous, binary
Crump--Mode--Jagers process). We further assume that all individuals, at birth
time, are equipped with independent exponential clocks with parameter .
We are interested in the genealogical tree stopped at the first time when
one of those clocks rings. This question has applications in epidemiology, in
population genetics, in ecology and in queuing theory.
We show that conditional on , the joint law of , where is the jumping contour process of the tree truncated
at time , is equal to that of conditional on
, where : is the number of visits of 0, before some single
independent exponential clock with parameter rings, by
some specified L{\'e}vy process without negative jumps reflected below its
supremum; is the infimum of the path defined as killed at its
last 0 before ; is the Vervaat transform of .
This identity yields an explanation for the geometric distribution of
\cite{K,T} and has numerous other applications. In particular, conditional on
, and also on , the ages and residual lifetimes of
the alive individuals at time are i.i.d.\ and independent of . We
provide explicit formulae for this distribution and give a more general
application to outbreaks of antibiotic-resistant bacteria in the hospital
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