130 research outputs found
Proliferating parasites in dividing cells : Kimmel's branching model revisited
We consider a branching model introduced by Kimmel for cell division with
parasite infection. Cells contain proliferating parasites which are shared
randomly between the two daughter cells when they divide. We determine the
probability that the organism recovers, meaning that the asymptotic proportion
of contaminated cells vanishes. We study the tree of contaminated cells, give
the asymptotic number of contaminated cells and the asymptotic proportions of
contaminated cells with a given number of parasites. This depends on domains
inherited from the behavior of branching processes in random environment (BPRE)
and given by the bivariate value of the means of parasite offsprings. In one of
these domains, the convergence of proportions holds in probability, the limit
is deterministic and given by the Yaglom quasistationary distribution.
Moreover, we get an interpretation of the limit of the Q-process as the
size-biased quasistationary distribution
On a model for the storage of files on a hardware II : Evolution of a typical data block
We consider a generalized version in continuous time of the parking problem
of Knuth. Files arrive following a Poisson point process and are stored on a
hardware identified with the real line, at the right of their arrival point. We
study here the evolution of the extremities of the data block straddling 0,
which is empty at time 0 and is equal to \RRR at a deterministic time
On a model for the storage of files on a hardware I : Statistics at a fixed time and asymptotics
We consider a generalized version in continuous time of the parking problem
of Knuth. Files arrive following a Poisson point process and are stored on a
hardware identified with the real line. We specify the distribution of the
space of unoccupied locations at a fixed time and give its asymptotics when the
hardware is becoming full.Comment: 25 page
Queuing for an infinite bus line and aging branching process
We study a queueing system with Poisson arrivals on a bus line indexed by
integers. The buses move at constant speed to the right and the time of service
per customer getting on the bus is fixed. The customers arriving at station i
wait for a bus if this latter is less than d\_i stations before, where d\_i is
non-decreasing. We determine the asymptotic behavior of a single bus and when
two buses eventually coalesce almost surely by coupling arguments. Three
regimes appear, two of which leading to a.s. coalescing of the buses.The
approach relies on a connection with aged structured branching processes with
immigration and varying environment. We need to prove a Kesten Stigum type
theorem, i.e. the a.s. convergence of the successive size of the branching
process normalized by its mean. The technics developed combines a spine
approach for multitype branching process in varying environment and geometric
ergodicity along the spine to control the increments of the normalized process
Large deviations for Branching Processes in Random Environment
A branching process in random environment is a
generalization of Galton Watson processes where at each generation the
reproduction law is picked randomly. In this paper we give several results
which belong to the class of {\it large deviations}. By contrast to the
Galton-Watson case, here random environments and the branching process can
conspire to achieve atypical events such as when is
smaller than the typical geometric growth rate and
when . One way to obtain such an atypical rate of growth is to have
a typical realization of the branching process in an atypical sequence of
environments. This gives us a general lower bound for the rate of decrease of
their probability. When each individual leaves at least one offspring in the
next generation almost surely, we compute the exact rate function of these
events and we show that conditionally on the large deviation event, the
trajectory converges to a
deterministic function in probability in the sense of
the uniform norm. The most interesting case is when and we
authorize individuals to have only one offspring in the next generation. In
this situation, conditionally on , the population size stays
fixed at 1 until a time . After time an atypical sequence
of environments let grow with the appropriate rate () to
reach The corresponding map is piecewise linear and is 0 on
and on $[t_c,1].
New approaches of source-sink metapopulations decoupling the roles of demography and dispersal
Source-sink systems are metapopulations of habitat patches with different,
and possibly temporally varying, habitat qualities, which are commonly used in
ecology to study the fate of spatially extended natural populations. We propose
new techniques that allow to disentangle the respective contributions of
demography and dispersal to the dynamics and fate of a single species in a
source-sink metapopulation. Our approach is valid for a general class of
stochastic, individual-based, stepping-stone models, with density-independent
demography and dispersal, provided the metapopulation is finite or else enjoys
some transitivity property. We provide 1) a simple criterion of persistence, by
studying the motion of a single random disperser until it returns to its
initial position; 2) a joint characterization of the long-term growth rate and
of the asymptotic occupancy frequencies of the ancestral lineage of a random
survivor, by using large deviations theory. Both techniques yield formulae
decoupling demography and dispersal, and can be adapted to the case of periodic
or random environments, where habitat qualities are autocorrelated in space and
possibly in time. In this last case, we display examples of coupled
time-averaged sinks allowing survival, as was previously known in the absence
of demographic stochasticity for fully mixing (Jansen and Yoshimura, 1998) and
even partially mixing (Evans et al., 2012; Schreiber, 2010) metapopulations.Comment: arXiv admin note: substantial text overlap with arXiv:1111.253
Some stochastic models for structured populations : scaling limits and long time behavior
The first chapter concerns monotype population models. We first study general
birth and death processes and we give non-explosion and extinction criteria,
moment computations and a pathwise representation. We then show how different
scales may lead to different qualitative approximations, either ODEs or SDEs.
The prototypes of these equations are the logistic (deterministic) equation and
the logistic Feller diffusion process. The convergence in law of the sequence
of processes is proved by tightness-uniqueness argument. In these large
population approximations, the competition between individuals leads to
nonlinear drift terms. We then focus on models without interaction but
including exceptional events due either to demographic stochasticity or to
environmental stochasticity. In the first case, an individual may have a large
number of offspring and we introduce the class of continuous state branching
processes. In the second case, catastrophes may occur and kill a random
fraction of the population and the process enjoys a quenched branching
property. We emphasize on the study of the Laplace transform, which allows us
to classify the long time behavior of these processes. In the second chapter,
we model structured populations by measure-valued stochastic differential
equations. Our approach is based on the individual dynamics. The individuals
are characterized by parameters which have an influence on their survival or
reproduction ability. Some of these parameters can be genetic and are
inheritable except when mutations occur, but they can also be a space location
or a quantity of parasites. The individuals compete for resources or other
environmental constraints. We describe the population by a point measure-valued
Markov process. We study macroscopic approximations of this process depending
on the interplay between different scalings and obtain in the limit either
integro-differential equations or reaction-diffusion equations or nonlinear
super-processes. In each case, we insist on the specific techniques for the
proof of convergence and for the study of the limiting model. The limiting
processes offer different models of mutation-selection dynamics. Then, we study
two-level models motivated by cell division dynamics, where the cell population
is discrete and characterized by a trait, which may be continuous. In 1
particular, we finely study a process for parasite infection and the trait is
the parasite load. The latter grows following a Feller diffusion and is
randomly shared in the two daughter cells when the cell divides. Finally, we
focus on the neutral case when the rate of division of cells is constant but
the trait evolves following a general Markov process and may split in a random
number of cells. The long time behavior of the structured population is then
linked and derived from the behavior a well chosen SDE (monotype population)
Random walk with heavy tail and negative drift conditioned by its minimum and final values
We consider random walks with finite second moment which drifts to
and have heavy tail. We focus on the events when the minimum and the final
value of this walk belong to some compact set. We first specify the associated
probability. Then, conditionally on such an event, we finely describe the
trajectory of the random walk. It yields a decomposition theorem with respect
to a random time giving a big jump whose distribution can be described
explicitly.Comment: arXiv admin note: substantial text overlap with arXiv:1307.396
Lower large deviations for supercritical branching processes in random environment
Branching Processes in Random Environment (BPREs) are the
generalization of Galton-Watson processes where in each generation the
reproduction law is picked randomly in an i.i.d. manner. In the supercritical
regime, the process survives with a positive probability and grows
exponentially on the non-extinction event. We focus on rare events when the
process takes positive values but lower than expected. More precisely, we are
interested in the lower large deviations of , which means the asymptotic
behavior of the probability as
. We provide an expression of the rate of decrease of this
probability, under some moment assumptions, which yields the rate function.
This result generalizes the lower large deviation theorem of Bansaye and
Berestycki (2009) by considering processes where \P(Z\_1=0 \vert
Z\_0=1)\textgreater{}0 and also much weaker moment assumptions.Comment: A mistake in the previous version has been corrected in the
expression of the speed of decrease in the case without
extinctio
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