129 research outputs found
Invasion and adaptive evolution for individual-based spatially structured populations
The interplay between space and evolution is an important issue in population
dynamics, that is in particular crucial in the emergence of polymorphism and
spatial patterns. Recently, biological studies suggest that invasion and
evolution are closely related. Here we model the interplay between space and
evolution starting with an individual-based approach and show the important
role of parameter scalings on clustering and invasion. We consider a stochastic
discrete model with birth, death, competition, mutation and spatial diffusion,
where all the parameters may depend both on the position and on the trait of
individuals. The spatial motion is driven by a reflected diffusion in a bounded
domain. The interaction is modelled as a trait competition between individuals
within a given spatial interaction range. First, we give an algorithmic
construction of the process. Next, we obtain large population approximations,
as weak solutions of nonlinear reaction-diffusion equations with Neumann's
boundary conditions. As the spatial interaction range is fixed, the
nonlinearity is nonlocal. Then, we make the interaction range decrease to zero
and prove the convergence to spatially localized nonlinear reaction-diffusion
equations, with Neumann's boundary conditions. Finally, simulations based on
the microscopic individual-based model are given, illustrating the strong
effects of the spatial interaction range on the emergence of spatial and
phenotypic diversity (clustering and polymorphism) and on the interplay between
invasion and evolution. The simulations focus on the qualitative differences
between local and nonlocal interactions
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)
Uniform estimates for metastable transition times in a coupled bistable system
We consider a coupled bistable N-particle system driven by a Brownian noise,
with a strong coupling corresponding to the synchronised regime. Our aim is to
obtain sharp estimates on the metastable transition times between the two
stable states, both for fixed N and in the limit when N tends to infinity, with
error estimates uniform in N. These estimates are a main step towards a
rigorous understanding of the metastable behavior of infinite dimensional
systems, such as the stochastically perturbed Ginzburg-Landau equation. Our
results are based on the potential theoretic approach to metastability.Comment: 20 page
Speed of coming down from infinity for birth and death processes
We finely describe the speed of "coming down from infinity" for birth and
death processes which eventually become extinct. Under general assumptions on
the birth and death rates, we firstly determine the behavior of the successive
hitting times of large integers. We put in light two different regimes
depending on whether the mean time for the process to go from to is
negligible or not compared to the mean time to reach from infinity. In the
first regime, the coming down from infinity is very fast and the convergence is
weak. In the second regime, the coming down from infinity is gradual and a law
of large numbers and a central limit theorem for the hitting times sequence
hold. By an inversion procedure, we deduce that the process is a.s. equivalent
to a non-increasing function when the time goes to zero. Our results are
illustrated by several examples including applications to population dynamics
and population genetics. The particular case where the death rate varies
regularly is studied in details.Comment: 30 pages. arXiv admin note: text overlap with arXiv:1310.740
A host-parasite multilevel interacting process and continuous approximations
We are interested in modeling some two-level population dynamics, resulting
from the interplay of ecological interactions and phenotypic variation of
individuals (or hosts) and the evolution of cells (or parasites) of two types
living in these individuals. The ecological parameters of the individual
dynamics depend on the number of cells of each type contained by the individual
and the cell dynamics depends on the trait of the invaded individual. Our
models are rooted in the microscopic description of a random (discrete)
population of individuals characterized by one or several adaptive traits and
cells characterized by their type. The population is modeled as a stochastic
point process whose generator captures the probabilistic dynamics over
continuous time of birth, mutation and death for individuals and birth and
death for cells. The interaction between individuals (resp. between cells) is
described by a competition between individual traits (resp. between cell
types). We look for tractable large population approximations. By combining
various scalings on population size, birth and death rates and mutation step,
the single microscopic model is shown to lead to contrasting nonlinear
macroscopic limits of different nature: deterministic approximations, in the
form of ordinary, integro- or partial differential equations, or probabilistic
ones, like stochastic partial differential equations or superprocesses. The
study of the long time behavior of these processes seems very hard and we only
develop some simple cases enlightening the difficulties involved
Estimates for the density of a nonlinear Landau process
The aim of this paper is to obtain estimates for the density of the law of a
specific nonlinear diffusion process at any positive bounded time. This process
is issued from kinetic theory and is called Landau process, by analogy with the
associated deterministic Fokker-Planck-Landau equation. It is not Markovian,
its coefficients are not bounded and the diffusion matrix is degenerate.
Nevertheless, the specific form of the diffusion matrix and the nonlinearity
imply the non-degeneracy of the Malliavin matrix and then the existence and
smoothness of the density. In order to obtain a lower bound for the density,
the known results do not apply. However, our approach follows the main idea
consisting in discretizing the interval time and developing a recursive method.
To this aim, we prove and use refined results on conditional Malliavin
calculus. The lower bound implies the positivity of the solution of the Landau
equation, and partially answers to an analytical conjecture. We also obtain an
upper bound for the density, which again leads to an unusual estimate due to
the bad behavior of the coefficients
Quasi-stationary distributions and population processes
This survey concerns the study of quasi-stationary distributions with a
specific focus on models derived from ecology and population dynamics. We are
concerned with the long time behavior of different stochastic population size
processes when 0 is an absorbing point almost surely attained by the process.
The hitting time of this point, namely the extinction time, can be large
compared to the physical time and the population size can fluctuate for large
amount of time before extinction actually occurs. This phenomenon can be
understood by the study of quasi-limiting distributions. In this paper, general
results on quasi-stationarity are given and examples developed in detail. One
shows in particular how this notion is related to the spectral properties of
the semi-group of the process killed at 0. Then we study different stochastic
population models including nonlinear terms modeling the regulation of the
population. These models will take values in countable sets (as birth and death
processes) or in continuous spaces (as logistic Feller diffusion processes or
stochastic Lotka-Volterra processes). In all these situations we study in
detail the quasi-stationarity properties. We also develop an algorithm based on
Fleming-Viot particle systems and show a lot of numerical pictures
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