385 research outputs found
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)
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
Stochastic hybrid system : modelling and verification
Hybrid systems now form a classical computational paradigm unifying discrete and continuous system aspects. The modelling, analysis and verification of these systems are very difficult.
One way to reduce the complexity of hybrid system models is to consider randomization. The need for stochastic models has actually multiple motivations. Usually, when building models complete information is not available and we have to consider stochastic versions. Moreover, non-determinism and uncertainty are inherent to complex systems. The stochastic approach can be thought of as a way of quantifying non-determinism (by assigning a probability to each
possible execution branch) and managing uncertainty. This is built upon to the - now classical - approach in algorithmics that provides polynomial complexity algorithms via randomization.
In this thesis we investigate the stochastic hybrid systems, focused on modelling and analysis.
We propose a powerful unifying paradigm that combines analytical and formal methods. Its
applications vary from air traffic control to communication networks and healthcare systems.
The stochastic hybrid system paradigm has an explosive development. This is because of its
very powerful expressivity and the great variety of possible applications. Each hybrid system model can be randomized in different ways, giving rise to many classes of stochastic hybrid systems.
Moreover, randomization can change profoundly the mathematical properties of discrete and continuous aspects and also can influence their interaction. Beyond the profound foundational and semantics issues, there is the possibility to combine and cross-fertilize techniques from analytic mathematics (like optimization, control, adaptivity, stability, existence and uniqueness of trajectories, sensitivity analysis) and formal methods (like bisimulation, specification, reachability
analysis, model checking). These constitute the major motivations of our research. We
investigate new models of stochastic hybrid systems and their associated problems. The main difference from the existing approaches is that we do not follow one way (based only on continuous or discrete mathematics), but their cross-fertilization. For stochastic hybrid systems we introduce concepts that have been defined only for discrete transition systems. Then, techniques
that have been used in discrete automata now come in a new analytical fashion. This is partly explained by the fact that popular verification methods (like theorem proving) can hardly work even on probabilistic extensions of discrete systems. When the continuous dimension is added, the idea to use continuous mathematics methods for verification purposes comes in a natural
way.
The concrete contribution of this thesis has four major milestones:
1. A new and a very general model for stochastic hybrid systems;
2. Stochastic reachability for stochastic hybrid systems is introduced together with an approximating method to compute reach set probabilities;
3. Bisimulation for stochastic hybrid systems is introduced and relationship with reachability analysis is investigated.
4. Considering the communication issue, we extend the modelling paradigm
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