Letteratura scientifica selezionata sul tema "Fleming-Viot processes"

In 1991 Perkins [7] showed that the normalized critical binary branching process is a time inhomogeneous Fleming-Viot process. In the present paper we extend this result to jump-type branching processes and we show that the normalized jump-type branching processes are in a new class of probability measure-valued processes which will be called ‘jump-type Fleming-Viot processes’. Furthermore we also show that by using these processes it is possible to introduce another new class of measure-valued processes which are obtained by the combination of jump-type branching processes and Fleming-Viot processes.

In 1991 Perkins [7] showed that the normalized critical binary branching process is a time inhomogeneous Fleming-Viot process. In the present paper we extend this result to jump-type branching processes and we show that the normalized jump-type branching processes are in a new class of probability measure-valued processes which will be called ‘jump-type Fleming-Viot processes’. Furthermore we also show that by using these processes it is possible to introduce another new class of measure-valued processes which are obtained by the combination of jump-type branching processes and Fleming-Viot processes.

In this paper, a sample path large deviation for Fleming–Viot processes corresponding to the small time asymptotics is established. The rate function is identified as the energy functional of paths associated to the Bhattacharya metric, the intrinsic metric of Fleming–Viot processes.

Abstract Reversibility of the Fleming-Viot process with mutation, selection, and recombination is well understood. In this paper, we study the reversibility of a system of Fleming-Viot processes that live on a countable number of colonies interacting with each other through migrations between the colonies. It is shown that reversibility fails when both migration and mutation are non-trivial.

We consider a new type of lookdown processes where spatial motion of each individual is influenced by an individual noise and a common noise, which could be regarded as an environment. Then a class of probability measure-valued processes on real line ℝ is constructed. The sample path properties are investigated: the values of this new type process are either purely atomic measures or absolutely continuous measures according to the existence of individual noise. When the process is absolutely continuous with respect to Lebesgue measure, we derive a new stochastic partial differential equation for the density process. At last we show that such processes also arise from normalizing a class of measure-valued branching diffusions in a Brownian medium as the classical result that Dawson–Watanabe superprocesses, conditioned to have total mass one, are Fleming–Viot superprocesses.

The subject of this thesis is the study of certain stochastic models arising in Population Genetics. The study of biological evolution naturally motivates the construction and use of sometimes sophisticated mathematical models. We contribute to the study of the so-called Lambda models. Our work is divided into two parts. In Part I, we study non-spatial models, introduced in 1999. Although there is a very rich literature concerning the description of genetic diversity thanks to the genealogies arising in these models, we obtain new results by considering the dynamics of the full population. We also contribute by presenting the first Bayesian method that allows us to reconstruct the genealogies generated by these models from data. In Part II, we study a recent extension of these models to the spatial setting. In particular, we prove a non trivial result concerning the geographical dispersal of a new mutant under this model.

O processo de Fleming-Viot é um processo de Markov cujo espaço de estado é um conjunto de medidas de propabilidade. As funções-amostras do processo representam as prováveis possibilidades de transformação das freqüencias de tipos genéticos presentes numa população ao longo do tempo. Obtido como solução de um problema de martingala bem posto para um operador linear construído de forma a modelar diversas características importantes no estudo da genética populacional, como mutação, seleção, deriva genética, entre outras, o processo de Fleming-Viot permite, por meio de uma abordagem matemática unificadora, tratar problemas de complexidade variada. No presente trabalho, estudamos s processs de Fleming-Viot com saltos, introduzidos por Hiraba. Interpretamos biologicamente esse saltos como mudanças abruptas que podem ocorrer, num curto espaço de tempo, durante a evolução de uma população de indivíduos, causadas por epidemias, desastres naturais ou outras catástrofes, e que levam a descontinuidades nas frequências dos tipos gênicos. Apresentamos uma forma de incluir um fator de seleção no processo com saltos, através da aplicação de uma transformação de medida do tipo Girsanov. Em seguida, fazemos uma análise do comportamento assintótico do processo utilizando técnicas de dualidade e acoplamento.

This thesis focus on the effect that selection has on the ancestry of a spatially structured population. In the absence of selection, the ancestry of a sample from the population behaves as a system of random walks that coalesce upon meeting. Backwards in time, each ancestral lineage jumps, at the time of its birth, to the location of its parent, and whenever two ancestral lineages have the same parent they jump to the same location and coalesce. Introducing selective forces to the evolution of a population translates into branching when we follow ancestral lineages, a by-product of biased sampling forwards in time. We study populations that evolve according to the Spatial Lambda-Fleming-Viot process with selection. In order to assess whether the picture under selection differs from the neutral case we must consider the timescale dictated by the neutral mutation rate Theta. Thus we look at the rescaled dual process with n=1/Theta. Our goal is to find a non-trivial rescaling limit for the system of branching and coalescing random walks that describe the ancestral process of a population. We show that the strength of selection (relative to the mutation rate) required to do so depends on the dimension; in one and two dimensions selection needs to be stronger in order to leave a detectable trace in the population. The main results in this thesis can be summarised as follows. In dimensions three and higher we take the selection coefficient to be proportional to 1/n, in dimension two we take it to be proportional to log(n)/n and finally, in dimension one we take the selection coefficient to be proportional to 1/sqrt(n). We then proceed to prove that in two and higher dimensions the ancestral process of a sample of the population converges to branching Brownian motion. In one dimension, provided we do not allow ancestral lineages to jump over each other, the ancestral process converges to a subset of the Brownian net. We also provide numerical results that show that the non-crossing restriction in one dimension cannot be lifted without a qualitative change in the behaviour of the process. Finally, through simulations, we study the rate of convergence in the two-dimensional case.

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O processo de Fleming-Viot é um processo de Markov cujo espaço de estado é um conjunto de medidas de propabilidade. As funções-amostras do processo representam as prováveis possibilidades de transformação das freqüencias de tipos genéticos presentes numa população ao longo do tempo. Obtido como solução de um problema de martingala bem posto para um operador linear construído de forma a modelar diversas características importantes no estudo da genética populacional, como mutação, seleção, deriva genética, entre outras, o processo de Fleming-Viot permite, por meio de uma abordagem matemática unificadora, tratar problemas de complexidade variada. No presente trabalho, estudamos s processs de Fleming-Viot com saltos, introduzidos por Hiraba. Interpretamos biologicamente esse saltos como mudanças abruptas que podem ocorrer, num curto espaço de tempo, durante a evolução de uma população de indivíduos, causadas por epidemias, desastres naturais ou outras catástrofes, e que levam a descontinuidades nas frequências dos tipos gênicos. Apresentamos uma forma de incluir um fator de seleção no processo com saltos, através da aplicação de uma transformação de medida do tipo Girsanov. Em seguida, fazemos uma análise do comportamento assintótico do processo utilizando técnicas de dualidade e acoplamento.

La théorie des dynamiques adaptatives est une branche de la biologie de l'évolution qui étudie les liens entre Écologie et Évolution. Les hypothèses biologiques qui définissent son cadre sont celles de mutations rares et petites et de grande population asexuée. Les modèles de dynamiques adaptatives décrivent la population au niveau des individus, lesquels sont caractérisés par leurs phénotypes, et visent à étudier l'influence des mécanismes d'hérédité, de mutation et de sélection sur l'évolution à long terme de la population. Le succès de cette théorie vient notamment de sa capacité à fournir une description de l'évolution à long terme du phénotype dominant dans la population comme solution de "l'Équation Canonique des Dynamiques Adaptatives'' dirigée par un gradient de fitness, où la fitness décrit la possibilité d'invasions mutantes, et est construite à partir de paramètres écologiques. Deux approches mathématiques principales portant sur l'équation canonique ont été développées à ce jour: une approche basée sur des EDP et une approche stochastique. Malgré son succès, l'approche stochastique est critiquée par des biologistes puisqu'elle est basée sur une hypothèse non-réaliste de mutations trop rares. Le but de cette thèse est de corriger cette controverse biologique en proposant des modèles probabilistes plus réalistes. Plus précisément, le but est de s'intéresser mathématiquement, sous une double asymptotique de grande population et de petites mutations, aux conséquences d'une nouvelle hypothèse biologique de mutations fréquentes sur l'équation canonique. Il s'agit de déterminer, à partir d'un modèle stochastique individu-centré, le comportement en temps long du trait phénotypique moyen de la population. La question que l'on se pose se reformule en une analyse asymptotique lent-rapide agissant sur deux échelles de temps éco-évolutives. Une échelle lente correspondant à la dynamique du trait moyen et une rapide correspondant à la dynamique d'évolution de la distribution recentrée et dilatée des traits. Cette analyse asymptotique lent-rapide repose sur des techniques de moyennisation. Cette méthode requiert d'identifier et de caractériser le comportement asymptotique de la composante rapide et que cette dernière possède des propriétés d'ergodicité. Plus précisément, le comportement en temps long de la composante rapide est non-classique et correspond à celui d'une diffusion à valeurs mesures originale qui s'interprète comme un processus de Fleming-Viot recentré que l'on caractérise comme l'unique solution d'un certain problème de martingale. Une partie de ces résultats repose sur une relation de dualité portant sur ce processus non-classique et nécessite des conditions de moments sur les données initiales. Au moyen de techniques de couplage et de la correspondance entre les processus particulaires de Moran et les généalogies de Kingman, on établit que le processus de Fleming-Viot recentré satisfait une propriété d'ergodicité avec résultat de convergence exponentielle en variation totale. La mise en œuvre des méthodes de moyennisation, inspirée par Kurtz, est fondée sur des arguments de compacité-unicité. L'idée consiste à prouver la compacité des lois du couple constitué de la composante lente et de la mesure d'occupation de la composante rapide puis d'établir un problème de martingale pour tous points d'accumulation de la famille des lois de ce couple. La dernière étape consiste à identifier ces points d'accumulation. Cette méthode requiert notamment l'introduction de temps d'arrêt pour contrôler les moments de la composante rapide et de prouver qu'ils tendent vers l'infini à l'aide d'arguments de grandes déviations, de réduire le problème posé initialement sur la droite réelle au cas du tore afin de prouver la compacité, d'identifier la limite de la composante rapide en adaptant un argument basé sur la dualité de Dawson, d'identifier la limite de la composante lente puis de passer du tore à la droite réelle
Adaptive dynamics theory is a branch of evolutionary biology which studies the links between ecology and evolution. The biological assumptions that define its framework are those of rare and small mutations and large asexual populations. Adaptive dynamics models describe the population at the level of individuals, which are characterised by their phenotypes, and aim to study the influence of heredity, mutation and selection mechanisms on the long term evolution of the population. The success of this theory comes in particular from its ability to provide a description of the long term evolution of the dominant phenotype in the population as a solution to the “Canonical Equation of Adaptive Dynamics” driven by a fitness gradient, where fitness describes the possibility of mutant invasions, and is constructed from ecological parameters. Two main mathematical approaches to the canonical equation have been developed so far: an approach based on PDEs and a stochastic approach. Despite its success, the stochastic approach is criticised by biologists as it is based on a non-realistic assumption of too rare mutations. The goal of this thesis is to correct this biological controversy by proposing more realistic probabilistic models. More precisely, the aim is to investigate mathematically, under a double asymptotic of large population and small mutations, the consequences of a new biological assumption of frequent mutations on the canonical equation. The goal is to determine, from a stochastic individual-based model, the long term behaviour of the mean phenotypic trait of the population. The question we ask is reformulated into a slow-fast asymptotic analysis acting on two eco-evolutionary time scales. A slow scale corresponding to the dynamics of the mean trait, and a fast scale corresponding to the evolutionary dynamics of the centred and dilated distribution of traits. This slow-fast asymptotic analysis is based on averaging techniques. This method requires the identification and characterisation of the asymptotic behaviour of the fast component and that the latter has ergodicity properties. More precisely, the long time behaviour of the fast component is non-classical and corresponds to that of an original measure-valued diffusion which is interpreted as a centered Fleming-Viot process that is characterised as the unique solution of a certain martingale problem. Part of these results is based on a duality relation on this non-classical process and requires moment conditions on the initial data. Using coupling techniques and the correspondence between Moran's particle processes and Kingman's genealogies, we establish that the centered Fleming-Viot process satisfies an ergodicity property with exponential convergence result in total variation. The implementation of averaging methods, inspired by Kurtz, is based on compactness-uniqueness arguments. The idea is to prove the compactness of the laws of the couple made up of the slow component and the occupation measure of the fast component and then to establish a martingale problem for all accumulation points of the family of laws of this couple. The last step is to identify these accumulation points. This method requires in particular the introduction of stopping times to control the moments of the fast component and to prove that they tend to infinity using large deviation arguments, to reduce the problem initially posed on the real line to the torus case in order to prove compactness, to identify the limit of the fast component by adapting an argument based on Dawson duality, to identify the limit of the slow component and then to move from the torus to the real line

L'objet de la thèse est d'étudier des processus stochastiques coalescents modélisant la généalogie d'une population échangeable avec immigration. On représente la population par l'ensemble des entiers N:={1,2,. . }. Imaginons que l'on échantillonne n individus dans la population aujourd'hui. On cherche à regrouper ces n individus selon leur ancêtre en remontant dans le temps. En raison de l'immigration, il se peut qu'à partir d'une certaine génération, certains individus n'aient pas d'ancêtre dans la population. Par convention, nous les regrouperons dans un bloc que nous distinguerons en ajoutant l'entier 0. On parle du bloc distingué. Les coalescents distingués échangeables sont des processus à valeurs dans l'espace des partitions de Z_+:={0,1,2,. . . }. A chaque temps t est associée une partition distinguée échangeable, c'est-à-dire une partition dont la loi est invariante sous l'action des permutations laissant 0 en 0. La présence du bloc distingué implique de nouvelles coagulations, inexistantes dans les coalescents classiques. Nous déterminons un critère suffisant (et nécessaire avec conditions) pour qu'un coalescent distingué descende de l'infini. C'est-à-dire qu'immédiatement après 0, le processus n'ait plus qu'un nombre fini de blocs. D'autre part, nous nous intéresserons à une relation de dualité entre ces coalescents et des processus à valeurs dans l'espace des mesures de probabilité, appelés processus de Fleming-Viot généralisés avec immigration. Dans le cas "simple", on parle de M-coalescents et de M-processus de Fleming-Viot. Nous établissons ensuite des liens entre ces derniers et les processus de branchement à temps continu avec immigration.