A microscopic probabilistic description of a locally regulated population and macroscopic approximations

We consider a discrete model that describes a locally regulated spatial population with mortality selection. This model was studied in parallel by Bolker and Pacala and Dieckmann, Law and Murrell. We first generalize this model by adding spatial dependence. Then we give a pathwise description in terms of Poisson point measures. We show that different normalizations may lead to different macroscopic approximations of this model. The first approximation is deterministic and gives a rigorous sense to the number density. The second approximation is a superprocess previously studied by Etheridge. Finally, we study in specific cases the long time behavior of the system and of its deterministic approximation.Comment: Published at http://dx.doi.org/10.1214/105051604000000882 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system

International audienceTo describe population dynamics, it is crucial to take into account jointly evolution mechanisms and spatial motion. However, the models which include these both aspects, are not still well-understood. Can we extend the existing results on type structured populations, to models of populations structured by type and space, considering diffusion and nonlocal competition between individuals? We study a nonlocal competitive Lotka-Volterra type system, describing a spatially structured population which can be either monomorphic or dimorphic. Considering spatial diffusion, intrinsic death and birth rates, together with death rates due to intraspecific and interspecific competition between the individuals, leading to some integral terms, we analyze the long time behavior of the solutions. We first prove existence of steady states and next determine the long time limits, depending on the competition rates and the principal eigenvalues of some operators, corresponding somehow to the strength of traits. Numerical computations illustrate that the introduction of a new mutant population can lead to the long time evolution of the spatial niche

Stochastic models for a chemostat and long time behavior

We introduce two stochastic chemostat models consisting in a coupled population-nutrient process reflecting the interaction between the nutrient and the bacterias in the chemostat with finite volume. The nutrient concentration evolves continuously but depending on the population size, while the population size is a birth and death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long time behavior of the bacterial population conditioned to the non-extinction. We prove the global existence of the process and its almost sure extinction. The existence of quasi-stationary distributions is obtained based on a general fixed point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities

Stochastic approximations of the solution of a full Boltzmann equation with small initial data

This paper gives an approximation of the solution of the Boltzmann equation by stochastic interacting particle systems in a case of cut-off collision operator and small initial data. In this case, following the ideas of Mischler and Perthame, we prove the existence and uniqueness of the solution of this equation and also the existence and uniqueness of the solution of the associated nonlinear martingale problem. 
Then, we first delocalize the interaction by considering a mollified Boltzmann equation in which the interaction is averaged on cells of fixed size which cover the space. In this situation, Graham and Méléard have obtained an approximation of the mollified solution by some stochastic interacting particle systems. Then we consider systems in which the size of the cells depends on the size of the system. We show that the associated empirical measures converge in law to a deterministic probability measure whose density flow is the solution of the full Boltzmann equation. That suggests an algorithm based on the Poisson interpretation of the integral term for the simulation of this solution

Modélisation probabiliste et éco-évolution d'une population diploïde

0n s'intéresse à la modélisation probabiliste pour l'évolution génétique de populations diploïdes, dans un contexte d'éco-évolution. La population considérée est modélisée par un processus de naissance et mort multi-types, avec interaction, et dont les taux de naissance modélisent la reproduction mendélienne. En particulier, la taille de la population considérée n'est pas constante et peut être petite. Une première partie du travail est consacrée à l'étude probabiliste du vortex d'extinction démo-génétique, un phénomène au cours duquel la taille d'une petite population décroît de plus en plus rapidement suite à des fixations de plus en plus fréquentes de mutations délétères. Nous donnons notamment une formule pour la probabilité de fixation d'un allèle légèrement délétère en fonction de la composition génétique de la population et nous prouvons l'existence d'un vortex d'extinction sous une hypothèse de mutations rares. Nous donnons par ailleurs des résultats numériques et une analyse biologique détaillée des comportements obtenus. Nous étudions en particulier l'impact du vortex sur la dynamique de la taille moyenne de population, et nous quantifions ce phénomène en fonction des paramètres écologiques. Dans une deuxième partie, sous une asymptotique de grande taille de population et événements de naissance et mort fréquents, nous étudions d'abord la convergence vers une dynamique lente-rapide et le comportement quasi-stationnaire d'une population diploïde caractérisée par sa composition génétique à un locus bi-allélique. Nous étudions en particulier la possibilité de coexistence en temps long de deux allèles dans la population conditionnée à ne pas être éteinte. Ensuite nous généralisons cette dynamique lente-rapide à une population présentant un nombre fini quelconque d'allèles. La population est alors modélisée par un processus à valeurs mesures dont nous prouvons la convergence lorsque le nombre d'allèles tend vers l'infini vers un superprocessus de Fleming-Viot généralisé, avec une taille de population variable et une sélection diploïde additive.We study the random modeling and the genetic evolution of diploid populations, in an eco-evolutionary context. The population is always modeled by a multi-type birth-and-death process with interaction and whose birth rates are designed to model Mendelian reproduction. In particular, the population size is not assumed to be constant, and can be small. In a first part, we provide a probabilistic study of the mutational meltdown, a phenomenon in which the size of a small population decreases more and more rapidly due to more and more frequent fixations of deleterious mutations. We give a formula for the fixation probability of a slightly deleterious allele, as a function of the initial genetic composition of the population and we prove the existence of a mutational meltdown under a rare mutations hypothesis. Besides, we give numerical results and detailed biological interpretations of the observed behaviors. In particular, we study the impact of the mutational meltdown on the mean population size dynamics and we quantify this phenomenon in terms of demographic parameters. In a second part, under an approximation of large population size and frequent birth and death events, we first study the convergence toward a slow-fast stochastic dynamics and the quasi-stationary behavior of a diploid population characterized by its genetic composition at one bi-allelic locus. In particular, we study the possibility of a long time coexistence of the two alleles in the population conditioned on non extinction. Next, we generalize this slow-fast dynamics for a population presenting an arbitrary finite number of alleles. The population is finally modeled by a measure-valued stochastic process that converges when the number of alleles goes to infinity, toward a generalized Fleming-Viot superprocess with randomly evolving population size and diploid additive selection.PALAISEAU-Polytechnique (914772301) / SudocSudocFranceF

A propagation of chaos result for a system of particles with moderate interaction

This paper is concerned with the asymptotic behaviour of a system of particles with moderate interaction. The main result is a propagation of chaos result which generalizes a convergence result of Oelschläger. A trajectorial propagation of chaos result is also given.system of particles moderate interaction propagation of chaos martingale problem

Interprétation probabiliste de l'équation de Landau

Cette thèse porte sur une approche probabiliste de l'équation de Landau, aussi appelée équation de Fokker-Planck-Landau. Cette équation aux dérivées partielles a été obtenue comme limite asymptotique d'équations de Boltzmann lorsque les collisions rasantes deviennent prépondérantes dans un gaz. Elle décrit le comportement de la densité de particules qui ont la même vitesse au même instant (on considère ici le cas spatialement homogène). Cette équation a été jusqu'à maintenant étudiée avec des méthodes d'analyse, ce travail donne une nouvelle approche. La première partie de la thèse est consacrée à l'étude de l'existence de solution de l'équation de Landau pour des gaz dit de 'potentiels modérément mous'. L'existence de mesures de probabilité solutions est obtenue par des outils du calcul stochastique. Pour des gaz plus particuliers, il y a en fait unicité de la solution et grâce au calcul de Malliavin, on en déduit l'existence d'une densité solution de l'équation de Landau. L'approche probabiliste permet d'avoir des conditions initiales assez générales. La seconde partie de la thèse donne une interprétation probabiliste du lien entre les équations de Boltzmann et de Landau. Tout d'abord, les résultats d'existence de solutions, au sens probabiliste, de l'équation de Boltzmann sont étendus aux potentiels 'modéréments mous'. Puis, on montre la convergence de ces solutions vers une solution de l'équation de Landau lorsque les collisions rasantes deviennent prépondérantes dans le gaz. Enfin, dans le cas particulier d'un gaz de Maxwell, la convergence ponctuelle des densités est obtenu en utilisant les techniques du calcul de Malliavin. L'approche probabiliste permet une meilleure compréhension du passage Boltzmann - Landau et permet de le simuler à l'aide d'un système de particules. Quelques simulations sont présentées dans cette thèse.The aim of this PhD thesis is to give a probabilistic interpretation of the Landau equation, also called the Fokker-Planck-Landau equation. This partial derivatives equation has been derived from the Boltzmann equation when the grazing collisions prevail in a gas. It describes the behaviour of the density of particles having the same velocity at the same time (we assume here that the density is spatially homogeneous). This equation has been studied from now with analytic tools, we give here a new approach. In the first part of this thesis, we prove the existence of a probability measure solution of the Landau equation for some gas, called 'moderately soft' potential gas, using some stochastic tools. Moreover, for some particular gas, we prove the uniqueness of the solution, and we deduce the existence of density solution of the Landau equation thanks to the Malliavin Calculus. The probabilistic approach allows general initial data, which can be degenerated as Dirac measures. The second part of this thesis gives a probabilistic interpretation of the convergence of the Boltzmann equation to the Landau equation in the asymptotic of grazing collisions. We first extend the existence results, in a probabilistic sense, of the Boltzmann equation to the case of 'moderatly soft' potential gas. Then, we state the convergence of these solutions to a solution of the Landau equation when grazing collisions prevail. At last, using the Malliavin calculus, we obtain the pointwise convergence of the densities for a Maxwell gas. The probabilistic approach gives a good understanding of the convergence of Boltzmann to Landau and allows to model it with a particle system. Some simulations are given in this thesis.NANTERRE-BU PARIS10 (920502102) / SudocSudocFranceF