The lecture courses of the CIME Summer School on Probabilistic Models for Nonlinear PDE's and their Numerical Applications (April 1995) had a three-fold emphasis: first, on the weak convergence of stochastic integrals; second, on the probabilistic interpretation and the particle approximation of equations coming from Physics (conservation laws, Boltzmann-like and Navier-Stokes equations); third, on the modelling of networks by interacting particle systems. This book, collecting the notes of these courses, will be useful to probabilists working on stochastic particle methods and on the approximation of SPDEs, in particular, to PhD students and young researchers.

Le but du livre est de definir et developper une grande gamme d'outils probabilistes pour la modelisation en biologie des populations, afin de decrire des dynamiques temporelles de quantites biologiques telles que la taille d'une ou plusieurs populations, la proportion d'un allele dans une population ou la position d'un individu. En partant de modeles markoviens discrets (marches aleatoires, processus de Galton-Watson), nous abordons progressivement le calcul stochastique et les equations differentielles stochastiques, puis les processus markoviens de saut, tels les processus de branchement a temps continu et les processus de naissance et mort. Nous etudions egalement les processus discret et continu pour l'evolution genetique et les genealogies: processus de Wright-Fisher et coalescent. Le livre detaille systematiquement les calculs de quantites d'interet pour les biologistes. De nombreux exercices d'application sont proposes. Le dernier chapitre montre l'apport de ces outils pour des problematiques biologiques actuelles. Il developpe en detail des travaux de recherche tres recents. This book defines and develops probabilistic tools for the modeling of populations in order to describe the dynamics of biological quantities such as population size, allele proportion in a population and individual location. From discrete Markovian models (random walks, Galton-Watson processes), it gradually introduces the stochastic calculus and the stochastic differential equations, as well as the jump Markov processes, such as the branching processes in continuous time and the birth and death processes. It also discusses the discrete and continuous processes of genetic evolution, genealogies and the Wright-Fisher processes and coalescent. The book systematically details the computation of quantities of interest to biologists and provides a number of exercises. The last chapter shows the use of probabilistic tools for real-world biological problems, and discusses recent research in detail.