Sylvie Meleard – författare

In this contribution, several probabilistic tools to study population dynamics are developed. The focus is on scaling limits of qualitatively different stochastic individual based models and the long time behavior of some classes of limiting processes.

Structured population dynamics are modeled by measure-valued processes describing the individual behaviors and taking into account the demographic and mutational parameters, and possible interactions between individuals. Many quantitative parameters appear in these models and several relevant normalizations are considered, leading  to infinite-dimensional deterministic or stochastic large-population approximations. Biologically relevant questions are considered, such as extinction criteria, the effect of large birth events, the impact of  environmental catastrophes, the mutation-selection trade-off, recovery criteria in parasite infections, genealogical properties of a sample ofindividuals.

These notes originated from a lecture series on Structured Population Dynamics at Ecole polytechnique (France).

Vincent Bansaye and Sylvie Méléard are  Professors at Ecole Polytechnique (France).  They are a specialists of branching processes and random particle systems in biology. Most of their research concerns the applications of probability to biodiversity, ecology and evolution.

Le but du livreest de définir et développer une grandegamme d''outils probabilistes pour la modélisation en biologie des populations,afin de  décrire des dynamiquestemporelles de quantités biologiques telles que la taille d''une ou plusieurs populations, la  proportion d''un allèle dans une population oula position d''un individu. En partant de modèles markoviens discrets (marchesaléatoires, processus de Galton-Watson), nous abordons progressivement lecalcul stochastique et les équations différentielles stochastiques, puis lesprocessus markoviens de saut, tels les processus de branchement à temps continuet les processus de naissance et mort. Nous étudions également les processusdiscret et continu pour l''évolution génétique et les généalogies: processus deWright-Fisher et coalescent. Le livredétaille systématiquement les calculs  dequantités d''intérêt pour les biologistes. De nombreux exercices d''application sont proposés.  Le dernier chapitre montre  l''apport de ces outils pour desproblématiques biologiques actuelles. Il développe en détail des travaux derecherche très récents.

This book defines and develops probabilistic tools forthe modeling of populations in order to describe the dynamics of biologicalquantities such as population size, allele proportion in a population and individual location. Fromdiscrete Markovian models (random walks, Galton-Watson processes), it gradually introduces thestochastic 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 deathprocesses. It also discussesthe discrete and continuous processes of genetic evolution, genealogies and the Wright-Fisher processes andcoalescent. The book systematically detailsthe computation of quantities of interest to biologists and provides a number ofexercises.  The last chapter shows the use of probabilistic tools for real-world biologicalproblems, and discusses recentresearch in detail.