Stochastic Models for Structured Populations

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.

• Birth and death processes
• Branching processes in random environment
• Cell division dynamics
• Continuous state branching processes
• Large population approximations
• Martingale properties
• Measure-valued Markov processes
• Population models
• Two-level models

“It deals mainly with the study of the limiting behavior under different scaling limits of several types of continuous-time individual based models of population dynamics subjected to demographic stochasticity … . The reader interested in the subject finds in this book very interesting results and a very well organized account on the state of the art, which integrates the main contributions of the authors and many other researchers in the (mostly recent) literature.” (Carlos A. Braumann, zbMATH 1333.92004, 2016)

• CMAP UMR 7641, École Polytechnique CNRS, Palaiseau Cedex, France

  Sylvie Meleard

• École Polytechnique CNRS, Palaiseau Cedex, France

  Vincent Bansaye

Vincent Bansaye is Associate Professor at Ecole Polytechnique in France and is a specialist of branching processes, particularly branching processes in random environments.

Sylvie Méléard is Full Professor at Ecole Polytechnique in France and is a specialist of random particle systems and their large number approximations models for physics and biology.

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