Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior

✍ Scribed by Sylvie Meleard, Vincent Bansaye (auth.)

• Publisher: Springer International Publishing
• Year: 2015
• Tongue: English
• Leaves: 111
• Series: Mathematical Biosciences Institute Lecture Series 1.4
• Edition: 1
• Category: Library

✦ Synopsis

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 of individuals.

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.

✦ Table of Contents

Front Matter....Pages i-x
Introduction....Pages 1-3
Front Matter....Pages 5-5
Birth and Death Processes....Pages 7-17
Scaling Limits for Birth and Death Processes....Pages 19-27
Continuous State Branching Processes....Pages 29-38
Feller Diffusion with Random Catastrophes....Pages 39-46
Front Matter....Pages 47-47
Population Point Measure Processes....Pages 49-60
Scaling limits for the individual-based process....Pages 61-77
Splitting Feller Diffusion for Cell Division with Parasite Infection....Pages 79-87
Markov Processes along Continuous Time Galton-Watson Trees....Pages 89-98
Back Matter....Pages 99-107

✦ Subjects

Probability Theory and Stochastic Processes; Genetics and Population Dynamics; Theoretical Ecology/Statistics