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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).
- Discrete Monotype Population Models and One-dimensional Stochastic Differential Equations
- Birth and Death Processes
- Scaling Limits for Birth and Death Processes
- Continuous State Branching Processes
- Feller Diffusion with Random Catastrophes
- Structured Populations and Measure-valued Stochastic Differential Equations
- Population Point Measure Processes
- Scaling limits for the individual-based process
- Splitting Feller Diffusion for Cell Division with Parasite Infection
- Markov Processes along Continuous Time Galton-Watson Trees
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.