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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
This course concerns the stochastic modeling of population dynamics. In the first part, we focus on monotype populations described by one-dimensional stochastic differential equations with jumps. We consider their scaling limits for large populations and study the long time behavior of the limiting processes. It is achieved, thanks to martingale properties, Poisson measure representations, and stochastic calculus. These tools and results will be used and extended to measure-valued processes in the second part. The latter is dedicated to structured populations, where individuals are characterized by a trait belonging to a continuum.
Vincent Bansaye, Sylvie Méléard

### Chapter 2. Birth and Death Processes

Abstract
The global jump rate for a population with size i ≥ 1 is λ i +μ i . After a random time distributed according to an exponential law with parameter $$\ \lambda _{i} +\mu _{i}$$, the process increases by 1 with probability $$\frac{\lambda _{i}} {\lambda _{i}+\mu _{i}}$$ and decreases by − 1 with probability $$\frac{\mu _{i}} {\lambda _{i}+\mu _{i}}$$. If $$\ \lambda _{i} +\mu _{i} = 0$$, the process is absorbed at i.
Vincent Bansaye, Sylvie Méléard

### Chapter 3. Scaling Limits for Birth and Death Processes

Abstract
If the population is large, so many birth and death events occur that the dynamics becomes difficult to describe individual per individual. Living systems need resources in order to survive and reproduce and the biomass per capita depends on the order of magnitude of these resources.
Vincent Bansaye, Sylvie Méléard

### Chapter 4. Continuous State Branching Processes

Abstract
In this part, we consider a new class of stochastic differential equations for monotype populations, taking into account exceptional events where an individual has a large number of offspring. We generalize the Feller equation (3.​12) obtained in Subsection 3.​2 by adding jumps whose rates are proportional to the population size. The jumps are driven by a Poisson point measure, as already done in Subsection 2.​4 This class of processes satisfies the branching property: the individuals of the underlying population evolve independently. Combining this property with the tools developed in the first part, we describe finely the processes, their long time behavior, and the scaling limits they come from.
Vincent Bansaye, Sylvie Méléard

### Chapter 5. Feller Diffusion with Random Catastrophes

Abstract
We deal now with a new family of branching processes taking into account the effect of the environment on the population dynamics. It may cause random fluctuations of the growth rate [12, 33] or catastrophes which kill a random fraction of the population [7].
Vincent Bansaye, Sylvie Méléard

### Chapter 6. Population Point Measure Processes

Abstract
In the previous sections, the models that we considered described a homogeneous population and could be considered as toy models. A first generalization consists in considering multitype population dynamics. The demographic rates of a subpopulation depend on its own type. The ecological parameters are functions of the different types of the individuals competiting with each other. Indeed, we assume that the type has an influence on the reproduction or survival abilities, but also on the access to resources. Some subpopulations can be more adapted than others to the environment.
Vincent Bansaye, Sylvie Méléard

### Chapter 7. Scaling limits for the individual-based process

Abstract
As in Chapter 2, we consider the case where the system size becomes very large. We scale this size by the integer K and look for approximations of the conveniently renormalized measure-valued population process, when K tends to infinity.
Vincent Bansaye, Sylvie Méléard

### Chapter 8. Splitting Feller Diffusion for Cell Division with Parasite Infection

Abstract
We now deal with a continuous time model for dividing cells which are infected by parasites. We assume that parasites proliferate in the cells and that their lifetimes are much shorter than the cell lifetimes. The quantity of parasites (X t : t ≥ 0) in a cell is modeled by a Feller diffusion (see Chapter 3 and Definition 4.1). The cells divide in continuous time at rate τ(x) which may depend on the quantity of parasites x that they contain. When a cell divides, a random fraction F of the parasites goes in the first daughter cell and a fraction (1 − F) in the second one. More generally, splitting Feller diffusion may model the quantity of some biological content which grows (without resource limitation) in the cells and is shared randomly when the cells divide (for example, proteins, nutriments, energy or extrachromosomal rDNA circles in yeast).
Vincent Bansaye, Sylvie Méléard

### Chapter 9. Markov Processes along Continuous Time Galton-Watson Trees

Abstract
Let us note that an individual may die without descendance when p 0 > 0. Moreover, when X is a Feller diffusion and p 2 = 1, we recover the splitting Feller diffusion of Chapter 8 In the general case, the process X is no longer a branching process and the key property for the long time study of the measure-valued process will be the ergodicity of a well-chosen auxiliary Markov process. A vast literature can be found concerning branching Markov processes and special attention has been payed to Branching Brownian Motion from the pioneering work of Biggins [10] about branching random walks, see, e.g., [28, 60] and the references therein. More recently, non-local branching events (with jumps occurring at the branching times) and superprocesses limits corresponding to small and rapidly branching particles have been considered and we refer, e.g., to the works of Dawson et al. and Dynkin [26].
Vincent Bansaye, Sylvie Méléard

### Backmatter

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