# Processus de branchement, génétique des populations et généalogies aléatoires

Abstract : We seek to construct genetics of branching populations, based upon continuous-state branching processes, also called CB-processes.

We get stationary versions of branching trees, by modifying them in two fashions: introducing competition-type interactions between individuals, to regulate population size (logistic branching process'); applying various conditionings, in the sens of Doob's h-process: conditioning the associated Lévy process to remain in a finite interval, conditioning CB-processes to non-extinction (Q-process), but also CB-processes with interactions, in their diffusion form.

We study the fixation probability of a mutant. Using diffusion theory, we propose a unifying framework to compare two classical models and the logistic branching model. Then, we endow each individual with a quantitative trait subject to mutations, and by a micro--macro approach, we follow the evolution of the resident trait (canonical diffusion of adaptive dynamics').

We study the genealogy of CB-processes with immigration, among which lies the aforementioned $Q$-process. We construct branching trees (whose width process is not Markovian), called splitting trees, on which can directly be seen both types of genealogies associated with CB-processes, that were discovered by J.-F. Le Gall and collaborators. We also provide a proof of the Lamperti representation of CB-processes as time-changed Lévy processes.

We give a retrospective description of the genealogical structure of CB-processes, and then of splitting trees, as does coalescence theory in modern population genetics.

Collaborations in various fields of population biology are also displayed : classical population genetics, ecology of invasions, conservation biology.
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Cited literature [136 references]

https://tel.archives-ouvertes.fr/tel-00252415
Contributor : Amaury Lambert <>
Submitted on : Tuesday, February 12, 2008 - 5:52:57 PM
Last modification on : Thursday, December 10, 2020 - 12:37:49 PM
Long-term archiving on: : Monday, May 17, 2010 - 10:50:06 PM

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• HAL Id : tel-00252415, version 1

### Citation

Amaury Lambert. Processus de branchement, génétique des populations et généalogies aléatoires. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2007. ⟨tel-00252415⟩

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