Dualities and genealogies in stochastic population models

by Tibor Mach
Doctoral thesis
Date of Examination:2017-12-20
Date of issue:2018-02-26
Advisor:Prof. Dr. Anja Sturm
Referee:Prof. Dr. Anja Sturm
Referee:Dr. Jan M. Swart
Persistent Address: http://dx.doi.org/10.53846/goediss-6734

 

 

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Abstract

English

In the thesis, population processes are studied in two different settings. In Part I, which arose in collaboration with Dr. Jan Swart, a so-called cooperative branching process is considered. We construct this process as an interacting particle system which consists of a population of individuals living on a discrete space who reproduce cooperatively, i.e. in order to produce a new individual it is necessary that two "parents" meet. The individuals also die independently of each other and in some special cases we consider a version where they can also move in the space and coalesce. In the thesis, variants of the process on a number of different graphs are studied, namely the finite complete graph and its so called mean-field limit as the size of the graph grows beyond all bounds, a regular tree of a finite degree and the finite-dimensional lattice. In Part II we consider a population of individuals which evolves according to a so called Moran model and in which every individual consists of a chromosome with a finite number of genes such that one gene has an effect on the fitness of the individual and other so called neutral genes do not. We assume that the population is further affected by mutation and recombination which, roughly speaking, is a phenomenon which causes two chromosomes to split and form new chromosomes out of their parts during reproduction. We then study the genealogy of a sample of these neutral genes in a setting where the population has evolved for a long time and has reached stationarity. This is a generalization of a model introduced by Barton, Etheridge and Sturm in the paper "Coalescence in a random background" in which only a single neutral gene is considered.
Keywords: Markov Processes; Interacting Particle Systems; Cooperative Branching Process; Genealogy of neutral loci; Wright-Fisher Diffussion
 
Statistik