Institute of Information Theory and Automation

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Dualities and genealogies in stochastic population models

Tibor Mach
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Institut für mathematische Stochastik, Georg-August-Universität Göttingen, Goldschmidtstrasse 7, Göttingen
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In this population processes in two different settings are studied. First, we consider a so-called cooperative branching process. We construct this process as an interacting particle system which consists of a population of individuals living on a discrete space who reproduce cooperatively - that is to say that 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. We study the process in some of its variants on a number of different graphs, namely the finite complete graph (and in the mean-field limit obtained from this process), a regular tree of a finite degree and a finite-dimensional lattice. In the second part of the thesis 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 their paper Coalescence in a random background in which only a single neutral gene is considered.
2018-05-03 08:01