PUB - Publikationen an der Universität Bielefeld

The Moran model is a widespread model for a finite population in the field of population genetics. In the first part of the thesis, we consider a Moran model with recombination. Models of recombination take a special role between linear and nonlinear models. Although there is abundant interaction and hence nonlinearity, the deterministic system that describes the frequencies of all possible types may be exactly transformed into a linear one. The underlying linear structure even allows an explicit solution. In certain important special cases (notably, in so-called single-crossover dynamics in continuous time), this solution is surprisingly simple and immediately plausible. Elucidating underlying linear structures in the corresponding stochastic system has only started very recently. In the single-crossover case, Baake and Herms (2006) observed that the expected type frequencies in the finite system follow those in the deterministic model. This and other results now lead to the question whether in the general recombination scheme the dynamics of the expectations may be embedded into a higher but finite dimensional space, such that they are given by a finite system of differential equations. We show that the system of moments closes here after a finite number of steps, without any need for approximations. Surprisingly, this property is lost when resampling is included. In the second part of the thesis, we consider a Moran model with selection and mutation. Starting from this, we trace back the ancestral lines of single individuals. We are interested in the stationary distribution of the corresponding ancestral types. Fearnhead and Taylor already showed two approaches to this problem. Fearnhead's (2002) is based on the ancestral selection graph (Krone/Neuhauser 1997), Taylor's (2007) relies on a description of the full population backward in time by means of a diffusion equation. Kluth (2010) derived a recursion that describes this stationary distribution directly from the model. For the diffusion limit, this leads to the same equation as Taylor's approach. We can solve this recursion explicitly and may solve Taylor's equation directly from this recursion and thus link his solution to the Moran model and its graphical representation. This leads to a better understanding of the model and allows us to propose a new particle model.