PUB - Publikationen an der Universität Bielefeld

Mutation-selection models are traditionally either deterministic or stochastic. Deterministic approaches assume that the size of the population is such that a law of large numbers applies so that random fluctuations may be neglected. The resulting models are (ordinary or partial) differential equations or (discrete-time) dynamical systems, which describe the evolution in the usual forward direction of time. In contrast, stochastic approaches take into account the fluctuations due to random reproduction; the resulting stochastic processes have a firm place in probability theory. Here, the corresponding ancestral processes, which describe the ancestry of a sample of individuals from a population at the present, play an eminent role in the analysis. Deterministic models of population genetics and their stochastic counterparts have largely led separate lives. It is the purpose of this thesis to bring these two areas of research closer together by extending the backward point of view, so far reserved for stochastic models of population genetics, to deterministic mutation-selection equations. The corresponding ancestral processes describe the history of a finite sample of individuals and remain random; although the type-frequency process of the entire population evolves deterministically. Tailored versions of the genealogical processes yield stochastic representations of the solutions of the deterministic equations. The analysis sheds new light on the deterministic dynamic and its long-term behaviour. Special emphasis is placed on the connection between bifurcation phenomena and ancestral structures. The genealogical approach allows the notion of a (random) ancestral type also in the deterministic setting and provides the framework to determine its distribution. We illustrate the underlying ideas by applying them to a special case of frequency-dependent selection. The ancestral processes for such models are largely unexplored territory. We first establish appropriate structures and then make them tractable by applying our aforementioned concepts. The tailored processes allow an explanation of the richer bifurcation structure by genealogical means and lead to expressions for the ancestral type distribution.