PUB - Publikationen an der Universität Bielefeld

In population genetics, the branch of biology concerned with the evolution of the genetic composition of populations, models relative to ancestral structures have proven to be invaluable tools of research. In particular, in the case of so–called genic selection, the ancestral selection graph (ASG) has provided great insight into the type-frequency evolution of a population, both backwards and forwards in time. The aim of this thesis is to capitalize on these ideas further in order to tackle the case of mutation and frequency–dependent selection, which we represent as an analytic function of the current type–frequency. This endeavor proves immediately satisfactory only under some particular selective regimes, which we denote as *nonlinear dominance*, and represent with two different Moran models, which will turn out to be equivalent. For this type of selection, we are able to the develop a corresponding killed and pruned lookdown ASG, obtaining results about the asymptotic behaviour of the type-frequency of the population at the current time and in the past. In particular, we connect these processes to the Moran model and a relative thereof through (factorial) moment duality. All the results are then extended to the diffusion limit. Outside the regime of nonlinear dominance, we ostensively show that the previously used methods are not effective. Nevertheless, we exploit the graphical intuition provided by the ASG to define a continuous-time Markov chain, the *(monomial) coefficient process*, parallel to the Bernstein coefficient process by Cordero, Hummel and Schertzer [CHS22]. We show the connection of this object with the diffusion limit of the Moran model with polynomial frequency–dependent selection, mutation and large offspring events through a new duality. We show the potential of this approach by obtaining a first result under appropriate parameter conditions, which we expect to be sharpened by forthcoming research.