Cultural transmission of fitness in mathematical population genetics

Thesis

Cultural transmission of fitness is the observed phenomenon in human populations that the number of a parent's siblings in uences the number of their own offspring. We propose an extension of the classical Wright-Fisher model for constant size haploid populations to allow for setting the magnitude of these `in uences' for each family size separately. This is achieved by assigning a reproductive weight to each individual in the parental generation depending on their family size.

The model is a robust extension in the sense that both the forward-in-time model and its dual coalescent process behave similarly to Wright-Fisher and the latter's own limit dual, the n-coalescent, respectively, provided the weight sequence has a constant tail, that is, individuals with many siblings receive the same reproductive weight. Indeed, in the limit of infinite population size, the coalescent dual of our model converges weakly to a constant scaled n-coalescent.

Second, we present results of numerical simulations of the extended Wright- Fisher model with strictly increasing weight sequences of the form w = (j^α), α ≥ 0. Here we observe a phase transition in family size frequency at the point α = 1; and we report multiple mergers occurring in the model's coalescent dual. We also use the data to approximate the coalescent's natural time scale. Such a rapidly increasing weight sequence as this one seems to lack biological relevance, however.

Third, we study a near-critical multi-type branching process model of the phenomenon. A sequence of rescaled Galton-Watson branching processes weakly converges to a Feller-type diffusion. The drift and the diffusion coeficients are expressed in terms of the reproductive weight sequence.

Fourth, we describe a branching diffusion model for cultural transmission of fitness whose dual process is the Bolthausen-Sznitman U-coalescent demonstrating that the phenomenon can naturally lead to a whole variety of dual processes with different qualitative features.

Authors: Kvasznicza, I

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English