Branching processes with spatial structure in population models

Thesis

We consider three different settings for branching processes with spatial structure which appear in population models.

Firstly, we consider the effect of adding a competitive interaction between nearby individuals in a branching Brownian motion. Each individual has a mass which decays when other individuals are nearby. We study the front location: the location at which the local mass density drops to o(1). We show that there are arbitrarily large times t at which the front location is order of t^(1/3) behind the maximum displacement of a particle from the origin.

Secondly, we study the strength of selection in favour of a particular allele in a spatially structured population required to cause a detectable trace in the patterns of genetic variation observed in the contemporary population. We suppose that the effective local population density is small. We show that whereas in dimensions at least three, selection is barely impeded by the spatial structure, in the most relevant dimension, d=2, selection must be stronger (by a factor of log(1/m) where m is the neutral mutation rate) if we are to have a chance of detecting it.

Finally, we model the behaviour of what are known in population genetics as hybrid zones. These occur when two genetically distinct groups are able to reproduce, but the hybrid offspring have a lower fitness. We prove that on an appropriate time and space scale, the hybrid zone in our model evolves approximately according to mean curvature flow. We also give a probabilistic proof of a (well-known) analogous result for a special case of the Allen-Cahn equation.

In the last two cases, we use the spatial Lambda-Fleming-Viot process to model the population (with different selection mechanisms), and our proofs rely on a duality with a system of branching and coalescing particles.

Authors: Penington, S

• DOI: 10.5287/ora-jnevmo7m5
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford