Title:

Selection in a spatially structured population

This thesis focus on the effect that selection has on the ancestry of a spatially structured population. In the absence of selection, the ancestry of a sample from the population behaves as a system of random walks that coalesce upon meeting. Backwards in time, each ancestral lineage jumps, at the time of its birth, to the location of its parent, and whenever two ancestral lineages have the same parent they jump to the same location and coalesce. Introducing selective forces to the evolution of a population translates into branching when we follow ancestral lineages, a byproduct of biased sampling forwards in time. We study populations that evolve according to the Spatial LambdaFlemingViot process with selection. In order to assess whether the picture under selection differs from the neutral case we must consider the timescale dictated by the neutral mutation rate Theta. Thus we look at the rescaled dual process with n=1/Theta. Our goal is to find a nontrivial rescaling limit for the system of branching and coalescing random walks that describe the ancestral process of a population. We show that the strength of selection (relative to the mutation rate) required to do so depends on the dimension; in one and two dimensions selection needs to be stronger in order to leave a detectable trace in the population. The main results in this thesis can be summarised as follows. In dimensions three and higher we take the selection coefficient to be proportional to 1/n, in dimension two we take it to be proportional to log(n)/n and finally, in dimension one we take the selection coefficient to be proportional to 1/sqrt(n). We then proceed to prove that in two and higher dimensions the ancestral process of a sample of the population converges to branching Brownian motion. In one dimension, provided we do not allow ancestral lineages to jump over each other, the ancestral process converges to a subset of the Brownian net. We also provide numerical results that show that the noncrossing restriction in one dimension cannot be lifted without a qualitative change in the behaviour of the process. Finally, through simulations, we study the rate of convergence in the twodimensional case.
