Branching processes with selection

Abstract

The main focus of this thesis is a branching particle system with selection, called the N-particle branching random walk (N-BRW), which was first proposed by Brunet and Derrida. The N-BRW is a discrete time stochastic process, which can be viewed as a toy model of an evolving population affected by natural selection. In the N-BRW we have N particles located on the real line at all times. At each time step, each of the N particles has two offspring, which have a random displacement from the location of their parent according to some fixed jump distribution. Then among the 2N offspring particles, only the N rightmost particles survive to form the next generation.

The most interesting questions about the N-BRW concern the following properties. First, the speed at which the particle cloud is moving to the right on the real line; second, the shape of the particle cloud; and finally the genealogy or family tree structure of the population.

The study of the N-BRW and related branching particle systems with selection has been of great interest in recent years. Existing results and conjectures show that the long-term behaviour of the N-BRW heavily depends on the jump distribution.

For the N-BRW with `light-tailed' (roughly means exponentially decaying tails) jump distribution, Brunet and Derrida made conjectures about the behaviours of the speed and shape of the particle cloud, and about the genealogies of the population of particles. These conjectures inspired several mathematical results in this area; for example, Bérard and Gouéré proved the conjecture concerning the speed of the particle cloud.

For the N-BRW with `heavy-tailed' (meaning polynomially decaying tails) jump distribution, Bérard and Maillard described the behaviour of the speed and made predictions about the genealogies and spatial distribution of the population. These results and conjectures all showed substantially different behaviour from those in the case when the jump distribution is `light-tailed'.

The first main result of this thesis proves the conjectures of Bérard and Maillard about the `heavy-tailed' case of the N-BRW. We prove that at a typical large time the genealogy of the population is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale.

Furthermore, motivated by the fact that in the `light-tailed' and `heavy-tailed' cases the N-BRW shows very different behaviour, we studied an intermediate case, where the jump distribution has stretched exponential tails. The second main result of this thesis describes the behaviour of the speed of the particle cloud in the stretched exponential case, filling a gap between the `light-tailed' and `heavy-tailed' regimes.

Our third result is on the genealogy of the N-BRW when the jump distribution has stretched exponential tails. We give a summary on the proof of this result rather than a full proof. We also mention some of the remaining open questions about the genealogies in this case, which we intend to study in the future.

Date of Award	2 Nov 2022
Original language	English
Awarding Institution	University of Bath
Supervisor	Sarah Penington (Supervisor) & Matthew Roberts (Supervisor)