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Title: Long term behaviour of spatial population models with heterozygous or asymmetric homozygous selection
Author: Gooding, Mitchell
ISNI:       0000 0004 7652 7482
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2018
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We investigate the long term behaviour of two models for the spatial distribution of alleles in a diploid population, one with asymmetric selection in favour of the homozygotes, and one with selection in favour of the heterozygote. We model the population with asymmetric homozygous selection using a version of the spatial Λ-Fleming-Viot process. We identify three regimes. For very small values of the asymmetry, the limiting behaviour of the process is the same as for the case with symmetric selection. This case was studied in Etheridge, Freeman and Penington (2017), and they showed, under certain rescaling and initial conditions, that the hybrid zone, the interface between two homogeneous regions of each homozygote, evolves according to mean curvature flow. However, for larger, but not that much larger, values of asymmetry, we show a new behaviour, that the hybrid zone evolves according to a different type of flow, which we call constant curvature flow. Furthermore, there is a strength of asymmetry for which elements of both types of curvature flow are present in the limit. This suggests that the behaviour found in Etheridge, Freeman and Penington (2017) is more sensitive to perturbations than first thought. We then go on to investigate the fluctuations of this process about its limit. To do this, we specialise to the one-dimensional case. We show that, when time, space and the strength of the asymmetry are appropriately rescaled, the hybrid zone, which is a single point in one dimension, evolves according to a Brownian motion, with drift proportional to the asymmetry. Finally, we turn to the model with heterozygous selection. We restrict ourselves to two dimensions, and investigate this process through its dual, the branching annihilating random walk. We show that, up to an arbitrary time, and with arbitrarily high probability, there exists a branching rate such that we may couple a branching annihilating random walk to a ternary branching Brownian motion. While interesting in its own right, this result lends support to a conjecture of Blath, Etheridge and Meredith (2007), that a branching annihilating random walk in two dimensions has a positive probability of survival for all time for any positive branching rate.
Supervisor: Etheridge, Alison Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available