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Title: Invariant manifolds of models from population genetics
Author: Seymenoglu, Belgin
ISNI:       0000 0004 7660 5073
Awarding Body: UCL (University College London)
Current Institution: University College London (University of London)
Date of Award: 2019
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Many models in population genetics feature some form of convergence of the genetic state of the population, typically onto a globally attracting invariant manifold. This allows one to effectively reduce the dynamical system to a problem with fewer dimensions, making it easier to investigate the stability of the steady states in the model, as well as to predict the long-term evolution of the population. Moreover, along this manifold, there is a balance between multiple processes, such as selection and recombination. For some models, restrictive assumptions such as small selection coefficients or additivity of fertilities and mortalities has helped show global contraction of dynamics onto a manifold which is close to the well-known Hardy-Weinberg manifold, and on this `quasiequilibrium' manifold the dynamics can be written in terms of allele frequencies (which is of more practical interest to geneticists than the genotype frequencies). This thesis focuses on proving the existence of an invariant manifold for two continuous-time models in population genetics: one is proposed by Nagylaki and Crow and features fertilities and mortalities (death rates), while the other is the selection-recombination model. Common themes in both proofs include a change of coordinates such that the dynamical system is monotone with respect to a certain cone. As a result, it is possible to construct an equicontinuous sequence of functions which has a convergent subsequence. We show this limiting function is indeed invariant. In fact, for the latter model, we show the manifold is globally attracting by proving the phase volume is contracting. The conditions obtained from the proofs are less restrictive than the use of parameters that are small or additive, hence our work is more widely applicable. For the former model, numerical examples are also provided in which the manifold need not be smooth, convex, unique or globally attracting.
Supervisor: Baigent, S. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available