## Departmental PhD Thesis Exam – Todd Parsons

Everybody welcome.  Refreshments will be served in the Math Lounge before the exam.

Tuesday, July 3, 2012, 2:10 p.m., in BA 6183, 40 St. George Street

PhD Candidate:  Todd Parsons

PhD Advisor:  Peter Abrams (Ecology and Evolutionary Biology) and Joe Repka

Thesis Title:  Asymptotic Analysis of Some Stochastic Models from Population
Dynamics and Population Genetics
http://www.math.toronto.edu/~parsons/pdf/ToddParsons_Thesis.pdf

Thesis Abstract:

Near the beginning of the last century, R. A. Fisher and Sewall Wright devised an elegant, mathematically tractable model of gene reproduction and replacement that laid the foundation for contemporary population genetics. The Wright-Fisher model and its extensions have given biologists powerful tools of statistical inference that enabled the quantification of genetic drift and selection. Given the utility of these tools, we often forget that their model – for reasons of mathematical tractability – makes assumptions that are violated in many real-world populations. In particular, the classical models assume fixed population sizes, held constant by (unspecified) sampling mechanisms.

Here, we consider an alternative framework that merges Moran’s continuous time Markov chain model of allele frequencies in haploid populations of fixed size with the density dependent models of ecological competition of Lotka, Volterra, Gause, and Kolmogorov.  This allows for haploid populations of stochastically varying — but bounded — size. Populations are kept finite by resource limitation. We show the existence of limits that naturally generalize the weak and strong selection regimes of classical population genetics, which allow the calculation of fixation times and probabilities, as well as the long-term stationary allele frequency distribution. Unlike the classical theory, we find that life-history strategy shapes fixation, so that the expected lifetime reproductive success, and the corresponding selection coefficient, s, is no longer sufficient to capture the essential features of genic selection.