Wednesday, March 22, 2023, 10:00am, 113 Physics

Hwai-Ray Tung (Duke University, Mathematics)

- This dissertation consists for two projects in mathematical biology. The
first is on tumor heterogeneity. Recent work of Sottoriva, Graham, and
collaborators have led to the controversial claim that exponentially
growing tumors have a site frequency spectrum that follows the $1/f$ law
consistent with neutral evolution. This conclusion has been criticized
based on data quality issues, statistical considerations, and simulation
results. Here, we use rigorous mathematical arguments to investigate the
site frequency spectrum in the two-type model of clonal evolution. If the
fitnesses of the two types are $\lambda_0 < \lambda_1$, then the site
frequency spectrum is $c/f^\alpha$ where $\alpha=\lambda_0/\lambda_1$. This
is due to the advantageous mutations that produce the founders of the type
1 population. Mutations within the growing type 0 and type 1 populations
follow the $1/f$ law. Our results show that, in contrast to published
criticisms, neutral evolution in an exponentially growing tumor can be
distinguished from the two-type model using the site frequency spectrum.
The second project is on the lack of coexistence in a three species two seasons resource competition model. Investigating how temporal variation in environment affects species coexistence has been of longstanding interest. The competitive exclusion principle states that $n$ niches can support at most $n$ species, but what constitutes a niche is not always clear. For example, Hutchinson in 1961 drew attention to the diversity of phytoplankton coexisting despite the small number of resources in ocean water. Hutchinson then suggested that this could be explained by a changing environment; times when different species are favored would be considered different niches. In this paper, we examine a model where three species interact with each other solely through the consumption of one resource. The growth per resource rates, death rates, resource rates, and methods of resource consumption vary periodically through time. We give a necessary and sufficient condition for the coexistence of all three species. In particular, this condition rules out coexistence for the mean field limit of a three species two seasons model studied by Chan, Durrett, and Lanchier in 2009.

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