Two Problems in Mathematical Biology

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This dissertation consists of two projects in mathematical biology. The first project studies tumor heterogeneity through the site frequency spectrum, the expected number of mutations with frequency greater than $f$. 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 deviation from the $1/f$ law 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 still 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 considers whether three species can coexist in a resource competition model with two seasons. 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.






Tung, Hwai-Ray (2023). Two Problems in Mathematical Biology. Dissertation, Duke University. Retrieved from


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