I have a Ph.D. I jointly published one paper as an undergraduate. The other two are papers I published as part of my thesis.
- Upper Bound on the Rate of Adaptation in an Asexual Population
- A Hierarchical Probability Model of Colon Cancer
- On Polya's Orchard Problem
ABSTRACT: We consider a model of asexually reproducing individuals. The birth and death rates of the individuals are affected by a fitness parameter. The rate of mutations that cause the fitnesses to change is proportional to the population size, \(N\). The mutations may be either beneficial or deleterious. In a paper by Yu, Etheridge and Cuthbertson [Ann. Appl. Probab. 20 (2010) 978ā1004] it was shown that the average rate at which the mean fitness increases in this model is bounded below by \(log^{1ā\delta} N\) for any \(\delta > 0\). We achieve an upper bound on the average rate at which the mean fitness increases of \(O(log N/(log log N)^2)\).
ABSTRACT: We consider a model of fixed size \(N = 2^l\) in which there are \(l\) generations of daughter cells and a stem cell. In each generation \(i\) there are \(2^{iā1}\) daughter cells. At each integral time unit the cells split so that the stem cell splits into a stem cell and generation \(1\) daughter cell and the generation \(i\) daughter cells become two cells of generation \(i + 1\). The last generation is removed from the population. The stem cell gets first and second mutations at rates \(u_1\) and \(u_2\) and the daughter cells get first and second mutations at rates \(v_1\) and \(v_2\). We find the distribution for the time it takes to get two mutations as \(N\) goes to infinity and the mutation rates go to \(0\). We also find the distribution for the location of the mutations. Several outcomes are possible depending on how fast the rates go to \(0\). The model considered has been proposed by Komarova (2007) as a model for colon cancer.
ABSTRACT: In 1918 Polya formulated the following problem: ``How thick must the trunks of the trees in a regularly spaced circular orchard grow if they are to block completely the view from the center?" (Polya and Szego [2]). We study a more general orchard model, namely any domain that is compact and convex, and find an expression for the minimal radius of the trees. As examples, solutions for rhombus-shaped and circular orchards are given. Finally, we give some estimates for the minimal radius of the trees if we see the orchard as being \(3\)-dimensional.