Doctor of Philosophy (PhD), Ohio University, 2015, Mathematics (Arts and Sciences)

This thesis is devoted to using the techniques of dynamical systems to studying clustering in the cell division cycle of yeast through the medium of a feedback model in which cells in one part of the cell division cycle influence the rate of cells in a different part of the cell division cycle. In Chapter 1, we introduce the model and see that it causes clustering under both positive and negative feedback. In Chapter 2 we pass to a clustered submanifold. We factorize a Poincare map and use this factorization to prove that asymptotically stable cyclic solutions exist under negative feedback. We partition parameter space in a way that allows us to quickly investigate stability of solutions under a wide range of parameter values. In Chapter 3 we prove many of the observations of Chapter 2, in particular that positive feedback gives rise to unstable solutions. We also prove that asymptotic stability in the clustered submanifold implies asymptotic stability in the full phase space. In Chapter 4, we contrast the effect of adding biologically motivated bounded noise against the effect of zero-mean Gaussian white noise, and see that the various noise models act essentially similarly under a variety of metrics. In Chapter 5, we consider a modification of the system wherein certain previously disjoint intervals are allowed to overlap. We observe that in the clustered subspace, the dynamics of this system are similar to those of the original system. However, we observe complicated behavior in the full phase space, where solutions that are asymptotically stable in the clustered space are merely stable, or even unstable, in the full phase space.

Committee: Todd Young Ph.D. (Advisor); Winfried Just Ph.D. (Committee Member); Alexander Neiman Ph.D. (Committee Member); Sergiu Aizicovici Ph.D. (Committee Member) Subjects: Applied Mathematics; Cellular Biology; Mathematics