Persistence for “Kill the Winner” and Nested Infection Lotka-Volterra Models by Daniel Korytowski A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Approved November 2016 by the Graduate Supervisory Committee: Hal Smith, Chair

In recent decades, marine ecologists have conducted extensive field work and experiments to understand the interactions between bacteria and bacteriophage (phage) in marine environments. This dissertation provides a detailed rigorous framework for gaining deeper insight into these interactions. Specific features of the dissertation include the design of a new deterministic Lotka-Volterra model with n + 1 bacteria, n/n + 1 phage, with explicit nutrient, where the j phage strain infects the first j bacterial strains, a perfectly nested infection network (NIN). This system is subject to trade-off conditions on the life-history traits of both bacteria and phage given in an earlier study Jover et al. (2013). Sufficient conditions are provided to show that a bacteria-phage community of arbitrary size with NIN can arise through the succession of permanent subcommunities, by the successive addition of one new population. Using uniform persistence theory, this entire community is shown to be permanent (uniformly persistent), meaning that all populations ultimately survive. It is shown that a modified version of the original NIN Lotka-Volterra model with implicit nutrient considered by Jover et al. (2013) is permanent. A new one-to-one infection network (OIN) is also considered where each bacterium is infected by only one phage, and that phage infects only that bacterium. This model does not use the trade-offs on phage infection range, and bacterium resistance to phage. The OIN model is shown to be permanent, and using Lyapunov function theory, coupled with LaSalle’s Invariance Principle, the unique coexistence equilibrium associated with the NIN is globally asymptotically stable provided that the interand intra-specific bacterial competition coefficients are equal across all bacteria. Finally, the OIN model is extended to a “Kill the Winner” (KtW) Lotka-Volterra model of marine communities consisting of bacteria, phage, and zooplankton. The zooplankton acts as a super bacteriophage, which infects all bacteria. This model is shown to be permanent. i