UC Irvine

Biological populations are heterogeneous, spatially structured, and responsive to changing environments. These features are especially important in tumors, where treatment alters the relative fitness of susceptible and resistant cells, and in bacterial biofilms, where environmental flow affects cells differently according to their phenotype and local surroundings. This dissertation develops deterministic and spatial stochastic models to investigate how growth, evolution, and environmental change shape population dynamics and control. The first part of the dissertation considers treatment scheduling for melanoma containing treatment-susceptible and treatment-addicted resistant cells. A bilinear ordinary differential equation model is formulated in which treatment suppresses susceptible cells but promotes resistant cells, while treatment withdrawal reverses these effects. An optimal control problem balances cumulative and terminal tumor burden against treatment cost. Pontryagin’s maximum principle is used to derive the switching function and characterize the possible structure of optimal protocols. For biologically relevant initial conditions with a small resistant population, the resulting schedules have a monotone structure: full treatment may be followed by an intermediate singular regimen and ultimately by permanent treatment cessation. Repeated transitions between treatment and no treatment do not arise within this model. A sequential quadratic Hamiltonian method with finite-step convergence is then used to compute the controls numerically. Parameter estimation from published melanoma data provides a biologically informed application of the analytical and computational framework. The dissertation next examines how spatial competition modifies melanoma dynamics. A two-dimensional stochastic agent-based model represents susceptible and resistant cells undergoing division and death during alternating treatment and no-treatment phases. Because division requires neighboring space, suppression of susceptible cells can release spatial constraints and facilitate resistant-cell expansion. Time-series data generated by the model are analyzed using sparse identification of nonlinear dynamics. Effective logistic equations capture important features of aggregate cell growth, providing a connection between individual-based spatial simulations and lower-dimensional models that may be suitable for future control formulations. Finally, the same event-driven spatial stochastic framework is adapted to model producer and non-producer phenotypes in Vibrio cholerae biofilms. Matrix production imposes a reproductive cost but protects producer cells from flow-induced removal, particularly within producer-rich neighborhoods. Density-dependent phenotypic switching couples changes in population size to changes in composition. Simulations show that flow preferentially removes non-producers, reduces population density, and shifts phenotypic conversion toward the adhesive producer state, thereby maintaining producer-rich clusters. An extension to producer-cheater competition demonstrates that non-producing cells persist more effectively when they can benefit from matrix generated by neighboring producers. These studies show how environmental conditions can reverse relative fitness, how spatial interactions generate population-level dynamics not captured by well-mixed models, and how mathematical analysis and computation can connect biological mechanisms to questions of population control.