Traveling waves, relaxation, and oscillations in a model for biodegradation
AuthorMurray, Regan E.
AdvisorXin, Jack X.
MetadataShow full item record
PublisherThe University of Arizona.
RightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
AbstractIn-situ bioremediation is a promising biotechnology for removing aqueous phase contaminants from groundwater. Utilizing indigenous bacteria to degrade organic contaminants into non-toxic components, bioremediation is relatively inexpensive, fast, and complete. Making predictions about its applicability and success is difficult because of the complexity and variability intrinsic to the subsurface environment. Analytical studies of models, independent of this detailed subsurface data, are essential to finding accurate quantitative results, yet few have been obtained. This dissertation is a collection of three mathematical reports on a one-dimensional model for bioremediation. Using degree theory, the elliptic maximum principle, and comparison theorems, existence of traveling wave solutions to the biodegradation model is proved, a formula for the speed of the traveling concentration front is derived, and bounds on the biomass concentration are obtained. In the second section, the model is shown to reduce to a single equation in the relaxation limit by using properties of systems of hyperbolic conservation laws. In the third section, a formula is found for the parameters at which an unstable traveling wave solution bifurcates to a stable limit cycle (oscillatory solution). These results provide practical information about the structure of concentration fronts for the contaminant, nutrient, and biomass. The fronts travel at speeds that are either constant or time-periodic, depending on the kinetic parameters of the bacteria and the sorption properties of the contaminant. When there is little growth in biomass, many critical properties of the concentrations are derived. For aquifers with low permeability, the model is reduced to a much simpler system, also allowing the derivation of many analytical properties. Though comparisons with experimental data have not yet been done, numerical simulations support these results.
Degree ProgramGraduate College