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Item A biofilm model that avoids a tragedy of the commons(Montana State University - Bozeman, College of Letters & Science, 2021) Dayutis, Seth Aaron; Chairperson, Graduate Committee: Jack D. DockeryThe study of competition between multiple species is of great significance in biology. Competitive behavior is often observed to occur in biofilms and understanding cooperation between multiple species in a single biofilm is the center of much research. The species that grow in biofilms are frequently studied in chemostats, which have a rich history in mathematical modeling. In this thesis, a review of a mathematical chemostat model is presented in which a tragedy of the commons occurs. The chemostat model is then developed into a biofilm model to see if a tragedy occurs in a biofilm under similar conditions. The biofilm and chemostat model consist of two species, a cooperator and a cheater. The cooperator produces an enzyme that combines with a substrate to produce a nutrient. The nutrient is then consumed by the cooperator and cheater. The cooperator is at a disadvantage since it must allocate some of its nutrient uptake towards enzyme production. A one dimensional biofilm model is developed with reaction advection equations governing the behavior of the species and reaction-diffusion equations governing the behavior of the substrate, nutrient ,and enzyme. A set of numerical methods is then outlined on how to solve the system of equations. It is found that a tragedy of the commons is avoided in the biofilm and both species can persist when numerical simulations are run for a finite amount of time. It is then argued that the cooperative behavior exhibited by the two species is a stable equilibrium by approximating the steady state solutions. Further evidence is provided for the existence of a stable equilibrium by perturbing the system and finding that the perturbed system tends back to the equilibrium. Finally, the eigenvalues of the discretized linear system are computed and the results suggest that either the equilibrium is stable or moves away from the equilibrium slowly.Item Stress tensor symmetry preserving model applied to the 2-D viscoelastic flow of a biofilm(Montana State University - Bozeman, College of Letters & Science, 2016) Kanewske, Daniel Bert; Chairperson, Graduate Committee: Tianyu ZhangThe symmetry of the numeric representation of the stress tensor has been shown to be important for maintaining stability, in the sense of Hadamard, of the numeric method. Also, the viscoelastic behavior of biofilms is well documented. A 2D model for the viscoelastic flow of a biofilm using a modified Navier-Stokes equation (NSE) with a novel elastic stress term are presented. The elastic stress is modeled using a numeric stress tensor symmetry preserving scheme that is based on the numeric solution to the Lie derivative and its equivalent counterpart in the form of a symmetric matrix Riccati differential equation (SMRDE). In addition, a coupled advection equation (AE) is applied to the biofilm volume fraction. Solutions to the NSE and AE are found by applying the finite element method (FEM) to the Eulerian-Lagrangian method (ELM). The ELM is solved by first determining the 'characteristic foot' for each Gaussian quadrature point and node point in the mesh. The advection equation is solved using a modified Galerkin Least Squares (GLS) method. Computations are made using the Trilinos iterative sparse matrix solver library called AztexOO which has built in matrix preconditioners and support for parallel processing. The resulting model is used to predict the deformation of a biofilm in a 2D channel. In addition, the accompanying distribution of the pressure and stresses over the evolving velocity field is presented.Item Separating the EPS in a biofilm : models and simulations of movement of the EPS within(Montana State University - Bozeman, College of Letters & Science, 2016) McClanahan, Nathan James; Chairperson, Graduate Committee: Tianyu ZhangIn this dissertation two models are investigated for describing movement of different components within a biofilm. The first model uses a single fluid three component formulation of the biofilm to model the movement within the biofilm in 1D. This is done using a system partial differential equations to model the expansion or contraction of the biofilm in order for the three components to reach an ideal concentration. The model is further refined to include separate velocities for each component as well as using zeroth order kinetics for the growth. In order to solve this system of partial differential equations a finite difference method with an upwind scheme was used to solve the system numerically. The second model is an energy based approach done in both 1D and 2D. An energy, in the case the Flory-Huggins free energy density, is used to describe the interactions of different components within the biofilm. The Cahn-Hilliard equation with the Flory-Huggins free energy density is used to model the separation of the biofilm into two phases. A brief derivation of both the Flory-Huggins equation and the Cahn-Hilliard equation is given using a lattice model and thermodynamic properties. The Flory-Huggins equation is modified slightly for simplicity. A movement energy is also added to the Flory-Huggins equation in order to allow the polymers within the biofilm to move around the domain. In the 1D case the numerical solution was found using finite differences with an upwind scheme similar to the first model. The 2D case is more difficult to solve due to the extra dimension. Due to this the projection method was used to solve part of the system of equations and finite difference using central difference instead of upwind is used to solve the rest.Item Transport of dissolved and particulate material in biofilm-lined tubes and channels(Montana State University - Bozeman, College of Letters & Science, 2015) Jackson, Benjamin David; Chairperson, Graduate Committee: Tianyu Zhang; Isaac Klapper (co-chair)This dissertation develops two models for biofilm-lined channels. The first model seeks to address the rate at which cells move in or out of the flow in a natural hot spring drainage channel. This is done by building a one- and then two-dimensional partial differential equation model of the stream. The model is parameterized using data gathered at Mushroom Spring in Yellowstone National Park in 2011 and 2012. Using this data, we predict erosion and adhesion rates at steady state in upper and lower regions of the stream. The second model describes the utilization of urea by biofilms in an artificial tube flow reactor. The goal of this model is to determine kinetic parameters for ureolytic biofilms. The model is created by deriving two coupled steady state ordinary differential equations, which are parametrized using experimental data. Once the model is fully described, an inverse problem is formulated and solved using a Markov Chain Monte Carlo method. From this model we obtain first order kinetic parameters for a particular strain of E. coli, and discuss results for Michaelis-Menten kinetics. These two model systems are linked by a set of intersecting elements. First, both models concern biofilm-lined channels. Second, in each model these biofilms are found in a streamflow system in which some component transfers from the flow to the biofilm or vice-versa. Third, both systems are represented by low dimensional mathematical models which seek to summarize complex physical behaviors using broad, summarizing parameters. Fourth, in both scenarios the parameters of interest are estimated by combining experimental measurements and mathematical modeling. Finally, error plays an important role in model efficacy. The effects of error are implicit in the first model, but explicitly analyzed in the second.Item Analysis of a one dimensional biofilm model(Montana State University - Bozeman, College of Letters & Science, 2000) Pritchett, Lori AnneItem Biofilm growth in a homogeneous porous medium(Montana State University - Bozeman, College of Letters & Science, 1997) Tiwari, Sunil KumarItem Analysis and control of a biofilm disinfection model(Montana State University - Bozeman, College of Letters & Science, 2006) Szomolay, Barbara; Chairperson, Graduate Committee: Isaac KlapperThe goal of this dissertation is to study a complex biofilm model with a phenotypic structure presented in [34]. The model in [34] is extended - growth and detachment is added, making the new model more interesting in applications. The crucial feature of our model is that cells are able to enter an adapted resistant state when challenged with antimicrobials (adaptation). We study this model in both a qualitative and quantitative manner. Existence and uniqueness of solutions is shown as well as the existence and non-uniqueness of steady-state solutions. Another question of interest is the effective dosing of biocide, i.e. exploring dosing strategies that could minimize the number of living cells or biofilm thickness. Constant and periodic dosing regimes are modeled numerically and studied analytically. One of our main results is that on and off dosing is significantly better than the other dosing types. The model presented in this dissertation contributes to a better understanding of one of the resistant mechanisms in biofilms.