Analytical and Numerical Investigation of Long-Term Behavior of Microbial Flocculation Equations

Flocculation is the reversible combination and separation of suspended particles in a fluid. It is a phenomenon ubiquitous in a wide variety of fields such as meteorology, marine science, astronomy, polymer science, and biotechnology. Flocculation is an efficient liquid-solid separation technique and has a broad range of industrial applications including fermentation, biofuel production, mineral processing, and wastewater treatment. A common mathematical model for the microbial flocculation is a 1D nonlinear partial integro-differential equation, which has been used successfully in matching many flocculation experiments. In this dissertation, we rigorously investigate the long-term behavior of the microbial flocculation equations. When the long-term behavior of biological populations is considered, many populations converge to a stable time-independent state. Towards this end, using results from fixed point theory, we first derive conditions for the existence of continuous, non-trivial stationary solutions. We further apply the principle of linearized stability and semigroup compactness arguments to provide sufficient conditions for local stability of stationary solutions as well as sufficient conditions for instability. Consequently, we develop a numerical framework for computing approximations to stationary solutions of the microbial flocculation equations, which can also be used to produce approximate existence and stability regions for steady states. Furthermore, this numerical framework can be used to numerically investigate stationary solutions of general evolution equations. We develop several efficient and high-precision numerical schemes based on Finite Difference and Spectral Collocation methods to approximate stationary solutions of the microbial flocculation equations. We exploit spectral accuracy of the Spectral Collocation method for the numerical spectral analysis, which in turn allows us to heuristically deduce local stability of numerically computed steady states. We explore the stationary solutions of the model for various biologically relevant parameters and give valuable insights for the efficient removal of suspended particles. Lastly, we investigate the inverse problem of identifying a conditional probability measure in measure-dependent evolution equations arising in size-structured population modeling. We illustrate that a particular form of the microbial flocculation equations is one realization of a system satisfying the hypotheses in our framework