Impulsive Differential Equations with Applications to Self-Cycling Fermentation

Abstract

<p>Self-cycling fermentation is a computer-aided process used for culturing microorganisms. Cells feeding off a growth-limiting nutrient in a tank grow and reproduce until a computer determines the time to end the cycle. At this point, a fraction of the volume of the tank is removed and replaced with an equal volume of fresh nutrient. The remaining cells then consume the fresh medium, grow, and reproduce. The process continues, releasing a fraction of the tank containing only a small amount of nutrient at the end of each cycle. Applications include sewage treatment, toxic waste cleanup, the production of antibiotics, and the examination of cell evolution. A basic model of growth will be formulated in terms of a system of impulsive differential equations. The predictions of the dynamics of the model will be discussed. The model will be refined to better describe the process for nutrient-minimizing applications such as sewage treatment. Criteria to ensure a positive, orbitally attractive periodic orbit with one impulse per period will be given. The predictions of the analysis will be supported by numerical simulations. The model will also be refined to include size-structured populations and a more accurate description of cell reproduction. The predictions of these refined models will be described. Competition of populations will also be considered and criteria for coexistence of more than one population on a single nonreproducing growth limiting nutrient will be given, and supported by numerical simulations.</p>