The talk will begin by providing a short, apparently overlooked, proof of competitive exclusion for a model of n-species competition in the chemostat when species specific death rates are ignored. Then a simplified model of anaerobic digestion will be considered. Anaerobic digestion is a complex naturally occurring process used for waste and wastewater treatment to produce biogas as a renewable source of energy. The socalled Anaerobic Digestion Model No. 1 (ADM1) includes 32 state variables and is not mathematically tractable. Bornh\"{o}ft, Hanke-Rauschenbach, Sundmacher, [Nonlinear Dynamics, 73:535--549, 2013] proposed a simplified model that they claimed captures most of the qualitative dynamics of the ADM1 model, including the possibility of bi-stability and the bifurcation dynamics when substrate concentration is used as the bifurcation parameter. Our analysis shows that not all of the dynamics possible in ADM1 are captured. Our analysis of this model also required studying a chemostat model without ignoring the species specific death rates for a general class of response functions, that includes nonmonotone response functions, that had not been analyzed. I will provide a proof in this case for competition between two populations using a Liapunov function. The talk will conclude with a consideration of the sensitivity of the predictions of the model to stochastic perturbations, and a discussion of the implications of the model predictions for successful operation of the process of anaerobic digestion.