Competition in the Unstirred Chemostat with Periodic Input and Washout

The model of an unstirred chemostat is generalized to that of a chemostat with time-dependent input/washout rates. The novelty of the new model is that time periodicity appears in the boundary conditions. The asymptotic dynamics of the competition between two microbial populations is determined in terms of the corresponding period map, which is shown to preserve the standard competitive ordering. It is shown that the dynamics of competition is similar to that of a chemostat with constant boundary conditions. Simple criteria for coexistence versus competitive exclusion are presented. 1. Introduction. The chemostat represents a basic model of an open system in microbial ecology. In its simplest form, it consists of three vessels. The rst, called the feed bottle, contains medium with all of the nutrients needed for growth in sur- plus except one, which hereafter is simply called the nutrient. The contents of the feed bottle are pumped at a constant rate into the second vessel, called the culture vessel or bioreactor. The culture vessel is charged with one or more populations of microorganisms. The contents of the culture vessel are pumped into the remaining vessel, called the over∞ow vessel, at a constant rate, keeping the volume of the reactor constant. The organisms compete for the nutrient in a purely exploitative manner. Basic assumptions include that the vessel is well mixed and that all other parame- ters (pH, temperature, etc.) are strictly controlled. The ∞ow rate is assumed to be sucient to preclude wall growth or the accumulation of metabolic products. Let S(t) denote the concentration of the nutrient in the culture vessel and xi(t), i =1 ; 2, denote the concentration of the competitors. LetS 0 denote the concentration of the input nutrient and let D denote the dilution rate (∞ow rate/volume). If growth is assumed to be proportional to consumption then the basic equations take the form