A Discrete-Time Chemostat Model

Abstract

The chemostat is an important laboratory apparatus used for the continuous culture of microorganisms. In ecology it is often viewed as a model of a simple lake system, of the wastewater treatment process, or of biological waste decomposition. Mathematical models of microbial growth and competition for a limiting substrate in a chemostat have played a central role in population biology. See [334] for a treatment of chemostat models. However, the classical model ignores the size structure of the population and the observation that many microbes roughly double in size before dividing. Size-structured chemostat models formulated by Metz and Diekmann [248] and by Cushing [76] lead to hyperbolic partial differential equations with nonlocal boundary conditions. A conceptually simpler approach to modeling size structure was taken by Gage, Williams and Horton [127], who formulated what is now referred to as a nonlinear matrix model for the evolution, in discrete-time steps, of a finite set of biomass classes. Smith [327] modified this model and showed that competitive exclusion holds for two competing microbial populations. The purpose of the present chapter is to give a thorough mathematical analysis of this model of any number of competing populations.