Commensalism and syntrophy in the chemostat: a unifying graphical approach

The aim of this paper is to show that Tilman's graphical method for the study of competition between two species for two resources can be advantageously used for the study of commensalism or syntrophy models, where a first species produces the substrate necessary for the growth of the second species. The growth functions of the species considered are general and include both inhibition by the other substrate and inhibition by the species' limiting substrate, when it is at a high concentration. Because of their importance in microbial ecology, models of commensalism and syntrophy, with or without self-inhibition, have been the subject of numerous studies in the literature. We obtain a unified presentation of a large number of these results from the literature. The mathematical model considered is a differential system in four dimensions. We give a new result of local stability of the positive equilibrium, which has only been obtained in the literature in the case where the removal rates of the species are identical to the dilution rate and the study of stability can be reduced to that of a system in two dimensions. We describe the operating diagram of the system: this is the bifurcation diagram which gives the asymptotic behavior of the system when the operating parameters are varied, i.e., the dilution rate and the substrate inlet concentrations.