On the Use of the Logistic Equation in Models of Food Chains

Abstract

In food chain models the lowest trophic level is often assumed to grow logistically. Anomalous behaviour of the solution of the logistic equation and problems with the introduction of mortality have recently been reported. As predation on the lowest trophic level is a kind of mortality, one expects problems with these food chain models. In this paper we compare two formulations for the lowest trophic level: the logistic growth formulation and the mass balance formulation with resources modelled explicitly. We examine the effects of both models on the dynamic behaviour of a tri-trophic microbial food chain in a chemostat. For this purpose bifurcation diagrams, which give the existence and stability of the equilibria of the nonlinear dynamic system, are used. It turns out that the dynamic behaviours differ in a rather large region of the control parameter space spanned by the dilution rate and the concentration of the resources in the reservoir. We urge that mass balance equations should be used in modelling food chains in chemostats as well as in ecosystems.