Structural instability of the theory of simple competition

Abstract

"It is well known that two microbial populations competing for a single nutrient in a homogeneous environment with time-invariant inputs cannot coexist in a steady-state" (Baltzis & Fredrickson, 1983). It is perhaps less well known that an arbitrarily small deviation from the single nutrient dependence, homogeneity or time-invariance renders this statement invalid. This paper establishes that the theory of simple competition of two species for a single growth-limiting substrate is structurally unstable. Conclusions based on such a theory therefore cannot be expected, a priori, to apply to systems for which there exist even small deviations from simple competition. It then becomes necessary to establish how the conclusions are modified under the introduction of such perturbations. The presence of an additional substrate, for which both species exhibit a weak affinity of O(e{open}), gives rise to new phenomena, but only within an interval of dilution rates whose extent decreases with e{open}. However, within this interval either stable coexistence or conditional exclusion occurs, depending on the relative affinities for the additional substrate. Such states persist for arbitrarily small e{open}, though their observability and significance decreases with e{open}. These conclusions are independent of the particular functional forms used to model the substrate dependence, and characterize an entire class of competitive systems in which specific models based on Michaelis-Menten, Monod, Holling or sigmoidal type kinetics arise as particular instances. Despite the impossibility of ever establishing with limitless accuracy whether an observed system is purely one of simple competition for a single growth-limiting substrate, conclusions based on the theory of simple competition, such as the principle of competitive exclusion, have been universally applied to understand species diversity and community structure. This paper formulates those predictions of the theory of simple competition which survive small perturbations in the structure of the system, enabling conclusions such as the principle of competitive exclusion to be applied in a consistent manner. © 1988 Academic Press Limited.