Density compensation, isocline shape and single-level competition models

Abstract

Criticism of the Lotka-Volterra competition model implies that the theory of competition should be based upon more general concepts. It is suggested that the shape of the competitor isoclines can provide this basis. The relationship between the total density of a competitive community and species number depends crucially upon isocline shape. This has immediate relevance to the interpretation of the excess density compensation seen in some island communities, since if isoclines are sufficiently concave (curved towards the origin) then this phenomenon is expected to be the rule rather than the exception. These observations do not depend upon any specific model, but in order to determine the shape of isoclines in natural communities a link must be made between the biological processes of competition and isocline shape. To this end three types of single-level competition model are distinguished (additive, multiplicative and temporal resource models) depending upon how gains from resources interact in determining individual fitness. The models are based upon resource availability functions (RAFs), which are decreasing functions of the level of competition and determine the availability of each resource to each species. Provided that the argument of these functions is always a weighted sum of the number of competitors then in the case of the additive resource model the shape of the RAFs determines directly the shape of the isoclines. For the multiplicative model, the shape of the logarithm of the RAFs adopts this role. Analysis of a special case of the additive resource model suggests that concave isoclines are likely to predominate, and that the degree of concavity is of an order which minimizes the tendency of total numbers to increase with species number. In some circumstances, involving “scramble”-type competition and habitat selection, the expected concavity is sufficient to cause a decrease. In any event the expected occurrence of concave competition isoclines predicts a much higher incidence of excess density compensation (due to carrying capacity differences) than expected from any model having linear isoclines. The effect of shifting from an additive to a multiplicative resource model is to make the existence of purely concave isoclines less probable and to raise the possibility of purely convex isoclines. On the other hand, shifting from an additive resource model to a temporal resource model apparently has no such simple interpretation and specific predictions must await further analysis.