Generalized Logarithmic Species-Area Relationship resolves the Arrhenius-Gleason debate

The species-area relationship (SAR) is widely applied in ecology. Mathematically, it is usually expressed as either a semi-log or power-law relationship, with the former being introduced by Gleason and the latter by Arrhenius. We here resolve the dispute about which form of the SAR to prefer by introducing a novel model that smoothly transforms between the Gleason semi-log (GSL) and Arrhenius power law (APL) forms. The model introduced has the form of lnq (S) = a + z ln A, with lnq being a generalized logarithmic function, which is a linear map (y = x) for q = 0 and a logarithmic map (y = ln x) for q = 1 and q can take any intermediate value between 0 and 1. We applied this model to 100 datasets (mostly islands), linking species richness to island area. The APL was the preferred model in 68% of head-to-head comparisons with the GSL. Both models were supported in 40% of cases. In just under half (44%) of the cases, an intermediate model best explained the data. The results demonstrate the utility of a simple intermediate SAR model. Visualizing the profile of the range of model fits for all q ∈ [0, 1] (a q chart) allows us to gain extra insight into SARs not yielded by head-to-head comparisons of GSL and APL. The mathematics related to the generalized logarithmic function introduced here should have applications to other areas of ecological modelling.