Arrhenius and Gleason revisited: new hybrid models resolve an old controversy

Aim Studies have typically employed speciesarea relationships (SARs) from sample areas to fit either the power relationship or the logarithmic (exponential) relationship. However, the plots from empirical data often fall between these models. This article proposes two complementary and hybrid models as solutions to the controversy regarding which model best fits sample-area SARs. Methods The two models are S-A = (c(1) + b logA)(dA/A+n), (c(2)A(z))1-(dA/A+n) and S-A = (c(1) + b logA)(1-dA/A+n).(c(2)A(z))(dA/A+n), where SA is number of species in an area, A, where z, b, c1 and c2 are predetermined parameters found by calculation, and where d and n are parameters to be fitted. The number of parameters is reduced from six to two by fixing the model at either end of the scale window of the data set, a step that is justified by the condition that the error or the bias, or both, in the first and the last data points is negligible. The new hybrid models as well as the power model and the logarithmic model are fitted to 10 data sets. Results The two proposed models fit well not only to Arrhenius' and Gleason's data sets, but also to the other six data sets. They also provide a good fit to data sets that follow a sigmoid (or triphasic) shape in log-log space and to data sets that do not fall between the power model and the logarithmic model. The logtransformation of the dependent variable, S, does not affect the curve fit appreciably, although it enhances the performance of the new models somewhat. Main conclusions Sample-area SARs have previously been shown to be convex upward, convex downward (concave), sigmoid and inverted sigmoid in log-log space. The new hybrid models describe successfully data sets with all these curve shapes, and should therefore produce good fits also to what are termed triphasic SARs.