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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. [ABSTRACT FROM AUTHOR]

Aim: The General Dynamic Model of island biogeography (GDM) explains patterns of species richness on volcanic hotspot archipelagos with respect to island age (T) and area (A), most commonly employing what we here call semi‐log ATT2 and LogATT2 models. These can be considered direct extensions of semi‐log and power law species‐area relationships (SARs), with a hump‐shaped function of T modelling drivers of biodiversity over and above area per se. We compare these and related models and investigate their conceptual and mathematical connections. Location: The Azores, Galapagos, Hawaii, Marquesas, Society Islands, Cape Verde Islands, Canary Islands and Aeolian archipelagos. Taxa: Vascular plants, birds, bryophytes, arthropods and molluscs. Methods: We gathered species richness data from databases and published sources, resulting in 34 datasets for various taxa. We used the Akaike information criterion corrected for small sample size (AICc) to compare the relative performance of models. We analysed the data at archipelago level and across archipelagos by using mixed effect models. We also calculated the relative time of expected peak diversity (Tpeak) from parameter estimates. Results: In almost three‐fourth of the cases the best model overall was one that contained only A or T rather than both predictors. There was greater variance between taxa than between archipelagos. Better dispersed taxa tended to reach peak diversity earlier. The most generally applicable of the models including T were the LogATT2 and LogTT2 models. These have important, previously unrecognized mathematical properties. Main Conclusions: SARs and species–time relationships should always be considered as potentially more parsimonious options. Considering the LogATT2 model as a modification of the power law S = cAz, where a lognormal function in T replaces the constant, c allows the construction of an extended framework for the GDM with incorporation of additional predictors such as measures of environmental heterogeneity and application beyond hotspot archipelagos. [ABSTRACT FROM AUTHOR]

Aim To determine if habitat diversity, as estimated by climatic and topographic variables, can predict patterns of orchid diversity on different islands and archipelagos with similar explanatory power to biogeographical variables, such as area, isolation and age of an island. Location Sixty-three islands on eight archipelagos (Solomon Islands, Vanuatu, New Caledonia, Fiji, Samoa, Tonga, Niue and the Cook Islands) in the south-west Pacific. Methods For each island, we determined the orchid species present, age, area, isolation, and indicators of topographic heterogeneity and climatic variability. We then determined the power of various biogeographical, climatic and topographic variables to predict the number of indigenous and endemic species on archipelagos, and on islands within archipelagos, using generalized linear models ( GLMs) and generalized linear mixed models ( GLMMs) respectively. Results We identified a total of 552 species in 110 genera. Area was the only significant biogeographical variable for predicting patterns of orchid species diversity on archipelagos and islands. However, climatic and topographic predictors of habitat diversity performed similarly well. The range in curvature was the best indicator of species richness from the topographic variables, while the range of temperature was the best climatic predictor. These key variables were often strongly correlated with area. Main conclusions Climatic and topographic variables are useful indicators of habitat diversity. The high explanatory power of area and climatic and topographic predictors, and the strong correlation among these variables, suggests that increasing habitat diversity with increasing area may be the major driver of the species-area relationship. Using climatic and topographic variables as predictors of species richness therefore allows determining the key environmental factors and processes driving species diversity. [ABSTRACT FROM AUTHOR]