## Introduction

The species–area relationship (SAR) is one of the most fundamental patterns in macroecology and an important tool in assessing species diversity, extinction rates and species hotspots. SARs are at the heart of understanding the distribution of diversity in space because they address the manner in which the total number of species (*S*) increases with area (*A*). The observation that *S* almost always increases with *A* (i.e. grows monotonically) has been dubbed one of the few true laws in ecology (Lawton, 1999).

Preston (1962) had early on distinguished between curves compiled from ‘isolates’ and those compiled from ‘samples’, and both types are commonly encountered in the literature, though this distinction is seldom made explicit. Isolates are areas where some less hospitable surrounding matrix reduces biological interactions with other isolates (of the same type) and where biotic and abiotic factors define their boundaries and sizes. Examples of isolates include forest remnants, ponds, mountaintops and islands. Sample areas, on the other hand, are survey areas where sampling design defines boundaries and sizes. Examples of samples include survey units (e.g. quadrats and latitude–longitude blocks) and political units (e.g. counties and states). This distinction is important because processes producing SARs should influence the two in different ways. Sample area curves and isolate curves should, therefore, have different shapes and consequently require different interpretations. Sample area curves are expected to be constantly decelerating in arithmetic space, whereas isolate curves may become sigmoid and include an inflection point (Tjørve, 2003). Also, sampling design affects curve shape (Hill *et al.*, 1994; Keeley & Fotheringham, 2005; Ulrich & Buszko, 2007).

The purpose of this paper is to review developments since my earlier paper on shapes and functions of species–area curves (Tjørve, 2003) and to discuss the criteria for applying different models. In this short time, several new models have been proposed or applied to species–area data sets, most of which have been extensions (i.e. inclusion of new parameters) or parameterizations of earlier models. In particular, one group of candidate models that has been proposed recently is the group of modified or extended power models (EPMs).

Accordingly, the first aim of this paper is to review these extensions and parameterizations and to discuss their usefulness. Whatever its theoretical and practical limitations may be, the power function remains the most frequently preferred model for fitting curves to species–area data. Yet, the realization of the shortcomings of the power model in describing the growth curve of SARs has inspired a search for extensions and alternative candidate models (Tjørve, 2003). This search has not been unproblematic. This paper thus brings attention to the effects of over-parameterization and the use of location parameters. Also, the use of the same name for alternative parameterizations in the literature further confuses matters. The second aim of the paper is to review and assess models with two predictor variables, i.e. one predictor variable in addition to area. The extra variable is included in the regression model either as a weighting of area or as an additional predictor variable for a multiple regression. By reviewing and assessing models, it should be possible both to identify models that are more likely to be useful candidates and to discuss when different models may be better alternatives than others.