Biological invasions by alien species (in this context, species transported by human actions to areas in which they do not naturally occur) represent one of the primary ways in which people are changing global biodiversity. Native species richness is a positive function of geographic area – the well-known species–area relationship. An interesting question is whether alien species recapitulate this pattern, broadly and/or in detail. Here, we review the increasing body of research exploring alien species richness in the context of geographic area. We assess both the similarities and differences between the species–area relationships of alien and native species groups and how adding alien species to areas affects the overall species–area relationship. Then, we assess how analysis of data for alien species informs our understanding of the mechanisms controlling species–area relationships more generally. We finish with some broad conclusions on the basis of the previous sections.

Hotspot identification is a crucial strategy for setting conservation priorities. Since both the total number of species and the number of endemic species tend to increase with area, prioritizing sites according to their overall species richness or endemic species richness can produce rankings that simply mirror the sizes of the sites. Thus, it is important to control for the dependence of species number on site area. For this reason, some authors have proposed that the species–area relationship (SAR) and/or the endemics–area relationship (EAR) should be modelled and then the sites located above the fitted curve(s) (i.e. those having positive residuals) designated as hotspots. However, (1) there may be large uncertainties about which model provides the best fit to the SARs/EARs, (2) the use of residuals may lead to sites being identified as hotspots when they only have very few species and (3) there is no guarantee that the sites selected as hotspots by the SAR really include a large fraction of the overall diversity. Thus, it is important to evaluate the ability of the hotspots designated by these procedures to really conserve total and endemic species diversity; the best strategy may in fact be to use a combination of approaches.

It is widely acknowledged that we are in the midst of an extinction crisis and habitat loss is generally considered the primary driver. However, providing accurate estimates of extinction rates has proven to be problematic and a range of extinction estimates have been published. Arguably, the most commonly used method for predicting extinctions resulting from habitat loss has been application of the species–area relationship (SAR). The purpose of this chapter is to provide a review of the many ways in which the SAR has been used to predict the number of extinctions resulting from habitat loss. By doing so, we highlight the pitfalls of using the SAR in such a way and discuss how the SAR has been argued to both over-predict and under-predict extinctions. We also provide examples of the myriad ways in which studies have extended and built on standard SAR models and approaches to better model and predict extinctions. We conclude by arguing that there is a need to recognize that any approach based on a single variable (i.e. area), such as the SAR, is unlikely to provide a perfect extinction prediction, regardless of the specific details.

Although the species–area relationship (SAR) is commonly presumed to be either a power law or to follow the logarithmic relationship, a large number of other mathematical expressions have been proposed to describe the relationship. These models can be divided into four general categories, distinguishing between asymptotic and non-asymptotic, and between convex upward and sigmoid models (in arithmetic space). The choice of regression model should not be determined by best fit alone; rather, the choice should relate to the purpose of fitting mathematical models to SAR data: either descriptive, explicative or predictive. Therefore, we should choose models that are likely to result from expected ecological patterns. We argue that neither (accumulative) sample-area SARs (saSARs) nor island SARs (ISARs) have upper asymptotes and ISARs may be sigmoid if the smallest islands (finest scales) are included. Amongst the 30 different models we review here, few are non-asymptotic. Both the power model and logarithmic model return convex non-asymptotic curves, whereas the second persistence (P2) model and the quadratic logarithmic model consistently return sigmoid curves without asymptotes. We add the Tjørve-hybrid to this shortlist, as it can be useful when neither the power nor the logarithmic model provides a good fit to saSAR data.