The species±area relationship does not have an asymptote! Mark Williamson 1 *, Kevin J. Gaston 2 and W. M. Lonsdale 3 1 Department of Biology, University of York, YO10 5DD, UK, 2 Biodiversity and Macroecology Group, Department of Animal and Plant Sciences, University of Shef®eld, Shef®eld S10 2TN, UK, and 3 CSIRO Entomology, GPO Box 1700, Canberra, ACT 2601 Australia Abstract Aim To attack a widespread myth. Location World-wide. Methods Simple mathematical logical and empirical examples. Results As both species and area are ®nite and non-negative, the species±area relationship is limited at both ends. The log species±log area relationship is normally effectively linear on scales from about 1 ha to 10 7 km 2 . There are no asymptotes. At the intercontinental scale it may get steeper; at small scales it may in different cases get steeper or shallower or maintain its slope. Main conclusion The species±area relationship does not have an asymptote. Keywords Birds, continents, islands, lumbricids, plants, Species±area relationship. INTRODUCTION In one of a stimulating set of Millennium guest editorials in the Journal of Biogeography, there is a remarkable statement by Lomolino (2000) `The actual form of the (species±area) relationship may differ fundamentally from that predicted by conventional models'. The two conventional models he has in mind are the Arrhenius [log S  c+z(log A)] and the Gleason [S  k 0 +k 1 (log A)]. It is well known that several surveys have shown that the majority of data ®t the Arrhenius relationship, a minority the Gleason, and some neither (Williamson, 1988). Lomolino's criticism is based on the observation that `Two critical shortcomings of such models are that they lack an asymptote (for the larger ecosystems) and that they ignore the possibility of what has been termed the small island effect'. Here we concentrate on the ®rst of these `shortcomings' but comment also on the second. Lomolino's argument for species±area relationships having an asymptote is `because isolated faunas are ultimately derived from a limited pool of species, the species area relationship should asymptotically approach or level off at that maximum value of richness'. In other words, the number of species is ®nite. But so too is the area; we live on a ®nite planet. In fact, the mathematical function describing the species±area relationship must be limited at both ends. Many people think that mathematical functions of the sort that could be used to describe the species±area relationship inevitably go on to in®nity. This is not so; a function may be as legitimately de®ned between any set of limits as between none. For the species±area relationship neither variable can be negative, and indeed it makes little sense at an area less than the size of an individual organism (which will, of course, in many cases be extremely small). The function is thus limited at the left hand side. It is also limited at the right hand side because of the ®nite number of species living on a ®nite (part of the) earth. With these necessary limits, there is no need whatever for the species± area relationship to have an asymptote and it does not have one, as we will demonstrate. We agree with Lomolino that it is important to be clear what shape species±area relationships take and when, before considering the causes of such shapes. THE SPECIES±AREA RELATIONSHIP AT LARGE SCALES Figures 1±3 are three species±area relationships plotted on a logarithmic scale on both axes. Straight lines on such *Correspondence: Department of Biology, University of York, York, YO10 5 DD, UK. E-mail: mw1@york.ac.uk Journal of Biogeography, 28, 827±830 Ó 2001 Blackwell Science Ltd