An asymptote is an asymptote and not found in species–area relationships

Abstract

A comment on M.V. Lomolino (2002) ‘…there are areas too small, and areas too large, to show clear diversity patterns…’ R.H. MacArthur (1972: 191). Journal of Biogeography, 29, 555–557. Species–area relationships (SARs) can be all sorts of shapes, as Lomolino's (2000) title indicates, but are most commonly effectively straight on a log–log plot (Williamson, 1988; Williamson et al., 2001). In his response (Lomolino, 2002) to our comment (Williamson et al., 2001) on his millennium essay (Lomolino, 2000), Lomolino claims again to be putting forward an unappreciated paradigm for the SAR. As he says, his view is not new, but our objection to his view on the form of the upper end of the SAR is that it is empirically not, in general, true. The Lomolino view is that the SAR is sigmoidal. He does not specify the scaling of his axes and in Lomolino & Weiser (2001), he uses both Gleason (spp./log–area) and Arrhenius (log–spp./log–area) plots, so presumably he means that the SAR is sigmoidal (flat at both the right and left hand ends) irrespective of the scaling used. What we say is that, empirically, the SAR hardly ever flattens at the right hand end and has all sorts of shapes from flat to perpendicular at the left hand end. That is, his paradigmatic shape is a scarcity amongst empirical SARs. Although the adjectives are perhaps inappropriate, we regard our comment as ‘the slaying of a beautiful hypothesis by an ugly fact’ (Huxley, 1893–94). For the left hand end we did not ‘assert… that there is no small island effect’. What we actually said was ‘at small scales it may in different cases get steeper or shallower or maintain its slope’. A small island effect may sometimes be seen, sometimes not—not a good basis for a paradigm. To that extent, we can agree with part of the quote from MacArthur in Lomolino's (2002) title. However, we were mainly concerned about the right hand end and the statement in Lomolino (2000) that ‘the species area relationship should asymptotically approach or level off’. Although asymptote is a standard and well-defined concept, Lomolino (2002) now says ‘whether we call it an asymptote or boundary’, he still seems to think some value must be approached in a flattening off way. There is, of course, for any area (in a defined time span), a finite number of species. That defines a point, not an asymptote or a boundary, and that point may theoretically be approached more or less horizontally or more or less vertically or in any diagonal way. In practice, up to continental scales, the highest point is usually, but by no means always, approached diagonally on a log–log plot. Lomolino claims we base our view on three plots. That is not so. They were just examples of SARs with many points over a wide range of areas: ‘All three clearly show that there is no asymptote at the right hand end, nor have we ever seen such an asymptote in other published data at these large scales.’ We presumed that he and most readers of the Journal would be familiar with the numerous SARs in the literature. There are, for instance, more than seventy empirical SAR plots in Williamson (1981), Rosenzweig (1995) and Hubbell (2001), none of which shows a noticeably flat right hand end; nor for that matter do the eleven in Lomolino (2000) and Lomolino & Weiser (2001). Most of these have a considerably smaller range of areas than the three we figured. We have worked with considerably more than those. There are, for instance, 2406 SARs calculated, but naturally not plotted, in Lennon et al. (2001). What we said, and what we mean, is that the SAR does not have an asymptote, to which we should perhaps add that by asymptote we mean asymptote.