The species–area relationship does not have an asymptote!

Authors

  • Mark Williamson,

    1. Department of Biology, University of York, YO10 5DD, UK
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    • BIOSKETCHES

      Kevin Gaston is a Royal Society University Research Fellow, with research interests in biodiversity, macro-ecology and conservation biology.

      Mark Lonsdale is a plant ecologist who leads the Weeds Program of the CSIRO Division of Entomology.

      Mark Williamson is a population biologist particularly involved with biological invasions and the analysis of distribution data.

  • Kevin J. Gaston,

    1. Biodiversity and Macroecology Group, Department of Animal and Plant Sciences, University of Sheffield, Sheffield S10 2TN, UK
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    • BIOSKETCHES

      Kevin Gaston is a Royal Society University Research Fellow, with research interests in biodiversity, macro-ecology and conservation biology.

      Mark Lonsdale is a plant ecologist who leads the Weeds Program of the CSIRO Division of Entomology.

      Mark Williamson is a population biologist particularly involved with biological invasions and the analysis of distribution data.

  • W. M. Lonsdale

    1. CSIRO Entomology, GPO Box 1700, Canberra, ACT 2601 Australia
    Search for more papers by this author
    • BIOSKETCHES

      Kevin Gaston is a Royal Society University Research Fellow, with research interests in biodiversity, macro-ecology and conservation biology.

      Mark Lonsdale is a plant ecologist who leads the Weeds Program of the CSIRO Division of Entomology.

      Mark Williamson is a population biologist particularly involved with biological invasions and the analysis of distribution data.


Mark Williamson 1 Department of Biology, University of York, York, YO10 5 DD, UK. E-mail: mw1@york.ac.uk

Abstract

Aim

To attack a widespread myth.

Location

World-wide.

Methods

Simple mathematical logical and empirical examples.

Results

As both species and area are finite 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 107 km2. 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.

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=k0+k1(log A)]. It is well known that several surveys have shown that the majority of data fit 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 first 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 finite. But so too is the area; we live on a finite 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 infinity. This is not so; a function may be as legitimately defined 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 finite number of species living on a finite (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 plots fit the standard Arrhenius relationship. It can be seen at a glance that all three are essentially linear throughout. The areas covered are, for vascular plants (Fig. 1), from just under 1 km2 to just over 107 km2. The upper end is almost at the continental scale, the major continents being roughly 107.5 km2, the ice free land area of the world being about 1.3 × 108 and the whole world 5 × 108 km2. For land birds, the islands in Fig. 2 cover a slightly smaller range, from 5 to 106 km2. The largest island is New Guinea, the largest isolated one is Madagascar. Both plots give regressions with slopes around 0.3 (0.27 and 0.33 for all points, similar values for subsets). There is of course a scatter round the regression lines, caused by other factors and stochasticity.

Figure 1.

 The relationship between the number of native species and site area for 104 sites around the world, broken down into island reserves (○), island non-reserves (bsl00084), mainland reserves (●) and mainland non-reserves (bsl00066). The fitted line is log S=1.96+0.27(log A). From Lonsdale (1999).

Figure 2.

 Species–area plot for the land birds on individual island in the warmer seas. (bsl00066) Islands more than 300 km from the next largest land mass, or in the Hawaiian or Galapagos archipelagos. From Williamson (1981) and (1988), from data in Slud (1976).

Figure 3.

 Species–area relationships for Lumbricidae (Annelida, Oligochaeta) in Europe. From Gaston & Spicer (1998) and Gaston (2000), from data in Judas (1988).

Figure 3, for a smaller group of organisms in all senses, lumbricid earthworms, goes to a much smaller size, 10−4 km2 (10−2 ha, 102 m2), and the upper end is <106 km2 (France). It is also much flatter, with a regression slope of only 0.09, for reasons that remain obscure. Using standardized residuals, the top right hand point and all the points with S=1 are deviant. Two points with S=0 have been omitted. As a logarithmic plot can only cope with S=0 by some fudge, it is not surprising that results with S=1 are a bit odd. None of the three plots shows evidence of limitation by the pool of species that Lomolino assumes exists.

Plots covering such a wide range of orders of magnitude and going to almost continental scales are unusual. 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. The species–area relationship does not have an asymptote.

THE RIGHT HAND END

Species numbers for an even wider span of areas are given by Williams (1964; reproduced in Heywood (1995) as Figs 3.2–4) for flowering plants, from 10 cm2 (10−9 km2) to a number of points near 107 km2 and a point for the whole land surface. The species–area plot is remarkably linear over 16 orders of magnitude, considering how heterogeneous the data are, from a wide variety of climates (which accounts for much of the variance; Woodward, 2001), but there is a sharp turn up to the whole land surface. Rosenzweig (1995), on the basis of a single nested set of points (his Fig. 2.15) and a dubious wider set (his Fig. 2.16), says that such a turn up is normal.

Looking at the three figures and remembering that extant globally there may be c. 250,000 species of angiosperm, c. 9500 species of birds (or double that if some cladists have their way) and c. 500 species of lumbricid, it is easy to see that all would need a steep increase at some point to reach the world total. This is the very opposite of Lomolino’s asymptote. But these increases may be spurious. World totals for species are dominated in almost all groups by the number of continental tropical species. In the data sets used here the lumbricids are all temperate, the birds all insular and the plants on the whole non-tropical. Similarly, the data of Williams (1964) include few figures from tropical rain forest.

However, the existence of Wallace’s zoogeographical regions (or any other such scheme for that matter) tells us that the biota of each is largely distinct. When each subarea has a different set from every other subarea, then the slope of the species–area relationship becomes one (Williamson, 1988). So, on classical biogeographical grounds, an increase of slope between continental and world data points is to be expected. There is no reason to suppose that there is an up turn elsewhere.

THE LEFT HAND END

The second shortcoming that Lomolino claimed to see in standard accounts of the species–area relationship is that there is a small island effect, that is, that the relationship flattens out at the left hand side. We agree that it may sometimes be so, but it may continue linearly or get steeper too.

As already noted, the Arrhenius relationship runs into trouble once the areas become small enough for zero values to be expected in some quadrats (but the Gleason relationship is unaffected). Even so, the small island effect would be obvious. It is not apparent in our figures. Indeed Fig. 3, which goes to the smallest areas, suggests if anything a steepening at small areas; with the four samples which each contain just one species all having large standardized residuals (and the two at zero species even larger ones using either log(s+1) or sinh−1(s) for the whole data).

There have been claims from time to time for both a steepening and a flattening at the left hand side. May (see May & Stumpf, 1999) claims that steepening follows from the log-normal species-abundance distribution. But the maths is not exact, for several reasons the log-normal cannot be exactly true (Williamson, 1981), and significant departures from log-normal distributions are commonly observed (Gaston & Blackburn, 2000). For remarkable nested samples from tropical rain forests (Plotkin et al., 1999) show a steepening from 1 ha downwards on 50 ha plots. Crawley & Harral (2001) on the other hand show a flattening below about 0.2 ha. None of these nested or contiguous surveys may be strictly comparable with the independent points shown in Figs 1 and 2, and by Williams (1964) or, by extension, with the small islands that Lomolino (2000) had in mind.

What is certain is that samples below a hectare or so will be much more strongly affected by local habitat heterogeneity, a heterogeneity that normally is averaged out at larger scales. The species–area relationship has long been known to be affected by distance, habitat, climate and history (Williamson, 1988). These effects need to be considered, the studies above reconciled, and the appropriate functional extension of the Arrhenius relationship identified, before we can say that the species–area relationship in areas less than a hectare is understood. But the studies summarized by Dony (1970) (see Williamson, 1988), and the data collected by Williams (1964), show that linear Arrhenius relationships can be found down to 1 m2, and that these can be continuous with the relationship at larger areas.

CONCLUSIONS

Because the species–area relationship is necessarily limited at both ends, there is no need for it to come to an asymptote. Empirically, at scales from about 1 ha to 107 km2 it is typically straight on an Arrhenius plot. Slopes are about half the time between about 0.15 and 0.40 (Williamson, 1988), but are probably usually steeper than this at intercontinental scales (107–108 km2). At scales less than about a hectare even more variable slopes have been reported and the various studies have yet to be compared critically. What is certain is that the species–area relationship does not have an asymptote.

Footnotes

  1. BIOSKETCHESKevin Gaston is a Royal Society University Research Fellow, with research interests in biodiversity, macro-ecology and conservation biology.Mark Lonsdale is a plant ecologist who leads the Weeds Program of the CSIRO Division of Entomology.Mark Williamson is a population biologist particularly involved with biological invasions and the analysis of distribution data.

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