## Effect of Area

The species-area relation, whereby the number of species in a spatial unit increases with that unit's area, well predates the MacArthur and Wilson theory of island biogeography, having been documented for about 150 years. Two general kinds of models for this relation have been proposed. The first has number of species predicted from an assumed species-abundance distribution and the total number of individuals of all species combined (assumed proportional to area). The second develops species-area relations from MacArthur and Wilson's species-equilibrium approach.

May's paper coalesces the literature for the first sort of model. Two species-abundance distributions are of particular importance.

The first, a log-series distribution, has been used to describe light-trap data and other collections. It leads exactly to the following species-area relation:

where a is a parameter of the abundance distribution, and where p is the density of individuals. For A sufficiently large, this can be written

Notice that this is an exponential or semilog-linear relation of species to area, that is,

as opposed to the log-log-linear relation that the lognormal implies (below), that is, log 5 = log c + z log A ) 5 = cAz [5]

Note also that eqn [3] is inexact for small A (or 5) and that 5 flattens out as A approaches zero.

Various more or less plausible ways to arrive at a log-series distribution from hypothetical biological processes have been given, perhaps the most common of which is not a biological mechanism but rather a property of the sampling procedure: species-area data in which samples of different areas are taken randomly from some homogeneous large area should have a semilog-linear plot for sample areas sufficiently large.

The second species-abundances distribution, the lognormal, is expected when (1) per-individual population growth rates vary randomly over some substantial period of time or (2) the relative abundances of each species is governed by many factors acting on the per-individual growth rate (and therefore, on the logarithm ofpopulation size) independently of one another. Both follow from the Central Limit Theorem of statistics. For Preston's (the originator of this approach) one-parameter ('canonical') distribution, if we assume that J, the total number of

Interval (years)

Figure 3 Population persistence curves for orb spider on 108 islands of the Bahamas. Top: All species combined. Bottom: Individual species curves (see text for further explanation).

Interval (years)

Figure 3 Population persistence curves for orb spider on 108 islands of the Bahamas. Top: All species combined. Bottom: Individual species curves (see text for further explanation).

individuals in all species combined divided by the number of individuals in the rarest species, is proportional to island area, then species number (S) increases as approximately the 0.26 power of area, that is, S = cA02 , a particular example of eqn [5]. However, the true relationship is not a power function but approaches one with power 0.25 as S gets large. For small S, the relation bends downward and approaches a linear relation of

o c ii 101

Protozoa

Sessile marine organisms

Arthropods

Birds

Lizards

Vascular plants

10-2 10-1 100 Generation time (years)

Figure 4 Relative turnover (eqn [1]) as a function of generation time in six types of organisms. Reproduced from Schoener TW (1983) Rate of species turnover decreases from lower to higher organisms: A review of the data. Oikos 41: 372-377, with permission from Blackwell Publishing Ltd.

species to area (thus being quite different from the curves generated by the log-series distribution just discussed). Note this derivation assumes that something is constant about the shape of the distribution from small to large islands. Preston argues that what is constant is the number of individuals in the rarest species and the density of all individuals combined. It is fairly plausible that total number of individuals increases linearly with area for some well-defined taxon, although evidence bearing on this is not entirely supportive. While one study found that total density of birds increased with total species diversity, other results are more in accord with the assumption. On the other hand, while the assumption that number of individuals in the rarest species is constant is a natural one; given the mathematics of the distribution, it is perhaps less plausible biologically. For the two-parameter distribution, some other feature (e.g., standard deviation) also varies. However, the power of the species-area relation is fairly insensitive through the range of reasonable biological variation in the distribution (Engen has derived a species-area relation from yet a third species-abundance distribution, the broken stick; it is again a power function).

Which description is better, eqn [4] or [5]? Connor and McCoy interpret their review of 100 data sets to say that the two fit about equally. Clear examples of each of the two are given in Figure 5.

The second type of species-area mathematical theory explicitly takes into account the ingredients of the MacArthur-Wilson equilibrium model. The model of Schoener that assumes abundances at equilibrium are complementary (summing over all species to pA, where

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Log-log Semilog Log-log
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