An extreme-value function model of the species incidence and species-area relations

Ecology, Dec, 1995 by Matthew R. Williams

INTRODUCTION

The relation between the area of a region and the number of resident species has long fascinated ecologists (Arrhenius 1921, 1923a, Gleason 1922, 1925, MacArthur and Wilson 1967). Much of the interest in the relation between species number and area has focused on islands. Islands are particularly suitable for study because for most terrestrial organisms, except some bird species, presence may, in general, be equated with residence. Empirically, the power function model S = C[A.sup.z] has been widely applied as a model of the species-area relation, usually in the linear form:

log S = log C z log A,

where C and z are parameters characteristic of an archipelago, S is the number of resident species, and A the area of the island. Originally proposed by Arrhenius (references in Arrhenius 1921), since its endorsement by Preston (1962) the power function model has dominated studies of the species-area relation (Tokeshi 1993). However, both Arrhenius (1923b) and Preston (1962) specifically referred to the power function model as an "approximation formula" or "first approximation." It is therefore noteworthy that widespread use of the power function model, and the large amounts of empirical data that appear to fit this model, have led to it recently being described by Wissel and Maier (1992) as a "law of island biogeography." This is in contrast to the findings of Connor and McCoy (1979, but see also Sugihara 1981), who concluded that the power function model is merely a computationally convenient method of fitting a curve and has been applied primarily because of its ability to fit observed data, despite some undesirable properties.

In this paper I draw attention to Coleman's (1981) finding that the cumulative extreme-value function (EVF) is an appropriate model of species incidence when individuals are assumed to be randomly distributed in space, and demonstrate that it is appropriate even when this assumption is severely violated. I show that the EVF model of species incidence is superior to the logistic function, which has usually been applied as the model of species incidence, and that the parameters have biologically meaningful interpretations. I also show that the EVF is an alternative model of the species-area relation that is as good as, and in some respects superior to, the power function model.

PROBLEMS WITH THE POWER FUNCTION MODEL

Although now widely used, the power function model has a well-known shortcoming: the power function is unbounded, so that as area increases, S increases without limit. However, it has long been known that species-area curves become asymptotic at large areas (Gleason 1922, Schoener 1976, Connor and McCoy 1979, Buys et al. 1994). Similarly, the equilibrium theory of island biogeography (MacArthur and Wilson 1967) defines S to be the result of a balance between immigration from some source pool, and local extinction; in the absence of in situ speciation, logically S cannot exceed the size of the source pool (Connor and McCoy 1979). For most empirical species-area studies this unboundedness is of little consequence, since few islands are of sizes approaching the area of the mainland or have species numbers approaching that of the source pool. Nonetheless, for studies examining large islands, or where the species of interest require only small areas and the size of island faunas approaches that of the source pool, this is an undesirable property. The power function model could be expected to overestimate S at such large areas. A number of authors (Gilpin and Diamond 1976, Schoener 1976, Connor and McCoy 1979, Martin 1981) have found that within an archipelago, the slope parameter z varies with area, thus confirming the inaccuracy of the power function model at some spatial scales.

Further problems with the power function model that have not received detailed attention relate to the methods employed to estimate the parameters of the relation. Estimation is usually performed using ordinary least squares regression, which makes two implicit assumptions about the nature of S or log S. First, S is presumed to have a normal or lognormal distribution, and hence also be a continuous variable. Second, either S or log S is assumed to have constant variance at all areas, i.e., be homoscedastic. Ordinary least squares regression is inappropriate where these assumptions fail (Myers 1990, Trexler and Travis 1993).

It is instructive to consider the two assumptions of normality and homoscedasticity in turn, to determine whether each is likely to be true for species-area data. Particularly useful in this consideration is an understanding of the behavior of S at very small areas. First, the distribution of S cannot be either normal or log-normal, since at a sufficiently small area S must be zero without variation. Similarly, at somewhat larger areas S could be expected to vary between zero and one. Such a distribution is inconsistent with either the normal or lognormal distributions. Similarly, S is clearly a discrete rather than continuous variable. Second, by again considering small islands it is clear that S is heteroscedastic, since the variance at very small areas is zero and increases with area. While logarithmic transformation of S is likely to reduce this heteroscedasticity, it will not remove it. At least at small areas, then, the two assumptions implicit in ordinary least squares regression of S on A (or log S on log A) are untrue. Wright (1981) has been the only author to discuss the variance of S in the context of the species-area relation. He concluded that the error variance of S is constant at all areas, but had confused the error of measurement of S with the error variance of S. However, Barton and David (1959, and discussed in Pielou 1974) showed that the distribution of S is approximately binomial. In the context of the species-area relation it is only by considering the variance of S at small areas, as above, that it becomes apparent that S has a distribution more consistent with the binomial, than with the normal or lognormal distributions. The binomial distribution of S has not been previously recognized in species-area studies, as most species-area data consist of relatively small numbers of observations inadequate to determine empirically the distribution of S.