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In our reply to Scheiner (2003), we argued (Gray et al., 2004) that there were only three and not six types of relationships between plots of the cumulative number of species and area. This is disputed by Scheiner (2004), in an extensive reply. Scheiner (2004) contends that the species accumulation curve is a relatively new term. Actually the species accumulation curve, also known to taxonomists as the collectors’ curve, has a long history in ecology, dating back to the 1920s at least. In such curves, the sampling units do not have to be area and for many studies, such as number of birds observed in a forest, the sampling unit may be time or number of individuals. The key point is that there is a major distinction between species accumulation curves, which measure the rate at which new species are found with increasing sampling effort, and species–area curves, which simply plot the number of species found in areas of different sizes. Thus, as they measure fundamentally different aspects of species richness, to include such types of plots within a family of species–area relationships is incorrect. It is better to define the two types of curves as different species accumulation and species–area. In this context, we disagree with Scheiner's (2004) statement that ‘accumulation curves can be translated into area curves’; they cannot. Species accumulation curves can be divided into two types, those that are plotted according to some geographical pattern and those that are randomised, making up the family of three curves.

In our reply to Scheiner (Gray et al., 2004), we derived a simple model to illustrate why we believe there are only three major types of curve expressing the species accumulation and species–area relationship. Scheiner (2004), in commenting on the model states that ‘because the samples differ in diversity from 10 to 23, they have already assumed there is no effect of area on species diversity’. This statement misses the point as the islands differ in size and the number of species increases with increasing island size. Scheiner then uses heterogeneity as a possible explanation for the model results. In our model, all islands are assumed to be homogeneous so heterogeneity is not needed as an explanation. Furthermore, our Fig. 3 shows that whilst the species area curves are identical for the two patterns of species distribution, the two basic types of species accumulation curves are different (Fig. 4). Thus the three types of curves alone show the key determinants of the differences between the species distribution patterns. In his original paper, Scheiner (2003, Fig. 2) shows only three types of curve (his Type II A and B and Type III A and B curves all have the same shape!).

We therefore continue to disagree with Scheiner's defence of his six types of species–area curves. Scheiner reiterates that he believes it is the sampling design that determines the types of curves. The sampling design is irrelevant as such; it is the randomisation process that leads to the semilogarithmic curve and such randomisation is usually now achieved using multiple computer runs. We have shown mathematically that whatever the underlying sampling scheme, an accumulation curve based on randomisation may be very closely approximated by a semilogarithmic curve (Ugland et al., 2003). [In fact the curve is sigmoid, as the smallest possible sample size is one individual of one species. However, because the smallest units actually sampled are usually much larger and contain many species, the semilogarithmic curve is a sound generalization. Interestingly, if the smallest possible sampling unit is taken into account, the sigmoid shape of the curve will apply to both the species accumulation and the species–area curve.]

We stress again that species accumulation curves can be made either by successive additions of new samples or areas along a gradient (see Fig. 10.2 in Henderson (2003) for another illustrative example) or by randomisation (e.g. Figure 10.3 in Henderson, 2003). In both types of curve, each species is only counted once. In species–area curves, each species may be counted several times, because in each area, all species present are included irrespective of their presence in other areas.

Scheiner fails to recognize that in our model, it is the total species number in all samples from the island that gives the number of species found. It is not the nested sample design that is important, neither is it the number of species within the single samples that matters but the total number of species within all the samples. For simplicity in the model, we made all samples identical and took island size into account by increasing sample numbers as 2, 4, 8, 16, 32. A more sophisticated sampling regime would be to assume that on the smallest island, the first sample could contain species number 2, 5, 7, 9 and 10 while the second sample contained species number 1, 3, 4, 6 and 8. The observed number of species in the two samples is 10 exactly, thus the answer is the same and clearly has nothing to do with the nested sampling design, as Scheiner believes.

We agree with Scheiner that the species-area relationship may have a variety of shapes and from sigmoid to semilogarithmic to log-log. However, the hypothesis that there is a relationship between species number and area is the simplest explanation for patterns of species presence and absence and hypotheses encompassing environmental heterogeneity and other explanations for locally restricted patterns are secondary. We do not understand how the three basic types of curves can be used to infer differences in environmental heterogeneity as Scheiner claims.

In relation to turnover (or beta diversity), we refer Scheiner to Koleff et al.'s (2003) review, which defines beta diversity as follows: ‘the spatial turnover or change in identities of species, is a measure of the difference in species composition either between two or more local assemblages or between local and regional assemblages’. Note that they use the word ‘difference’ and not ‘increase’. Scheiner's suggested measure of beta diversity (the slope of the species-area curve) takes into account only increases in numbers of species with increased sample size and omits species decreases as species drop out. Traditional definitions (and measures) of beta diversity incorporate both species gain and species loss. Of course, Scheiner is at liberty to add to the already large number of measures of beta diversity by redefining the concept but at the risk of adding more confusion.

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