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The answer to this question is highly relevant to the questions posed in the title of this chapter because these two relative-abundance hypotheses yield profoundly different predictions for the total number of tree species in the Amazon as well as for how many of these species are likely to go extinct. The logseries hypothesis predicts a far larger number of species—and that a far larger fraction of these species are rare to very rare—than does the lognormal hypothesis. This is because Preston’s (1962) canonical lognormal hypothesis postulates a fixed variance or spread in the distribution of log abundance of species irrespective of sample size. The result of this assumption is that the number of octaves of log2 abundance separating the commonest and rarest species does not increase with increasing sample size. Consequently, as the abundance of common species increases in larger samples, so the sample abundance of rare species must also increase in logarithmic proportion. The canonical lognormal hypothesis, in turn, implies that if one takes a large enough sample, as for example, the entire Amazon, the number of absolutely very rare species ought to be extremely small because the total abundance of the most common Amazonian tree species is very large.

In contrast, Fisher’s logseries makes no such fixed-variance assumption, and the variance in log species abundance increases steadily with increasing sample size. This is because extremely rare species not previously encountered are continually discovered as sample sizes increase, even as previously discovered species become ever more common in the larger samples. In the logseries, the expected number of species having abundance n is given by

where α is a fitted diversity parameter, and x is a parameter whose value is close to but less than unity (if x > 1, then the series does not converge). Fisher’s α, as parameter α is now known, has become one of the most widely used measures of species diversity because its value changes only slowly in the face of increasing sample sizes of individuals drawn from communities and sorted into species. Why Fisher’s α should be relatively constant, and the biological significance of both parameters α and x, was not understood until the development of neutral theory.

APPLYING NEUTRAL THEORY: FISHER’S LOGSERIES OR PRESTON’S LOGNORMAL?

How do we estimate Amazonian tree-species richness and extinction risk due to habitat loss? We can begin by using the framework of neutral theory to estimate the total tree diversity in the Amazonian meta-



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