Log-Series and Log-Normal Parameters as Diversity Discriminants for the Lepidoptera

When H. W. Bates (1863) and A. R. Wallace (1853) first wrote about the diversity of tropical butterflies, they were deeply impressed by the variety in a visual sample. '. . . no description can convey an adequate notion of the beauty and diversity in form and colour of this class of insects. . .' (Bates 1892 edn, p. 351). 'I do not speak of the quantity of individuals . . . but the variety or in other words, the number of species, is very great' (p. 52). Nevertheless, some species were excessively common; 'They assembled in denselypacked masses ... so that the beach looked as though variegated with beds of crocuses''The greater portion of them belonging to the genus Callidryas' (p. 128). But many species were rare; 'An infinite number of curious and rare species may then be taken, most diversified in habits, mode of flight, colours and markings ... .' (p. 52). This property of diversity, recognized by all the great naturalists to distinguish some faunas, or fioras, from others, is undoubtedly real. Nevertheless it has been quantitatively elusive enough to lead some later writers to question, not only its form, but even its existence. There is no doubt in our minds that species diversity, divested of the encumbrance of species names, responds sensitively to both the physical and biological environments. However, to investigate this response requires the formulation of a diversity measure that behaves as intuition demands; the proof of this pudding is entirely in the eating. Nearly a century after Bates' and Wallace's expedition, when R. A. Fisher required a descriptive function for species frequency in the tropical butterfly collections of A. S. Corbet and the temperate moth collections of C. B. Williams, he derived the logarithmic series and suggested that one of its parameters, a, might be useful as a measure of species richness (Fisher, Corbet & Williams 1943). Although Fisher, in his paper, developed the log-series as a consequence of a gamma distribution of species abundance in the population, it is likely that he originally conceived it as a natural extension of the hyperbolic series which had been used previously by Corbet (1941); the idea of an underlying model came later. Dr C. B. Williams shares this view, and is further convinced that Fisher had no conception of a biological model at the level of species interaction. However that may be, Fisher's mathematical intuition enabled him to show that the data could be described by a two parameter model, with one of the natural defining parameters, x, devoted to sample characteristics, and the other, a, characterizing the required population quality in the way the naturalists' intuition sought. Williams saw the possibilities of oc as an index of diversity and later (Williams 1964) explored the wide range of material apparently capable of description by the log-series. However, he did not dig deeply into the behaviour of a in response to environmental difference and change, except in its seasonal cycles.