Some applications of the logarithmic series and the index of diversity to ecological problems

The paper describes the application of a logarithmic series to a number of problems of the division of individuals into species and of species into genera, the range of which is best seen from the table of contents. The series, first suggested by R. A. Fisher in this connexion, is n1, n1/2 x, n1/3 x2, n1/4 x4, etc., where n1 is the number of groups with one unit and x is constant less than unity. Unlike the hyperbolic series, which had previously been considered to apply to some of the cases discussed, the logarithmic series is convergent: both the number of groups (e.g. species) and the number of units (e.g. individuals) can be summed. When several samples are taken from a population containing a number of species it is found that the ratio n1/x is constant and, as it is therefore a characteristic of the population, it has been called the index of diversity. The logarithmic series is found to fit extremely well to a large number of frequency series drawn from insects, birds, butterflies and plants, except that there is a slight tendency for the calculated n1 to be below the observed. It also fits well, sometimes extremely well, to the number of genera with different numbers of species in standard classifications of groups of both animals and plants. The conception of the index of diversity is applied to problems of the number of species of plants on different areas, and to the comparison of floras of different areas with interesting results. A classification is given of the 171 families of dicotyledons according to their index of diversity to stimulate a discussion as to which may be the factors which bring about differences and resemblance in this Index. In general, the families with large numbers of species and of genera have large index of diversity, but there may be a very big range of index in families of approximately the same size.