The log series
Also called the logarithmic series, the log series is the oldest description of SADs sensu stricto in the ecological literature (Fisher et al., 1943). The log series has been flagged as fitting tropical tree inventory data sets particularly well (Ulrich et al., 2015) – but see below. The neutral model of biodiversity (Hubbell, 2001) was developed in part to justify this belief, and it predicts the log series as a result of steady immigration, continuous point speciation, and exactly balanced birth and death: each individual lost is replaced immediately. The assumption of constant turnover was used in earlier population dynamical models that predicted the log series (e.g., Kendall, 1948; Caswell, 1976).
A particular algorithm that produces the series (Figs. 1A, B) is as follows: (1) the total population size for a community of 10,000 species is fixed at some value N (here, 100 x 10,000); (2) at each time step, all N individuals are removed; (3) and the Nreplacements are drawn from a probability distribution that is a linear function of the preceding population sizes of individual species, so the relative probability of drawing an individual of a species with nindividuals (i.e., its weight) is just n . The prevent permanent loss of species, at the beginning of each step immigration is simulated. Whenever z species have zero counts, z individuals are randomly assigned to increase the values for those species. So ifz is two, a missing species could end up with 0, 1, or 2 individuals.
The basic form of the log series, meaning the probability mass function (PMF), is a proportion p between 0 and 1 raised to an integer series k and then divided by the same series:
P(X = x ) = –1/ln(1 – p )pk /k (1)
where the left-hand term is a scaling constant that causes the sum of probabilities to be 1 and p is a constant fitted by a standard recursive equation (Chatfield, 1969; May, 1975). This form is a special case of the negative binomial, but otherwise unique. It implies semi-log plotted RADs that each start with a rapid drop in the counts of common species and then asymptote quickly on a straight falling line (Figs. 1A, B).