The lognormal distribution is not an appropriate null hypothesis for the species–abundance distribution

Abstract

Summary Of the many models for species–abundance distributions (SADs), the lognormal has been the most popular and has been put forward as an appropriate null model for testing against theoretical SADs. In this paper we explore a number of reasons why the lognormal is not an appropriate null model, or indeed an appropriate model of any sort, for a SAD. We use three empirical examples, based on published data sets, to illustrate features of SADs in general and of the lognormal in particular: the abundance of British breeding birds, the number of trees > 1 cm diameter at breast height (d.b.h.) on a 50 ha Panamanian plot, and the abundance of certain butterflies trapped at Jatun Sacha, Ecuador. The first two are complete enumerations and show left skew under logarithmic transformation, the third is an incomplete enumeration and shows right skew. Fitting SADs by χ2 test is less efficient and less informative than fitting probability plots. The left skewness of complete enumerations seems to arise from a lack of extremely abundant species rather than from a surplus of rare ones. One consequence is that the logit‐normal, which stretches the right‐hand end of the distribution, consistently gives a slightly better fit. The central limit theorem predicts lognormality of abundances within species but not between them, and so is not a basis for the lognormal SAD. Niche breakage and population dynamical models can predict a lognormal SAD but equally can predict many other SADs. The lognormal sits uncomfortably between distributions with infinite variance and the log‐binomial. The latter removes the absurdity of the invisible highly abundant half of the individuals abundance curve predicted by the lognormal SAD. The veil line is a misunderstanding of the sampling properties of the SAD and fitting the Poisson lognormal is not satisfactory. A satisfactory SAD should have a thinner right‐hand tail than the lognormal, as is observed empirically. The SAD for logarithmic abundance cannot be Gaussian.