Bimodality in occurrence classes: a direct consequence of lognormal or logarithmic series distribution of abundances a numerical experimentation

In communities with adequate species richness, Raunkiaer's rule states that species occurrences form a bimodal distribution when plotted in a series of 20% frequency intervals. Specifically, most species occur in the 0-20 and 81-100% frequency intervals, and the minimum is usually in the 61-80% interval. Raunkiaer's (1918) original paper attracted little attention. Du Rietz's (1930) comprehensive work in German, and Raunkiaer's (1934) collected papers eventually published in English led to the use of Raunkiaer's law by vegetation ecologists to assess the homogeneity of communities (McIntosh 1962). Balogh (1953) was among the first zoologists to present "bimodal constancy graphs" of animal populations based on Du Rietz's publication. Recently, papers on the dynamics of metapopulations, in particular those based on Hanski's (1982) "core and satellite species" hypothesis have renewed the debate regarding the measurement and meaning of bimodal frequency distributions (Brown 1984, Gaston and Lawton 1988, Nee et al. 1991). In a study of coleopteran and dipteran assemblages of sheep-run droppings in Hungary, we also found that species frequency distributions were bimodal. In addition, we also found that these frequency distributions corresponded well to the lognormal distribution. Preston (1948) was the first to relate Raunkiaer's rule to the lognormal abundance distribution. He demonstrated the relationship between the probability of the "average expectation" of a given species and the probability of the occurrence in a sample quadrat. According to Preston (1948), a bimodal frequency distribution of occurrences may be a consequence of a lognormal distribution of species abundances. Williams (1950) demonstrated a relationship between bimodality and a logarithmic series model of species abundances. Both works are cited in the literature on species occurrence bimodality. Since that time rather few papers have been published concerning the statistical relationship between bimodality and lognormal and logarithmic abundance distributions (see e.g. Tokeshi 1992). The purpose of our analyses was to use simulation models to test for a relationship between the lognormal and logarithmic series frequency distributions and bimodality. Simulation models can generate more quantitative details on each frequency class of the bimodal distribution and their variances. Furthermore, by using model distributions with different parameters we can detect systematic changes in the occurrences of each frequency class. We argue that our simulations cast doubts on the relevance of the formal statistical approach to bimodality. We conducted simulations and numerical experimentation based on the truncated lognormal and logarithmic series distributions. The simulations are based on an urn model. As a first step n1, . , n5Oor n, , noo frequencies (E ni = 106) were assigned to 50 or 100 "species" in such a way that the frequency series could be taken for a representation of a truncated lognormal or a logarithmic series distribution. Using this approach we had to choose a leading frequency or greatest value that can influence the simulation results (leading frequencies were slightly modified at the end of the procedure in order to get 106 as the sum of frequencies). Therefore we used different leading frequencies in our representations of the model distributions. These leading frequencies are in the case of lognormal distribution 600 145 and 555 345, respectively (50 species models) and 480 362 (100 species model), and 199 112 in the case of the logarithmic series distribution (100 species model). Below we identify the models by their leading frequencies. The medians of models 600 145, 555 354 and 480 362 are 101.5, 99.0 and 308.5, respectively. Based on a Wilcoxon test, the differences between model 600 145 and model 555 354 were not significant. Significant differences occurred between model 480 362 and the other two models. Fig. 1 shows the log rank