Frequency Distribution Models in the Ecology of Plankton and Other Organisms

Abstract

Mathematical frequency-distribution models for the distribution of organisms in space are discussed.. Appropriate models for random distribution and underdispersion are respectively the Poisson and binomial series. The negative binomial is the formal mathematical opposite of the binomial and provides a relatively good model for overdispersion, although there does not appear to be any biological situation which leads unequivocally to this distribution. The discrete log-normal distribution often gives a good fit to large-sample data, but has usually been regarded only as an approximation. Other "contagious" models are discussed, but do not appear to have any unique properties which are particularly appropriate to plankton. At least for the data under consideration, the best solution would be a distribution with positive skewness, intermediate between the negative binomial and discrete log-normal. For small samples the negative binomial tends to overestimate and the discrete log-normal to underestimate the frequency of zero counts. A model, the "Poisson-log-normal" distribution is proposed, in which the means of a continuous series of Poisson variates are distributed as a log-normal. This has the required degree of skewness and appears to estimate zero counts without bias. A method is shown for separating the parameters of different populations from a polymodal distribution, using log-probability paper.