Ever since Newton and Kepler, science has made extensive use of mathematical models for systems, at times in fields where they are probably simpler than useful. Ecology is among the fields where formulae are useful, but can never be truly precise. One of the more common metrics calculated about an ecosystem is the rarefaction curve. This curve approximates how many species within a specified group will be found, with sample size as the independent variable. They may be used to compare species richness between sites with differing levels of sampling, approximate the total diversity of a site, estimate the total global diversity of a clade, or perform other similar calculations. They are generally assumed to follow either an inverse decay curve—(\hat{s}=a(1-e^{-bx}))—or a logarithmic curve—(\hat{s}=a\ln({bx})).

Much of my research work has been focused on documenting the fauna of the early Pleistocene Carolinian Waccamaw Formation. As part of this, I have also done systematics/taxonomy and paleoecology. Since much of the material for the faunal documentation consists of bulk samples, this allows for calculation of a rarefaction curve across a very wide range of sample sizes. Plotting these data points reveals that they match neither of the expected types of curves, but have a much longer slow increase than expected. The closest match found so far is the integral of a cumulative lognormal, however, suggestions of other possible curves are welcome.