Species Abundance Distribution and Species Accumulation Curve: A General Framework and Results

We build a general framework which establishes a one-to-one correspondence between species abundance distribution (SAD) and species accumulation curve (SAC). The appearance rates of the species and the appearance times of individuals of each species are modeled as Poisson processes. The number of species can be finite or infinite. Hill numbers are extended to the framework. We introduce a linear derivative ratio family of models, $\mathrm{LDR}_1$, of which the ratio of the first and the second derivatives of the expected SAC is a linear function. A D1/D2 plot is proposed to detect this linear pattern in the data. By extrapolation of the curve in the D1/D2 plot, a species richness estimator that extends Chao1 estimator is introduced. The SAD of $\mathrm{LDR}_1$ is the Engen's extended negative binomial distribution, and the SAC encompasses several popular parametric forms including the power law. Family $\mathrm{LDR}_1$ is extended in two ways: $\mathrm{LDR}_2$ which allows species with zero detection probability, and $\mathrm{RDR}_1$ where the derivative ratio is a rational function. Real data are analyzed to demonstrate the proposed methods. We also consider the scenario where we record only a few leading appearance times of each species. We show how maximum likelihood inference can be performed when only the empirical SAC is observed, and elucidate its advantages over the traditional curve-fitting method.