John Alroy 

1. Counts of species in ecological samples are of interest when they tell us about community assembly processes. Older process-based models of count distributions are either complex, widely rejected, or not able to predict high unevenness.

2. I leverage a general strategy for deriving simple one-parameter models. A distribution of abundances x on a continuous scale is predicted from a transform of a uniform distribution U; U is solved for to yield one minus a cumulative distribution function (CDF) for x; and the result is differenced and rounded to down to yield a probability mass function. The same workflow has long been used to derive the geometric series from the exponential distribution. Three variants are proposed, respectively based on the transforms μ/U – μ = (μ – U)/U where μ is a fitted constant (a scaled odds); [–ln(U)/λ]² where –ln U is just an exponential random variate and λ is the constant; and [–ln(2/U – 1)/γ]⁴ where γ is the constant. They collectively cover the range of functions that lead from some U to a non-negative real number.

3. The distributions are all consistent with simple population dynamical models in which recruitment rates, and sometimes death rates, vary randomly amongst species and are fixed for each species. The number of recruited offspring produced during each interval by each species is Poisson-distributed, and death rates are per-capita. Population counts are equilibrial, allowing co-existence in the absence of competition.

4. Large-scale surveys of corals, fishes, butterflies, and trees are consistent with the distributions, as are local-scale inventories of trees and assorted vertebrate and insect groups. Each inventory is used to predict the counts of another one that is matched based on group representation, biogeography, and richness. Based on examining decisive differences between the resulting likelihoods, the new models routinely outperform eight different rivals.

5. Thanks to their simplicity, grounding in non-competitive equilibrial population dynamics, and predictive power, the new approaches have considerable relevance throughout ecology.