The study of patterns in species distributions has long been a major focus of ecology. The distribution of a species’ abundance across many plots of a fixed area, known as the quadrat count distribution (QCD), is commonly used to describe spatial aggregation within a species. The QCD is often modeled as a negative binomial distribution, with the clustering parameter k used to describe the degree of intraspecific aggregation. Knowledge of the shape of the QCD across multiple spatial scales is important for cross-scale analysis of species populations, including the construction of species-area relationships, the prediction of species’ abundance from presence-absence data, and the derivation of other similar metrics important to conservation.
Despite its importance, the shape of the QCD across scales has not been extensively characterized empirically, and little is known about the mechanisms that might generate different shapes of this distribution. In this talk, I explore the relationship between a well-known birth-death-immigration stochastic process model and the shape of the QCD. By extending this model to a spatial context, I suggest a simple plausible mechanism for the generation of negative binomial QCD’s and relate the QCD to Taylor’s Law, a well-known pattern relating the mean and variance of a population inhabiting a fixed area.
A spatial extension of a negative binomial birth-death-immigration stochastic process model shows that for this simple population model, the clustering parameter k of the QCD may scale either linearly with area or as area to the 0.5 power, in the two limits in which immigration into a plot is proportional to plot area (a “target” model) or plot perimeter (a “diffusion” model). Immigration is found to be the dominant control on the scaling relationship within this framework, with birth and death rates for a species playing a negligible role. The findings from a recent review of Taylor’s Law across a variety of empirical systems are shown to correspond to this same negative binomial QCD with k scaling to the power of 0.5.
Based on both the simplicity of the underlying population model and the empirical evidence, I conclude that a model in which the negative binomial QCD value of k scales to the power of 0.5 is both empirically and theoretically supported, and should be used as a first-pass null model for spatial scaling applications. This recommendation stands in contrast to several existing spatial scaling theories, which predict either a constant or linearly scaling k.