The ubiquity of some of the most widely observed ecological patterns goes beyond the field of ecology. One notable example is the distribution of individuals among species in a community (a.k.a. the species abundance distribution, or the SAD), which often exhibits a highly uneven, hollow-curve shape. While the shape of the SAD can be (and has been) explained by various ecological processes, distributions of similar shapes have also been observed in many non-ecological systems including distributions of wealth, earthquakes, and even college basketball wins. Identifying the statistical nature of the SAD and other such patterns would allow better predictions to be made with minimal information on the details of each specific ecological system. Moreover, deviation from the statistical expectation can be used to identify when important ecological processes come into play.

### 1. The feasible set and Taylor’s Law

In a recent study, Locey & White (2013) proposed the concept of the feasible set, or the set of all possible configurations of a system given the way it is constrained. They further showed that the feasible set of an SAD was dominated by hollow curves, providing an *a priori* explanation for the ubiquity of the pattern.

In an collaborative project with Kenneth Locey and Ethan White, we found that the same concept could be applied to explain another general pattern in ecology, Taylor’s Law (i.e., the power-law relationship between the variance and the mean of one or multiple populations across space or through time). Most configurations within the feasible set show a power-law relationship between the variance of the mean with an exponent between 1 and 2, well matching the empirical patterns (see Figure below). However, the approach does not predict the exact parameters of the relationship in any given system, which suggests that this could be where ecological signals come into play. This work is published in the American Naturalist (DOI: 10.1086/682050).

### 2. Species range size distribution drives spatial biodiversity

It is well recognized that common species with broad ranges have a disproportionately large effect on the spatial diversity patterns. For example, Jetz & Rahbek (2002) showed that among the sub-Saharan African birds, the diversity of the top 25% species with the largest ranges had a high correlation with overall diversity of the group (r = 0.91), while the 25% species with the smallest ranges had a more different diversity pattern across space (r = 0.56). This observation appears to be generally true across geographical locations and taxonomic groups (e.g., Lennon et al. 2004, Kreft et al. 2006).

In an ongoing project, I work with my postdoc mentor, Brian McGill, to explore the statistical null expectation of the pattern. We have found with simulations that the high correlation between overall diversity of a group and the diversity of its most widely-distributed members is mostly driven by the highly skewed range size distribution (i.e., a few species have enormous ranges while most other ranges are quite small), which exists even when species are allowed to distribute randomly in space. On the other hand, correlation between the diversity of rare species and overall diversity is highly non-random with the empirically observed correlation much higher than expected, which implies that small-ranged species are aggregated at environmentally suitable habitats and/or limited by barriers.