Ecology Letters, (2007) 10: 1037–1045 doi: 10.1111/j.1461-0248.2007.01101.x IDEA AND PERSPECTIVE A statistical theory for sampling species abundances Abstract Jessica L. Green1 and The pattern of species abundances is central to ecology. But direct measurements of Joshua B. Plotkin2* species abundances at ecologically relevant scales are typically unfeasible. This limitation 1Center for Ecology and has motivated a long-standing interest in the relationship between the abundance Evolutionary Biology, University distribution in a large, regional community and the distribution observed in a small of Oregon, Eugene, OR, USA sample from the community. Here, we develop a statistical sampling theory to describe 2 Department of Biology, how observed patterns of species abundances are influenced by the spatial distributions University of Pennsylvania, 433 of populations. For a wide range of regional-scale abundance distributions we derive S. University Ave., Philadelphia, exact expressions for the sampled abundance distributions, as a function of sample size PA 19104, USA *Correspondence: E-mail: and the degree of conspecific spatial aggregation. We show that if populations are [email protected] randomly distributed in space then the sampled and regional-scale species-abundance distribution typically have the same functional form: sampling can be expressed by a simple scaling relationship. In the case of aggregated spatial distributions, however, the shape of a sampled species-abundance distribution diverges from the regional-scale distribution. Conspecific aggregation results in sampled distributions that are skewed towards both rare and common species. We discuss our findings in light of recent results from neutral community theory, and in the context of estimating biodiversity. Keywords Species-abundance distribution, random sampling, negative-binomial sampling, spatial aggregation, biodiversity, community. Ecology Letters (2007) 10: 1037–1045 distribution of a large, regional community and the observed INTRODUCTION abundance distribution when sampling a small proportion The distribution of species abundances is a fundamental of the community. topic in ecological research. Species-abundance distributions Efforts to develop a sampling theory of species abun- have been used to examine the influence of niche dances have utilized several approaches. The most widely differentiation, dispersal, density dependence, speciation utilized approach, which dates back to Fisher et al. (1943) and extinction on the structure and dynamics of ecological assumes that individuals are randomly sampled from an communities (Tokeshi 1993; Hubbell 2001; Chave et al. ecological community. Fisher et al. (1943) sought to 2002; Magurran 2004; McGill et al. in press). In conservation understand patterns of species abundance in butterfly, biology, knowledge of the species-abundance distribution beetle and moth communities sampled throughout the helps one to predict the likelihood of population persistence world. They found that species abundances in random and community stability in face of global change. Despite samples were well described by a logseries distribution, a the theoretical and practical importance of species-abun- distribution they mathematically derived by Poisson sam- dance distributions, it is difficult to directly measure all pling from a gamma distribution. Their analyses laid the speciesÕ abundances at ecologically relevant scales. For foundation of biodiversity sampling theory (May 1975; micro-organisms, this poses a challenge at the scale of a Pielou 1975) and remain at the forefront of literature on single environmental sample (e.g. < 1 g of soil; Prosser et al. species-abundance distributions (Hubbell 2001; Chave 2004; 2007). In plant and animal communities as well, the task of Magurran 2004; McGill et al. in press). exhaustively sampling a full community is typically impos- Engineer and ornithologist Frank Preston (1948) sible. Therefore, ecologists have a long-standing interest in popularized the lognormal species-abundance distribution the relationship between the underlying species-abundance in ecology by demonstrating that the lognormal had Ó 2007 Blackwell Publishing Ltd/CNRS 1038 J. L. Green and J. B. Plotkin Idea and Perspective scale-invariant properties upon random sampling. He In this study, we present a general statistical framework showed by tabulation (Preston 1948) that Poisson sampling for understanding the effect of spatial heterogeneity (or, individuals from a lognormal species-abundance distribution equivalently, heterogeneity in the sampling scheme) on the results in a sample distribution that is approximately species-abundance distribution observed in a sample. Our lognormal with identical variance. Preston (1962, p. 186) sampling framework does not assume a particular type of recognized that Poisson sampling individuals was analogous population aggregation (e.g. fractal theory) or community to Ôa situation in space and time where the individuals, or dynamics (e.g. neutral theory). We begin by analysing the pairs, are distributed at random, not clumped on one hand simple case in which individuals are randomly sampled from or over-regularized on the otherÕ. He noted that contagion, the larger regional community, corresponding to random or conspecific aggregation, would likely to result in a spatial distributions across the landscape. We then examine Ôsomewhat skewedÕ sample distribution (Preston 1962, p. a more realistic scenario of negative-binomial sampling, 203); however, he did not rigorously explore the influence of which models spatial clustering of conspecific individuals. this contagion on his samples. Biodiversity sampling theory We apply our techniques to a wide range of abundance has since predominantly assumed random sampling (e.g. distributions, deriving exact expressions for the sampled Pielou 1975; Dewdney 1998; Gotelli & Colwell 2001; Chao abundance distributions as a function of sample size and the & Bunge 2002), while less is known about the sampling degree of conspecific clustering. We also demonstrate two properties of species-abundance distributions for spatially important, generic properties of how spatial aggregation aggregated populations. affects the sampled species-abundance distribution. Efforts to understand the sampled abundance distribu- tion of aggregated populations have focused primarily on a STATISTICAL FRAMEWORK specific type of aggregation. The assumption of fractal, or self-similar spatial distributions has been leveraged to Species-abundance distributions are measured in the labo- explore how species-abundance distributions scale with ratory or field by counting the number of species in a sampling area (Banavar et al. 1999; Harte et al. 1999). Recent community represented by n individuals. For the purpose of analyses, however, suggest that such fractal models are our analysis, we will characterize species abundances in a biologically unrealistic (Green et al. 2003; Pueyo 2006). An large region using a continuous probability density function alternative statistical model known as the Hypothesis of /(n). The expression /(n)dn represents the fraction of Equal Allocation Probabilities (HEAP) allocates individuals species whose abundanceR falls between n and n + dn. Proper 1 across a landscape according to a set of assembly-rules, normalization requires 0 /ðnÞdn ¼ 1. In reality speciesÕ yielding a scale-dependent species-abundance distribution abundances are discrete. We use continuous distributions to that matches empirical vegetation data relatively well (Harte provide consistency with a sampling theory of b-diversity et al. 2005). Although the HEAP framework allows for a (Plotkin & Muller-Landau 2002). Aside from offering range of aggregation patterns, from random to highly analytical tractability, there is a long-standing precedent clustered, this model has not been utilized to explore how for using continuous distributions to describe species the degree of aggregation influences sampled species- abundances (Pielou 1975). abundance distributions. Our interest lies in the relationship between the species- Neutral community theory (Hubbell 2001) provides a abundance distribution at the regional scale, /(n), and the mechanistic approach for modelling heterogeneity by abundance distribution observed in a sample that constitutes accounting for the effect of dispersal limitation. Alonso & a proportion a of the larger ambient region, denoted /a(y). McKane (2004) applied Poisson sampling to the metacom- Let wa(y|n) denote the probability that a species will be munity multinomial relative-abundance distribution to represented by y individuals in the sample, given that it has derive an analytical expression for the sample distribution abundance n in the larger region. Then, the sampled species- under the assumption of zero dispersal limitation. Etienne & abundance distribution may be expressed as: Alonso (2005) later invoked dispersal limitation by replacing Z1 random sampling with the dispersal-limited binomial / ðyÞ¼ w ðyjnÞ/ðnÞdn ð1Þ (actually, hypergeometric) sampling. This allowed for a a sampling heterogeneity by modelling the probability for a 0 dispersal-limited species to be present in a sample with a Equation 1 provides a general expression for the scaling of given abundance under the assumptions stipulated by the species-abundance distribution with sample size, given neutral community theory. These recent developments, an arbitrary sampling scheme wa(y|n). Equation 1 follows while