## Introduction

One of the most widely used relationships in spatial biodiversity studies is the distance–decay, which describes how the similarity in species composition between two communities varies with the geographic distance that separates them. This relationship received the early interest of Whittaker in his seminal study of vegetation in the Siskiyou mountains (Whittaker 1960, 1972) and Preston when examining the Galapagos flora (Preston 1962). The distance–decay relationship became increasingly popular after Nekola & White (1999) formalized its ability to describe, compare and understand biodiversity patterns. Considered one of the few ‘distributions of wealth’ characterizing communities (Nekola & Brown 2007), distance–decay curves have now been studied across a wide range of organisms, geographic gradients and environments (Nekola & White 1999; Condit *et al.* 2002; Tuomisto *et al.* 2003; Green *et al.* 2004; Novotny *et al.* 2007; Qian & Ricklefs 2007; Soininen & Hillebrand 2007).

There are many reasons to explain the success of the distance–decay relationship in ecology. Data required to plot the distance–decay curve are readily obtained by sampling at local scales across a landscape, making large-scale biodiversity studies empirically tractable (Harte *et al.* 1999; Condit *et al.* 2002; Green *et al.* 2004; Krishnamani *et al.* 2004). Because the distance–decay relationship reflects patterns of spatial distribution and autocorrelation, it is likely sensitive to key spatial processes such as dispersal limitation, making it a powerful tool for testing mechanistic ecological theories (Chave & Leigh 2002; Condit *et al.* 2002). Even in the absence of theoretical derivations, distance–decay data can be used to understand the forces driving community turnover patterns such as dispersal limitation and environmental heterogeneity (Tuomisto *et al.* 2003; Ferrier *et al.* 2007; see Legendre *et al.* (2005) and Tuomisto & Ruokolainen (2006) for discussion of statistical approaches). Finally, the recent incorporation of species’ evolutionary history in distance–decay approaches offers a novel perspective for investigating the spatial turnover of phylogenetic composition across landscapes (Ferrier *et al.* 2007; Bryant *et al.* in press).

Despite a longstanding interest in the distance–decay relationship, its theoretical foundations remain poorly understood. The first theoretical derivation of the distance–decay relationship was based on dimensional analyses and the assumption of fractal species’ spatial distributions (Harte & Kinzig 1997; Harte *et al.* 1999). More recent analyses stemming from the neutral theory of biodiversity provide predictions for the distance–decay relationship in an environmentally homogeneous landscape, under the assumption that species are demographically identical (Hubbell 2001; Chave & Leigh 2002; Condit *et al.* 2002). However, a theoretical framework for the distance–decay relationship free of assumptions about the spatial organization of individuals or community dynamics is still lacking. Such a general framework is necessary to interpret distance–decay curves observed in nature, where no particular clustering or assembly processes can be assumed *a priori*.

Sampling theory provides a foundation for understanding the spatial scaling of diversity with minimal assumptions (McGill *et al.* 2007). Sampling theory has been used to derive scaling relationships for many macroecological patterns including the species–area and endemics–area relationships (He & Legendre 2002; Green & Ostling 2003), the species-abundance distribution (Green & Plotkin 2007) and species turnover (Plotkin & Muller-Landau 2002). Plotkin & Muller-Landau (2002) paved the way for integrating the distance–decay relationship into sampling theory by deriving the compositional similarity between two samples randomly drawn from a landscape, independent of their spatial location. However, the distance–decay relationship requires understanding how community similarity varies as a function of the geographic distance separating samples, and there currently exists no general sampling formula for this spatial pattern.

In this paper, we merge sampling theory and spatial statistics to develop a framework for understanding the distance–decay relationship. We begin by deriving a general formula for distance–decay as a function of the landscape-scale species-abundance distribution and intraspecific spatial autocorrelation. This general framework does not assume a particular type of population clustering or community dynamics. To illustrate the utility of this framework, we examine a specific model of clustering: the Poisson Cluster Process. This spatial-point process was chosen due to its mathematical tractability (Cressie 1993; Diggle 2003), its ability to reproduce species–area curves (Plotkin *et al.* 2000) and its potential to characterize the dispersal capacity of species (Seidler & Plotkin 2006). We compare our theoretical predictions to empirical data from three tropical forests with distance–decay curves that differ widely in their compositional similarity values, rate of decay and functional form. We conclude by discussing the implication of our results for biodiversity and biogeography studies.