LETTER A new sampling formula for neutral biodiversity Rampal S. Etienne Community and Conservation Ecology Group, University of Groningen, PO Box 14, 9750 AA Haren, The Netherlands Correspondence: E-mail: r.s.etienne@rug.nl Abstract The neutral model of biodiversity, proposed by Hubbell (The Unified Neutral Theory of Biodiversity and Biogeography, Princeton University Press, Princeton, NJ, 2001) to explain the diversity of functionally equivalent species, has been subject of hot debate in community ecology. Whereas Hubbell studied the model mostly by simulations, recently analytical treatments have yielded expressions of the expected number of species of a particular abundance in a local community with dispersal limitation. Moreover, a formula has been offered for the joint likelihood of observing a given species-abundance dataset in a local community with dispersal limitation, but this formula is too complicated to allow practical applications. Here, I present a much simplified expression that can be regarded as an enhanced version of the famous Ewens sampling formula. It can be used in maximum likelihood methods for quick estimation of the model parameters, using all information in the data, and for model comparison. I also show how to rapidly generate examples of species-abundance distributions for a given set of model parameters and how to calculate Simpson’s diversity index. Keywords Biodiversity, community, Ewens sampling formula, Hubbell, neutral model, urn scheme. Ecology Letters (2005) 8: 253–260 INTRODUCTION Community ecology has recently been endowed with the neutral model of biodiversity (Hubbell 1997, 2001) that aimed at giving a simple explanation of biodiversity patterns such as species-abundance distributions and species–area curves. According to this model these patterns solely result from the stochastic processes of birth, death, speciation and immigration. This model has been heavily debated (e.g. Yu et al. 1998; Abrams 2001; Brown 2001; De Mazancourt 2001; Bengtsson 2002; Chave & Leigh 2002; Clark & McLachlan 2003; Fargione et al. 2003; Harte 2003; Ricklefs 2003), but it is more or less accepted as a useful null model that merits further study. In this paper I introduce a new sampling formula for the neutral model that has important applications in the confrontation of the neutral model to data. For a complete understanding of this formula, I first briefly review the neutral model and recent advances in neutral theory. In Hubbell’s (2001) model, when individuals in a local community die, they are immediately replaced by offspring of other local individuals or by immigrants from the regional species pool (the metacommunity in Hubbell’s terminology), keeping the total number of individuals constant (the zero-sum assumption). The replacement probability is proportional to each species abundance in the local community (when replaced by a local individual), or in the regional species pool (when replaced by an immigrant); this proportionality is the neutrality assumption. The regional species pool is in a balance between speciation and extinction. The model contains two parameters: the immigration probability m and the fundamental biodiversity number h that is a measure of species diversity in the regional species pool. Hubbell (2001) defines h and h :¼ 2J M m¢, where J M is the number of individuals in the regional species pool and m¢ is the speciation probability per unit birth. Vallade & Houchmandzadeh (2003) define h :¼ mð J M 1Þ 1v , which is, apart from a factor of 2, equivalent to Hubbell’s (2001) definition in the limit that Hubbell takes in his derivation ( J M  1), because m 0 :¼ m 1v where m is the speciation probability per time-step. The factor of 2 results from whether or not one allows multiple speciations to occur within one time-step. Regarding the immigration probability, m <1 means that immigration is limited (dispersal limitation in Hubbell’s terminology). The recogni- tion of the potentially crucial role of dispersal limitation in determining biodiversity is one of the main achievements of Hubbell’s work (Hubbell 1997, 2001; Hubbell et al. 1999), Ecology Letters, (2005) 8: 253–260 doi: 10.1111/j.1461-0248.2004.00717.x Ó2005 Blackwell Publishing Ltd/CNRS