Studying ecological communities from a neutral standpoint: A review of models’ structure and parameter estimation

Ecological Modelling 220 (2009) 2603–2610

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Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel

Review

Studying ecological communities from a neutral standpoint: A review of models’ structure and parameter estimation Champak R. Beeravolu a,b,∗ , Pierre Couteron a , Raphaël Pélissier b,a , Franc¸ois Munoz c a

IRD, UMR-AMAP, TA A-51/PS2, 34398 Montpellier cedex 05, France Institut Franc¸ais de Pondichéry (IFP), UMIFRE MAE-CNRS 21, 11 St Louis Street, 605001 Puducherry, India c Université Montpellier 2, UMR-AMAP, TA A-51/PS2, 34398 Montpellier cedex 05, France b

a r t i c l e

i n f o

Article history: Received 8 April 2009 Received in revised form 19 June 2009 Accepted 24 June 2009 Available online 28 July 2009 Keywords: Community ecology Community models Dispersal limitation Population genetics models Parameter inference Unified neutral theory of biodiversity and biogeography

a b s t r a c t Neutral models provide an alternative to niche-based assembly rules of ecological communities by assuming that communities’ properties are shaped by the stochastic interplay between ecological drift, migration and speciation. The recent and ongoing interest about neutral assumptions has produced many developments on the theoretical side, with nevertheless limited echoes in terms of analyses of real-world data. The present review paper aims to help bridge the widening gap between modellers and field ecologists through two objectives. First, to provide a multi-criteria typology of the main neutral models, including those from population genetics that have not yet been transposed to ecology, by considering how the fundamental processes of ecological drift, speciation and migration are modelled and, specifically, how space is taken into account. Second, to review methods recently proposed to estimate models parameters from field data, a point that should be mastered to allow for broader applications. © 2009 Elsevier B.V. All rights reserved.

Contents 1. 2.

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6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2603 Ecological drift and speciation in neutral models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2604 2.1. The fundamental process of ecological drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2604 2.2. Speciation events: the origin of species in neutral models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2604 Spatial patterns of migration and speciation in neutral community models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2605 3.1. Non-hierarchical neutral models in continuous space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2605 3.2. Non-hierarchical neutral models in discrete space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2606 3.3. Hierarchical spatially implicit models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2606 Parameter estimation: a critical step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2607 4.1. Parameter estimation with the spatially continuous neutral model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2607 4.2. Estimation of SINM parameters from a single sample plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2607 4.3. Estimation of SINM parameters from multi-sample data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2608 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2608 5.1. Using spatially implicit or explicit models: a matter of scale and data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2608 5.2. What is at stake through estimation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2609 Conclusion and prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2609 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2609 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2609

1. Introduction ∗ Corresponding author at: IRD, UMR-AMAP, TA A-51/PS2, 34398 Montpellier cedex 05, France. Tel.: +33 4 67 61 75 23; fax: +33 4 67 61 56 68. E-mail address: champak [email protected] (C.R. Beeravolu). 0304-3800/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2009.06.041

“What are the biological processes responsible for the maintenance of species diversity in natural communities?” is a question that crops up as a leitmotiv in ecological literature (Chesson, 2000).

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To tackle this issue, one of the fundamental ecological principles is competitive exclusion (Gause, 1934; Hardin, 1960), which states, provided that ecological factors are invariant, that two species competing for the same limited resource cannot coexist forever. To get around this constraint, the niche partitioning theory (Grinnell, 1917; Elton, 1927) states that each species has a specific fundamental niche, i.e. the grouping of all conditions that allows a species to survive, and a comparatively smaller realized niche, resulting from the biotic interactions amongst individual organisms (Hutchinson, 1957), which let species coexist in equilibrium over a variety of niche conditions. It is however increasingly recognised that stochastic processes (especially, metapopulation and metacommunity dynamics, see Pulliam, 2000; Leibold et al., 2004) also contribute to shaping the realized niche of species. Indeed, the pervasive influence of stochastic processes on species assembly have long been underlined by Gleason (1926) and refer to a random variation in species establishment success and survival. It has also been the inspiration for MacArthur and Wilson’s (1967) work on island biogeography, which was further developed by Hubbell (2001) in the form of the unified neutral theory of biodiversity and biogeography (UNTB). Hubbell’s UNTB relies on concepts from population genetics (Kimura, 1983), as it likens the alleles’ non-selective dynamics at a single multi-allelic locus (in a haploid population) to species dynamics in an ecological community (Watterson, 1974; Caswell, 1976; Hubbell, 1997; Chave, 2004; Leigh, 2007). It assumes that individuals at a certain trophic level are ecologically equivalent (Chave, 2004) insofar as they have equal prospects of birth, death, dispersal and speciation, so that the stochastic interplay of these processes is seen as the primary driver of species diversity patterns. The historical development of neutral community ecology, emphasizing the connections with closely related approaches in population genetics, starting from Watterson (1974) and Caswell (1976) has been reviewed by several authors (Chave, 2004; Hu et al., 2006; Leigh, 2007). Ever since, Hubbell’s UNTB has caused much ink to flow, either for criticizing/supporting the theory from real-world case-studies (McGill, 2003b; Gilbert and Lechowicz, 2004; Latimer et al., 2005; Dornelas et al., 2006; Volkov et al., 2007; Kelly et al., 2008) or for proposing more or less substantial modifications of the model (Volkov et al., 2003, 2005; He, 2005; Chave et al., 2006; Haegeman and Etienne, 2008). Most ecologists would probably agree that both niche differentiation and dispersal limitation do occur in natural systems (Barot, 2004; Gravel et al., 2006) and that neutral community models should be seen as first-step null models in ecology (Alonso et al., 2006). But to date there has been no clear way to disentangle the effects of niche and stochastic community dynamics, and to measure their relative influence from empirical data. In order to achieve this goal, the most general lessons of neutral theory should be firmly integrated as part of the common conceptual background of community ecology, as happened, for instance, to the Hardy–Weinberg principle in population genetics or the Lotka–Volterra model in population dynamics. Since the last most general syntheses (Chave, 2004; Leigh, 2007), the field of neutral community ecology has burgeoned on the theoretical side and many variants of the fundamental UNTB have been proposed. But applications to the diversity of real-world data remain limited as many ecologists today probably find it hard to grasp if and how neutral models can be useful to their field of investigation. We, thus, propose a review and discussion of neutral models, initially considering how the fundamental neutral processes of ecological drift, speciation and migration are modelled and, specifically, how space is taken into account in these models (see also McGill et al., 2006). This has led us to propose a multi-criteria typology of the neutral models, which includes some models of population genetics that have not yet been transposed in ecology but may inspire future developments. We also identify

parameter estimation to be a key issue, as it is still insufficiently mastered to permit a wider application of neutral models to currently available species composition data. 2. Ecological drift and speciation in neutral models 2.1. The fundamental process of ecological drift The zero-sum assumption is fundamental to neutral theory. It states that the number of locally coexisting individuals saturating space or key resources is limited, and that the replacement for any newly dead individual is determined through a competition between the offspring of surviving individuals. While all the individuals may have the same prospect of survival and reproduction (the ecological equivalence hypothesis), because of the zero sum limitation, the random nature of replacement in birth and death events will ultimately lead to the predominance of some lineages and to the extinction of others. Thus, in the absence of any speciation event or arrival of immigrants, the species richness is due to decrease (Hubbell, 2001) and the local community may even become monospecific (“fixation”) after a long enough time lag, inversely proportional to the population size (Leigh, 2007). The effect of ecological drift in a local community also depends on how birth and death processes are defined, and particularly models relying on either overlapping or non-overlapping generations correspond to differing drift intensities. Wright (1950) considered non-overlapping generations that is, all individuals in a given population are simultaneously replaced by their descendents, while Moran (1958) alternatively used an overlapping generation model for which the time step is the time for a single death to occur among the individuals, with immediate replacement of any suppressed individual as per the zero-sum game. When overlapping generations are assumed, the drift-induced decrease in Simpson’s diversity (in analogy to genetic heterozygosity) is twice as fast as that for non-overlapping generations (Malécot, 1969; Leigh, 2007:3–4). In his UNTB, Hubbell (2001) assumed overlapping generations but slightly departed from the original idea of Moran (1958) by stating that a suppressed individual does not provide offspring. It is to be noted that beyond the question of generation overlapping, modelling of neutral communities may benefit in the future from the growing debate about event vs. time-driven modelling (Gronewold and Sonnenschein, 1998; Ewing et al., 2002). 2.2. Speciation events: the origin of species in neutral models A second fundamental aspect of individual equivalence is that speciation may occur at each event of individual replacement according to a fixed per capita probability (in the UNTB this principle only applies to the metacommunity, see below). Hence the larger a population, the more frequent is the apparition of a new species. New species are supposed to have never been present in the population (“infinite species” models) as for new alleles in the infinite allele model of population genetics. As a consequence, speciation is a force counteracting ecological drift and the speciation-drift equilibrium that directly rules species richness in a non-subdivided population is a strict analogue of the mutationdrift equilibrium, which has been extensively studied under the infinite alleles model (Kimura and Crow, 1964; Ewens, 1972). The difference is that speciation is far more complex and less understood a process than that of genetic mutation of elementary alleles, and there is currently no real consensus on which mode of speciation is prevalent in nature. Neutral models in ecology have so far mainly supposed a “point speciation” process according to which a dead individual is replaced by the first, only specimen of a totally new species (Hubbell, 2001). The only alternative is a variant of the

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Table 1 Characterization of neutral models according to spatial patterns of speciation and migration: SCNM: spatially continuous neutral model; SDNM: spatially discontinuous neutral model; SINM: spatially implicit neutral model. Model properties

Non-hierarchical SCNM

Hierarchy of scales Discrete local communities Spatial dependence of migrant fluxes Explicit inter-community fluxes Speciation at local scale Model diagram

No No Distance dependent (Yes) Yes Fig. 1a

Hierarchical SDNM

SINM

Stepping stone

Island

Graph matrix

UNTB (2L-SINM)

3L-SINM

He (2005)

No Yes Topology (nearest-neighbours) Yes Yes Fig. 1b

No Yes None Yes Yes Fig. 1c

No Yes Topology (variable) Yes Yes

Yes Yes None No No Fig. 1d

Yes Yes None No No Fig. 1e

Yes Yes None Yes Yes

UNTB proposed in Hubbell’s book (2001:264) as a “random fission” speciation mode in which a given species is split into two isolates of random size in analogy to the process of allopatric speciation. According to Hubbell (2001:271), the random fission mode leads to a zero-sum multinomial species abundance distribution (SAD) in a non-subdivided population (Hubbell’s metacommunity) instead of the log-series SAD found for the point mutation model. There has been so far neither further development nor application of random fission speciation. In fact, point speciation may appear acceptable as a first approximation of real-world plant communities as Leigh et al. (2004) pointed out that most plant species seem to have descended from small local populations, making the approximation of a single ancestor acceptable (Leigh, 2007:10).

ative influence of these two processes on the species abundances in ecological community models can be paralleled to the relative influence of gene flow and mutation on the distribution of allelic frequencies in subpopulations, which has been historically highlighted by population genetic models (Slatkin, 1985). How migration and speciation can possibly interact with ecological drift is closely dependent on how neutral models account for space. We try to expose the specificities of the various models by including more criteria starting with two main categories, namely hierarchical models inspired by the UNTB that features several levels (or scales) leading from the biogeographic region to the local community, and non-hierarchical models in either continuous or discrete space, which only deal with individuals or local communities, respectively (Table 1).

3. Spatial patterns of migration and speciation in neutral community models

3.1. Non-hierarchical neutral models in continuous space

Under the zero sum and constant community size assumptions, either migration from outside the local community or speciation events are needed to counterbalance drift and prevent the community from becoming monodominant. Consequently, the rel-

Spatially continuous neutral models (SCNM) assume a homogeneous habitat and a fixed density of individuals whose offspring are dispersed with a probability of success decreasing with distance (Wright, 1943; Malécot, 1969; Nagylaki, 1974; Slatkin, 1985).

Fig. 1. The underlying spatial structure of some of the main neutral models used in community ecology and/or population genetics. Arrows denote fluxes of migrant organisms between communities (circles) or spatial locations: (a) the spatially continuous neutral model (SCNM) as studied by Chave and Leigh (2002) using an individual-based model; here each species has a different colour (right) using a common dispersal kernel, i.e. an arbitrary decreasing function of distance (left); (b) the stepping stone model (Kimura and Weiss, 1964); (c) the island model (Wright, 1931); (d) the mainland–island model corresponding to two-level spatially implicit neutral model (2L-SINM) of Hubbell (2001) and (e) the three-level spatially implicit neutral model (3L-SINM) of Munoz et al. (2008) with the metacommunity and the regional pool denoted by ellipses.

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Dispersal is assumed to be stationary in space and time and there are no distinct communities, just a continuum of individuals (Fig. 1a). Although these models have generally not been presented or analyzed as such, the conceptual relationship with individual-based models can be shown as the spatial location and the fate of each individual are explicit. Simulation experiments for SCNM are indeed individual-based as in Chave and Leigh (2002), while these authors emphasized the close link with voter models (Bramson et al., 1998). For simulating a SCNM scenario, individuals have so far been placed at the nodes of a lattice, which diverges from the strict representation of a continuous space, but constraint could probably be relaxed in the future. The published SCNM are defined by the shape of a dispersal kernel (i.e. a probability curve in function of distance) along with two parameters, the mean square dispersal distance ( 2 ) and the speciation rate per individual replacement event (). An infinite population and boundary conditions are assumed to allow spatial stationarity and avoid strong edge effects. From the pioneering works in population genetics, Chave and Leigh (2002) have derived an analytical expression for the expected decrease of similarity (i.e. the probability that two individuals be conspecific) with increasing distance under the assumptions of a SCNM. 3.2. Non-hierarchical neutral models in discrete space In contrast to SCNM, spatially discontinuous neutral models (SDNM) do not accurately model the physical location of individuals and the whole set of observations are subdivided into discrete local communities. In a SDNM the species abundance distributions within communities as well as inter-community similarity can be predicted from the pattern of inter-community fluxes of migrants. Among the simplest and the most classical models following this principle in population genetics are the island (Wright, 1931) and stepping stone models (Kimura and Weiss, 1964). In the second of these models, local communities are arranged in a regular lattice of one, two or three dimensions with constant, non-zero migration fluxes only between adjacent sites (usually, the four nearest neighbours in two dimensions). Thus this model represents an extreme case of short distance migration (Malécot, 1969; Slatkin, 1985) and can be seen as a spatially explicit SDNM (Fig. 1b) since the fluxes of migrants depends on the relative positions of communities. Applications to real life situations may be relevant in cases when the species dispersal curve is known to be a thin tailed distribution (Clark et al., 1999) indicating that long distance dispersal events are infrequent. In analogy to SCNM, assumptions of infinite number of communities or of periodic boundary conditions are required in order to eliminate strong edge effects (Maruyama, 1972). The stepping stone model has not yet had any major application in community ecology although recently (Chave and Norden, 2007) it has been used to model a large community on a grid with adjacent cells interacting with each other. In the island model (Wright, 1931), migration is irrespective of inter-community distance so that analytical predictions are more easily tractable than for the stepping stone model (Latter, 1973). Such a model is spatially implicit and de facto assumes that long distance dispersal is not rare (Fig. 1c). The original island model depicts the dynamics of communities receiving a constant flux of migrants from other communities, irrespective of the distance. The infinite island assumption ensures that all the communities receive the same fluxes of migrants, with no edge effect (Slatkin, 1985). Another variation of this approach with a finite number of islands called the n-island model has also been studied in population genetics, including its asymptotic limit when n becomes large (Latter, 1973). One of the most general SDNM is the matrix model of population genetics (Bodmer and Cavalli-Sforza, 1968) which represents

the migration pattern by a matrix mij whose values express the portion of individuals in a given subpopulation i that have migrated from community j in the previous generation (published models all deal with non-overlapping generations). Since the matrix traces back the migrants’ origin, it is called a “backward matrix” and its rows are normalized to unity. Such a matrix can also be associated to a weighted graph and edge weights, where mij values reflect the underlying spatial (or connectivity) structure of the community network by letting neighbouring or connected communities exchange more migrants. Both island and stepping stone models can be modelled as specific cases via appropriate migration matrices and network structures (Bodmer and Cavalli-Sforza, 1968). The potential of this conceptual framework for community ecology was first underlined by Economo and Keitt (2008) who provided a theoretical investigation about how different network structures influence within (alpha-) and between (beta-) diversity patterns in neutral communities. This work echoes the recent introduction of graph theory in spatially explicit population models (Minor and Urban, 2007). 3.3. Hierarchical spatially implicit models The now classical Hubbell’s (2001) unified neutral theory of biodiversity and biogeography (UNTB) differentiates itself from the island model of population genetics as it introduces hierarchy by distinguishing the local communities and their source of migrants. It also decisively departs from MacArthur and Wilson’s (1967) theory of island biogeography by defining neutral processes (migration and speciation) at an individual level rather than at a species level. In close analogy with the mainland–island model of population genetics (Wright, 1931; Rannala and Hartigan, 1995) the UNTB assumes a large regional source pool (the metacommunity) that is modelled as independent from the dynamics of local communities. In particular, speciation is only modelled at metacommunity scale. Moreover, this model is a spatially implicit neutral model (SINM) since neither speciation nor the fluxes of migrants are influenced by the spatial configuration of local communities (Fig. 1d) and the dynamics of the metacommunity is thus termed “well-mixed” (Economo and Keitt, 2008) or “panmictic” (Leigh, 2007). The metacommunity is assumed to be at a speciation-drift equilibrium, which is summarized by a single parameter, called the fundamental biodiversity number,  (Hubbell, 2001). At local scale a given community is conditioned by , and by the parameter, I (introduced by Etienne (2005) as the “dispersal number”), which is composed of the migration probability from the metacommunity, m, and the size of the local community, J. The parameter I features the number of potential immigrants from the metacommunity competing with local offspring to replace a newly dead individual of the local community. It also summarizes the migration-drift equilibrium, in analogy with  for the speciation-drift equilibrium at the metacommunity level (Etienne and Alonso, 2005). Among recent theoretical attempts (see Hu et al., 2007) to generalize the UNTB, He (2005) introduced inter-community migration by using an additional process (emigration) and by allowing for speciation in the local community. This essentially defines a local community as a specific case of the metacommunity in order to establish conceptual links between the mainland–island approach defining the UNTB and the island model. Speciation here is defined on a per capita basis, thus the rate of local speciation events is far smaller than for the metacommunity and moreover converges to zero for small sized local communities. Another attempt at enriching Hubbell’s two level model (2LSINM) is the three level spatially implicit neutral model (3L-SINM) proposed by Munoz et al. (2008), which introduces an intermediate common source pool, of smaller extent than the metacommunity that directly conditions the species composition of the local

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communities under study (Fig. 1e). This permits a greater flexibility in applying the neutral theory to real-world data (see Section 4) by relaxing Hubbell’s assumption that the source pool (i.e. the metacommunity) is at a speciation-drift equilibrium. Another advantage of such a framework is that it resembles the hierarchical schemes for the additive apportioning of total and beta diversity (Whittaker, 1972; Couteron and Pelissier, 2004; Pelissier and Couteron, 2007), which could be an approach worth developing in the future. 4. Parameter estimation: a critical step Using neutral models to investigate the structure and dynamics of real-world communities requires reliable methods to estimate models’ parameters from field data (calibration step), before comparing the model outputs with additional observations, ideally independent from the calibration dataset. Whatever the objectives of the second step (validation/refutation or study of specific departures), the quality of parameter estimation is critical to ensure that subsequent analyses are meaningful. In spite of the overall interest drawn by the UNTB and its variants, estimation has proved to be technically difficult due to complex maths and simplistic interpretations from unwarranted results have probably reinforced the scepticism of many ecologists about the usefulness of a neutral approach of community composition. Although in many instances neutral community models can rely on straightforward analogies from population genetics, the intrinsic difference between the data used in the two fields (in terms of sampling designs, number of alleles vs. species, etc.) along with the conceptual specificities has led to the proposal of several new estimation techniques (McKane et al., 2000; Chave and Leigh, 2002; Etienne, 2005; Jabot et al., 2008; Munoz et al., 2008). In the following, we review the estimation procedures relative to the SCNM of Chave and Leigh (2002) as well as the SINM that descend from Hubbell’s (2001) UNTB. For the latter, we will consider the relevance of estimation methods on the basis of the two main categories of sampling schemes currently used in forest ecology, namely either one single large, permanent sampling plot (e.g. 50 ha, as in Barro Colorado Island, Panama), or a network of several small plots (e.g. ≤1 ha). 4.1. Parameter estimation with the spatially continuous neutral model Chave and Leigh (2002) and Chave et al. (2002) have used individual-based models (DeAngelis and Mooij, 2005; Strigul et al., 2008) to simulate a neutral model in continuous space (SCNM) for which Chave and Leigh (2002) have also provided an analytical expectation for the probability that two trees (individuals) separated by a distance r are of the same species (pairwise similarity), which is a spatially explicit formulation of beta-diversity (species turnover). From this, one may estimate model parameters (the mean square dispersal distance,  2 , and the speciation rate, ) by minimizing least squared deviations from the observed similarity function. From three datasets in Ecuador, Peru and Panama, Condit et al. (2002) have produced comparative results for their estimated mean dispersal distance  that ranges between 40 and 73 m. These values may appear reasonable in the light of field studies investigating dispersal distances in the tropical rainforests (see Turner, 2001 for a synthesis) but the range of variation of the estimates of the speciation parameter is very large (10−8 –10−14 ), while  does not lend itself to field measurements. Considering that this is, to our knowledge, the only application of the method and that results of the estimation covariance between  and  were not provided, it is still unclear to what extent inter-site variations in ␴ estimates can sustain meaningful interpretations.

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4.2. Estimation of SINM parameters from a single sample plot Most parameter estimation methods for Hubbell’s UNTB are typically based on a large field plot assumed to be embedded in a single local community, and the probability of successful installation of external immigrants is the same throughout the plot irrespective of any edge effects (these assumptions may be problematic for very large plots). From this premise, the information on species composition is fully contained in the species abundance distribution (SAD) of the plot, which plays a central role for several estimation techniques. Initially, Hubbell (2001) proposed to estimate the two fundamental UNTB parameters,  and m, through a sequential fit, which basically amounts to SAD curve fitting under neutral assumptions. Ever since, the expectation of a neutral SAD has been available analytically using a mean-field approach and used to estimate neutral parameters from observed SAD (McKane et al., 2000, 2004; Vallade and Houchmandzadeh, 2003; Volkov et al., 2003). The same approach can yield an analytical expectation of the SAD even if the zero-sum assumption is relaxed (Volkov et al., 2007). Another approach to estimate parameters from SAD information uses an innovative coalescence approach investigating all the possible genealogies of individuals within a local community undergoing migration and drift (Etienne and Olff, 2004; Etienne, 2005). These authors provide an exact expression, conditional to model parameters , m and sample size J, for the mean probability over all the possible species-ancestry abundance distributions that are compatible with the observed SAD. Etienne’s (2005) sampling formula (SF) derived from this expression has paved the way to the maximum-likelihood estimation of  and I, the “dispersion” number introduced by Etienne (2005). Although parameter estimation from SAD-values is now firmly grounded on analytical results (through either mean-field or coalescence approaches), questions can be raised about the statistical efficiency of the proposed estimators since little is known about their estimation variances. In the context of the infinite alleles model of population genetics, Ewens (1972) demonstrated that using the alleles abundance distribution, homologous to the SAD, is a statistically inefficient way to estimate . For this reason, Ewens (1972) maximum likelihood estimate (MLE) of  is based on the number of alleles in a population sample and not on Ewens SF established by Karlin and McGregor (1972), which results, to use ecological wording, in the expected SAD of a metacommunity sample (Caswell, 1976; Hubbell, 2001). Since Etienne SF logically converges to Ewens SF under unrestricted migration (as I tends to infinity; Etienne, 2005), one may wonder on which ground its generalization to the case of restricted migration (finite I) is immune to low statistical efficiency. Whatever the method used, it is still not clear whether the SAD from a single sample conveys sufficient information for a robust estimation dealing simultaneously with  and I. The doubts expressed by Harte (2003) and McGill (2003a) may well be relevant. Indeed, the strong intrinsic dependence between  and I through the pairwise similarity (i.e. the complementary to one of Simpson’s diversity, see Eq. (8) in Etienne, 2005) makes it difficult to discriminate between high-–low-I situations and low-–high-I situations, especially when using the SAD of a small particular plot (Etienne et al., 2006; Munoz et al., 2007). For methods that fit the SAD from binned species abundance data (e.g. Volkov et al., 2003), this problem is further amplified by the probable influence on the results of how bins of species frequencies are a priori defined, while there is no consensus on a standardized binning process for SAD (Gray et al., 2006). Moreover, it has long been recognised that different assumptions either purely statistical or ecological (including nonneutral effects) can also result in a non-discernable SAD (McGill, 2003a; Volkov et al., 2003, 2005; Chave, 2004; Purves and Pacala, 2005; Chave et al., 2006; McGill et al., 2007).

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4.3. Estimation of SINM parameters from multi-sample data

5. Discussion

Multi-sample data are made of many field plots scattered across a study area. This sampling strategy favours a larger spatial coverage, at the expense of the size of local samples. When samples are small, the estimation of neutral parameters from individual samples is quite unstable and inconsistent (Munoz et al., 2007). Lumping samples in order to apply any of the single-sample approaches evoked in the previous section would be erroneous and could result in serious biases (Etienne, 2007; Munoz et al., 2007). Instead, relying on methods designed for multi-sample data provides opportunities for decoupling the estimation of  (at the biogeographic scale of the metacommunity) and I (at the local level). An analytical formula that generalizes Etienne SF for multiple samples from the same metacommunity was provided by Etienne (2007), but a tractable algorithm to compute this formula has so far only been found in the case where all samples are assumed to have the same I value (Munoz et al., 2007). However, it appears rather desirable in terms of applicability to allow the variation of I values across plots as to account for the possible variation of the migrant fluxes through space. This is currently possible using three distinct methods. The “two-stage” method of Munoz et al. (2007) estimates  by applying Ewens (1972:98) MLE to a subset of the individuals, sampled in the network of available plots (one individual is drawn in each plot as a surrogate to a random sample of the metacommunity). In the second step, I values are estimated for each community sample using Etienne SF with  a priori set to the mean value provided by Ewens MLE for many random draws of one individual per plot. It is to be noted here that using Ewens MLE means estimating  from the number of species found in a random sample of the metacommunity, as recommended by Ewens (1972) for the infinite allele model, and not from the SAD of the lumped plots. Estimation of  at metacommunity level are circumvented in both Munoz et al. (2008) and Jabot et al. (2008), who directly estimated I values by conceptually referring to a regional source pool from which migrants originate. The existence of a common source pool at speciation-drift equilibrium is indeed a debatable assumption of the UNTB as well as of the island model. Jabot et al. (2008) introduced a novel adaptation of Etienne SF directly applicable to the SAD of the regional pool (approached by lumping the plot data) to get an estimate of the migrant flux via I(k) values in each plot k. An alternative approach was developed by Munoz et al. (2008) by revisiting the Gst statistic of population genetics (Nei, 1973), which combines intuitive similarity statistics (intra- and intersample probabilities of species’ identity) into the ratio of beta to gamma diversity (Whittaker, 1972). Neutral expectations of the Gst statistic have already been shown in population genetics to be independent of mutation and to rapidly approach steady state compared to the constituting similarities (Crow and Aoki, 1984). Munoz et al. (2008) proposed a new statistic conditional to each community sample k, called Gst(k), to measure the relative dissimilarity between k and the regional species pool approximated by the pooled set of samples. They established that the expectation of Gst(k) is a simple function of the local immigration number I(k). This analytical result, enabling to estimate I(k) from each observed Gst(k) value, has been derived under weaker neutral assumptions than the UNTB, in reference to the three level spatially implicit neutral model (3L-SINM) with no assumption on the SAD of the source pool. This estimation technique applies strictly to multiple field plots under the assumption that they are distant enough from one another in order to belong to distinct local communities. On simulated neutral community samples it has been verified by Munoz et al. (2008) that I(k) estimates are congruent with their MLE-based homologues obtained by the two-sage method of Munoz et al. (2007).

The recent and ongoing effervescence about neutral theory has produced many developments on the theoretical side, with nevertheless limited echoes in terms of analyses of real-world data. The aim of the present paper is to help bridge the widening gap between modellers and field ecologists by providing a typology of the main neutral models, including those from population genetics that have not yet been transposed to ecology, along with an up-to-date synthesis on how the parameters of the most classical models (UNTB and SCNM) are estimated. 5.1. Using spatially implicit or explicit models: a matter of scale and data The recent developments in neutral community theory have been mainly related to variants of Hubbell’s initial UNTB, which are spatially implicit models (SINM). Parameters of these models are to such extent phenomenological since they neither refer to the spatial configuration of the communities nor to the fundamental process of dispersal limitation in space (Etienne, 2005, 2007; He, 2005; Munoz et al., 2007, 2008; Economo and Keitt, 2008). As a consequence, estimated values of the migration probability m (or equivalently of the dispersal number I) are likely to reflect not only genuine dispersal limitation but also several additional processes limiting the migration of organisms between communities (habitat filtering, anthropogenic or physiographical barriers). For example, reciprocal values of I(k) of Munoz et al. (2007, 2008) could be interpreted as measuring how isolated from the regional species pool each community is, whatever the causes of isolation may be. Hence a broader concept of limitation of migration seems necessary to correctly interpret values of the SINM parameters. Environmental barriers or gradients are also likely to violate the assumption of stationary dispersal process on which the SCNM is built, since some locations may be more isolated than others. Moreover, Condit et al. (2002) acknowledged that if non-neutral processes significantly influence the data, estimates of  2 and  may be biased and not interpretable. As a consequence, spatial stationarity of the dispersal kernel may not be less unrealistic than the hotly criticised assumption of a common source pool (Leigh, 2007), making the SINM framework an alternative to examine the variation of I(k) values across plots (Munoz et al., 2008). This does not contradict that the SCNM is very suitable for studying the fine scale mechanistic process of dispersal limitation, for instance within the boundaries of a large plot and to compare with dispersal ranges estimated from dedicated protocols (Clark et al., 1999; Turner, 2001). But there is no point in explicitly modelling fine-scale processes if the data does not provide for information on fine scale spatial patterns of species (e.g. if the data is based on a network of distant plots and there is no mapping of species within the plots). The common sense idea that both models and estimation methods should be adapted to the nature of the data and the scale of the study has been somewhat overlooked during these last few years. As illustrated by Economo and Keitt (2008), migration matrix models could become a means of integrating spatial structures (or at least topology) in SINM without stringent assumptions about stationarity of the dispersal process in space. Moreover, such models let us explore how migration patterns between local communities could possibly control diversity patterns at the metacommunity level, which is the feed-back process that is not modelled in the UNTB (Bellemain and Ricklefs, 2008) although Hu et al. (2007) have handled the case of the influence of a single local community. Unfortunately, there has been so far limited experience on how to estimate the main parameters of migration matrix models (i.e. inter-communities migration rates) from a network of sampling plots, and this topic could appear as a priority of future

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research. Stepping stone and island models, as specific cases of the migration matrix frame (Economo and Keitt, 2008), offer an interesting possibility to grasp species migration in natural communities from two extreme and complementary standpoints, i.e. very limited and unlimited dispersal ranges, respectively. One interesting perspective could be to investigate how the prevalence of short vs. unlimited dispersion may interact with a simple modelling of environmental heterogeneity and filtering effects on species composition. Few would indeed disagree about the need to progressively reconcile niche and neutral standpoints (e.g. Gravel et al., 2006; Thompson and Townsend, 2006), although only a parsimonious increase in models’ complexity has a reasonable chance to be echoed by empirical achievements. If research on conceptual models has its own legitimacy, only models featuring parameters straightforwardly translatable into field measures or indirect estimation procedures can influence the manner in which species assemblages are studied in the field. 5.2. What is at stake through estimation? We believe that parameter estimation is a crucial step, which has so far been insufficiently investigated and is not fully mastered. Therefore building models with fine structure with parameters that cannot be identified from available data is of limited relevance. Nevertheless, methods of parameter estimation now exist although their respective domains of validity and statistical properties have been poorly documented. One may reasonably demand that a given method should (i) yield asymptotically unbiased estimates with a quantifiable estimation of errors for data complying with model assumptions and (ii) allow for the investigation of its robustness against departures from fundamental assumptions. Even the first, basic condition, has not yet been thoroughly addressed, since only a few papers (Jabot et al., 2008; Munoz et al., 2008), have been tested on simulations. Apart from simulations, it may be desirable to establish analytical results on estimation variances at least for the less computationally complex methods. The goal seems achievable since recent papers from population genetics (Rottenstreich et al., 2007) have reached analytical results about the variance of the Gst statistic under the island model. The robustness of estimation techniques in a probably nonneutral world is of course a more challenging issue, which has close ties with how one may subsequently interpret and use parameter estimates. To be efficient, a direct test on whether the reality is neutral would better avoid the noise induced by parameter estimation and directly rely on the data. A test of this kind is still awaited in ecology while experience from population genetics has taught us that tests would probably be of limited power (Ohta and Gillespie, 1996). Moreover, we believe that ‘yes or no’ tests would probably not be the most useful contribution of neutrality in the development of quantitative community ecology, and that much more is to be learnt from detailed analyses of the departures (e.g. plot by plot or species by species) subsequent to model calibration. For instance, looking for covariation between parameter values and ecological variables (as attempted by Jabot et al., 2008), which is unexpected under the most common set of neutral assumptions is a very intuitive, albeit potentially fruitful way to apply a neutral approach to real-world data. 6. Conclusion and prospects Community ecology is progressively equipping itself with a coherent quantitative theory applicable at different scales with the introduction of neutral dynamics. Although the relevance of neutral models is still hotly debated in the light of available empirical evidence (Gilbert and Lechowicz, 2004; Alonso et al., 2006; Dornelas et

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al., 2006; Leigh, 2007; Volkov et al., 2007), there is little doubt that improved versions of the neutral theory may help increase awareness about the role of both stochasticity and spatial constraints in community dynamics. Furthermore, as underlined by Chave (2004), one cannot neglect that competitive tradeoffs in certain highly diverse communities (e.g. tropical rainforests), can make most of the species appear equivalent where per capita demographic characteristics is concerned and neutral models can be relevant tools to understand the dynamics of their relative abundances. An equilibrium theory in community ecology, similar to the Hardy–Weinberg principle that is found in the field of population genetics, can be an effective null hypothesis, which in spite of its apparently simplistic assumptions, may yield many non-trivial comparative results if properly applied to diversified biogeographical contexts through well-mastered calibration methods. However, the most available data type that we have emphasized in this paper, namely species composition in field plots bear intrinsic limitations. And, fortunately, enriched datasets featuring information on species’ phylogenies and/or life traits along with enhanced environment measures are being progressively assembled and collected. This ongoing, highly needed effort of data acquisition may help develop new approaches to understand community structure. But any substantial progress has to be accompanied by increasingly focussing attention towards sampling designs and to the nature of information to be collected. The introduction of parsimoniously refined versions of existing neutral community models could also play a big role in orienting data acquisition by providing new insights into processes shaping the taxonomic composition of diversity-rich biomes.

Acknowledgements This research was supported by grants from the Ministère de l’Ecologie et du Développement Durable (EcoTrop 05 OSDA) to CBR, PC, RP and FM, and partly by the program BRIDGE (ANR Biodiv 06) to PC and RP. CBR’s Ph.D. is funded by a BSTD grant from the IRD (DSF Dept.), France. We would like to thank two anonymous reviewers for their helpful comments.

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