The zero-sum assumption in neutral
biodiversity theory
Rampal S. Etienne a,, David Alonso b
, Alan J. McKane c
a Community and Conservation Ecology Group, Centre for Ecological and Evolutionary Studies, University of Groningen, P.O. Box 14, 9750 AA Haren, The Netherlands
b Ecology and Evolutionary Biology, University of Michigan, 830 North University Av, Ann Arbor, MI 48109-1048, USA
c Theory Group, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK
Abstract
The neutral theory of biodiversity as put forward by Hubbell in his 2001 monograph has received much criticism for its unrealistic simplifying assumptions. These are the assumptions of functional equivalence among different species (neutrality), the assumption of point mutation speciation, and the assumption that resources are continuously saturated, such that constant resource availability implies constant community size (zero-sum assumption). Here we focus on the zero-sum assumption. We present a general theory for calculating the probability of observing a particular species-abundance distribution (sampling formula) and show that zero-sum and non-zero-sum formulations of neutral theory have exactly the same sampling formula when the community is in equilibrium. Moreover, for the nonzero-sum community the sampling formula has this same form, even out of equilibrium. Therefore, the term ‘‘zero-sum multinomial (ZSM)’’ to describe species abundance patterns, as coined by Hubbell [2001. The Unified Neutral Theory of Biodiversity and Biogeography, Princeton University Press, Princeton, NJ], is not really appropriate, as it also applies to non-zero-sum communities. Instead we propose the term ‘‘dispersal-limited multinomial (DLM)’’, thus making explicit one of the most important contributions of neutral community theory, the emphasis on dispersal limitation as a dominant factor in determining species abundances. r 2007 Elsevier Ltd. All rights reserved.
Keywords: Biodiversity; Neutral model; Species abundances; Allele frequencies; Ewens sampling formula; Etienne sampling formula
1. Introduction
Biodiversity is not only determined by the number of different entities (species in community ecology or alleles in population genetics), but also by the abundance of these entities (Magurran, 2004). Traditionally, species abundances have been mostly studied for competitive communities (Hubbell, 2001). Recently, they have also been heralded as important determinants in food webs (Cohen et al., 2003). Abundance data are relatively easy to collect, particularly in community ecology, and thus are good candidates to provide information on the processes that determine biodiversity. For many decades models have been proposed to describe and explain observed patterns in species abundances or allele frequencies (Fisher et al., 1943; Preston, 1948, 1962; MacArthur, 1957, 1960; Pielou, 1969, 1975; Ewens, 1972; Sugihara, 1980; Tokeshi, 1990, 1993, 1996; Engen and Lande, 1996a,b; Diserud and Engen, 2000; Dewdney, 1998, 2000; Hubbell, 2001; Harte et al., 1999; Etienne and Olff, 2005), but consensus on a single adequate model has not been reached. This has even led some scientists to claim that abundance data cannot distinguish between different models (Volkov et al., 2005; but see Etienne et al., 2007). Although it is true that patterns never uniquely imply process (Cohen, 1968; Clinchy et al., 2002; Purves and Pacala, 2005), scientific progress can still be made by testing whether different hypotheses on plausible processes can predict observed patterns. As for all scientific tests, no theory can be proved to be true, but inadequate theories can certainly be rejected. However, in comparisons of alternative abundance models, a solid sampling theory that provides the likelihoods of these models given the data (i.e. sampling formulas) has been lacking (Chave et al., 2006). In this paper, we present such a sampling theory. For convenience we will speak only of species abundances in an ecological community, but our results also apply to allele abundances (frequencies) in a population.
Our sampling theory is inspired by neutral theory, as this theory provides a null model for abundance distributions, but we stress that the theory extends beyond neutral theory. Nevertheless, we use our sampling theory here to settle a debate in neutral theory concerning the zero-sum assumption (Hubbell, 2001). Neutral theory has three basic assumptions: the neutrality assumption (functional equivalence among different species), the point mutation assumption (speciation takes place by point mutation) and the zero-sum assumption, which have all received much criticism. Here we do not enter the debate concerning the first two assumptions, but focus on the third. The weak formulation of the zero-sum assumption states that individuals of different species in a community that are limited by their environment (either because of limited shared resource, e.g. space or light, or because of natural enemies that restrain the otherwise unbridled growth of the species) will always saturate the environment, i.e. there are never fewer individuals than allowed by the environment. The strong formulation states that the limitation by the environment (e.g. the number of resources or the number of natural enemies) is constant and that therefore the total number of individuals competing for the resource is also constant. While there is some evidence for this assumption (Hubbell, 1979, 2001), this assumption (in either form) has been felt to be too restrictive (Volkov et al., 2003; Poulin, 2004). Here we show, however, that the zero-sum assumption is not crucial for one of the most important predictions of the theory, the sampling formula, i.e. the formula that gives the probability of observing abundances n1; ... ; nS in a sample of J individuals. This is not trivial, as the characteristic time scales of zero-sum and non-zero-sum models are very different (see below).
We will first present our general sampling theory, particularly its fundamental assumptions and the general sampling formula based on these assumptions. We will then formulate a general master equation for a single species and show how it can lead to a general sampling formula for multiple neutral species that have either fully independent (non-zero-sum) or fully dependent (zero-sum) dynamics. For the latter case we will introduce the subsample approach. Up to this point, our treatment will be quite general in the sense that it does not rely on any assumptions concerning the dynamics of each species such as intraspecific competition, or the source of new species such as speciation or immigration. From this point we will proceed by inserting such details of the Hubbell model for the metacommunity and for the local community and we will subsequently link the two community scales. We do this for the non-zero-sum case both in and out of equilibrium and refer to the Appendix for the zero-sum equilibrium case. We will demonstrate that the sampling formula for the zero-sum case in equilibrium also applies to the non-zero-sum case, both in equilibrium and out of equilibrium. We end with a discussion of our results. For the reader’s convenience the mathematical symbols used in this paper are summarized in Table 1.
2. A sampling theory
The metacommunity concept represents a powerful framework to study the relative importance of local species interactions and dispersal in determining community composition and dynamics (Leibold et al., 2004; Holyoak et al., 2005). Our sampling theory is based on a classical metacommunity structure. First, we assume that local communities form a metacommunity by global migration (global dispersal, propagule mixing and propagule rain (see Fig. 1)). Such a migration assumption has a long tradition in metapopulation ecology (Gotelli and Kelley, 1993; Bascompte et al., 2002), and has been used to gain insight into the processes underlying the distribution of species diversity from local to global spatio-temporal scales (MacArthur and Wilson, 1967), has inspired current neutral theory (Hubbell, 2001), and underlies recent studies on community similarity under neutrality (Dornelas et al., 2006). Second, as in Hubbell’s (2001) approach, our theory builds on the assumption that ecological and evolutionary time scales are very different and can be decoupled. Under these assumptions, our general framework requires the following three general steps:
1. We assume that ecological and evolutionary processes have determined metacommunity composition (species abundances) at the largest spatio-temporal scale. Neutral speciation, adaptive speciation and trade-off invariance are possible mechanisms. We usually evaluate the stationary distribution of species abundances emerging from these processes. We do not require that this distribution is frozen-stable. We do require, however, that processes at the biogeographic level are at a much longer temporal scale relative to the process affecting the ecological assembly of local communities. (See Vallade and Houchmandzadeh, 2006 for a relaxation of this assumption.)
2. In local communities (or islands) mainly local ecological processes, but not speciation, are at play. Examples of these processes are dispersal-limitation, density dependence, habitat heterogeneity and species differential adaptation to different habitats.
3. Sampling formulas contain information about both levels of description and about the sampling process. They enable us to empirically evaluate the imprint of evolutionary processes at the biogeographic scale and ecological processes at the local scale on the observed diversity in our local samples.
Fig. 1. Classical metacommunity structure. Propagules from all local communities are assumed to completely mix in the metacommunity and return to the local areas in the form of a propagule rain.
4. Discussion
Since Hubbell published his monograph in 2001, the evaluation of neutral theory has encountered three main difficulties. First, neutral theory in community ecology has been formulated using a variety of models (Chave, 2004; Alonso et al., 2006). Second, most studies based on abundance data have focused on the expected abundance curve, better known as the species abundance distribution (McGill, 2003; Volkov et al., 2003; Turnbull et al., 2005; Volkov et al., 2005; Harpole and Tilman, 2006). Although this curve is a signature of community structure (Pueyo, 2006), recent work has emphasized the difficulty of identifying the underlying processes leading to this pattern if a snapshot of the abundance curve is the only information available (Chave and Leigh, 2002; Bell, 2005; Purves and Pacala, 2005; Volkov et al., 2005; Etienne et al., 2006). Third, alternative niche theories were only loosely defined (McGill, 2003); they did not generally provide the probability of obtaining data through sampling assuming the alternative niche model at play. As a consequence, it has been practically impossible to perform sound model selection (Chave et al., 2006). The sampling theory presented in this paper aims to find a solution to these three difficulties. We discuss them in order.
First, we have stressed that Hubbell’s basic model (Hubbell, 2001) plays a central role in the theory. This model was initially understood as a mainland–island model (MacArthur and Wilson, 1967; Hubbell, 2001) with zerosum dynamics. The metacommunity was simply regarded as a regional species pool (Etienne, 2005) with the local community either separate from (real mainland–island) or embedded in the metacommunity (continuous landscape), as pointed out by Alonso et al. (2006). Here we have carefully examined the assumptions of this model in two ways. (i) Most importantly, we have demonstrated that all analytical results on the species abundance distribution apply regardless of zero-sum dynamics. This was already suggested by Caswell’s simulations (Caswell, 1976) simulations, argued analytically by Rannala (1996) for the birth–death–immigration model in a population genetics context and hinted at by Volkov et al. (2003, 2005) for the expected abundance curve, but we have provided a more rigorous proof for the full sampling formula (that involves both immigration to the local community and speciation in the metacommunity) using a general sampling theory. Because natural communities are not constrained by a strict zero-sum rule, this result increases the degree of First, we have stressed that Hubbell’s basic model (Hubbell, 2001) plays a central role in the theory. This model was initially understood as a mainland–island model (MacArthur and Wilson, 1967; Hubbell, 2001) with zerosum dynamics. The metacommunity was simply regarded as a regional species pool (Etienne, 2005) with the local community either separate from (real mainland–island) or embedded in the metacommunity (continuous landscape), as pointed out by Alonso et al. (2006). Here we have carefully examined the assumptions of this model in two ways. (i) Most importantly, we have demonstrated that all analytical results on the species abundance distribution apply regardless of zero-sum dynamics. This was already suggested by Caswell’s simulations (Caswell, 1976) simulations, argued analytically by Rannala (1996) for the birth–death–immigration model in a population genetics context and hinted at by Volkov et al. (2003, 2005) for the expected abundance curve, but we have provided a more rigorous proof for the full sampling formula (that involves both immigration to the local community and speciation in the metacommunity) using a general sampling theory. Because natural communities are not constrained by a strict zero-sum rule, this result increases the degree of Second, although we agree that the abundance curve may not be enough to elucidate underlying processes, our work on species abundances is based instead on what we call a general sampling formula, the central expression of the theory (1). This is a multivariate abundance distribution that may encode much more information than the simple abundance curve, the expected number of species at each abundance level. Different processes can lead to similar, perhaps indistinguishable, average abundance curves (Volkov et al., 2005). Because this curve is theoretically obtained by averaging the full sampling formula (Etienne and Alonso, 2005), it can be seen as a first moment of a multivariate distribution. But the first moment of a distribution does not describe the distribution completely. So, it is possible that two processes lead to the same average abundance curve (Volkov et al., 2005), but the underlying multivariate distributions may be different (Chave et al., 2006). Moreover, because the multivariate distribution keeps track of individual abundances, it can be extended to multiple samples across space or time. To date, most studies on species abundances dealing with neutral theory have only used the abundance curve (McGill et al., 2006; Pueyo, 2006), rather than this powerful multivariate representation of the community; hence conclusions that species abundance distributions contain little information are premature. The sampling formula under neutrality for a single sample is the Etienne sampling formula (Etienne, 2005). It has recently been extended to multiple samples across space which conveys more information (Etienne, 2007; Munoz et al., 2007). Also, most studies have analyzed only snapshots of the community rather than dynamical data (but see Gilbert et al., 2006), which may contain more information as well. Our framework can be easily extended to deal with this type of data (38). When we sample the same system at T different times (where the sample size may vary), yielding a time series of multivariate abundance observations Dt1 ; Dt2 ; ... ; DtT , we can calculate the likelihood of this time series for the model that is assumed to describe the assembly process, by simply multiplying conditional transitional probabilities: ...
Outline
Abstract
Keywords
1. Introduction
2. A sampling theory
2.1. Independent species
2.2. Dependent species: the subsample approach
3. Non-zero-sum neutral communities
3.1. Metacommunity
3.2. Local community
3.3. Linking the local community to the metacommunity
4. Discussion
Acknowledgments
Appendix A. The subsample approach: from binomial to multinomial
Appendix B. The subsample approach: zero-sum communities
B.1. Metacommunity
B.2. Local community
References
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