From genetic neighbourhood to local community : Estimating a key parameter of the Unifi ed Neutral Theory of Biodiversity

Issues related to the size and nature of what is termed a ‘‘local community’’ in a continuous metacommunity under unifi ed neutral theory are explored. Developments in the parallel fi eld of neutral genetics theory allow the size of genetic neighbourhoods in the context of isolation by distance processes to be calculated. These suggest a way to estimate the geographical size of an equivalent form of local community in ecology. It is derived from the estimated spatial autocorrelation value at zero geographical distance and this is obtained using the distribution of the relative frequencies of species in fi eld samples separated by varying geographical distance. A data set of 62000 beetles collected for other purposes, and consisting of beetle samples collected by fogging 52 trees in Nothofagus spp. forest and 24 trees in Araucaria forest distributed in southern Chile were used to demonstrate the proposed methods. Local communities had estimated effective community sizes (Je) of ~12000 individuals in each forest type for predators and ~30000 for phytophages/xylophage beetle communities. These numbers can be used in available software to estimate the number of species and the distribution of specimens between species in a local community. The geographical size of a local community can be used in other ways. Firstly, because it is related to the distance moved between birth and breeding, it provides information on the geographical scale of community processes; it gives a measure of the scales over which speedy recolonization is possible and scales over which breaks in the habitat cannot be easily crossed. Secondly, replicate samples geographically separated by less than the area of a local community are not independent, but pseudoreplicate measures of the characteristics of the metacommunity.


INTRODUCTION
Over the past decade explorations of the conceptual basis of community ecology have been enriched by the development of a theoretical model called the unified neutral theor y of biodiversity and biogeography (hereafter, unified neutral theor y).This seeks to predict the number of species and relative distribution of individuals between species in a community (Hubbell 2001, Chave 2004, Bell et al. 2006, Leigh 2007) assuming only stochasticity in the birth and death of individuals and the functional equivalence of species are relevant.Unified neutral theor y as developed by Hubbell can be used to make mathematical predictions of the number of species and the shape of the species relative abundance curve of both the metacommunity and of a 'local community' within this metacommunity (Hubbell & Borda de Agua 2004).
In theor y, predictions of the shape of the metacommunity cur ve can be made from knowledge of the underlying variables, the fundamental biodiversity number (θ), migration rate (m) from the metacommunity into local communities, and community size (J; number of individuals).θ and m can be estimated using maximum likelihood methods and the obser ved species abundance cur ves (Etienne 2007(Etienne , 2009)).As well as examining the fit of the theoretical cur ve for the metacommunity to field data, it is possible to predict the shape of the cur ve for a local community (Hubbell & Borda de Agua 2004) from the shape of the cur ve for the metacommunity and the effective size of the local community, if this is known.
The origin of unifi ed neutral theory is the mathematically equivalent theory in population genetics, the neutral theor y of molecular evolution (Kimura 1983).Two main models of the geographical structure of a species' local populations were used in this development: fi rstly, a metacommunity structure consisting of a series of interconnected but distinct local ('island') populations, and, secondly, a continuous population in which gene fl ow was restricted by isolation by distance processes.The island model was used by Hubbell (2001) in the development of his form of the unifi ed neutral theory and will provide the framework used here.The isolation by distance model in a continuous community is also relevant to the study of str ucturing in widespread natural communities (Chave & Leigh 2002).Of critical interest is the effective size of a local community in both scenarios as it is an essential parameter, for example in modelling community abundance curves.
In genetics the issue of estimating the size of a local population was addressed by Wright (1946Wright ( , 1969) ) and Malecot (1948), who developed the concept of a 'genetic neighbourhood'.This was defined as the 'population of a region in a continuum from which the parents of individuals born near the centre may be treated as if drawn at random'.That is, it is related to the distance individuals move between birth and breeding (in most animals, more generally, between the same point in the life cycle in consecutive generations).Wright showed that in a two dimensional continuum it was an area with a radius that could be measured as the variance of the distances moved by individuals between birth and breeding.When the size of such an area is multiplied by the density of individuals within it, the result is an estimate of N e , the ef fective population size, a key concept in neutral genetics theor y.These concepts, though developed in the field of population genetics, can be transferred to community ecology within the framework of unifi ed neutral theory as the mathematical basis is the same (Hubbell 2001).In this context, N e would become 'J e ', following Hubbell's terminology, the effective size of a local community within the metacommunity, and the geographical size of a 'genetic neighbourhood' becomes the 'geographical size of a local community'.
Thus, in unified neutral theor y, while J m , the size of the metacommunity, remains unchanged, J, the size of a local community or patch under an island/metacommunity model is replaced by J e the effective size of a local community at a particular location in a continuous community.Mathematically, J e is equivalent to N e in the development of the original genetic theory (Hu et al. 2006), though 'community size' (actually sample size) has been used as if equivalent, but incorrectly, in fi eld studies of continuous communities.

Measuring the size of a local community
Actually measuring the geographical size of a local community by measuring the variance of the distances moved by individuals of many species between birth and breeding is ver y diffi cult, if not impossible.Other approaches to measuring neighbourhood size have been developed in genetics (e.g., Broquet & Petit 2009, Guillot et al. 2009), and these can be transfer red to community ecology as the mathematical assumptions are the same (Hu et al. 2006).One approach is based on the obser vation that the isolation by distance processes that underlie the str ucturing of communities result in geographically adjacent samples being more similar than would be expected if they were samples taken at random across the geographical range of the metacommunity.As a consequence, increased spatial autocorrelations will be found between samples of species as the geographical distance separating the samples is reduced.
While various ways of estimating the geographical size of a genetic neighbourhood from this relationship have been explored (Epperson & Li 1996, Epperson et al. 1999, Hardy & Vekemans 1999, Rousset 2000, 2008, Vekemans & Hardy 2004, Epperson 2005, 2007), recently an equation that predicts the size of a neighbourhood from the level of autocor relation (as Morans I) in allele frequencies in the 'zero' geographical distance class has been developed (Epperson 2005, Rousset 2008).The geographical size of a local community can be measured in the same fashion by using species frequencies in each of a series of fi eld samples rather than allele frequencies in the analysis.
The purpose of this paper, then, is to describe a method for estimating the geographical size and effective size of a local community and to demonstrate it using data on the str ucture of beetle communities in Nothofagus and Araucaria forest in Chile (Arias et al. 2008, Richardson & Arias 2011).

METHODS
The data set developed by Arias et al. (2008) and Richardson & Arias (2011) and used to analyse the community structures of the coleopteran faunas of temperate Chilean forests was used in this study.
Data sets of the beetles collected from 24 Araucaria (Araucaria araucana [Molina]) trees and 52 Nothofagus trees were used.The Nothofagus trees were of a combined data set from the very similar Nothofagus dombeyi (Mirb.)Blume, N. obliqua (Birb.)Blume, and N. nitida (Phil.)Krasser.The analyses were carried out using beetles from two different feeding guilds (predators and xylophages plus phytophages) considered separately.
To calculate the geographical size of a local community, the data for each species was transformed to a relative proportion of the total number of specimens collected in the sample (i.e.equivalent to calculating allele frequencies).To ensure the data were independent, only species present on at least two trees, but two or more less than all trees, were used as recommended by Epperson (2005).Spatial autocorrelations were calculated for each beetle community using Moran's I and the program PASSaGE 2 (Rosenberg 2007).For the Nothofagus data, ten distance classes containing equal numbers (~57 pairs/class) were used, except for the zero distance class where only pairs separated by 0-1 km were used.For the Araucaria data six equal classes were used (~47 pairs/class), again except for the zero distance class where only pairs separated by 0-1 km were used.Correlograms showing means and 95 % confi dence intervals of I were calculated for each geographical distance class over all species, with the longest distance class being excluded from the statistical analysis (Legendre & Legendre 1998).
The number of classes does not affect the I value for the zero class however the number of pairs and the geographical size of this class can radically affect the estimate of I and its CI.Zero class pairs should not be geographically separated by more than the diameter of a local community area.A pilot study using sampling points selected to give a series of logarithmically increasing distance classes may be necessary to give a fi rst estimate the diameter of a local community.
The value of Morans I for the zero distance class was then used to calculate the geographical size of the local community as described by Rousset (2008, after Epperson, 2005) by using the formula: ln (Nb) = 1.11-1.13ln (I).
Where I is the value of Moran's I in the zero distance class and Nb is the area of the local community.In the demonstration study, where the sampling unit was a fogged tree, local community area would be estimated as 'number of trees' (actually the area covered by this many trees).In most studies, it would be 'number of quadrats' (actually the total area of this number of quadrats).Local community size (J e ) was then calculated as density (specimens per tree or quadrat) times local community area (number of trees or quadrats in the area).
Values for the fundamental biodiversity number (θ) and fundamental dispersal number (I) were estimated as described by Etienne (2007) from the data sets using maximum likelihood methods via the software provided as a supplement to Etienne (2007).This package assumes constant local population size and migration rates: more recent software packages take account of variation in these parameters (Etienne 2009).Using the combined sample size (N) and the value of I, m can be estimated from the formula: m = I/(I + N-1) (Etienne 2007).
Given these estimated values for θ, m, and J e , programs developed by Hubbell & Borda de Agua (2004), allow the number of species and the shape of the species abundance curve to be predicted for the metacommunity and thence for the local community.The local community program also provides confi dence limits for the estimates.It was assumed that 14000 generations had passed since the last ice-age (Ashworth & Hoganson 1993), and the metacommunity size was 99000000 (the largest the software allowed, though this is a gross underestimate) and this, along with a presumed turnover of half of the individuals per year, were used in the modelling.Sensitivity analysis using the ± 95 % CI values for J e and 10 % variation in metacommunity size showed only minor effects on the apparent distribution of individuals between species (results not shown).The predicted distribution of specimens between species was then compared with the fi eld data.

RESULTS
The relationship between Morans I and geographical distance for predators in the Nothofagus beetle fauna is shown in Fig. 1.It can be seen that, as predicted, I is most positive in the fi rst (0-1 km distance) class.A similar shaped cur ve (not shown) was found for each of the other three sample sets.The zero distance means and confidence values for I are given in Table 1, as are the estimates of the number of trees in a local community and the effective size of J e .The confi dence inter vals are wider in the Araucaria forest estimates because of the smaller number of pairs available.The numbers of trees in a local community are similar in all four cases, as are the estimated sizes of J e for predators in the two forest types.Because of differences in density (specimens per tree), the number of specimens in a local community for phytophages/xylophages is lower in the Nothofagus forest than in Araucaria forest.
The estimated values for θ and m for each community are given in Table 1.It can be seen that the value of θ for the Araucaria communities is less than half that found for the Nothofagus communities whilst the estimates of migration rates from metacommunities into local communities are similar in the two forest types but, as expected (Arias et al. 2008), higher for predators than for phytophages and xylophages.
A comparison of the obser ved species abundance cur ve for the metacommunity for predators living on Nothofagus and the shape of the cur ve predicted from θ shows they are similar (Fig. 2).The shape of the predicted local community species abundance cur ve was then calculated based on the shape of the predicted metacommunity cur ve and the estimated values of J e and m.These results are also shown in Fig. 2. It can be seen that the predicted structure of the local community  1.

Fig. 2: Fit of observed predator community data for
Nothofagus forest to predicted metacommunity and local community species abundance cur ves.These were developed by modelling using the values for θ, m and J e given in Table 1 and the programs of Hubbell & Borda de Agua (2004).
Ajuste de los datos observados de la comunidad de coleópteros depredadores para los bosques de Nothofagus, y de las curvas de abundancia de las comunidades locales.Estos fueron desarrollados con el modelado utilizando los valores de θ, y Je m en la Tabla 1 y los programas de Hubbell & Borda de Agua (2004).
dif fers from that of the metacommunity and of the field data.Unfor tunately the sampling sites, which were selected for other reasons, did not include a set of trees from within the same local community that would provide sufficient specimens to allow a species abundance cur ve for an actual local community to be estimated.It should be noted that, though the predicted metacommunity cur ve was similar to that for the field data, the predicted number of species was a five fold over-estimate (Richardson & Arias-Bohar t 2011).Similar patterns were found in the results for all four analyses.

DISCUSSION
The method proposed here to estimate the geographical size of a local community works as expected and is straight-for ward to apply.The estimates have narrow confi dence limits (Table 1) allowing them to be used in modelling and practical management (see below).As methods of estimating J e as well as θ and m are now available, it is possible, using the programs provided in Hubbell & Borda de Agua ( 2004), to calculate the metacommunity and local community species abundance curves predicted under unifi ed neutral theor y in a continuous community (Fig. 2) and the number of species expected in each case.The methods used here therefore would allow the null predictions of the neutral model to be tested against fi eld data.It is noteworthy that, though the theoretical species abundance curve was similar to that of the fi eld data, the predicted number of species was grossly in error (modelled 1451, estimated using Chao 1362).
There are, however, further issues to be considered when comparing the predicted and observed curves.All the relative species abundance curves based on fi eld data showed a ver y large excess of species represented by single specimens when compared to equivalent curves for most other groups (e.g., Hubbell 2001).Similar results have been found, however, in studies of large tropical arthropod faunas (e.g., Morse et al. 1988, Stork 1997).The causes of this anomaly relate to failures to meet the underlying assumptions made when collecting and interpreting such field data sets.These failures include: undersampling bias, community disequilibria, and combining data for species that are not ecologically equivalent (Coddington et al. 2009, Richardson & Arias 2011).It is likely, for example, that the sample from Nothofagus forest (already ~46000 individuals) would need to be fi ve times larger before the number of species and the shape of the species accumulation cur ve stabilised (Richardson & Arias 2011).These issues would need to be taken into consideration when collecting data sets to test neutral theory predictions.
The use of spatial autocor relations to calculate neighbourhood size in this way has been considered in the genetic literature and there are at least two further issues that need to be considered in designing such studies.Firstly other factors may affect the structure of a community with increasing geographical distance, for example clinal variations in rainfall or temperature.Such factors would increase the value of I in the zero distance class, reducing the predicted geographical size of a local community.Secondly, Rousset (1997) argues that there are theoretical reasons for restricting the geographical distances between the furthest apart samples in such an analysis to approximately twenty times the size of a neighbourhood.While the Chilean forests used might be considered to constitute single entities, the sample sets used in the present study to explore the methodology were collected for other reasons and were separated by distances of up to 600 km.The experimental design was not therefore suitable for other than a demonstration study and to obtain fi rst estimates of local community size.In the future, a targeted study of such communities would need to be based on a series of samples collected over much shor ter distances, distributed so as to give reasonable numbers of comparisons in each distance class, including the zero distance class and each of a suffi cient size to remove the problem of undersampling (Coddington et al. 2009, Richardson & Arias 2011).It also will need to include suffi cient material from a single local community to provide the curve necessary to test the validity of the species abundance curve predicted for a local community.The distribution of sample sites should also be chosen so as to reduce the ef fects of confounding variables like temperature, rainfall or soil type.The result of such a study would likely be larger local community sizes (i.e.lower values for I) for Chilean forests than those estimated here.
The geographical size of a local community has several attributes that can be conveniently used in practical ways unrelated to the estimate of species abundance curves.Firstly, because it is related to the distance moved between birth and breeding, it provides information on the geographical scale of community processes; it gives, for example, a measure of the scales over which speedy recolonization is possible and scales over which breaks in the habitat cannot be easily crossed.Such information may also be useful in deciding the size of areas that should be burnt in hazard reduction burns or in setting cutting coupe sizes and the gaps between them.Secondly, replicate samples geographically separated by less than the size of a local community, are not independent estimates of characteristics of the metacommunity but of a local community.That is, if samples are taken from within the same local community they are pseudoreplicate measures of the characteristics of the metacommunity.If samples are taken fur ther apar t than the geographical size of a local community, they are not replicate measures of the characteristics of a particular local community.Similarly the distances separating samples when estimating beta diversity need to take account of local community size.
The design of sampling regimes and the estimation of neighbourhood size from spatial structuring are matters actively being explored in genetics at present and the results will be relevant to the design of community studies.At this stage no clear conclusions have been reached, though, no doubt, matters will be clarifi ed over the next few years (Broquet & Petit 2009, Guillot et al. 2009).

Fig. 1 :
Fig. 1: Relationship between spatial autocorrelation (as Moran's I) and geographical distance for the predator beetle communities of Nothofagus forest.Mean and 95 % C.I. shown.Information for the zero distance class is summarised inTable 1.

TABLE 1
Summary of the results for beetle communities in Nothofagus spp.andAraucaria forests in Chile.Resumen de los resultados de comunidades de coleópteros en bosques de Nothofagus spp.y en Araucaria en Chile.