Measuring biodiversity across spatial scales in a raised bog using a novel paired-sample diversity index

Summary

1. Biodiversity is an inherently spatial phenomenon. It is determined by environmental heterogeneity and by spatially structured ecological processes such as disturbance, competition and dispersal. However, conventional biodiversity indices are based on discrete samples or pooled sets of samples without sufficient consideration of the spatial relationships between the samples.

2. Here we describe a new method for spatial analysis of species diversity, based on a paired-sample version of the widely used Gini–Simpson diversity index and its numbers equivalent. The index and its numbers equivalent are plotted as a function of lag distance between two samples along spatial and/or environmental gradients.

3. We demonstrate the potential of this approach by applying it to two transects of fine-scale (5 × 5 cm quadrat) vegetation data from sites with contrasting hydrology within a raised bog, where the location of each quadrat is accurately recorded and the height of the bog surface above the water table is measured using a terrestrial laser scanner.

4. Both transects have similar alpha-diversity as measured using the Gini–Simpson index, and the transition between alpha- and gamma-diversity occurs at similar length scales, suggesting that species aggregate at similar scales along both transects. However, the transect from the central bog dome has higher gamma-diversity than that from the bog margin, and shows more marked significant spatial structure at a length scale of 135–140 cm, corresponding to the typical hummock–hollow microtopography at the site. We show that beta-diversity at both transects can be attributable to both species clustering along the hydrological gradient, consistent with niche partitioning, as well as independent spatial aggregation of species that is not explained by hydrology.

5. Synthesis. The paired-sample diversity index described here is a potentially useful tool in detecting and attributing patterns of beta-diversity along both spatial and environmental gradients.

Introduction

The causes of patterns of biodiversity are a topic of considerable research interest in ecology and conservation science, and many studies seek to explain observed spatial variation in diversity in various landscapes through correlation with a range of different factors such as species pool, land cover types, disturbance regime, heterogeneity, patch geometry and area and anthropogenic influences (Pärtel & Zobel 1999; Münzbergová 2004; Pierce et al. 2007; Klimek et al. 2008; Marini et al. 2008; Reitalu et al. 2008). There is a growing awareness in such studies that spatial scale is a critical determinant of which drivers are found to influence diversity (Gering & Crist 2002; Økland, Rydgren & Økland 2008; Lang et al. 2009), and that conservation methods seeking to maintain diversity within regions or landscapes must target management at the appropriate scale. Functional explanations for diversity, such as niche differentiation, disturbance, or neighbourhood recruitment limitation are also expected to have characteristic spatial scales in different systems (Hurtt & Pacala 1995; Tilman 1999; Økland, Rydgren & Økland 2008; Murrell 2010).

Measures of diversity

Conventionally, measures of biodiversity (Whittaker & Woodwell 1969; Whittaker 1972) recognize a distinction between local, or alpha (α-) and regional, or gamma (γ-) diversities. Beta (β-) diversity is then the diversity between habitats or sampling units, a conceptual bridge between these two discrete scales. β-diversity is caused by the tendency of individuals of the same species to be aggregated together in space, whether individualistically or in clearly defined communities. It is also clearly a function of both the grain size at which α-diversity is measured and the degree of ‘patchiness’ of the sampled domain within which γ-diversity is measured. Jurasinski et al. (2009) have suggested the term ‘inventory diversity’ for both α- and γ-diversity, as they are identical measures differing only in the spatial scales at which they are applied. These scales are usually the smallest sampling unit of a survey, or ‘grain’ as defined by Dungan et al. (2002) and the entire sampled domain or ‘extent’. In most studies, α-diversity is assumed to be sampled comprehensively at the grain scale, whereas γ-diversity is approximated by pooling samples within the extent. For example, Martin, Moloney & Wilsey (2005) measured plant species α-diversity in 40 × 100 cm quadrats in prairie remnants and restoration sites and pooled data from eight quadrats from each site to estimate γ-diversity for each site.

Despite the widespread use of the concept, methods for quantifying β-diversity vary and the appropriateness of different methods, and of the concept itself, is much debated (Veech et al. 2002; Jost 2006; Jurasinski et al., 2009). Jurasinski et al. (2009) recommend the terminology ‘differential diversity’ to describe measures of compositional difference between samples, and ‘proportional diversity’ to describe comparisons between scales. As species distributions in natural environments tend to aggregate both in space and along environmental gradients, β-diversity may have two components:

• 1
  environmental variation, because of species aggregation along environmental or disturbance gradients within the domain, which may or may not have spatial structure at the given grain size;
• 2
  spatial aggregation that is independent of environmental variation, due to the inherent patchiness of species distributions, for example, determined by biotic processes such as competitive exclusion and recruitment limitation, clonality or stochastic population processes.

In the original definitions of Whittaker (1972)β-diversity was defined as the ratio β = γ/α. An alternative definition of γ = α + β has become widely used, allowing the additive partitioning of components of diversity (Lande 1996; Veech et al. 2002). This approach has the advantage that β shares the same units as α and γ, but suffers from the disadvantage that β is not independent of α, and therefore cannot be compared between sites (Jost 2007), and that β-diversity defined in this way does not recognize the degree of difference or similarity between samples. Diversity in this context is conventionally measured as species richness (simply the total number of species recorded in a sample), or with indices such as the Gini–Simpson index (Simpson 1949) or Shannon index (Weaver & Shannon 1949). However, both of the latter indices are constrained to values between 0 and 1, and may give non-intuitive results (Jost 2006). It has been argued that transforming such indices into true measures of diversity, with units commensurate with species richness, as ‘Hill numbers’, or ‘numbers equivalent’ (Hill 1973; Jost 2006) provides a more intuitive measure of diversity. Following Jost (2007), the Gini–Simpson index (and its paired-sample spatial equivalent) is here referred to as a diversity index and denoted by the symbol H; Jost’s numbers equivalent, or ‘true diversity’ (and its paired-sample equivalent) is denoted by the symbol D.

The concept of additive partitioning of diversity has been used to compare diversity across discrete spatial scales, and for attributing determinants of species diversity within different sized sampling units (Klimek et al. 2008). In the additive partitioning framework, α-diversity across a hierarchy of nested spatial scales can be expressed as α_x = α_x-1 + β_x-1, where diversity α_x measured at the xth level of a scale hierarchy is the sum of the mean α-diversity and the β-diversity between samples at the x - 1th level (Wagner et al. 2000). However, in addition to the theoretical drawbacks with the additive definition of β-diversity (Jost 2007), one clear drawback with this method is that the scales at which diversity is measured are subjective and defined by the sampling design. If species are spatially aggregated at a scale greater than the distance between samples at which α-diversity is measured, then samples may not be independent and pooled-sample estimates of γ-diversity will be biased. Furthermore, a sampling design in which samples is selected from within clearly identifiable communities or habitats will under-sample ecotones and edge habitats and potentially underestimate the average diversity of a landscape sector. This is particularly important as in many ecosystems transition zones between communities are regions of increased biodiversity (Smith et al. 1997; Kark et al. 2007).

The concept of β-diversity is often considered as a measure of species turnover along spatial or environmental gradients. Vellend (2001), however, has shown that the most frequently used methods of calculating it are independent of the distributions of species on either spatial or environmental gradients and recommends plotting similarity-distance graphs to show the rate of species turnover per unit distance.

Ecologists have long made use of spatial statistical methods including semivariogram analysis (Garrigues et al. 2006) and spatial autocorrelation (Fortin, Drapeau & Legendre 1989) to provide spatial equivalents of statistical properties of populations (in these cases correlation coefficients or variance), and to describe spatial structure and pattern in ecosystems. This approach has been extended to spatial patterns of diversity by plotting the spatial covariance of species richness (‘variogram of complimentarity’; Wagner 2003; Bacaro & Ricotta 2007). Condit et al. (2002) and Chave & Leigh (2002) calculate a ‘similarity function’ representing the probability that two trees separated by a given distance in a forest plot belong to the same species. This function is essentially a paired-sample version of the Simpson concentration, which is equal to unity minus the Gini–Simpson index of diversity. We propose a similar method for spatially explicit analysis of species diversity, based on a paired-sample version of the Gini–Simpson diversity index (Simpson 1949; Pielou 1969) and its numbers equivalent (Jost 2006). This method has the advantage that it is in units directly comparable to a commonly used measure of α- and γ-diversity. The method is suitable for systematic or random sampling designs and requires no prior assumptions about the spatial arrangement of communities other than the choice of appropriate grain size and spacing of the sampling unit and the domain of the sample. The method is also similar in approach to spatial statistical techniques such as variography (Fortin & Dale 2005), paired-quadrat variance methods (Schaefer & Messier 1994; Guo & Kelly 2004), similarity-distance graphs (Vellend 2001) and ‘variograms of complementarity’ (Wagner 2003) in that a characteristic of paired samples is plotted against the lag, or distance between samples in space and/or along a environmental gradient. Unlike the former approaches, the proposed method is based on analysing the spatial structure of diversity rather than patterns in the distribution of a single species or variable and measurements are in units commensurate with α- and γ-diversity.

Study system

To demonstrate the potential of this method we apply it to a fine-scale vegetation survey and hydrological dataset collected for a separate study. Vegetation data were collected at 5-cm intervals along two 10-m transects at plots of contrasting hydrology at a raised bog in Cumbria, UK. The main environmental gradient at the site, variation in surface height above the water table, was measured at each plot using hydrological data loggers in conjunction with a terrestrial laser scanner to accurately measure surface topography (Anderson, Bennie & Wetherelt 2010a). Intact peat bogs typically show distinct patterns in hummock-hollow topography which is associated with fine-scale patterns in species composition, particularly of Sphagnum species (Nordbakken 1996; Økland, Rydgren & Økland 2008). Interactions between individual Sphagnum plants necessarily occur at a fine spatial scale (<10 cm) and experimental studies have shown that whether species are grown in monocultures or mixtures (and hence the relative importance of inter- and intra-specific interactions) has marked effects on height increment and cover changes (Robroek et al. 2007). Previous studies at this site have shown that the characteristic scale of remotely sensed surface (shrub canopy structure and Sphagnum microtopography) reflects hydrological status (Anderson et al. 2010b).

Materials and methods

Calculation of diversity

We start with the definition of Simpson’s diversity or the Gini coefficient (Simpson 1949; Pielou 1969):

• (eqn 1)

where p_k is the proportion of species k in an infinite population, and S is the total number of species present. When applying the index to vegetation data, it is common practice to use the proportional cover or biomass of each species rather than the proportion of individuals; and hence an adjustment to prevent bias in a finite sample of individuals (Simpson 1949) is not applied here.

The Gini–Simpson diversity index, ranging from 0 to 1, represents the probability that two individuals chosen at random from the sample are from different species or, in the case of data expressed as proportional cover, the probability that two randomly chosen points within the sample area are occupied by different species. As with other frequently used indices of diversity it is a function of both the number of species present (species richness) and the frequency distribution of these species within the sample (species evenness). The average index of α-diversity across a set of n samples is:

• (eqn 2)

The diversity index of a pooled set of samples (γ-diversity) is

• (eqn 3)

Here we define an new index of diversity H(h) which is a function of separation (‘lag’) distance h. For vegetation cover data this can be easily interpreted as the probability that a pair of points within two samples a distance h apart are occupied by different species. For a pair of samples i and j, this is defined as:

• (eqn 4)

where p_ik is the proportional cover of species k in sample i and p_jk the proportional cover of species k in sample j. For a finite set of samples, the mean value of H at a separation distance h is

• (eqn 5)

where N(h) is the number of pairs of samples that are a separation distance h apart. When samples are equally spaced, for example along a transect or grid-based sampling system, H(h) is calculated at intervals of h corresponding to the sample spacing. Where sample spacing varies, for example in a randomized sampling strategy, H(h) is calculated at a series of lag distances h with N(h) the set of pairs of samples that are within a given tolerance of h. Plotting H(h) against h gives a plot analogous to a (semi)variogram in geostatistics (Fig. 1). Following Jost (2006), the ‘true’ diversity, in units commensurate with the ‘numbers equivalent’ of the diversity index, can be calculated as:

• (eqn 6)

Figure 1.  Conceptual plot of paired-sample diversity index D(h) against lag distance h for a hypothetical domain with fine-scale spatial structure. When h = 0, D(h) is equal to the mean α diversity within samples, D. As the lag distance between pairs of samples increases, the samples tend to become less similar and hence between-sample diversity increases. D(h) levels off to an approximation of γ diversity within the domain, D.

Key features of these plots can be identified:

When h = 0, i = j and N(h) = n; eqn 5 simplifies to eqn 2. The values of D(h) and H(h) when h is zero, are therefore equal to and , which is the mean within-sample or α-diversity (and conceptually equivalent to the variogram ‘nugget’, or non-spatial variance; Issacs & Srivastava 1989). This value is a function of the size and shape of the sampling unit and hence grain size (i.e. quadrat size). D(h) will usually tend to increase with h due to spatial segregation of species – as the distance between paired samples increases, their species composition becomes less similar. At a separation distance h, where the species composition of plots are truly independent, i.e. there is no spatial autocorrelation at this length scale, the probability of resampling a species from paired samples is equal to the probability of resampling from an infinitely large pooled set of samples; H(h) becomes an unbiased sample of H and D(h) an estimate of γ-diversity. In a domain where spatial patterns in species distribution consist solely of aggregated pseudo-random patches (and not, for example, regular patterns or more complex spatial pattern), the value of h at which D(h) reaches the sill (analogous to the ‘range’ of a variogram) represents the characteristic length scale above which no further pattern in β-diversity is recognized.

We assessed the significance of spatial patterns in the plots using a Monte-Carlo bootstrap method. For each lag distance h, N(h) pairs of samples were selected at random from the entire data set, and values of H and D were calculated for these sets of paired samples. As these pairs of samples were selected randomly, without reference to the distance between samples, the calculated value of H, the average probability that two points within the samples are occupied by different species, is an unbiased estimate of H, the probability that any two points in the domain are occupied by different species. These random-pair values of H and D therefore provide estimates of γ-diversity. This procedure was repeated 1000 times for each value of h and the 2.5% and 97.5% quantiles of this distribution were plotted on the diversity–distance graphs. Where the calculated values of H(h) and D(h) fall outside these limits, the paired-sample diversity differs significantly (P < 0.05) from γ-diversity. At these lag distances spatial structure exists and contributes towards β-diversity.

The method can be generalized to incorporate distances between paired samples along further spatial or environmental axes. In the case of raised bogs, niche differentiation between hummock and hollow species occurs along a gradient in the vertical position of samples (i.e. the height of the surface above the water table; Soro, Sundberg & Rydin 1999; Økland, Rydgren & Økland 2008). In this study we also plot paired-sample diversity as a function of both the lag distance between samples h and the measured difference between samples in height above the water table between samples z; hence h represents the separation between paired samples in horizontal space, and z the separation in vertical space (the hydrological gradient):

• (eqn 7)

Similarly,

• (eqn 8)

In this case H(h,z) may be plotted against both h and z to examine the effects of spatial and environmental components of diversity separately (Fig. 2).

Figure 2.  Conceptual plot of paired-sample diversity index H(h,z) against lag distances along the spatial (h) and environmental (z) axes for three hypothetical domains. (a) Spatial structure, as in Fig. 1, with no species aggregation along environmental gradient z; (b) species aggregation along the environmental gradient, but no spatial aggregation independent of this gradient (c) aggregation along both the environmental gradient and independent spatial structure.

Fitting surfaces

As with (semi)variograms, empirically derived models may be fitted to plots of H(h,z) and D(h,z). In this study we estimate H(h,z) as the linear sum of log-transformed variables h and z (in cm) with an interaction term:

• (eqn 9)

Values of a, b, c and d were fitted to each transect using the GLM procedure in R. v2.2.1, with H(h,z) as the dependent variable and log(z + 1), log(h + 1), and the interaction term as fixed factors. All terms and combinations of terms were tested for significance at P < 0.05, and final model selection between significant models was on the basis of the minimum model Akaike Information Criteria (AIC).

Field Site

Wedholme Flow (54.86^° N, 3.23^° W) is an ombrotrophic lowland raised bog of 780 ha situated on the Solway Plain, Cumbria, UK. Much of the site has been affected by historical peat cutting and recent commercial peat extraction, which ceased in 2002. As this date site management has included blocking ditches and other groundworks in order to maintain a stable high water table for the recovery and maintenance of peat-forming vegetation. This study focuses on a relatively intact peat dome of the north bog. Two plots with contrasting hydrological conditions were selected for this study; the first, the central bog dome transect, was located near the centre of the intact bog peat dome, where hummock–hollow topography is clearly defined. The second, the bog margin transect, has less clearly defined hummock–hollow topography and was located c. 30 m from, and roughly parallel to, the eastern edge of the bog dome where a sharp transition between raised bog vegetation and a modified lag, dominated by Betula pubescens carr woodland, occurs.

Vegetation survey

At each site, all species of vascular plant and bryophyte were identified and their fractional cover estimated in 200 5 × 5 cm adjacent quadrats along a 10-m transect running north to south. Quadrats were defined in a horizontal plane and cover was estimated with the ‘any-part’ system (Williamson, 2003). Nomenclature follows Stace (1999) and Hill (1992). The bog dome transect showed marked fine-scale hummock–hollow topography typical of an intact raised bog, with a low canopy of the dwarf shrubs Calluna vulgaris, Erica tetralix and Andromeda polifolia and the peat moss species Sphagnum capillifolium and S. magellanicum dominant on hummocks and S. tenellum and S. papillosum on lawns. The bog margin transect showed less marked hummock–hollow topography and a mixture of S. papillosum, S. magellanicum and S. tenellum under a low canopy of E. tetralix and C. vulgaris.

Measurement of surface depth to water table

The laser-scanning method used to determine the bog surface height above the water table has been described in detail elsewhere (Anderson, Bennie & Wetherelt 2010a) and is summarized briefly here.

At the mid-point of each transect a hydrological dipwell with a barometric pressure sensor was installed in March 2008. At each site, the depth to water table was calculated relative to the dipwell cap, whose horizontal and vertical location was determined from a differential global positioning system (DGPS) survey. This allowed water table depth to be expressed relative to other surveyed points in the vicinity. In order to accurately measure the height above the water table along the transect, fine-scale microtopographic data describing the peatland surface structure were collected using a close-range laser scanner (HDS3000; Leica Geosystems, San Ramon, CA, USA). The scanner tripod was elevated above the peatland surface on the flat-bed trailer of a tracked vehicle, with an instrument height of c. 3.5 m above the surface. At each test site scans were taken from three different viewpoints, towards a 10-m diameter region of interest centred on the dipwell (Fig. 3).

Figure 3.  (a) Plan view of laser scan. (b) Example 3D point cloud of bog surface.

During the laser scanning data capture, the tops of the dipwell caps were used as reference points to enable registration (linking up of scans) for each site. Three additional reference markers were positioned at the perimeter of the scan region. Their positions were measured using DGPS and proprietary software (Cyclone 5.4, Leica Geosystems) was used to register the three scans using the known position of the markers and the top of the dipwell as reference targets. This produced a combined point cloud referenced to the Ordnance Survey GB National Grid in the horizontal plane and metres above sea level in the vertical plane. Points within the 10-m diameter region of interest were selected and exported as ASCII text files for analysis.

A digital surface model (DSM) of the region of interest was created by selecting the minimum height of each point within a 5-cm grid, then using a smoothing function to smooth the surface using firstly the minimum value within a five-pixel window, then the mean value. The resultant DSM of the bog surface was used to derive the relative height above the dipwell water table of pairs of quadrats in this analysis. It should be noted that this value represents the height of the surface above the water table at the central dipwell – the actual height of the capitulum surface above the water table at a point may differ from these values due to fine-scale differences in capillary action associated with hummocks and hollows.

Results

Paired-sample diversity is plotted against lag distance h for both sites in Fig. 4. Calculated Gini–Simpson indices and numbers equivalent for α-diversity were similar between sites, H_α = 0.25, D_α = 1.33 for the central bog dome transect and H_α = 0.27, D_α = 1.37 for the bog margin transect. The calculated γ-diversity values were H_γ = 0.89, D_γ = 8.55 and H_γ = 0.81, D_γ = 5.24, respectively. Figure 4 shows the spatial versions of the Gini–Simpson index and Jost’s number’s equivalent for each site. The shaded band represents the Monte-Carlo estimate of γ-diversity in each case. Spatial structure is detected where the line falls outside the limits of the shaded area; in both cases spatial aggregation causes low diversity at length scales under 50 cm, as samples close together tend to be more similar in species composition. In the central bog dome transect the peak at 135–140 cm shows that samples this far apart tend to be highly differentiated in their species composition, a function of the well-developed hummock–hollow topography at this site.

Figure 4.  Spatial versions of Simpson concentration H(h) (a,c) and species-equivalent diversity D(h) (b,d) for the raised bog centre (top) and margin (bottom). Shaded areas show 95% intervals of Monte-Carlo random sampling of plots. H(h) represents the probability that two sampled points in space separated by distance h are occupied by different species; D(h) represents the equivalent in units of diversity. The shaded band is the Monte-Carlo estimate of γ-diversity; spatial structure is detected where the line falls outside the limits of the shaded area.

Three dimensional plots of H(h,z) against h and z, and fitted surfaces are shown in Fig. 5. In both transects, the lowest values (i.e. high probability of species reoccurrence) occur at low h lags, when samples are close in horizontal space, but for a given h lag values also increase with increasing z lags, showing that pairs of plots at similar heights above the water table share similar species, even when separated in horizontal space. Both sites show evidence of aggregation of species both in space and along the environmental gradient.

Figure 5.  Plots of paired-sample Gini–Simpson index H(h,z) against distance between samples in the horizontal (x) and vertical (z) for central bog dome (a,b) and bog margin (c,d). Scatter plots (a,c) show mean values for each combination of h and z; surfaces (b,d) show the fitted surface of the regression model.

Fitting a model surface to the empirical plot of D(h,z) gave a significant fit to all terms in eqn 9 for both transects (P < 0.05 in all cases; Table 1). In each case comparison of model AIC values suggested that the full model, including interaction terms, was the best model form. This implies that at both transects β-diversity is along both pure spatial and environmental gradients, and also influenced by an interaction between these two. The forms of the fitted surfaces are shown in Fig. 5.

Table 1.   Fitted coefficients, P-values and adjusted R² values for terms in fitted surfaces

	z	P-value	h	P-value	z × h	P-value	R^2(adj)
Column z represents the coefficient for log-transformed height above the water table; h represents log-transformed distance; z × h represents the interaction term.
Central bog dome	0.137	<0.001	0.107	<0.001	−0.029	<0.001	0.534
Bog margin	0.077	0.0328	0.106	<0.001	−0.021	0.0103	0.444

Discussion

Diversity patterns in bogs

In a study of Norwegian boreal swamp forests, Økland, Rydgren & Økland (2008) considered that the determinants of diversity within a given plot are (i) the size of the available species pool; (ii) the position of the plot along environmental gradients and (iii) the within-plot environmental heterogeneity, via β-diversity effects. They showed that fine-scale variation in microtopography was associated with β-diversity at fine scales for moss and liverwort species. In this study we have found that in both a central and a marginal bog community spatial structure at a length scale of <50 cm contributes to overall diversity through β-diversity effects. At both transect sites, species aggregation along a gradient in depth to water table was also identified as a component of β-diversity. Significant spatial structure also appears to emerge at a length scale of 130–140 cm in both transects (Fig. 4), although this is much more pronounced in the data from the central bog dome transect. Quadrats separated by this distance were less similar in their species composition than expected. The determined length scale of this patterning (130–140 cm) is characteristic of the size of hummock–hollow–pool complexes within this particular peatland, and hence probably reflects niche differentiation along the gradient in depth to water table. This is consistent with the more marked ‘hummock–hollow’ topography frequently observed at the centres of undisturbed bogs, with clear distinctions between patches of ‘hummock’- and ‘hollow’-adapted species of Sphagnum. As such the results presented shown here are consistent with the extensive survey of Soro, Sundberg & Rydin (1999), who found that undisturbed mires had not only higher species richness, but also lower niche overlap and a higher number of non-random associations than mires where the water table had been lowered by historical peat extraction.

Methodological issues

The novel paired-sample diversity index described here gives an insight into the spatial components of biodiversity and could be used to identify the characteristic scales (grain size and extent) at which α and β-diversity dominate. Furthermore, by incorporating the separation between paired samples along an environmental gradient (here height above water table) as well as horizontal distance, it becomes possible to distinguish between two components of β-diversity, species clustering along both spatial and environmental gradients. Further environmental gradients could be added to this type of analysis simply by increasing the number of environmental axes in eqn 7, although very large data sets would need to be used to populate matrices of H and D in more than two dimensions.

The choice of measure between H(h), the paired-sample version of the Gini–Simpson index, and D(h), the paired-sample version of its true diversity or numbers equivalent, will depend on the context in which it is used. H(h) has the disadvantage of being restricted to values between 0 and 1, but does have the advantage of being easily interpretable in terms of the probability of reoccurrence of a species at a sampling location in its neighbourhood. This makes it particularly applicable to investigations of functional explanations and null models of diversity involving spatial ecological processes such as inter- and intra-specific competition or dispersal (Chave & Leigh 2002), where the probability of an individual existing in proximity to another individual of the same species is critical. If diversity is maintained by environmental heterogeneity, then it may be reasonable to assume that the characteristic spatial scale is likely to be similar to that of the underlying pattern of the environmental gradient. Competitive interactions between plants also occur at distinctive scales, with individuals interacting at the scale at which their canopy and root systems overlap. The level of aggregation of species may determine the outcome of competitive interactions, and clumped distributions may reduce rates of competitive exclusion (Silvertown et al. 1992). The relative probabilities of nearby space being occupied by the same, or different, species, is relevant for the relative contribution of inter- and intra-specific competition in structuring plant communities. There are clear links to the concept of ecological neighbourhood (Antonovics & Levin, 1980) and species interactions (Silvertown et al. 1992; Murrell 2010). On the contrary, D(h) has the clear advantage of being expressed in comparable units to species richness and has the intuitive benefits as a measure of true diversity shared by its non-spatial analogues (Jost 2006).

In either case, the use of a paired-sample measure of diversity may give useful insights into the spatial structure of biodiversity and it links to the processes which generate and maintain diversity at different scales. It also provides a clear basis for integrating the concepts of ‘inventory’, ‘differentiation’ and ‘proportional’ diversity described by Jurasinski et al. (2002). The method has the potential for broad application and could be easily adapted and applied to the analysis of data from other ecosystems and taxonomic groups.

Acknowledgements

The authors are grateful for fieldwork support from Emma Rockall (University of Southampton) and Amy Cook (University of Exeter). We would like to thank Alasdair Brock at the South Solway Mosses Natural England Office (Kirkbride, Cumbria, UK), for assistance on site. Fiedwork was funded through a UK Natural Environment Research Council grant NE/F000421/1 ‘Remote Sensing of Peatland Responses to Hydrological Change’. We would also like to thank the editors and two anonymous reviewers for their insightful and constructive comments.