Measuring biodiversity across spatial scales in a

raised bog using a novel paired-sample diversity

index

Jonathan Bennie1*, Karen Anderson2 and Andrew Wetherelt3

1Centre for Ecology and Conservation, College of Life and Environmental Sciences, University of Exeter, Cornwall

Campus, Penryn, Cornwall TR10 9EZ, UK; 2Department of Geography, College of Life and Environmental Sciences,

University of Exeter, Cornwall Campus, Penryn, Cornwall TR10 9EZ, UK; and 3Camborne School of Mines, University

of Exeter, Cornwall Campus, Penryn, Cornwall TR10 9EZ, UK

Summary

1. Biodiversity is an inherently spatial phenomenon. It is determined by environmental heterogene-

ity 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 newmethod 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 spe-

cies 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 signifi-

cant spatial structure at a length scale of 135–140 cm, corresponding to the typical hummock–hol-

low 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 detect-

ing and attributing patterns of beta-diversity along both spatial and environmental gradients.

Key-words: beta diversity, biodiversity, determinants of plant community diversity and

structure, diversity index, laser scanning, peatland, scale, Simpson index, spatial ecology,

Sphagnum

Introduction

The causes of patterns of biodiversity are a topic of consider-

able 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 (Partel & Zobel 1999; Munzber-

gova 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 driv-

ers 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*Corresponding author. E-mail: j.j.bennie@exeter.ac.uk

� 2010 The Authors. Journal of Ecology � 2010 British Ecological Society

Journal of Ecology doi: 10.1111/j.1365-2745.2010.01762.x

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 & Woo-

dwell 1969; Whittaker 1972) recognize a distinction between

local, or alpha (a-) and regional, or gamma (c-) diversities.Beta (b-) diversity is then the diversity between habitats or

sampling units, a conceptual bridge between these two

discrete scales. b-diversity is caused by the tendency of indi-

viduals of the same species to be aggregated together in

space, whether individualistically or in clearly defined com-

munities. It is also clearly a function of both the grain size at

which a-diversity is measured and the degree of ‘patchiness’

of the sampled domain within which c-diversity is measured.

Jurasinski et al. (2009) have suggested the term ‘inventory

diversity’ for both a- and c-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, a-diversity is assumed to be sampled comprehen-

sively at the grain scale, whereas c-diversity is approximated

by pooling samples within the extent. For example, Martin,

Moloney & Wilsey (2005) measured plant species a-diversityin 40 · 100 cm quadrats in prairie remnants and restoration

sites and pooled data from eight quadrats from each site to

estimate c-diversity for each site.

Despite the widespread use of the concept, methods for

quantifying b-diversity vary and the appropriateness of dif-

ferent methods, and of the concept itself, is much debated

(Veech et al. 2002; Jost 2006; Jurasinski et al., 2009). Jura-

sinski 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 natu-

ral environments tend to aggregate both in space and along

environmental gradients, b-diversity may have two compo-

nents:

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 distribu-

tions, 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) b-diversitywas defined as the ratio b = c ⁄a. An alternative definition

of c = a + b has become widely used, allowing the additive

partitioning of components of diversity (Lande 1996; Veech

et al. 2002). This approach has the advantage that b shares

the same units as a and c, but suffers from the disadvantage

that b is not independent of a, and therefore cannot be com-

pared between sites (Jost 2007), and that b-diversity defined

in this way does not recognize the degree of difference or

similarity between samples. Diversity in this context is con-

ventionally 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 lat-

ter 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 diver-

sity, with units commensurate with species richness, as ‘Hill

numbers’, or ‘numbers equivalent’ (Hill 1973; Jost 2006) pro-

vides a more intuitive measure of diversity. Following Jost

(2007), the Gini–Simpson index (and its paired-sample spa-

tial 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 differ-

ent sized sampling units (Klimek et al. 2008). In the additive

partitioning framework, a-diversity across a hierarchy of

nested spatial scales can be expressed as ax = ax-1 + bx-1,where diversity ax measured at the xth level of a scale hierar-

chy is the sum of the mean a-diversity and the b-diversitybetween samples at the x - 1th level (Wagner et al. 2000).

However, in addition to the theoretical drawbacks with the

additive definition of b-diversity (Jost 2007), one clear draw-

back 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 a-diversity is measured,

then samples may not be independent and pooled-sample

estimates of c-diversity will be biased. Furthermore, a sam-

pling design in which samples is selected from within clearly

identifiable communities or habitats will under-sample eco-

tones 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 b-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 distribu-

tions 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 meth-

ods 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 vari-

ance), and to describe spatial structure and pattern in ecosys-

tems. This approach has been extended to spatial patterns of

2 J. Bennie, K. Anderson & A. Wetherelt

� 2010 The Authors. Journal of Ecology � 2010 British Ecological Society, Journal of Ecology

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 prob-

ability 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 com-

monly used measure of a- and c-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 sin-

gle species or variable and measurements are in units com-

mensurate with a- and c-diversity.

STUDY SYSTEM

To demonstrate the potential of this method we apply it to a

fine-scale vegetation survey and hydrological dataset col-

lected for a separate study. Vegetation data were collected at

5-cm intervals along two 10-m transects at plots of contrast-

ing 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 typ-

ically show distinct patterns in hummock-hollow topography

which is associated with fine-scale patterns in species compo-

sition, 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 micro-

topography) 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 coeffi-

cient (Simpson 1949; Pielou 1969):

Ha ¼ 1�Xsk¼1

p2k eqn 1

where pk 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 propor-

tional cover or biomass of each species rather than the propor-

tion 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 sam-

ple are from different species or, in the case of data expressed as pro-

portional 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 distri-

bution of these species within the sample (species evenness). The aver-

age index of a-diversity across a set of n samples is:

Ha ¼ 1� 1

n

Xn

i¼1

Xs

k¼1 pik2 eqn 2

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

Hc ¼ 1�Xsk¼1

1

n

Xn

i¼1 pik

� �2

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:

Hij ¼ 1�Xs

k¼1 pikpjk eqn 4

where pik is the proportional cover of species k in sample i and

pjk 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

HðhÞ ¼ 1� 1

NðhÞX

ði;jÞ2NðhÞ

Xsk¼1

pikpjk eqn 5

where N(h) is the number of pairs of samples that are a separa-

tion 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 sam-

pling 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 toler-

ance 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:

DðhÞ ¼ 1

ð1�HðhÞÞ eqn 6

Key features of these plots can be identified:

Measuring biodiversity across spatial scales 3

� 2010 The Authors. Journal of Ecology � 2010 British Ecological Society, Journal of Ecology

When h = 0, i = j and N(h) = n; eqn 5 simplifies to eqn 2. The val-

ues ofD(h) andH(h) when h is zero, are therefore equal toDaandHa,

which is the mean within-sample or a-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 resam-

pling 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 Hc and D(h) an estimate of c-diver-sity. 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 b-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 val-

ues 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 differ-

ent species, is an unbiased estimate of Hc, the probability that any

two points in the domain are occupied by different species. These ran-

dom-pair values of H and D therefore provide estimates of c-diver-sity. 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 signif-

icantly (P < 0.05) from c-diversity. At these lag distances spatial

structure exists and contributes towards b-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 hol-

low 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):

Hðh;zÞ ¼ 1� 1

Nðh; zÞX

ði;jÞ2Nðh;zÞ

Xsk¼1

pikpjk eqn 7

Similarly,

Dðh; zÞ ¼ 1

ð1�Hðh; zÞÞ eqn 8

In this caseH(h,z) may be plotted against both h and z to examine the

effects of spatial and environmental components of diversity sepa-

rately (Fig. 2).

FITT ING 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 inter-

action term:

Hðh; zÞ ¼ aþ b logðzþ 1Þþ c logðhþ 1Þ þ d logðzþ 1Þ logðhþ 1Þ

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

DY

D(h

)

Dα

h

Fig. 1. Conceptual plot of paired-sample diversity indexD(h) against

lag distance h for a hypothetical domain with fine-scale spatial struc-

ture. When h = 0, D(h) is equal to the mean a diversity within sam-

ples, Da. As the lag distance between pairs of samples increases, the

samples tend to become less similar and hence between-sample diver-

sity increases.D(h) levels off to an approximation of c diversity withinthe domain,Dc.

(a) (b) (c)

Fig. 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.

4 J. Bennie, K. Anderson & A. Wetherelt

� 2010 The Authors. Journal of Ecology � 2010 British Ecological Society, Journal of Ecology

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 ground-

works 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 contrast-

ing 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, domi-

nated byBetula pubescens carr woodland, occurs.

Vegetation survey

At each site, all species of vascular plant and bryophyte were identi-

fied 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 shrubsCalluna vulgaris, Erica tetralix

and Andromeda polifolia and the peat moss species Sphagnum capil-

lifolium 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 andC. 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 (Ander-

son, 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 differ-

ential 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).

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 posi-

tioned at the perimeter of the scan region. Their positions were mea-

sured 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 analy-

sis. It should be noted that this value represents the height of the sur-

face 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 a-diversity were similar between sites,

Ha = 0.25, Da = 1.33 for the central bog dome transect and

Ha = 0.27,Da = 1.37 for the bogmargin transect. The calcu-

lated c-diversity values were Hc = 0.89, Dc = 8.55 and

(a)

(b)

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

surface.

Measuring biodiversity across spatial scales 5

� 2010 The Authors. Journal of Ecology � 2010 British Ecological Society, Journal of Ecology

Hc = 0.81, Dc = 5.24, respectively. Figure 4 shows the spa-

tial versions of the Gini–Simpson index and Jost’s number’s

equivalent for each site. The shaded band represents the

Monte-Carlo estimate of c-diversity in each case. Spatial struc-ture is detected where the line falls outside the limits of the

shaded area; in both cases spatial aggregation causes low diver-

sity 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–hol-

low topography at this site.

Three dimensional plots ofH(h,z) against h and z, and fitted

surfaces are shown in Fig. 5. In both transects, the lowest val-

ues (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.

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, includ-

ing interaction terms, was the best model form. This implies

that at both transects b-diversity is along both pure spatial

and environmental gradients, and also influenced by an

interaction between these two. The forms of the fitted sur-

faces are shown in Fig. 5.

Discussion

DIVERSITY PATTERNS IN BOGS

In a study of Norwegian boreal swamp forests, Økland, Ryd-

gren & Økland (2008) considered that the determinants of

diversity within a given plot are (i) the size of the available spe-

cies pool; (ii) the position of the plot along environmental gra-

dients and (iii) thewithin-plot environmental heterogeneity, via

b-diversity effects.They showed that fine-scale variation inmic-

rotopography was associated with b-diversity at fine scales formoss and liverwort species. In this study we have found that in

both a central and amarginal bog community spatial structure

at a length scale of <50 cm contributes to overall diversity

throughb-diversity effects.At both transect sites, species aggre-

gation along a gradient in depth to water table was also identi-

fied as a component of b-diversity. Significant spatial structurealso 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 sepa-

rated by this distance were less similar in their species composi-

tion 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 peat-

land, and hence probably reflects niche differentiation along

(a) (b)

(c) (d)

Fig. 4. Spatial versions of Simpson concentrationH(h) (a,c) and species-equivalent diversityD(h) (b,d) for the raised bog centre (top) and mar-

gin (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

theMonte-Carlo estimate of c-diversity; spatial structure is detectedwhere the line falls outside the limits of the shaded area.

6 J. Bennie, K. Anderson & A. Wetherelt

� 2010 The Authors. Journal of Ecology � 2010 British Ecological Society, Journal of Ecology

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 distinc-

tions 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

thewater table had been lowered byhistorical 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 a and b-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 b-diversity, species clustering

along both spatial and environmental gradients. Further envi-

ronmental 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

populatematrices ofH andD inmore than two dimensions.

The choice of measure betweenH(h), the paired-sample ver-

sion 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 proba-

bility of reoccurrence of a species at a sampling location in its

neighbourhood. This makes it particularly applicable to inves-

tigations of functional explanations and null models of diver-

sity 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 envi-

ronmental gradient. Competitive interactions between plants

(a) (b)

(c) (d)

Fig. 5 Plots of paired-sample Gini–Simpson indexH(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 regressionmodel.

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 R2(adj)

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

Column z represents the coefficient for log-transformed height

above the water table; h represents log-transformed distance;

z · h represents the interaction term.

Measuring biodiversity across spatial scales 7

� 2010 The Authors. Journal of Ecology � 2010 British Ecological Society, Journal of Ecology

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 contri-

bution of inter- and intra-specific competition in structuring

plant communities. There are clear links to the concept of eco-

logical neighbourhood (Antonovics &Levin, 1980) and species

interactions (Silvertown et al. 1992;Murrell 2010). On the con-

trary, D(h) has the clear advantage of being expressed in com-

parable units to species richness and has the intuitive benefits

as a measure of true diversity shared by its non-spatial ana-

logues (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 taxo-

nomic groups.

Acknowledgements

The authors are grateful for fieldwork support fromEmmaRockall (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.

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Received 31 July 2010; accepted 21October 2010

Handling Editor:Hans Cornelissen

Measuring biodiversity across spatial scales 9

� 2010 The Authors. Journal of Ecology � 2010 British Ecological Society, Journal of Ecology