Robertson (1987), and Wagner (2003). Examples of the use of geostatistics in forest ecology include:
● Köhl and Gertner (1997), who used these methods in forest damage surveys
● Bebber et al. (2002), who used geostatistics to determine the spatial relation- ships between canopy openness and seedling performance in secondary low- land forest in Borneo
● Hohn et al. (1993), who used three-dimensional kriging in space and time to predict defoliation caused by gypsy moth in Massachusetts
● Nanos et al. (2004), who used geostatistics to analyse the stand characteristics of a pine forest in Spain
● Schumeet al. (2003), who carried out a spatiotemporal analysis of soil water content in a mixed Norway spruce–European beech stand in Austria.
interpolating and extracting species accumulation curves is described by Colwell et al. (2004) (see also Golicher et al. 2006).
Species accumulation curvesplot the cumulative number of species recorded as a function of sampling effort, and illustrate the increase in the total number of species encountered during the process of data collection. Species–area curves can be considered as one form of species accumulation curve, in which species richness is related to an increase in the area sampled. Smooth curves can be produced if samples are added randomly and the process repeated a number of times (at least 100 is recommended). Species accumulation curves are often plotted on a linear scale on both axes, although Longino et al. (2002) suggest that the x-axis should be log-transformed to enable easier differentiation between asymptotic and logarith- mic curves.
The total species richness within a particular area can be estimated by extrapolating from species accumulation curves. The best method of making this extrapolation has been the subject of some debate, as has the most appropriate way of obtaining samples (Magurran 2004). According to Colwell and Coddington (1994), random samples should be taken from areas of relatively homogeneous habitat. Rosenzweig (1995) states that a nested design should be used; in other words, subplots that are used to sample individuals for production of a species–area curve should be contiguous. However, in this case subplots are not statistically independent of one another; this could lead to the results being biased and statistical inferences being invalid (Crawley 1993).
Functions fitted to species accumulation curves may be either asymptotic or non-asymptotic. The equation most commonly used for fitting an asymptotic curve is the Michaelis–Menten equation (Magurran 2004):
where S(n) is the number of species observed in nsamples,Smaxis the total number of species in the assemblage, and Bis the sampling effort required to detect 50% of Smax.
Non-asymptotic curves that have been used include logarithmic transform- ations of the x-axis (a log–linear model) and of both axes (a log–log relation).
However, Colwell and Coddington (1994) suggest that non-parametric methods (see below) are preferable to either of these.
If information on the species abundance distributionis available, namely the rela- tion between the number of species and the number of individuals in those species, this can also be used to estimate species richness. A wide variety of different models have been used to characterize species abundance distributions, and these are considered in detail by Hubbell (2001) and Magurran (2004). Those with the greatest potential for estimating species richness are the log series and log normal distributions, which are evaluated by Colwell and Coddington (1994). Again, however, the use of non-parametric estimators is generally preferred to these methods (Magurran 2004).
S(n)Smaxn Bn
A range of different non-parametric estimatorsare described by Chazdon et al.
(1998) and Colwell and Coddington (1994). These include Chao 1, Chao 2, the abundance-based coverage estimator (ACE), the partner incidence-based coverage estimator(ICE), two methods based on the use of jackknife statistics (Jackknife 1 and 2), and a bootstrap estimator (see Box 3.1). These can all be calculated easily by using EstimateS software, which can be downloaded free of charge from 具http://viceroy.eeb.uconn.edu/EstimateS典(Colwell 2004a, b). A detailed manual is provided with this software, which should be consulted carefully before use.
Although this program is now widely used, it should be employed with caution because of potential errors (Golicher et al. 2006). Selection of which estimator is most appropriate depends on the characteristics of the forest being studied, includ- ing the sample size, the patchiness of the vegetation, and the total number of indi- viduals in the sample (Magurran 2004). Relatively few comparative studies of Species richness and diversity | 127
Box 3.1 Formulae for selected non-parametric estimators of species richness (after Magurran 2004)
● Chao 1, an abundance-based estimator of species richness
● Chao 2, an incidence-based estimator of species richness
● Jackknife 1; first-order jackknife estimator of species richness (incidence- based)
● Jackknife 2; second-order jackknife estimator of species richness (incidence- based)
● Boostrap estimatorof species richness (incidence-based)
● Abundance-based coverage estimator(ACE) of species richness SaceScommSrare
Cace F1
Cace␥ace 2
SbootSobsk1
兺
Sobs(
1pk)
mSJack2Sobs
冤
Q1(2m3)m Qm(m1)2(m2)2冥
SJack1SobsQ1
冢
m1m冣
SChao2SobsQ1 2
2Q2 SChao1SobsF1
2
2F2
these estimators have been made to date; examples include Chazdon et al. (1998), Colwell and Coddington (1994), Condit et al. (1996b), and Longino et al. (2002).
When estimating species richness, Gotelli and Colwell (2001) emphasize the importance of standardizing data sets that are to be compared to a common num- ber of individuals. This can be achieved by using species accumulation curves and rarefaction curves, which are produced by repeatedly resampling the pool of indi- viduals or samples at random and plotting the average number of species repre- sented as the sample size increases (Figure 3.14) (Gotelli and Colwell 2001).
Gotelli and Colwell (2001) make the following recommendations with respect to this approach:
● It is essential that a species accumulation curve or rarefaction curve is plotted when estimating species richness. Raw species richness counts can only be validly compared when such accumulation curves have reached a clear
● Incidence-based coverage estimator(ICE) of species richness
where:
Sest estimated species richness, where est is replaced in the formula by the name of the estimator
Sobs total number of species observed in all quadrats pooled
Srare number of rare species (each with 10 or fewer individuals) when all quadrats are pooled
Scomm number of common species (each with more than 10 individuals) when all quadrats are pooled
Sifreq the number of infrequent species (each found in 10 or fewer quadrats) Sfreq number of frequent species (each found in more than 10 quadrats) m total number of quadrats
Fi number of species that have exactly iindividuals when all quadrats are pooled (F1 is the frequency of singletons, F2 is the frequency of doubletons)
Qj the number of species that occur in exactly j quadrats (Q1 is the frequency of uniques, Q2is the frequency of duplicates)
Pk the proportion of quadrats that contain species k Cace sample abundance coverage estimator
Cice sample incidence coverage indicator
estimated coefficient of variation of the Fifor rare species estimated coefficient of variation of the Qifor infrequent species
␥ice 2
␥ace 2
SiceSfreqSifreq
Cice Q1
Cice␥ice 2
asymptote. Estimates of species richness should be reported together with information about the sampling effort involved.
● When sample-based approaches are used, the number of species should be plotted as a function of the accumulated number of individuals, not the accu- mulated number of samples, because data sets may differ in the mean number of individuals per sample.
● Individual-based rarefaction analysis is based on the assumption that the spa- tial distribution of individuals in the environment is random, that sample sizes are sufficient, and that assemblages being compared have been sampled in the same way. If these assumptions are not met, misleading results may be obtained.
● It is invalid to simply divide the number of species encountered by the number of individuals included in the sample to correct for unequal numbers of indi- viduals between samples. This is because such a correction assumes that richness increases linearly with abundance, which is rarely the case (Figure 3.15).
Rarefaction curves can be plotted by EstimateS software, as well as by other commercially available software packages such as Species diversity and richness (Pisces Software, 具www.pisces-conservation.com典). Kindt and Coe (2005) have Species richness and diversity | 129
25 0
10 20 30
Species
50 75 100 125
Individuals Samples 900
600 300
Individuals: rarefaction Individuals: accumulation
Samples: accumulation Samples: rarefaction
Fig. 3.14 Sample- and individual-based rarefaction and accumulation curves.
Accumulation curves (jagged curves) represent a single ordering of individuals (solid-line, jagged curve) or samples (open-line, jagged curve), as they are successively pooled. Rarefaction curves (smooth curves) represent the means of repeated re-sampling of all pooled individuals (solid-line, smooth curve) or all pooled samples (open-line, smooth curve). The smoothed rarefaction curves therefore represent the statistical expectation for the corresponding accumulation curves. The sample-based curves lie below the individual-based curves because of the spatial aggregation of species. Curves were produced by using EstimateS (see text). (From Gotelli and Colwell 2001. Quantifying biodiversity: procedures and pitfalls in the measurement and comparison of species richness. Ecology Letters, Blackwell Publishing.)
produced a useful software program and accompanying manual that can also be used to calculations a number of measures relating to tree diversity, including rarefaction. These can be downloaded from 具www.worldagroforestry.org/treesand- markets/tree_diversity_analysis.asp典.
Many ecological studies assess the number of species present in a particular area, or the species density, which depends both on species richness and on the mean density of individuals. To compare species density estimates from different locations, the x-axis of individual-based rarefaction curves can be rescaled from the number of individuals to area, by using a measure of average density (Gotelli and Colwell 2001). However, when comparing species density data for two unequal areas it is invalid to divide the number of species by the area measured, because the
Second growth Old growth
(a)
(b)
Slope = Spp/individual
Slope = Spp/individual Individuals
SpeciesSpecies
Individuals
Gaps
Non-gaps
Fig. 3.15 Pitfalls of using species : individual ratios to compare data sets. In (a), an old-growth and a second-growth forest stand are compared. The two stands have identical individual-scaled rarefaction curves and thus do not differ in species richness. The second-growth curve extends farther simply because stem density is greater, so that more individuals have been examined for the same number of samples. However, when the species : individual ratio is computed for each, the ratio is much higher for the old-growth stand. In (b), species richness in treefall gap quadrats is compared with richness in non-gap (forest matrix) quadrats. In this case, species : individual ratios are identical, yet the true species richness is higher in gaps. (From Gotelli and Colwell 2001. Quantifying biodiversity:
procedures and pitfalls in the measurement and comparison of species richness.
Ecology Letters, Blackwell Publishing.)
number of species increases non-linearly with area (Gotelli and Colwell 2001).
Again, species accumulation curves should be used to compare samples.
Which measure should be used, species richness or species density? For most conservation assessments, species density is more likely to be of value. Species richness may be preferred when testing ecological theories or models (Gotelli and Colwell 2001).
3.9.2 Species diversity
Species diversity is generally assessed by using some type of diversity index, which incorporates information on species richness and evenness. The term evenness refers to the variability in the relative abundance of species. A wide variety of dif- ferent indices are available, which are comprehensively described by Magurran (2004). Measures of evenness (or heterogeneity) can be divided into two groups:
those that are based on the parameter of a species abundance model, and non- parametric methods that make no assumptions about the underlying species abun- dance distribution. Examples of parametric measures of diversity include log series a, log normal l, and the Qstatistic. Examples of non-parametric measures include the Shannon evenness measure, Heip’s index of evenness, Simpson’s index, Simpson’s measure of evenness and the Berger–Parker index (Magurran 2004).
Some of these are given in Box 3.2.
How should an appropriate diversity index be selected? The following advice is provided by Magurran (2004):
● Rather than simply calculating a range of diversity measures, it is preferable to define in advance which measure is most appropriate to the objective of the investigation; do not simply select the measure that provides the most appeal- ing answer.
● Sample size must be adequate for the method selected.
● Replication is always recommended. Many small samples are generally prefer- able to a single large one. Replication permits statistical analyses, such as the calculation of confidence limits.
● Consider whether a diversity index is really necessary. A robust estimate of the number of species present, without any consideration of their relative abun- dance, may be sufficient to meet the objectives set.
● If some measure of evenness is required, consider using either Fisher’sor Simpson’s index (Box 3.2). These measures are relatively well understood and are relatively easy to interpret. Fisher’sis relatively unaffected by sample size once the sample size is greater than 1000. Simpson’s index provides an accurate estimate of diversity for relatively small sample sizes.
● The Shannon index is not recommended, despite its popularity, because of its sensitivity to sample size. Interpretation can also be difficult.
● The Berger–Parker index and the Simpson’s evenness measure provide rela- tively simple and easily interpretable measures of dominance (i.e. weighted by the abundance of the commonest species).
Species richness and diversity | 131
Based on their detailed assessment of diversity in highly diverse lowland tropical forests, Condit et al. (1998) made the following additional recommendations:
● Compare samples with approximately equal numbers of stems, but ignore area.
● Use the same sampling protocol across sites that are to be compared; use the same shape of field plot.
● Compare samples by using the same dbh limit.
● With samples of fewer than 3000 stems, do not use species richness as a diver- sity metric; instead use Fisher’s.
● Be aware of sampling error; confidence limits associated with diversity assess- ments in samples of fewer than 1000 stems extend about 30% above and below the estimate.
Box 3.2 Selected measures of diversity (after Condit et al. 1998, Magurran 2004 and Southwood and Henderson 2000).
● Shannon–Wiener function:
where piis the proportion of individuals in the ith species.
● Fisher’s␣:
where Sis the number of species and Nis the number of stems.
● Simpson’s index:
where fiis the proportion of individuals in the ith species.
● Simpson’s measure of evenness:
where Dis Simpson’s index, and Sis the number of species. The measure ranges from 0 to 1 and is not sensitive to species richness.
● Berger–Parker index:
where Nmaxis the number of individuals in the most abundant species and Nis the total number of individuals. This index has the great advantage of being very easy to calculate.
dNmax/N E1/D
(
1/D)
S D
兺
fi2S␣ln
冢
1N␣冣
H
兺
i1SobspilogepiConditet al. (1998) place particular emphasis on the importance of stem number in influencing diversity estimates in tropical forests, suggesting the following specific rules:
● Never use samples of fewer than 50 stems, and in very diverse forests use 100 stems or more. These are absolute minimum values; larger samples are preferable.
● In samples of more than 2000 stems, Fisher’s␣ can be used to compare samples, even if they differ substantially in size.
● For samples between 50 and 20 000 stems, either subsample stems to provide a common number of stems for all sites to be compared, or apply a correction factor to Fisher’s␣to adjust for sample size (for details of this approach see Conditet al. 1998).
Magurran (2004) further emphasizes the importance of sampling approach for determining the outcome of diversity measures. Most importantly, the sample size must be adequate. However, this can be difficult to determine in practice.
Assessment of the rate at which new samples are being encountered is one useful guide; the experience of knowledgeable field ecologists is another (Magurran 2004). In high-diversity sites, Sørensen et al. (2002) recommended the following rule of thumb: 30–50 individuals should be sampled per species. When using sample-based approaches, it is necessary to determine an appropriate number of replicates. A number of different studies have found 10 replicates to be adequate, but the optimum number varies with the scale of the sampling unit in relation to the size of the assemblage being surveyed (Magurran 2004). As additional samples are included in the analysis, their effects on the precision of diversity estimates can be measured. Unequal sample sizes should be avoided; rather, a consistent approach to sampling should be used across the entire investigation. It is also important to remember that samples should be independent; repeated samples of the same plot are not true replicates, and replicates should be located randomly and not grouped together (Crawley 1993).
3.9.3 Beta diversity and similarity
Beta diversity is a measure of the extent to which the diversity of two or more spatial units differs (Magurran 2004) and is generally used to characterize the degree of spatial heterogeneity in diversity at the landscape scale, or to measure the change in diversity along transects or environmental gradients. A variety of indices are available for describing beta diversity, most of which are based on the use of presence–absence data, although quantitative abundance data can also be used.
One of the most widely used indices is Whittaker’s measure (W), which is calculated as:
WS/␣
Species richness and diversity | 133
Box 3.3 Measures of similarity and complementarity (from Magurran 2004)
● Jaccard similarity index(CJ):
where ais the total number of species present in both quadrats or samples being compared, bis the number of species present only in quadrat 1, and cis the num- ber of species present only in quadrat 2.
The statistic can be adapted to give a single measure of complementarity across a set of samples or along a transect:
where UjkSjSk2Vjkand is summed across all pairs of samples; Vjkis the number of species common to the two lists jandk(the same value as ain the for- mula above); SjandSkare the number of species in samples jandk, respectively;
andnis the number of samples.
● Sørenson’s similarity index(CS):
(see above for definitions of a,b, and c).
● Sørenson’s quantitative index (CN):
This is a modified version of Sørenson’s index that takes into account the relative abundance of species, introduced by Bray and Curtis in 1957:
where Nais the total number of individuals in site a,Nbthe total number of individuals in site b, and 2jNis the sum of the lower of the two abundances for species found in both sites.
● Morisita–Horn index:
This is recommended because it is not strongly influenced by sample size and species richness (Henderson 2003):
where Nais the total number of individuals in site a,Nbis the total number of individuals in site b, aiis the number of individuals in the ith species in a,bithe
CMH 2
兺 (
aibi)
(
dadb)(
NaNb)
CN 2jN NaNb
CS 2a
2abc CT
兺
Ujkn
CJ a
abc
where Sis the total number of species recorded in the system, and ␣is the average sample species richness, where each sample is a standard size. This is one of the sim- plest and most effective measures of beta diversity (Magurran 2004). Where this measure is calculated between pairs of samples or between adjacent quadrats along a transect, values of the measure will range from 1 (complete similarity) to 2 (no overlap in species composition). Subtracting 1 from the answer enables results to be presented on a scale of 0–1. Often, beta diversity is assessed by using measures of similarity such as the Jaccard similarity index and Sørensen’s measure (Box 3.3).
Clarke and Warwick (2001) note that similarity measures can be markedly affected by the abundance of the commonest species, and recommend that data be transformed before calculating the similarity measure. Data should either be square-root transformed or transformed to log (x1). Vellend (2001) suggests that measures of beta diversity should be differentiated from measures of species turnover, which measure the extent of change in species composition along prede- fined gradients. According to Vellend (2001), Whittaker’s measure of beta diver- sity should not be used to assess species turnover; rather, matrices of compositional similarity and physical or environmental distances among pairs of study plots should be used.
One of the main problems with using such similarity measures is that they are based on the assumption that sites being compared have been completely censused (Magurran 2004). Often, this is not the case. Recently developed methods focus- ing on the use of ACE are designed to address this, which estimate the number of unobserved shared species (Chao et al. 2000).