Download A semantic taxonomy for diversity measures

Acta Biotheor (2007) 55:23–33
DOI 10.1007/s10441-007-9008-7
REGULAR ARTICLE
A semantic taxonomy for diversity measures
Carlo Ricotta
Received: 27 October 2006 / Accepted 6 March 2007 / Published online: 8 May 2007
Ó Springer Science+Business Media, B.V. 2007
Abstract Community diversity has been studied extensively in relation to its
effects on ecosystem functioning. Testing the consequences of diversity on
ecosystem processes will require measures to be available based on a rigorous
conceptualization of their very meaning. In the last decades, literally dozens of
measures of diversity have been proposed. However, rather than using unrelated
metrics, we need to identify their separate components so that possible links
between them and ecosystem functioning can be examined using an agreed-upon
language. In this paper, first, a short overview on new and old measures of community diversity is presented. Next, I propose a general framework in which most of
the existing measures of diversity are sorted into four interrelated semantic classes:
richness, abundance-weighted diversity, evenness and divergence. In this view, this
paper constitutes an attempt to organize the very large number of existing diversity
measures avoiding ambiguities in the meaning of the different facets of community
diversity.
Keywords Complexity Compositional diversity matrix Divergence Evenness Pairwise species distances Richness
1 Introduction
The compositional diversity of a species assemblage is a central concept in ecology
that has been studied extensively for many decades in relation to its possible
connection with ecosystem functioning and organization. Classical examples
include the diversity-productivity (McCann 2000) and the diversity-stability debate
C. Ricotta (&)
Department of Plant Biology, University of Rome ‘‘La Sapienza’’, Piazzale Aldo Moro 5,
00185 Rome, Italy
e-mail: [email protected]
123
24
C. Ricotta
(Tilman et al. 1996; Hooper and Vitousek 1997; Hector et al. 1999). More recently
the influence of species diversity on exotic invasions has received increasingly
attention (Stohlgren et al. 1999; Fridley et al. 2004; Fargione and Tilman 2005).
However, an unambiguous characterization of the concept of community diversity
has proved elusive (see Ricotta 2005a). This observation has led to the classical
comments by Hurlbert (1971) on the ‘‘non-concept of diversity’’ and by Poole
(1974) that diversity measures are ‘‘answers to which questions have not yet been
found’’.
One reason for this ambiguity is that traditional measures of diversity like the
Shannon index or the Simpson index combine in non-standard way the two
components of species richness and their relative abundances (called variously
evenness, equitability or dominance). High species richness and evenness, that
occurs when species are equal in abundance, are both equated with high diversity.
Also, traditional measures are computed solely from species relative abundances
without incorporating ecological differences between species. However, a community composed of ecologically dissimilar taxa is intuitively more diverse than a
community composed of more similar taxa. Therefore, recently, the overall
divergence (i.e., a measure of ecological dissimilarity) between the species within a
community has received increasing attention as a component of community
diversity (Ricotta and Avena 2005; Mason et al. 2005).
To avoid confusion, in this paper I will refer in a very generic manner to
measures of ‘diversity’ or ‘community diversity’ as to all measures that are used to
summarize different aspects of the compositional diversity of a given community or
species assemblage with scalars. By contrast, I will refer to the particular subset of
diversity measures that are computed solely from the species relative abundances of
a given community as to ‘abundance-weighted measures of diversity’.
In the literature, there are a number of measures designed to summarize
community diversity incorporating inter-species differences. Such differences may
be related to functional type or morphology, taxonomic relatedness or genetic
distances, as ecological differences between species are believed to be reflected in
each of these (Faith 1992; Izsak and Papp 1995; Warwick and Clarke 1995; Clarke
and Warwick 1998; Webb 2000; Shimatani 2001; Barker 2002; Ricotta 2004). All
these measures summarize community diversity from different perspectives and
motivations. However, rather than using unrelated metrics of community diversity,
we need to identify their separate components.
In this paper, first, a short overview on existing measures of community diversity
is presented. Next, based on an old observation of Juhasz-Nagy (1993) that
community diversity can be summarized at different levels of increasing abstraction, I propose to integrate old and new measures into one general semantic
framework comprising four main families of community diversity: richness,
abundance-weighted diversity, evenness, and divergence. Ideally, this paper
completes the ideas announced in Ricotta (2005a); using an agreed-upon language,
it constitutes an attempt to organize the rich but dispersed literature on diversity
measures avoiding ambiguities in the meaning of the different components of
community diversity.
123
A semantic taxonomy for diversity measures
25
Such an attempt may lead to clues when one has to choose among several
diversity functions. Nonetheless, this paper is not explicitly aimed at describing
which family of diversity measures is more appropriate for providing distinct and
relevant information on particular aspects of ecosystem processes. Examples of
cutting-edge reviews on diversity/ecosystem functioning relationships include
Naeem and Wright (2003); Legendre et al. (2005); Mason et al. (2005); Petchey and
Gaston (2006). Researchers who would like guidance regarding which index to use
and why are referred to these papers.
2 A short overview on measures of community diversity
Community diversity is generally considered a multi-level concept encompassing
all scales of natural variation from ecosystems and landscapes down to the
molecular level (Huston 1994; Sarkar and Margules 2002; Wehenkel et al. 2006).
Classical
P measures of compositional diversity, such
P as the Shannon entropy
H ¼ Si¼1 pi log pi or the Simpson index D ¼ 1 Si¼1 p2i (where S is species
richness) are traditionally
computed from the species relative abundances pi
P
(0 pi 1 and Si¼1 pi ¼ 1) to the exclusion of other information as the degree of
ecological resemblance between the species in the assemblage.
Although each index weights rare and abundant species differently, high species
richness and high equitability in species relative abundances jointly imply high
compositional diversity. For this reason, Whittaker (1965) considers that the
partition of abundance cannot be adequately summarized by one statistic, but should
be characterized both by species richness and by the ‘dominance concentration’ of
the community.
Within this context Lloyd and Ghelardi (1964), see also Pielou (1975) have
defined a notion of ‘evenness’, with indices of evenness being basically
normalized measures of community diversity (Kvålseth 1991; but see Gregorius
1990). Evenness measures quantify the equality of species abundances in the
community, maximum evenness (1.0) arising for an equiprobable species
distribution, and the more the relative abundances of species differ, the lower
the evenness is (Alatalo 1981).
Several evenness indices have thus far been proposed in the ecological literature,
the most celebrated of which is a normalized version of the Shannon entropy E = H/
log S (see Pielou 1975). Additional evenness measures can be found in Taillie
(1979); Smith and Wilson (1996); Ricotta et al. (2001); Ricotta (2003).
Rao (1982) was among the first authors to propose an index of community
diversity that incorporates inter-species differences (see also Hendrickson and
Ehrlich 1971). Quadratic diversity (Q) is defined as the expected dissimilarity
between two individuals selected randomly with replacement:
Q¼
S X
S
X
dij pi pj
ð1Þ
i¼1 j¼1
123
26
C. Ricotta
where dij is the dissimilarity (i.e., not necessarily a metric distance) between species
i and j.
For
P dij = 1 for all i = j, and dii = 0 for all i, Q reduces to the Simpson index
1 Si¼1 p2i . The mathematical properties of quadratic diversity have been
extensively studied elsewhere (Shimatani 2001; Champely and Chessel 2002;
Pavoine et al. 2005a, 2005b; Ricotta and Szeidl 2006) and the reader is referred to
these papers for details.
Additional measures of community diversity that combine species relative
abundances and inter-species differences can be found in Warwick and Clarke
(1995); Ricotta (2004); Ricotta and Szeidl (2006).
Regardless of how inter-species differences are computed, since quadratic
diversity incorporates not only species relative abundances, but also information
about the degree of dissimilarity between the species in the assemblage, it comes
closer to a modern notion of community diversity than more traditional measures of
evenness or compositional diversity.
More recently, the concept of compositional diversity as a summary statistic that
is obtained from the species relative abundances has been criticized in fields as
different as conservation biology and functional ecology (Vane-Wright 1991;
Crozier 1992; Faith 1992; Petchey and Gaston 2002; Mason et al. 2005).
Vane-Wright et al. (1991) suggested that, for conservation purposes, we should
quantify taxonomic distinctness based solely on the topology of the phylogenetic
relationships among species. Successively, various refinements of this basic
idea have been proposed. For instance, Faith (1992) suggested a measure of
phylogenetic diversity (PD) that is simply the cumulative branch length of the full
phylogenetic tree, while Crozier (1992) proposed to summarize the community
taxonomic distinctness based on genetic distances between species.
In the next paragraphs, a general framework is described in which most of the
above traditional and contemporary measures of community diversity are decomposed into four interrelated components: richness, abundance-weighted diversity,
evenness and divergence.
3 A matrix representation of diversity
Juhasz-Nagy (1993) first imagined the measure of compositional diversity as an
iterative process involving successive levels of increasing abstraction: ‘‘Composition, even in the simplest etymological sense of the word (con-positio—‘collective
position’), refers always to some ‘mutual positional relations’ of some sets of
components. Even this simplistic pseudo-definition is contrary to the pragmatic use
of the term in ecology, where ‘composition’ refers usually to some sets or weighted
sets (like frequency distributions). But, we must realize that any composition (let it
be either a piece of art or a master-piece of Nature) is much more than silly
‘%-spectra’ or similar representations. Clearly, some kind of iteration is needed. At
a start, we can think of the well-known scheme:
123
A semantic taxonomy for diversity measures
Flora ! Vegetation
27
ðIÞ
or, in a much more general context, of:
Basic sets (alphabets) ! Compositional structures
ðIIÞ
where (II) may refer to a [biological] situation of some kind, and where ‘large
arrows’ try to indicate some ‘epistemological way’ of our understanding.
In order to make ‘large arrows’ more specific, we may decompose (I)–(II) into
several and more articulate states or steps. One way of such a decomposition is
shown by Fig. 1, where certain objects and operations are arranged in a proper
order.
A point set (like flora, fauna, biota) is just a list of the components involved,
without any further specification. A simplex includes abundance estimates (usually
as discrete frequency distributions, called ‘species-abundance’ relations) but does
not include representation of ‘interactions’ among components. The last requirement is the job of a Venn-complex (used frequently in Set Theory texts). Some
properties of V-complexes are used in the construction of an S-complex where a
sorted complex may be either a topological tree, a dendrite, or, it may be some
ordination diagram. If an S-complex is somehow allocated into the topographical
(‘real’) space, then an A-complex, an allocated complex is gained (where
classification can be regarded as a special, ‘fortunate’ case)’’.
According to Juhasz-Nagy’s (1993) enlightening (though neglected) scheme,
traditional diversity measures that are computed solely from species relative
abundances are ‘simplex’ representations of community diversity, as opposed to
more modern, ‘complex’ representations. For example, Solow and Polasky (1994)
argue that measures of biodiversity are best computed from the ‘complex’ set of
pairwise species distances dij, while most taxonomic diversity measures are
calculated from phylogenetic or taxonomic trees, increasing the level of complexity
of community representation further on (see Faith 1992; Webb 2000; Petchey and
Gaston 2002).
Likewise, more recently, Podani et al. (2005) emphasize the necessity of a
complex approach to vegetation analysis at various levels of abstraction. The
suggested analytical steps include analysis at the data, derived variable, distance,
ordination and classification levels.
Fig. 1 Schematic representation of community composition at different levels of abstraction (redrawn
from Juhasz-Nagy, 1993)
123
28
C. Ricotta
In this framework, Rao’s quadratic diversity combines the information proper of
a ‘simplex’ community representation (i.e., the species relative abundances) with
the information of a Venn-complex (the pairwise species distances dij) into a single
summary statistics. Here, it is worth noticing that besides simple pairwise species
distances, a Venn complex may also show more complex multispecies relationships
as well (see Juhasz-Nagy and Podani 1983; Juhasz-Nagy 1993). In this sense,
pairwise species distances represent the most elementary among all possible
representations of inter-species relationships.
Based on the above observations, a basic ‘complex’ representation of community
diversity may be obtained trough an N x N quadratic matrix summarizing the
pairwise distances between the N elements sampled along with their relative
abundances (see Table 1).
In the remainder, I will call this matrix the ‘compositional diversity matrix’ C. In
this matrix, N represents the number of components (usually species) that structure
the community under study. However, in some cases, two or more species can be
aggregated into a higher-level entity with uniform properties, like higher-level
taxonomic groups, or functional groups. For example, if our aim consists in
measuring functional diversity, in the presence of species with identical functional
traits, like two microspecies of Hieracium or Taraxacum, functionally identical
species are grouped into one single ‘functional component’(Ricotta 2005b; see also
Fonseca and Ganade 2001).
For sake of simplicity, in analogy to the constituents of traditional diversity
measures, in the remainder, I will refer to N as to ‘component richness’ or ‘species
richness’.
The entries dij (i = j) of the quadratic matrix summarize the pairwise distances
between the N community components. In the simplest case, first, a set of properties
thought to be of significance for ecosystem functioning is measured for each
component (species) obtaining an N x p matrix of p properties measured on N
components (these properties may be functional traits, morphological or ecological
characters, etc.). Next, the N x p matrix is converted into a distance matrix C the
elements dij of which embody the distances between components i and j. In this
case, the distances dij are the elements of a Venn-complex sensu Juhasz-Nagy
Table 1 Illustratory example of the compositional diversity matrix C for an artificial five-species
community composed of Ostrya virginiana, Populus grandidentata, Prunus serotina, Quercus rubra
and Ulmus americana
Ov
Pg
Ps
Qr
Ua
Ostrya virginiana (Ov)
0.28
4.31
3.01
1.89
1.89
Populus grandidentata (Pg)
4.31
0.24
3.66
4.31
2.85
Prunus serotina (Ps)
3.01
3.66
0.27
3.34
2.21
Quercus rubra (Qr)
1.89
4.31
3.34
0.14
2.85
Ulmus americana (Ua)
1.89
2.85
2.21
2.85
0.07
Here, component richness N = 5, the entries dii on the principal diagonal are the relative abundances of
each species, and the pairwise distances dij are the genetic distances proposed by Shimatani (2001,
Appendix 2)
123
A semantic taxonomy for diversity measures
29
(1993) that is derived directly from a simplex (the N x p matrix). Nonetheless, the
distances dij may be also obtained by more sophisticated (i.e., abstract) methods. For
example, the distances dij may be obtained from a phylogenetic tree (i.e., a Sortedcomplex) as the metric/topological distance separating component i from j along the
tree (for a review of the different measures of pairwise species distance that can be
obtained from a dendrogram, see Podani (2000), pp. 324–325).
Since the distance from a given component (species) to itself is zero, in
traditional distance matrices, the entries dii on the principal diagonal are usually set
to zero. By contrast, in the compositional diversity matrix, the entries dii on the
principal diagonal harbor the relative abundances pi of each component.
Traditionally, the values of pi are obtained from data on biomass, cover,
or number of individuals. Nonetheless, more abstract properties, like measures
of species originality (see Pavoine et al. 2005a), species conservation values, etc.
that are transformed to a probability space may be adequately used for summarizing
the relative abundances of each component.
As a result, the matrix of Table 1 is composed of three different pieces of
information: component richness N, the relative abundances of each component, and
the pairwise distances between component i and component j.
Interestingly, a similar matrix representation is extensively used in mathematical
chemistry for investigating the structure-activity relationships of chemical compounds. In this case, N represents the number of vertices in the molecular graph, dij
are the topological distances between vertices i and j, and dii take into account the
physicochemical properties, like the number of valence electrons of atoms
symbolized by vertices i (see Basak et al. 2000).
4 Quantifying the components of biological diversity
Based on the compositional diversity matrix, it is possible to classify most existing
diversity measures into a few general categories that are obtained by the
combination of the three pieces of information, N, dij, and dii.
Following the traditional view (e.g., Pielou 1975), a measure of ‘abundanceweighted diversity’ is a measure, like the Shannon entropy or the Simpson diversity
that is computed solely from the relative abundances of community components pi
(or dii) to the exclusion of other information. If diversity measures are normalized
using either ‘component richness’ N or an approved function of N, we obtain an
index of ‘evenness’.
According to this classical framework, abundance-weighted diversity indices can
be decomposed into two orthogonal constituents, component richness N and
evenness in species abundance (Mouillot et al. 2005). All this is very classic.
By contrast, the measures that are computed from the inter-species distances dij
basically summarize the overall ‘divergence’ between the N community components.
Within this context, we can make a distinction between ‘extensive measures of
divergence’ sensu Izsak and Papp (2000) that are computed solely from the
distances dij, and ‘intensive measures of divergence’ that are normalized by
123
30
C. Ricotta
component richness N, or by an adequate function of N. As an example, a
straightforward way to obtain an extensive measure of divergence from the
compositional diversity matrix C is to sum the elements dij (i = j) in C:
X
dC ¼
dij
ð2Þ
dij2C
If the distances dij are computed from species’ functional traits, the quantity in
Eq. (2) is generally known as ‘Functional Attribute Diversity’ (FAD, Walker et al.
1999), while Izsak and Papp (2000) proposed a similar measure that is computed
from species’ pairwise taxonomic distances.
Next, dividing dC by N(N1), an intensive (average) measure of species
divergence is obtained.
Finally, measures like Rao’s Q that combine species relative abundances and
pairwise species distances into a single scalar, may be termed ‘abundance-weighted
indices of divergence’ (Mason et al. 2005). Note that Eq. (2) is basically a special
case of Rao’s Q where all components are given equal weight (i.e., Q ¼ N 2 dC ).
In a previous paper (Ricotta 2002), I defined the indices that depend on both
variables, dij and dii ‘weak diversity indices’. In this view, the term ‘weak diversity
index’ may be considered a synonym of ‘abundance-weighted measure of
divergence’.
5 Conclusions
Recent studies suggest that changes in community diversity can alter both the
magnitude and the stability of ecosystem processes (Naeem et al. 1999). Testing the
effects of community diversity on ecosystem processes will require measures to be
available based on a rigorous semantic classification of compositional diversity.
From an historical perspective, traditional abundance-weighted diversity measures are summarized at a lower level of abstraction than more modern metrics of
community diversity. According to our proposal, compositional diversity may be
decomposed into four basic families: abundance-weighted diversity, richness
evenness and divergence. In turn, divergence can be partitioned into three
subclasses: component-independent (i.e., extensive), component dependent (i.e.,
intensive), and abundance-weighted divergence.
Each of these families can be computed by a combination of the three pieces of
information contained in the compositional diversity matrix, component richness N,
the relative abundances of each component, and the pairwise distances between
component i and component j. Increasingly complex combinations of these pieces
of information correspond to increasingly higher levels of abstraction in the
conceptualization of community diversity.
Using these four main families of measures will accommodate most of the
existing traditional and contemporary indices of community diversity into one
single conceptual framework eliminating much of the confusion currently
123
A semantic taxonomy for diversity measures
31
surrounding the study of the relationship between community diversity and
ecosystem functioning. In this sense, this paper focuses on the description of
compositional diversity measures and has little to say on ecosystem functioning.
Finally, it is worth stressing that the arsenal of possible measures of diversity is
by far not exhausted by our proposal. For instance, the compositional diversity
matrix C combines the data of an elementary Venn-complex that represents pairwise
species distances with the data obtained from a ‘simplex’ representation of
community diversity. However, it ignores the information associated to more
abstract levels of community representation, like sorted or allocated complexes (see
Juhasz-Nagy 1993).
For example, as mentioned in the previous paragraphs, Faith (1992) suggested
measuring phylogenetic diversity as the cumulative branch length of the full
phylogenetic tree. Likewise, Petchey and Gaston (2002) proposed a method for
computing functional diversity, which is based on the total branch length of the
functional dendrogram that results from clustering the species in trait space.
Both methods use data proper of a dendrogram (by contrast, methods for
computing compositional diversity from ordinations are much scarcer; see, e.g., Ter
Braak 1983; Péllissier et al. 2003). As such, since dendrograms and ordinations are
both sorted complexes sensu Juhasz-Nagy (1993), they cannot be included in our
conceptual framework. Accordingly, I propose to define all measures that are
obtained from a ‘complex’ representation of community diversity, like S-complexes
or A-complexes, as ‘measures of biological complexity’. On one hand, this
definition is general enough to allow the inclusion of a wide number of measures; on
the other hand, the term ‘complexity’ immediately suggests a high level of
abstraction of the underlying community representations. I hope, the proposed
semantic taxonomy of different aspects of community diversity will prove fruitful
for avoiding confusion in their formalization and ecological application.
Acknowledgements Unfortunately, this paper is not aimed at providing guidance on which index to use
for testing different aspects of the diversity/ecosystem functioning relationships. So, I would like to thank
the anonimous referee that appreciated the primary objective of this paper without asking that something
be said about a more field-oriented approach to diversity measures.
References
Alatalo RV (1981) Problems in the measurement of evenness in ecology. Oikos 37:199–204
Barker GM (2002) Phylogenetic diversity: a quantitative framework for measurement of priority
and achievement in biodiversity conservation. Biol J Linn Soc 76:165–194
Basak SC, Balaban AT, Grunwald GD, Gute BD (2000) Topological indices: their nature and mutual
relatedness. J Chem Inf Compu Sci 40:891–898
Champely S, Chessel D (2002) Measuring biological diversity using Euclidean metrics. Environ Ecol Stat
9:167–177
Clarke KR, Warwick RM (1998) A taxonomic distinctness index and its statistical properties. J Appl Ecol
35:523–531
Crozier RH (1992) Genetic diversity and the agony of choice. Biol Conserv 61:11–15
Faith DP (1992) Conservation evaluation and phylogenic diversity. Biol Conserv 61:1–10
Fargione JE, Tilman D (2005) Diversity decreases invasion via both sampling and complementarity
effects. Ecol Lett 8:604–611
123
32
C. Ricotta
Fonseca CR, Ganade G (2001) Species functional redundancy, random exinctions and the stability
of ecosystems. J Ecol 89:118–125
Fridley JD, Brown RL, Bruno JF (2004) Null models of exotic invasion and scale-dependent patterns
of native and exotic species richness. Ecology 85:3215–3222
Hector A, Schmid B, Beierkuhnlein C, Caldeira MC, Diemer M, Dimitrakopoulos PG, Finn JA, Freitas H,
Giller PS, Good J, Harris R, Högberg P, Huss-Danell K, Joshi J, Jumpponen A, Körner C, Leadley
PW, Loreau M, Minns A, Mulder CPH, O’Donovan G, Otway SJ, Pereira JS, Prinz A, Read DJ,
Scherer-Lorenzen M, Schulze ED, Siamantziouras AS, Spehn EM, Terry AC, Troumbis AY,
Woodward FI, Yachi S, Lawton JH (1999) Plant diversity and productivity experiments in European
grasslands. Science 286:1123–1127
Hendrickson JA Jr, Ehrlich PR (1971) An expanded concept of ‘‘species diversity’’. Notulae Naturae of
the Academy of Natural Sciences of Philadelphia 439:1–6
Hooper D, Vitousek PM (1997) The effects of plant composition and diversity on ecosystem processes.
Science 277:1302–1305
Hurlbert SH (1971) The nonconcept of species diversity: a critique and alternative parameters. Ecology
52:577–586
Huston MA (1994) Biological diversity: the coexistence of species on changing landscapes. Cambridge
University Press, Cambridge
Gregorius HR (1990) A diversity-independent measure of evenness. Am Nat 136:701–711
Izsak J, Papp L (1995) Application of the quadratic entropy index for diversity studies on drosophilid
species assemblages. Environ Ecol Stat 2:213–224
Izsak J, Papp L (2000) A link between ecological diversity indices and measures of biodiversity. Ecol
Modell 130:151–156
Juhasz-Nagy P (1993) Notes on compositional diversity. Hydrobiologia 249:173–182
Juhasz-Nagy P, Podani J (1983) Information theory methods for the study of spatial processes and
succession. Vegetatio 51:129–140
Kvålseth TO (1991) Note on biological diversity, evenness, and homogeneity measures. Oikos 62:123–
127
Legendre P, Borcard D, Peres-Neto PR (2005) Analyzing beta diversity: partitioning the spatial variation
of community composition data. Ecol Monogr 75:435–450
Lloyd M, Ghelardi RJ (1964) A table for calculating the ‘‘equitability’’ component of species diversity.
J Anim Ecol 33:217–225
Mason NVH, Mouillot D, Lee WG, Wilson JB (2005) Functional richness, functional evenness and
functional divergence: the primary components of functional diversity. Oikos 111:112–118
McCann KS (2000) The diversity-stability debate. Nature 405:228–233
Mouillot D, Mason NWH, Dumay O, Wilson JB (2005) Functional regularity: a neglected aspect of
functional diversity. Oecologia 142:353–359
Naeem S, Wright JP (2003) Disentangling biodiversity effects on ecosystem functioning: deriving
solutions to a seemingly insurmountable problem. Ecol Lett 6:567–579
Naeem S, Chapin T, Costanza R, Ehrlich P, Golley D, Hooper FB, Lawton JH, O’Neil R, Mooney H, Sala
O, Symstad A, Tilman D (1999) Biodiversity and ecosystem functioning. Ecological issues, No. 4.
Ecological Society of America, Washington, DC
Pavoine S, Ollier S, Dufour AB (2005a) Is the originality of a species measurable? Ecol Lett 8:579–586
Pavoine S, Ollier S, Pontier D (2005b) Measuring diversity from dissimilarities with Rao’s quadratic
entropy: Are any dissimilarities suitable? Theor Popul Biol 67:231–239
Péllissier , R, Couteron P, Dray S, D Sabatier (2003) Consistency between ordination techniques and
diversity measurements: two strategies for species occurrence data. Ecology 84:242–251
Petchey OL, Gaston KJ (2002) Functional diversity (FD), species richness and community composition.
Ecol Lett 5:402–411
Petchey OL, Gaston KJ (2006) Functional diversity: back to basics and looking forward. Ecol Lett 9:741–
758
Pielou EC (1975) Ecological diversity. Wiley, New-York
Podani J (2000) Introduction to the exploration of multivariate biological data. Backhuys Publishers,
Leiden, NL
Podani J, Csontos P, Tamás J, Miklós I (2005) A new multivariate approach to studying temporal changes
of vegetation. Plant Ecology 181:1–16
Poole RW (1974) An introduction to quantitative ecology. McGraw-Hill, New York
123
A semantic taxonomy for diversity measures
33
Rao CR (1982) Diversity and dissimilarity coefficients: a unified approach. Theor Popul Biol 21:24–43
Ricotta C (2002) Bridging the gap between ecological diversity indices and measures of biodiversity with
Shannon’s entropy: comment to Izsak and Papp. Ecol Modell 152:1–3
Ricotta C (2003) On parametric evenness measures. J Theor Biol 222:189–197
Ricotta C (2004) A parametric diversity measure combining the relative abundances and taxonomic
distinctiveness of species. Divers Distrib 10:143–146
Ricotta C (2005a) Through the jungle of biological diversity. Acta Biotheor 53:29–38
Ricotta C (2005b) A note on functional diversity measures. Basic Appl Ecol 6:479–486
Ricotta C, Avena GC (2005) A ‘fast-food approach’ to the standardisation of quadratic diversity. Plant
Biosyst 139:411–413
Ricotta C, Szeidl L (2006) Towards a unifying approach to diversity measures: bridging the gap between
the Shannon entropy and Rao’s quadratic index. Theor Popul Biol 70:237–243
Ricotta C, De Zuliani E, Pacini A, Avena GC (2001) On the mutual relatedness of evenness measures.
Community Ecol 2:51–56
Sarkar S, Margules C (2002) Operationalizing biodiversity for conservation planning. J Biosci
27(S2):299–308
Shimatani K (2001) On the measurement of species diversity incorporating species differences. Oikos
93:135–147
Smith B, Wilson JB (1996) A consumer’s guide to evenness indices. Oikos 76:70–82
Solow AR, Polasky S (1994) Measuring biological diversity. Environ Ecol Stat 1:95–107
Stohlgren TJ, Binkley D, Chong GW, Kalkhan MA, Schell LD, Bull KA, Otsuki Y, Newman G, Bashkin
M, Son Y (1999) Exotic plant species invade hot spots of native plant diversity. Ecol Monogr 69:25–
46
Taillie , C (1979) Species equitability: a comparative approach. In: Grassle JF, Patil GP, Smith WK,
Taillie C (eds) Ecological diversity in theory and practice. International Cooperative Publishing
House, Fairland, MD, pp 51–62
Ter Braak CF (1983) Principal components biplots and alpha and beta diversity. Ecology 64:454–462
Tilman D, Wedin D, Knops J (1996) Productivity and sustainability influenced by biodiversity in
grassland ecosystems. Nature 379:718–720
Vane-Wright RI, Humphries CJ, Williams PM (1991) What to protect: systematics and the agony of
choice. Biol Conserv 55:235–254
Walker B, Kinzig AP, Langridge J (1999) Plant attribute diversity, resilience, and ecosystem function: the
nature and significance of dominant and minor species. Ecosystems 2:95–113
Warwick RM, Clarke KR (1995) New ‘biodiversity’ measures reveal a decrease in taxonomic distinctness
with increasing stress. Mar Ecol Progr Ser 129:301–305
Webb CO (2000) Exploring the phylogenetic structure of ecological communities: an example for rain
forest trees. Am Nat 156:145–155
Wehenkel C, Bergmann F, Gregorius HR (2006) Is there a trade-off between species diversity and genetic
diversity in forest tree communities? Plant Ecol 185:151–161
Whittaker RH (1965) Dominance and diversity in land plant communities. Science 147:250–260
123