Syst. Biol. 60(5):645–660, 2011
c The Author(s) 2011. Published by Oxford University Press, on behalf of the Society of Systematic Biologists. All rights reserved.
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DOI:10.1093/sysbio/syr026
Advance Access publication on April 6, 2011
Fuzzy Chorotypes as a Conceptual Tool to Improve Insight into Biogeographic Patterns
J ES ÚS O LIVERO ∗ , R AIMUNDO R EAL , AND A NA L. M ÁRQUEZ
Grupo de Biogeografı́a, Diversidad y Conservación, Departamento de Biologı́a Animal, Facultad de Ciencias,
Universidad de Málaga, 29071 Málaga, Spain;
∗ Correspondence to be sent to: Grupo de Biogeografı́a, Diversidad y Conservación, Departamento de Biologı́a Animal, Facultad de Ciencias,
Universidad de Málaga, 29071 Málaga, Spain; E-mail: [email protected].
Received 8 September 2010; reviews returned 14 February 2011; accepted 16 February 2011
Associate Editor: Adrian Paterson
Abstract.—Chorotypes—statistically significant groups of coincident distribution areas—constitute biogeographic units that
are fuzzy by nature. This quality has been referred to in the literature but has not been analyzed in depth or methodologically developed. The present work redefines chorotypes as fuzzy sets from a pragmatic perspective and basically focuses
on the methodological and interpretative implications of this approach. The amphibian fauna in the Iberian Peninsula was
used as an example to explore the fuzzy nature of chorotypes. The method on which this article is based is a widely used
technique to define chorotypes. This method involves the fuzziness that is inherent to the identification between degree of
similarity and degree of membership and includes a probabilistic analysis of the classification for the objective delimitation
of chorotypes. The main innovation of this paper is a procedure to analyze chorotypes as fuzzy biogeographic units. A
set of fuzzy parameters to deal with the biogeographic interpretation of fuzzy chorotypes is also described. A computer
program has been developed and is freely available. History may be related to the degree of fuzziness of chorotypes. In our
example, with amphibian distributions in Iberia, less fuzzy chorotypes could have a historical explanation, and the internal fuzziness of chorotypes increases with their distance to hypothetical Pleistocene refugia. [Biodiversity; biogeographic
patterns; distribution; fuzzy chorotypes; fuzzy logic; history; ecological factors; geographic attractors.]
Baroni-Urbani et al. (1978) defined a “chorotype” as a
distribution pattern, followed by one or several species,
which can be operatively recognized within an area.
Similar definitions are found in Vargas et al. (1997) and
in Zunino and Zullini (2003). This concept has also been
named “distributional type” (Dzwonko and Kornaś
1978), “biotic element” (Birks 1987; Hausdorf 2002),
“chorological category” (Zunino and Zullini 2003), and
“chorological type” (Ferrer-Castan and Vetaas 2003;
Ojeda et al. 2005). In recent times, the term chorotype
has also been used to name a typical region for a “biochore” (“group of similar biotopes, the largest division
of animal and plant ecosystems”): According to this,
a chorotype would be a biochore of lower rank that is
chosen in the same way as a type species is designated
for a genus in taxonomy (Westermann 2000). The subject
concerning this work deals with chorotypes as defined
by Baroni-Urbani et al. (1978).
Chorotypes may result from ecological causes, that
is, differential responses to environmental conditions
shared by several species, or from historical causes, that
is, past events that restricted or biased certain groups
of species to different parts of the Earth (Real, Olivero,
et al. 2008). If chorotypes are detected, then it is not necessary to invoke a different cause for explaining each
species distribution, but some factors may be responsible for the distributions shared by each group of species,
and the overall environmental interpretation may be
more comprehensible (Márquez et al. 1997). Because of
this, chorotypes enhance the search for phenomena that
influence the spatial variation of biodiversity from the
perspective of a global process in which multiple species
are involved with different degrees of interrelationship
(Real et al. 2002). In this regard, numerous studies have
been published that have objectively detected and analyzed chorotypes worldwide, for example, for algae
(Báez el al. 2005), vascular plants (Birks 1976; Myklestad
and Birks 1993; Márquez et al. 1997; Gómez-González
et al. 2004; Teneb et al. 2004; Finnie et al. 2007), insects
(Baroni-Urbani and Collingwood 1976; Baroni-Urbani
et al. 1978; Hausdorf and Henning 2003), fish (Carmona
et al. 1999), amphibians (Vargas and Real 1997; Flores
et al. 2004), reptiles (Real et al. 1997; Vargas and Real
1997; Real, Márquez, et al. 2008), birds (Muñoz et al.
2003; Real, Márquez, et al. 2008; Real, Olivero, et al.
2008), and mammals (Vargas et al. 1997; Sans-Fuentes
and Ventura 2000; Real et al. 2003; Real, Márquez, et al.
2008).
In a new approach, chorotypes have been recently
used for the biogeographic deconstruction of biodiversity. Marquet et al. (2004) proposed the deconstruction
of biodiversity patterns (as a “turning to their roots”
or a “disaggregation” to make apparent what is hidden in them) according to the attributes of species,
as a way of gaining a better insight into the causes
of biodiversity trends. Huston (1994) also suggested
that it is useful to decompose biological diversity into
components with consistent biogeographic patterns
and then analyze the processes that influence each
of them. Chorotypes, as consistent biogeographic responses among subsets of species, aided Real, Olivero,
et al. (2008) in tracking down the different spatial responses of waterbird species richness in Europe to
environmental energy, after Bárcena et al. (2004) had
detected, for this group, a general trend shaped by this
factor (for a deconstruction of vascular plant species
richness in Europe according to chorotypes, see Finnie
et al. 2007). Chorotypes have also been proposed by
Thuiller et al. (2005) as a way to explore the relationships between ecological and distributional properties of species and their projected sensitivity to climate
change.
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All environmental factors may affect all species, although not necessarily to the same extent. If a chorotype
is interpreted as a biogeographic response to an environmental factor, then the degree to which the distribution
of a species belongs to such a chorotype could be a measure of the degree to which this species is affected by the
factor. This is an approach that can clearly benefit from
the tools of fuzzy logic.
A fuzzy set is a class of objects with a continuum
of grades of membership, so that such a set is characterized by a membership function that assigns to each
object a real number in the interval [0, 1] (Zadeh 1965).
The difference between the logic behind fuzzy sets and
the traditional probabilistic approach is that in probability something is either true or false, whereas in fuzzy set
theory the membership describes the degree to which
an element meets the definition of the set: It is neither
absolutely true nor false but true to some extent. Thus,
the fuzzy membership values indicate that each class
exists for each object to some degree (Brown 1998).
The fuzzy sets approach recognizes that, in some cases,
none of the classes is appropriate for a certain object,
whereas other objects may be reasonably classified into
two or more categories (Townsend and Walsh 2001). As
chorotypes may partly overlap, each species could be
defined as having a degree of membership in the different chorotypes described. In this sense, the distributions
included in a chorotype constitute a group that is fuzzy
by nature, so that the principles and rules of fuzzy logic
may be easily applied to them (Real, Olivero, et al.
2008).
The fuzzy logic approach has become frequent in
studies belonging to different biological disciplines in
the two last decades. Fuzzy sets have been applied to
taxonomy (e.g., Pappas 2006), where even the fuzzy
nature of taxa, including species, has been postulated
(Hall 1997; Turner 1999). Within the environmental sciences, fuzzy sets have formed the core of studies aimed
at determining priority areas for conservation (e.g.,
Stoms et al. 2002; Estrada et al. 2008) and have provided tools for the management of ecosystems (e.g.,
Freyer et al. 2000; Adriaenssens et al. 2004; Prato 2005).
In biogeography, two main fields have strongly benefited from fuzzy logic: land classification according to
biotic parameters and investigation into the spatial distribution of species. Fuzzy classifications of land have
been used to identify plant associations (e.g., Brown
1998; Olano et al. 1998), identify animal communities
(e.g., Tepavčevic and Vujić 1996; Eyre et al. 2003), and
delimit ecosystems using remote sensing imagery (e.g.,
Townsend and Walsh 2001; Arnot et al. 2004). On the
other hand, distribution modeling with methods based
on fuzzy logic has been applied to animals and plants
(e.g., Robertson et al. 2004; Gevrey et al. 2006; Van der
Broekhoven et al. 2006; Real et al. 2009). Chorotypes are
in an intermediate position between biogeographic land
area classification and species distribution modeling.
Chorotypes are the result of classifying species distributions according to how they cover the same area and are
defined as types of distributions that can be submitted
to environmental modeling in the same way as species
distributions (e.g., Real, Olivero, et al. 2008).
A widely used technique for the detection of chorotypes is described in Márquez et al. (1997) (see also
Real et al. 2002; Real, Olivero, et al. 2008). It is based
on a quantitative classification according to similarities between objects (in this case species distributions),
which involves the fuzziness that is inherent in the
identification between degree of similarity and degree
of membership (Salski 2007). A later probabilistic analysis of the classification, which follows the approach
of McCoy et al. (1986), permits the objective detection
of chorotypes. The results of this method can be, however, expressed in possibilistic nonprobabilistic ways,
such that the degree to which a certain species belongs
to every chorotype is expressed, thus allowing a fuzzy
approach to the classification results. Our objectives are
1) to improve the method to detect chorotypes with a
procedure to handle chorotypes as fuzzy biogeographic
units and 2) to explore the interpretative possibilities
and implications of this new approach with an example
based on the distribution of amphibians in the Iberian
Peninsula.
M ATERIAL AND M ETHODS
Species and Study Area
Amphibian distribution data were taken from the
Atlas and Red Book of Amphibians and Reptiles of
Spain (Pleguezuelos et al. 2004) and from the Atlas of
Amphibians and Reptiles of Portugal (Loureiro et al.
2008). The 27 species inhabiting the Iberian Peninsula
were used (see Fig. 1) all of which have been traditionally considered indigenous species. Recent phylogeographic studies, however, state that the current Iberian
distribution of Discoglossus pictus and the northeastern
populations of Hyla meridionalis are the result of a modern dispersion from France, where they were probably
anthropogenically introduced at the end of the 19th
century (Martı́nez-Solano 2004; Recuero et al. 2007).
These populations have been considered in our analyses because their century-long occurrence has resulted
in populations that are now integrated into Iberian
ecosystems. Nevertheless, the northeastern range of H.
meridionalis is currently segregated from its southwestern range in Iberia, and both ranges have been analyzed
as different distributions. This choice has been made
because a common history and similar environmental involvement cannot be assumed for both ranges,
whereas keeping them together could create significant
noise in the search for common biogeographic patterns
among amphibians from Iberia.
The exploration of chorotypes was made on 50 × 50km UTM squares. All squares at this resolution have
been prospected to build the atlases (Pleguezuelos
et al. 2004; Loureiro et al. 2008), and so the problem
of false absences is minimized. For discussion purposes,
chorotypes were also explored separately in Portugal
and in Spain.
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OLIVERO ET AL.—FUZZY CHOROTYPES AND BIOGEOGRAPHIC PATTERNS
FIGURE 1. Classification tree of the amphibian distributions in
the Iberian Peninsula according to S, the Baroni-Urbani and Buser
(1976) similarity index. The species that constituted the four chorotypical clusters (CC.1, CC.2, CC.3, and CC.4) are enclosed in squares.
The degree to which each distribution belongs to each significant
chorotype (Ch) is shown. Black symbols (circles and squares) indicate all branches susceptible to be interpreted as fuzzy biogeographic
patterns with different levels of fuzziness of which a maximum of 11
nonnested patterns are identified with black squares.
Chorotype Identification
Real et al. (1992) proposed a probabilistic procedure
for recognizing biotic boundaries based on species distributions. This procedure was adapted in Márquez
et al. (1997) for its use in detecting chorotypes and was
later modified in Muñoz et al. (2003). This method allows the simultaneous detection of continuous and discrete biogeographic patterns that could coexist and even
partially overlap within the study area, which permits
the interpretation of chorotypes as fuzzy sets.
A matrix of geographic similarities between the distributions of each pair of amphibian species in Iberia was
obtained using Baroni-Urbani and Buser (1976) index:
S= p
p
(C × D) + C
(C × D) + A + B − C
,
647
where A is the number of grids in which the species a is
present, B is the number of grids in which the species b
is present, C is the number of grids where both species
a and b are present, and D is the number of grids from
which both species a and b are absent. A similarity
value 1 means completely coinciding distributions, and
0 means completely noncoinciding distributions. This
coefficient takes into account shared absences, that is,
grids in the Iberian Peninsula outside the distribution
area of both species, and so the similarities were considered in relation to the study area and not only in
relation to the two combined distribution areas (Real
et al. 1992). This characteristic makes the similarity
value increase with the study area if no new presences
are added, which is in tune with the fuzzy concept of
similarity. Shared absences are important because they
could be due to either ecological or historical reasons
that should be taken into account (Baroni-Urbani and
Buser 1976). However, this index gives more importance to shared presences and the possibility that two
distributions are considered similar only because of
their shared absences is avoided by multiplying shared
absences by shared presences. In addition, the BaroniUrbani and Buser’s (1976) index has a table of critical
values (Baroni-Urbani and Buser 1976), as the method
described below to search for clusters requires.
An important requisite of chorotypes is to maximize
similarity within groups. Because of this, an agglomerative method of classification, the unweighted pair-group
method using arithmetic averages (UPGMA), was used.
Among the phenetic classification methods, average
linkage produces less distortion in relation to the original similarities than complete or single linkages, and
UPGMA produces less distortion than other average
linkages like WPGMA and UPGMC (Sneath and Sokal
1973). Kreft and Jetz (2010) found that UPGMA was the
consistently best performing clustering algorithm for a
biogeographical classification across a set of different
methods including 7 agglomerative hierarchical clusters (UPGMA, UPGMC, WPGMA, WPGMC, Ward’s
method, single linkage, and complete linkage), an iterative algorithm (neighbor-joining trees), and a divisive
hierarchical method (DIANA algorithm). Our result
was expressed as a dendrogram.
As the null model is that similarities between distributions are not different from those expected at random, we used the table of critical values presented in
Baroni-Urbani and Buser (1976) to perform exact randomization tests (Sokal and Rohlf 1981, p. 788) where
the observed similarity values were compared with all
the possible outcomes (see also Real and Vargas 1996).
Values of the similarity index higher than 95% of outcomes were considered “significant similarities” (+),
values lower than 95% of outcomes were “significant
dissimilarities” (−), and the rest were considered values expected at random (0). These “significances” do
not reflect the statistical probability of committing a
type II inference error, that is, the probability of making a false inference about a population on the basis of
what is found in a sample, but the exact mathematic
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probability of finding a lower or higher similarity value
in the population.
To detect chorotypes, we examined the dendrogram
to identify branches that exhibited significant positive
within-branch shared distributions and which were significantly disjoint from adjoining branches. For each
merging of branches in the dendrogram, let A be the
set of species in the left branch, and let B be the set of
species in the right branch. The Cartesian product A × A
represents the pairwise comparison of every species in
set A with every other species in set A. If there are nA
species in set A, there will be (n2A − nA )/2 comparisons
in the product because the product would be symmetrical and we are only interested in one triangle. Similarly
for set B, the product B × B compares every species in
set B with every other species in set B, and there are
(n2B − nB )/2 pairwise comparisons. Finally, let the product A × B represent the comparison of every species
in set A with every species in set B; there are nA × nB
comparisons.
If set A represents a chorotype, we would expect to
find primarily significant similarities in the product A ×
A, with relatively few significant dissimilarities. If set A
is distinct from set B, we would expect to find primarily significant dissimilarities or similarities expected at
random in product A × B, with relatively few significant
similarities.
Let
P
1 if i × j = +,
i,j∈A αi,j
, where αi,j =
Pp = 2
0 if i × j =
/ +
(nA − nA )/2
represent the proportion of pairwise comparisons in the
product A × B that are significant similarities, and let
P
P
i∈A ωi +
j∈B ωj
0
Psp =
nA + nB
1 if ∀ j ∈ B any i × j = +,
where ωi =
0 otherwise
1 if ∀ i ∈ A any i × j = +
and ωj =
0 otherwise
represent the number of species in set A or B that have
at least one significant similarity with another species in
the other set.
Based on these values, we can calculate the extent
to which significant similarities and dissimilarities predominate where expected. Let
Pp × Psp
,
dp = q
P2p + P2sp
Pm × Psm
dm = p
,
P2m + P2sm
P0p × P0sp
d0p = q
,
P0p 2 + P0sp 2
where dp is the predominance of significant similarities
within set A, dm is the predominance of significant dissimilarities in set A, and d0p is the predominance of significant similarities in product A × B.
Finally, we calculate the degree to which set A meets
our criteria for representing a chorotype as
IH =
dp − dm − d0p
√
,
2/2
where IH is the index of
√internal homogeneity and distinctness. Division by 2/2 rescales the IH to [−1, 1]
(without
such a rescaling, IH is equal to DW(A × A) in
P
Márquez
et al. 1997). In this way, we computed the IH
∀
j
∈
A
any
i
×
j
=
+,
γ
i
1 if
Psp = i∈A , where γi =
values
for
every branch of the dendrogram. A cluster
0 otherwise
nA
was considered “chorotypical” if: 1) IH = 1, that is, all
represent the proportion of species in set A that have at the criteria were completely met or else 2) IH was posleast one significant similarity with another species in itive, higher than those of the other clusters including
the distributions involved, and statistically significant.
set A. Similarly, let
Our statistical approach is based on challenging the null
P
hypothesis that + similarities between the distributions
1 if i × j = −,
i,j∈A δi,j
grouped in the tested cluster are not more frequent than
Pm = 2
, where δi,j =
0 if i × j =
/ −
+ similarities between such cluster and the most similar
(nA − nA )/2
branch of the dendrogram, using a G-test of indepenrepresent the proportion of pairwise comparisons in dence (Sokal and Rohlf 1981), which only is applicable
if the cluster includes at least three species and A × B
A × A that are significant dissimilarities, and let
includes at least six comparisons. Distributions that did
P
ζ
i
1 if ∀ j ∈ A any i × j = −, not fulfil either of these conditions did not belong to any
Psm = i∈A , where ζi =
chorotypical cluster.
0
otherwise
nA
The abovementioned procedure reduces drastically
the number of clusters tested with the G-test of indepenrepresent the number of species in set A that have at dence (2 out of 55 branches in our case). Nevertheless,
least one significant dissimilarity with another species we dealt with the probability of erroneously rejecting
in set A. Finally, let
one of the true null hypotheses due to the familywise error rate involved in multiple hypotheses testing
P P
1 if i × j = +,
i∈A
j∈B ϕi,j
(Benjamini and Hochberg 1995; Hausdorf and Henning
0
Pp =
, where ϕi,j =
0
if
i
×
j
=
/
+
2003)
by controlling the false discovery rate (FDR) using
nA × nB
represent the proportion of pairwise comparisons A × A
that are significant similarities. Let
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OLIVERO ET AL.—FUZZY CHOROTYPES AND BIOGEOGRAPHIC PATTERNS
the procedure proposed by Benjamini and Hochberg
(1995) under an FDR value of q = 0.05. This procedure
orders the tested clusters according to decreasing significance (increasing P value), being i the position of each
cluster in this ordered list, and only accepts clusters up
to the highest i position whose P value is lower than
i ∗ q/V, where V is the total number of tested clusters.
Chorotype Fuzzification
The explained criteria identify the least fuzzy chorotypical clusters, as all the conditions to be chorotype are
maximized. However, applying the concepts of the
fuzzy set theory, IH represents the degree to which a
cluster meets the definition of the chorotypical cluster,
which is neither absolutely true nor false, but true to
some extent. Thus, every cluster with IH values higher
than 0 fulfils to certain degree the conditions to be a
chorotypical cluster. Certain fuzzy chorotypical clusters may be nested within others, although it is always
possible to identify the nonnested fuzzy chorotypes.
Nevertheless, we applied the following fuzzy parameters to the least fuzzy chorotypes identified as mentioned above, although they may be equally applied to
all fuzzy chorotypical clusters.
The average of the similarities (Sij ) between a certain
species distribution (di ) and all n distributions included
in a chorotypical cluster (ch) was considered a measure
of the degree of membership of di in such a chorotype:
Pn
j=1 (Sij )
, ∀ j ∈ ch.
μch (di ) =
n
649
ch2 is the largest fuzzy set which is contained in both
ch1 and ch2 (Zadeh 1965). The cardinality of the union
between two chorotypes measures how much, taken
together, they comprise the species degrees of membership. The cardinality of the intersection indicates
pairwise coincidences in the species membership, and
thus, how fuzzy the limit between these two chorotypes
is. The degree of membership in the union between two
chorotypes (ch1 ∪ ch2 ) is
μch1 ∪ch2 (di ) = max[μch1 (di ), μch2 (di )].
The degree of membership in the intersection between
two chorotypes (ch1 ∩ ch2 ) is
μch1 ∩ch2 (di ) = min[μch1 (di ), μch2 (di )].
The fuzzy overlap—in reality a fuzzy application of Jaccard’s (1901) similarity index—measures the proportion
of the union that is in the intersection:
— fuzzy overlap (i.e., similarity) between chorotypes:
O(ch1 , ch2 ) =
Card[ch1 ∩ ch2 ]
.
Card[ch1 ∪ ch2 ]
A modification of this overlap provides us with
a measure of how much a chorotype is included in
another:
— inclusion of one chorotype (ch2 ) in another (ch1 ):
I(ch1 /ch2 ) =
Card[ch1 ∩ ch2 ]
.
Card[ch2 ]
Starting with this measure, various fuzzy parameters were computed (after Zadeh 1965; Dubois and
Prade 1980; Kosko 1986; Kuncheva 2001) to describe the
fuzzy nature of every chorotype. The importance of a
chorotype in the context of all the species distributions
considered depends on how many species have a high
degree of membership in it. The parameter that measures the importance of a fuzzy chorotype is cardinality.
Relative cardinality and height, which are, respectively,
the average and the maximum degree of membership
observed in a fuzzy chorotype, provide a context to
evaluate the membership of any particular distribution.
N being the total number of species distributions considered in the study, these parameters may be expressed
as follows:
The fuzzy entropy of a chorotype is the degree of
similarity between “belonging-and-not-belonging” and
“belonging-or-not-belonging” to the chorotype (Kosko
1986) and is a measure of fuzziness:
— a chorotype’s fuzzy cardinality: Card(ch) =
PN
i μch (di ),
— a chorotype’s relative cardinality: RCard(ch) =
1
N × Card(ch),
— a chorotype’s height: H(ch) = max[μch (di )], ∀ di .
A key question is how to map the fuzziness of
chorotypes, so that their fuzzy nature may be, on the
one hand, made observable throughout the study area,
and on the other hand, considered in the search for historical and environmental causes that might underlie a
chorotype. The difficulty lies in the fact that elements
of a chorotype are species distributions, and because
of this, the degree of membership in a chorotype refers
to these and not to locations in space. In every square
in which the study area was divided, we represent
the maximum degree of membership shown by any
The species whose membership degree in a chorotype
coincided with its height may be considered the species
most representative of the chorotype. The union of two
chorotypes ch1 ∪ ch2 is the smallest fuzzy set containing both ch1 and ch2 , whereas their intersection ch1 ∩
— entropy (i.e., degree of fuzziness) of every
chorotype:
E(ch) =
Card[ch ∩ ch0 ]
,
Card[ch ∪ ch0 ]
where ch0 is the fuzzy complement of chorotype ch, the
membership function for it being:
μch0 (di ) = 1 − μch (di ).
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species reported in that square in a given chorotype.
Fuzzy membership in every chorotype is consequently
associated with every point in the space. Expressed as
a formula, the maximum degree of membership in a
given chorotype (MMDch ) in a certain square is
MMDch = max[μch (di ) × pi ],
∀ i,
where pi is either equal to 1 if the species distribution
di includes that grid or it is equal to 0 if the reported
distribution di does not include that grid.
The fuzzy nature of chorotypes has also been taken
into consideration as a criterion to deconstruct biodiversity, such that a measure of fuzzy species richness
of each chorotype (FSRch ) could be computed in every
square of the study area with the following formula:
FSRch =
N
X
i=1
[μch (di ) × pi ].
In essence, this is to weight every distribution—when
computing the species richness—according to every
species degree of membership in a given chorotype.
The MMDch can be seen as a local measure of the
chorotype’s height, whereas the FSRch can be seen as a
local measure of the chorotype’s cardinality (i.e., respectively, the chorotype’s height and cardinality by only
considering the species that were reported at a given
site). As a basis to discuss the biogeographic meaning
of these representations, we calculated geographical
Spearman correlations between the species richness of
each chorotypical cluster (SRch ), MMDch and FSRch for
each chorotype, and geographical Spearman correlations between chorotypes for each of these parameters.
Chorotype detection and the associated fuzzy approach are entirely performed with the RMACOQUI
v1.0 software, which is freely available on request to the
authors of this article.
Comparison with a Nonfuzzy Method to Obtain Chorotypes
To compare our results with those of a nonfuzzy
method, we used the procedure proposed by Hausdorf
and Henning (2003) for the objective detection of biotic elements, which are equivalent to our chorotypes.
The model-based Gaussian clustering method, as implemented in the “prabclust” function in the R-package
PRABCLUS v2.1-4, was used because it also provides a
decision about the number of meaningful clusters and
about ranges that cannot be assigned adequately to any
biotic elements (see Hausdorf and Henning 2003 for
details about this method). Prabclust performs a metric
multidimensional scaling on a matrix of distances between distributions. Following Hausdorf and Henning
(2003), we used the Kulczynski distance (dK , Shi 1993):
1 C C
,
+
dK = 1 −
2 A B
where A is the number of grids in which the species
a is present, B is the number of grids in which the
species b is present, and C is the number of grids where
both species a and b are present. Before performing the
Gaussian clustering, pradclust makes an estimation of
noise that detects the ranges that are not assigned to
any group. The “hprabclust” function that implements
a model-based hierarchical clustering (introduced as
experimental in the PRABCLUST manual) was also
applied to our data set.
R ESULTS
We identified 16 clusters of the dendrogram whose
IH are positive for amphibians in the Iberian Peninsula
(Fig. 1). These can be interpreted as biogeographic patterns with different levels of fuzziness, some of them
nested within others. Among them, a maximum of 11
nonnested fuzzy chorotypes may be distinguished. The
four significant chorotypes (Table 1; Figs. 1 and 2) represent the least fuzzy distribution patterns, where conditions to be chorotype are maximized. Their chorotypical
clusters included 22 distributions. The 13 distributions
of chorotypical Cluster 1 include widespread species,
although the highest concentration of them occurs in
the western half of the Iberian Peninsula, mainly in the
south-west, between the Central Mountain Range and
the Guadalquivir River. Chorotype 2 is based on the
distribution of Alytes dickhilleni, in southeastern Spain,
and chorotypical Cluster 3 comprises the distributions
of two species occurring in the eastern half of Iberia. On
the other hand, the six species of chorotypical Cluster
4 are mainly distributed over the northern half of the
Iberian Peninsula, and all of them have been reported
to the north of the northwestern mountain ranges.
The six distributions left were classified into three
clusters that were not significantly chorotypical (Table 1;
Figs. 1 and 3). Both Rana dalmatina and Mesotriton
alpestris show partially overlapping distributions northeast of the Cantabrian Mountains and are associated
with Chorotype 4. The four distributions left occur in
the north-east, around the Pyrenees (see Fig. 3).
The degree to which each amphibian distribution
in the Iberian Peninsula belongs to each chorotype is
shown in Fig. 1. Representations of fuzzy chorotypes
in space are shown in Fig. 2. Table 2 contains the set
TABLE 1. IH values and independence G-tests (if applicable) for
the clusters in Fig. 1 for which IH was positive and higher than those
of the other clusters including the distributions involved
Cluster
Chorotype 1
Chorotype 2
Chorotype 3
Chorotype 4
Rana dalmatina and Mesotriton alpestris
Discoglossus pictus and N-E Hyla meridionalis
Calotriton asper and Rana pyrenaica
IH
G
P
0.330
1.000
1.000
0.722
0.888
0.684
0.684
10.695
—
—
15.008
—
—
—
0.001
—
—
0.000
—
—
—
Note: P is the probability associated with the G-test (degrees of freedom = 1).
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OLIVERO ET AL.—FUZZY CHOROTYPES AND BIOGEOGRAPHIC PATTERNS
651
FIGURE 2. Geographical representations of the four significant chorotypes described for the amphibians in the Iberian Peninsula. The presence of at least one species of the chorotypical clusters, the species richness of the chorotypical clusters (SR ch ), the maximum membership degree
(MMDch ), and the fuzzy species richness (FSR ch ) are mapped. Main mountain ranges and rivers are represented below to aid comprehension
of Results and Discussion section.
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TABLE 2. Species richness of each chorotypical cluster; parameters that describe chorotypes as fuzzy sets: fuzzy cardinality, relative cardinality, height, and entropy; and parameters to quantify the
fuzzy relationships between chorotypes: the cardinalities of the pairwise union, intersection and fuzzy overlap, and the inclusion degree
of each chorotype into the others
Ch1
Species number
Fuzzy cardinality
Relative cardinality
Height
Entropy
13
11.639
0.416
0.683
0.391
Ch2
1
3.913
0.140
1.000
0.116
Ch3
2
9.577
0.342
0.807
0.423
Ch4
6
10.915
0.390
0.739
0.425
Intersection\Union cardinality
Ch1
Ch2
Ch3
Ch4
Ch1
Ch2
Ch3
Ch4
—
2.931
7.109
7.498
12.620
—
3.026
2.048
14.107
10.464
—
7.183
15.056
12.780
13.310
—
Fuzzy overlap
Ch1
Ch2
Ch3
Ch4
—
—
—
0.232
—
—
0.504
0.289
—
0.498
0.160
0.540
into: Ch1
Ch2
Ch3
Ch4
—
0.749
0.742
0.687
0.252
—
0.316
0.188
0.611
0.773
—
0.658
0.644
0.523
0.750
—
Ch1
Ch2
Ch3
Inclusion
of: Ch1
of: Ch2
of: Ch3
of: Ch4
Note: Chorotypes are named as Ch followed by a number as in Fig. 1.
FIGURE 3. Amphibian distributions in the Iberian Peninsula that
were not included in significant chorotypes. For Hyla meridionalis, only
the north eastern (N-E) ranges, resulting from a human introduction,
are drawn.
of parameters that describe chorotypes as fuzzy sets:
fuzzy cardinality, relative cardinality, height and entropy, and parameters to quantify the fuzzy relationships between chorotypes: pairwise union cardinality,
pairwise intersection cardinality and fuzzy overlap, and
the degree of inclusion of each chorotype in the others. Chorotype 1 obtained the maximum cardinality,
followed by chorotype 4, and then by chorotype 3. The
highest degree of fuzziness (entropy) corresponded to
chorotype 4, whereas the monospecific chorotype 2 was
the least fuzzy.
The fuzziest chorotype 4 showed also the highest union and intersection cardinalities (both with
chorotype 1) and the highest overlap (with chorotype 3).
On the other hand, the least fuzzy chorotype 2 showed
also the lowest union and intersection cardinalities
(with chorotypes 3 and 1, respectively) and the lowest
overlap (with chorotype 4). Fuzziness was thus highest
in the limits between chorotype 4 and both chorotypes
1 and 3 (see Table 2 and Fig. 2).
MMDch is clearly related to the areas where at least
one species of the chorotypical cluster is present (Fig.
2). A noticeable similarity was also observed between
FSRch and SRch maps within the area where the chorotypical cluster is present (Fig. 2). The relationship between
FRSch and SRch approximated a positive linear trend
(see Fig. 4). The correlation between FSRch and SRch
was always significant, and it was higher than 0.800 in
chorotypes where SRch was above 2 (Table 3). MMDch
rapidly increased as SRch increased from 0 to 1 (except in chorotype 1, where SRch values below 1 do
not exist), and it then stabilized around the value that
corresponded to the chorotype’s height (Fig. 4). The correlation between MMDch and SRch was always positive
and statistically significant (Table 3).
Correlations between the SRch of different chorotypes
and between their MMDch were lower than 0.250 and
often negative, whereas for FSRch , the pairwise correlation values between chorotypes ranged between 0.350
and 0.629 except for the negative correlation between
chorotypes 2 and 4 (see Table 4).
Figure 5 represents the chorotypes detected in Portugal and in Spain based on separate analyses.
The biotic elements detected using the model-based
Gaussian and hierarchical clustering methods are represented on the two first axes of the multidimensional
scaling plot (Fig. 6). In the Gaussian clustering, 12 distributions were considered noise and thus remained
ungrouped in the plot; the four biotic elements grouped
species differently from chorotypes: The southwestern
biotic element (upper-left in the plot of Fig. 6) included 4 of the 13 distributions of chorotypical Cluster 1;
both the northwestern biotic element (lower-left) and
the northern-widespread biotic element (lower-right)
combined distributions from chorotype 1, and the fourth
2011
653
OLIVERO ET AL.—FUZZY CHOROTYPES AND BIOGEOGRAPHIC PATTERNS
FIGURE 4. Scatter plots of the relationships, in each chorotype,
between the maximum membership degree (MMD ch ) and species
richness of the chorotypical cluster (SR ch ), and between fuzzy species
richness (FSRch ) and SRch . Subscript “ch” was replaced with the corresponding number when referred to a single chorotype.
biotic element included two man-introduced distributions that were ungrouped in our output. Maps with the
species richness of each biotic element reflect the combinations described (Fig. 6). However, the performance
of the hierarchical clustering provided a more similar
output of biotic elements to chorotypes: A first biotic
element coincided completely with chorotypical Cluster
1, a second one included some of the distributions in
chorotypical Cluster 3, and the other distributions left
remained ungrouped (Fig. 6).
D ISCUSSION
Methodological Framework of Fuzzy Chorotypes
Fuzzy methods in biogeography often start with the
definition of linguistic rules (i.e., fuzzy rules). Such is the
case of fuzzy distribution modeling, where fuzzy rules
describe a priori the relationship between species and
the environment (e.g., Van der Broekhoven et al. 2006),
often in the framework of environmental envelope techniques (e.g., Robertson et al. 2004). There are, nonetheless, alternative methodologies that do not use linguistic
rules for the model construction; instead, a fuzzy output
with the form of degree of membership is obtained starting from a crisp species distribution. Examples of this
are the self organizing map (SOM) models, based on
neural networks (Gevrey et al. 2006), and the favorability models, based on binary logistic regressions (Real
et al. 2006). Our approach and that of the Fuzzy-CMeans clustering algorithm (FCM, Bezdeck et al. 1984)
provide a fuzzy partition starting from a collection of
crisp data, the presences and absences, based on the
perception that relationships between these species
ranges are not crisp (Salski 2007).
FCM derives fuzzy memberships based on the minimization of similarities between objects (e.g., sample
points) and the mean values of classes located in multivariate space (see, e.g., Brown 1998; Arnot et al. 2004;
Eyre et al. 2003). In principle, FCM could be used
to search for fuzzy chorotypes. This technique, however, would present limitations when applied to such a
goal: 1) with FCM, it is necessary to know a priori the
number of clusters (thus, it would be necessary to predefine a number of chorotypes); 2) all classified elements
are forced to be included in a cluster (unclassified distributions would then not be allowed); and 3) a degree
of partition fuzziness is also required (i.e., a fuzzifier:
the weighting exponent m), so that different chorotypes
could be obtained, with the same classification method,
depending on this decision. The first problem could be
solved by comparing different classifications obtained
with different numbers of clusters by means of the partition efficiency indicators (Roubens 1982), whereas both
the first and the second limitations are solved by a modification of FCM: the Possibilistic C-Means algorithm
(PCM, Krishnapuram and Keller 1993). PCM avoids
the FCM’s need to specify the number of clusters to be
produced and also accepts unclassified elements, but it
is still necessary to consider a priori by which degree
of overlap two clusters are distinguishable as different
groups. This is not necessary with our method because
the number of chorotypes derives from the structure of
the data set.
Ordination techniques may also be used for the
detection of biogeographic patterns by locating distributions into a multidimensional space. In principle,
an ordination plot can show a picture of vague or
fuzzy limits between groups of distributions, but the
existence of these groups need to be explored by means
of partitioning procedures according to different clustering criteria (Birks 1987). Myklestad and Birks (1993),
for example, used two-way indicator species analysis
(Hill 1979), a divisive classification technique, to search
for floristic elements along the axes of a correspondence analysis of Salix species distributions in Europe.
The model-based Gaussian clustering (Hausdorf and
Henning 2003) incorporates a testing procedure to make
objective partitions between clusters within a multidimensional scaling plot. Figure 6 shows that the output
of the model-based Gaussian clustering is very different
from the chorotypes for Iberian amphibians proposed in
TABLE 3. Spearman correlations, for every chorotype, between species richness (SR ch ), maximum membership degree (MMDch ) and fuzzy
species richness (FSRch ), throughout the UTM squares into which the study area was divided (N = 257)
SRch
MMDch
MMD1
FRS1
MMD2
FRS2
MMD3
FRS3
MMD4
FRS4
0.571
—
0.935
0.568
0.792
—
0.477
0.803
0.886
—
0.330
0.227
0.770
—
0.828
0.734
Note: Subscript “ch” was replaced with the corresponding number when referred to a single chorotype. All correlations are significant (P <
0.01).
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SYSTEMATIC BIOLOGY
TABLE 4. Spearman correlations between chorotypes, according
to the species richness of the chorotypical cluster (SR ch ), the maximum degree of membership (MMDch ), and the fuzzy species richness
(FSRch ), throughout the UTM squares in which the study area was
divided. Subscripts refer to the corresponding chorotype
SR1
SR2
SR3
MMD1
MMD2
MMD3
FSR1
FSR2
FSR3
SR2
SR3
SR4
−0.126*
—
—
−0.442**
0.143*
—
MMD2
MMD3
−0.240**
−0.295**
−0.213**
0.030 n.s.
—
—
0.040 n.s.
0.233**
—
FSR2
FSR3
0.474**
—
—
0.629**
0.350**
—
MMD4
0.048 n.s.
−0.562**
−0.140*
FSR4
0.613**
−0.235**
0.569**
Note: ∗∗ , Significant correlation (P < 0.01); ∗ , significant correlation
(P < 0.05); n.s., nonsignificant correlation (P > 0.05).
this paper, but the performance of the hierarchical clustering provided a more similar output of biotic elements
to chorotypes. The use of a hierarchical classification
thus determined the convergence between chorotypes
and biotic elements.
Some published distribution patterns defined with
other taxa using hierarchical classification methods
show coincidences with our chorotype pattern for amphibians. For example, the distribution of chorotype 1 is
found in patterns of Iberian pteridophytes (Márquez
VOL. 60
et al. 1997), of Iberian Cytiseae (Fabaceae) species
(Gómez-González et al. 2004), and of European vascular
plants (Finnie et al. 2007); chorotype 2 appears also in
patterns of Iberian Cytiseae species and of European
vascular plants; and both chorotypes 3 and 4 coincide
with two chorotypes for pteridophytes. The biogeographical patterns of Iberian reptiles and amphibians
were analyzed by Sillero et al. (2009) by hierarchically
classifying species potential occurrences based on remote sensing variables. Three of the chorotypes they
found included all species in chorotypical Cluster 1,
A. dickhilleni was classified alone as in chorotypical Cluster 2, and two other chorotypes involved most species
in chorotypical Cluster 4. On the other hand, the modelbased Gaussian clustering also showed two biotic elements for Iberian Helicoidea landsnails (Hausdorf and
Henning 2006) similar to our chorotypes 3 and 4.
In any case, the main strength of our method, compared with other available classification and ordination
approaches, is that the former provides criteria to interpret the complexity of a biogeographical pattern
considering its fuzziness, and also to establish the limits
to be a chorotype, based on the distributions of species.
In this way, our method recognizes the least fuzzy clusters where conditions to be a chorotype are maximized
(i.e., in our example, the four significant chorotypes)
and also the maximum number of chorotypes possible throughout a fuzzy biogeographical pattern (in
the example, 16 chorotypes considering nested and
FIGURE 5. Classification trees and geographical representations of the significant chorotypes described separately for the amphibians in
mainland Spain and Portugal. The species richness of the chorotypical clusters (SR ch ) and the fuzzy species richness (FSRch ) are mapped. Thick
lines in the trees indicate chorotypical clusters.
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OLIVERO ET AL.—FUZZY CHOROTYPES AND BIOGEOGRAPHIC PATTERNS
nonnested clusters and 11 chorotypes considering only
nonnonnested clusters).
The Fuzzy Nature of Chorotypes
The resultant biogeographic pattern constitutes an
evidently fuzzy model of amphibian diversity in the
Iberian Peninsula (see Fig. 2): 1) there is a partial overlap between the chorotypes, 2) the geographic limits of
chorotypes are vague despite being statistically significant because the overlap between the distributions of
species composing a chorotype is not perfect, and 3) unclassified distributions also overlap partially with others
and with certain chorotypes. Any numerical classification of vegetation or fauna distributions should allow
for some degree of overlap and even allow for leaving
some elements unclassified (De Cáceres et al. 2009). Because of this, considerable idealization is required to
describe these chorotypes by using mechanistic models, whereas a more realistic interpretation of complex
ecological and biological systems is enabled by using a
fuzzy approach (Setnes et al 1997; Olano et al. 1998).
The degree of fuzziness of a chorotype, that is, the
entropy, is a sign of biogeographic complexity and of
having blurred geographic limits. Fuzziness cannot
be simply deduced from the species number of the
chorotypical cluster, from the cardinality, or from the
height of a chorotype. Chorotype 1, for example, could
be expected to be the fuzziest chorotype because it represents the most widespread distributions, obtained the
highest cardinality and was highly overlapped with
other chorotypes, but it is not, as several species had
either very high or very low membership degree in
chorotype 1. In contrast, chorotype 4 shows the highest fuzziness (with an entropy value of 0.425), which
comes together with a low height, and with the highest
intersection cardinalities and fuzzy overlap with two
655
other chorotypes. On the other hand, the lowest fuzziness is shown by chorotype 2 that has a single-species
chorotypical cluster (A. dickhilleni), the highest height,
and the lowest cardinality, unions, intersections and
overlap with the other chorotypes.
To find exactly where the fuzziness is highest within
the overall pattern, pairwise relationships between
chorotypes provide valuable information. According
to the pairwise intersections, overlapping and mutual
inclusion values, the fuzziest limits are those located
between chorotype 4 and both chorotypes 1 and 3
(see Table 2). Fuzziness, however, can be found inside and outside significant chorotypes, and so it is
not restricted to the limits between them. Distributions
that are not included in any chorotypical cluster are
sometimes grouped together so that, though they do
not constitute significant chorotypes, represent patterns in which species substitute gradually to each
other (e.g., D. pictus, the northeastern distribution of
H. meridionalis, C. asper, and R. pyrenaica, Fig. 1; see also
marine waterbirds in Real, Olivero, et al. 2008). These
“extrachorotypical” clusters may be seen as “gradual
patterns” or “very-fuzzy-chorotypes”, and it can be
worth analyzing them as complementary patterns to
chorotypes in which every distribution also has a degree of membership. Inside significant chorotypes, other
“infrachorotypical” clusters can be detected that can be
treated as “subchorotypes.” For example, chorotypical
Cluster 1 is comprised by three main clusters of the
dendrogram (Fig. 1) that are worth considering in the
same way as gradual patterns to get a deeper insight in
the overall biographic pattern. These gradual patterns
and subchorotypes are some of the set of 11 nonnested
fuzzy chorotypes that may be distinguished at the most
by allowing a high level of fuzziness.
An apparent paradox of fuzzy membership serves to
illustrate the biogeographic meaning of biogeographic
FIGURE 6. Two first axes of the multidimensional scaling plot of the amphibian distributions in the Iberian Peninsula. Distributions in the
same biotic element are rounded together by a line: a black line for the model-based Gaussian clustering and a gray line for the model-based
hierarchical clustering; distributions in the same chorotypical cluster are grouped together with gray shadows. The number of distributions in
each biotic element according to the Gaussian clustering is geographically represented.
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SYSTEMATIC BIOLOGY
fuzziness. Some species can have high degrees of membership in more than one chorotype. For example, two
species of chorotypical Cluster 1, Salamandra salamandra
(its membership degree being 0.663) and H. arborea (its
membership degree being 0.650), have also high degrees
of membership in chorotype 4 (0.537 and 0.569, respectively). Similarly, the species of chorotypical Cluster 4
A. obstetricans (its membership degree being 0.644)
has also high degree of membership in chorotype 3
(0.501). In fact, these species play a pivotal role between chorotype 4 and the others, which contribute
highly to fuzzify the limits between this chorotype
and both chorotypes 1 and 3. A more extreme example of this pivotal role arises from the definition of
amphibian chorotypes separately in Spain and in Portugal (Fig. 5). In the Spanish pattern, Pleurodeles waltl
was included in the widespread chorotypical cluster,
but it had, however, a higher membership degree in the
western chorotype; another species of the same cluster, Triturus marmoratus, had also a higher membership
degree in the northern chorotype. In the Portuguese
and in the Iberian patterns, however, these pivotal distributions were relocated, so that they were grouped
with the species of the chorotypical clusters in which
they belonged most in Spain. The contribution of fuzzy
logic to stabilize this extent-depending species oscillation for chorotype interpretation is discussed in the last
paragraph of this article.
A more biogeographic interpretation of the chorotypes’
fuzziness may be obtained by analyzing their respective
species worldwide ranges: Two out of the six species in
the fuzziest chorotype 4 are Iberian endemics (Chioglossa
lusitanica and R. iberica), and a third species spreads
also in western France (T. marmoratus), whereas the
other three spread either throughout central Europe
(A. obstetricans) and Great Britain (Lissotriton helveticus) or beyond the European boundaries (R. temporaria).
Five of the 13 species that configure chorotype 1 are
Iberian endemics, and two more, H. meridionalis (i.e.,
its nonintroduced populations) and P. waltl are endemic to Iberia and northern Africa. This chorotype
includes also species that are widespread throughout
Iberia and the surrounding territories (Pelobates cultipres,
Pelophylax perezi), and also species that spread eastward
(Bufo calamita) and far beyond the European boundaries (B. bufo). Finally, the only species in the least fuzzy
chorotype 2 is endemic (A. dickhilleni), and one of the
two species in chorotype 3 is endemic too (D. jeanneae).
The Pearson’s correlation coefficient between the entropy and the proportion of endemic species was −0.964
(P < 0.05), and it was −0.958 (P < 0.05) when distributions restricted to Iberia and closely surrounding
areas (i.e., southern France and northern Africa) were
included too. Thus, at least in our example, fuzziness
had an inverse relationship to endemicity. This might
constitute an indicator of the relative importance of
history in shaping the biogeographic pattern described
by a chorotype if, as proposed by Webb and Gaston
(2000) (see also Araújo et al. 2008), narrow ranging
species are less likely to be in equilibrium with cur-
rent climate conditions: When comparing chorotypes,
lower fuzziness would indicate that history played a
stronger role in a chorotype’s configuration. Interesting
conclusions may be obtained by comparing the fuzzy
chorotype concept with Morrone’s (1994) areas of endemism, biogeographic units that are supposed to be
related to historical events such as vicariance (Hausdorf
2002).
Fuzzy Chorotypes and History
If it were possible to relate the origin of chorotypes
to a historical hypothesis based on glacial refugia, then
an overall pattern with stronger historical antecedents
could be expected to show higher fuzziness in areas far
from refugia than in areas close to refugia. After the
last glaciation, and facing newly available lands, species
that had shared refugial areas may have shown different dispersal patterns driven by the variety of possible
responses to ecological factors. As a result of this, even
species groups that usually share distributions may
show certain degrees of divergence over space.
Fuzziness in the most important chorotypes detected
for amphibians in the Iberian Peninsula increases eastward (Fig. 2). Additionally, the six distributions that
did not constitute chorotypes represent the fuzziest
component in the overall biogeographic pattern, and
four of them are eastern distributions (Fig. 3). In contrast, two chorotypes included all the species with a
preferably westerly distribution in the study area, although the highest species richness was reached, by
each chorotype, at a different geographic latitude.
The simultaneous existence of both chorotypes and
ungrouped distributions in the same region has been
interpreted as either reflecting a historical origin (Real
et al. 1997) or being the result of different strategies in
the species use of space (Real, Olivero, et al. 2008). History is potentially an important factor behind amphibian distributions in the Iberian Peninsula. Geological
events in the Tertiary period and Pleistocene glaciations are considered decisive historical events in Europe
whose effects on the current vertebrate distributions are
still detectable (Lomolino et al. 2006). All the freshwater
systems in Europe disappeared or were strongly modified in the Pleistocene and their later recolonization
is so recent that lakes and river systems are relatively
very “young” and quite impoverished compared with
tropical systems (Lévêque and Balian 2005). Although
the Iberian Peninsula was considered to have been a
refugium for fauna and flora during the glaciations,
phylogeographical studies show that Iberia was, in fact,
a bio-climatically complex region within which refugial areas were heterogeneously distributed (Gómez
and Lunt 2006). The main Pleistocene glacial refugia
for amphibians were located in the west of the Iberian
Peninsula and were located specifically in the southwest
(Algarve, southern Portugal) and the central-west (central Portugal and the westerly Central Range) but also
around the Cantabrian (north) and the Baetic (southeast)
Mountains.
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OLIVERO ET AL.—FUZZY CHOROTYPES AND BIOGEOGRAPHIC PATTERNS
A possible relationship between fuzziness and distance to glacial refugia is highly feasible in relation to
chorotype 1, which contains 50% of Iberian endemic
distributions in its chorotypical cluster. The phylogeographical analyses available for two of them, D. galganoi (Garcı́a-Parı́s and Jockush 1999; Martı́nez-Solano
2004) and L. boscai (Martı́nez-Solano et al. 2006), place
the glacial refugia of these Iberian endemisms in both
the central-west and southwestern areas. These are
the northerly and southerly limits of the areas where
chorotype 1 showed the highest species richness (see
SRch and FSRch in Fig. 2), that is, where this chorotype is
less fuzzy.
In the case of the fuzziest chorotype 4, the relationship between refugia and the current chorotype distribution is not as clear as in chorotype 1. Chorotype 4 is
mainly distributed in the north-west of Iberia, though
most of the species in its chorotypical cluster show also
some southern populations in the west. It is there, in
the central-west of the Iberian Peninsula, where phylogeographical studies locate the possible glacial refugia
of its two endemic species, C. lusitanica (Alexandrino
et al. 2000, 2005) and R. iberica (Martı́nez-Solano et al.
2005), and also of A. obstetricans (Martı́nez-Solano et al.
2004). The current main distributions of these species
would then be a result of a northward spread from their
glacial refugia, with which they are still connected across
Portugal.
The biogeographic pattern in the eastern half of Spain
is fuzzier probably as a result of a multiplicity of historical antecedents and responses to the environment
(see Fig. 3). Regarding chorotype 3, phylogeographical
studies suggest a Pleistocenic origin in a Baetic glacial
refuge, in the south-east, for D. jeanneae (Garcı́a-Parı́s
and Jockush 1999; Martı́nez-Solano 2004). Among the
ungrouped species, Iberia represents for R. dalmatina
and M. alpestris a marginal region of worldwide ranges
that reach central Europe and the Balkan Peninsula;
D. pictus and the northern ranges of H. meridionalis seem
to be a very recent anthropogenic introduction from
France (Martı́nez-Solano 2004; Recuero et al. 2007); and
for the Pyrenean endemic Calotriton asper (and probably for R. pyrenaica), it has been suggested that its
current range is actually a recent interglacial refugium
that was colonized from peripheral areas (Pleguezuelos
et al. 2004). The only nonfuzzy element in the east is
chorotype 2, the Baetic endemic A. dickhilleni, which
probably occurs close to a Pleistocene refuge (Martı́nezSolano 2004). In this way, the amphibian eastern distributions in Iberia seem to be linked to a heterogeneous
historical scenario that does not affect the species as a
group and in which relatively recent processes are often
involved. Hence, the extreme fuzziness they represent
in the global pattern.
Fuzzy Chorotypes: A Criterion to Deconstruct
Biodiversity Patterns
History and ecological factors (including climate, topography and human influence) could have driven,
657
or still be driving, the configuration of the species
current distributions. These factors could be seen as geographic attractors dynamically delimiting the behavior
of biodiversity over time and throughout space. In such
a complex scenario, chorotypes would be the geographical responses of species, depending on the past and
present attractors acting in a certain area. The fuzzy
membership of a species in a chorotype would then be a
measure of how much such a species has been attracted
toward a certain biogeographic response. In this way,
the fuzzy approach helps to connect chorotypes with
these biogeographic attractors.
In previous studies, the environmental and historical
factors behind chorotypes have been explored using
two main approaches: either by analyzing the area with
at least one species of the chorotypical cluster or by
analyzing the response of the species richness of each
chorotypical cluster to the environmental conditions
(e.g., Real, Olivero, et al. 2008). The second option constituted a criterion for the biogeographic deconstruction
of biodiversity, as a way of gaining insight into the
causes of biodiversity trends (Marquet et al. 2004; Real,
Olivero, et al. 2008). Now the challenge is to integrate
the fuzzy nature of chorotypes in their explanatory
analysis. In principle, chorotype fuzziness is based on
degrees of membership that do not describe spatial units
(in contrast to both presence/absence and the species
number), but they describe species distributions. By
computing the maximum membership degree (MMD ch )
and the fuzzy species richness (FSRch ) of every location,
chorotype fuzziness is transferred to space and can thus
be related to the spatial trends of possible explanatory
factors.
MMDch values are actual degrees of membership of a
certain distribution at a given position in space. The area
where at least one species of the chorotypical cluster occurs is not only visually well delimited by them but also
the degree of membership represented inside this area
corresponded to one of these species (see Fig. 1). The
MMDch , thus, closely resembles the area with at least
one species of the chorotype, but the former introduces
a fuzzy component mainly related to the membership
of distributions that did not configure the chorotypical
cluster itself. We propose the MMDch as the fuzzy alternative for the area with at least one species of the
chorotype to perform historical and ecological analyses
within the study area (see Real, Olivero, et al. 2008).
FSRch , unlike MMDch , quantifies a kind of weighted
species richness, that is, a fuzzy alternative for the crisp
species richness as is expressed in the chosen name
for FSRch . Visually, the general structure of SRch is preserved in the map of FSRch , but fuzziness is added
inside and outside the area where at least one species
of the chorotypical cluster is present. Although a linear relationship is conserved between SR ch and FSRch
for any SRch value multiple FSRch values can exist (see
Fig. 4). As a result of this, the biogeographically deconstructed pattern for amphibian diversity in the Iberian
Peninsula is blurred by including all those species that
are susceptible to being affected by any geographical
658
SYSTEMATIC BIOLOGY
attractor. This is the reason for the higher correlation between chorotypes according to FSRch than according to
SRch (these pairwise correlations were higher in highly
intersected chorotypes: compare Tables 2 and 4). This
consequence of including fuzziness in the species richness deconstruction is not unexpected but indicates the
close interrelation between biogeographic responses,
despite being significantly different according to a statistical criterion.
The deconstruction of biodiversity benefits also from
one of the most relevant advantages of the fuzzy approach to chorotypes, which is illustrated with the
comparative examination of Figs. 1–5. As a result of
the choice of the geographical extent of the analysis,
some species are grouped in different chorotypical clusters in the separate analyses of Iberia, Portugal, and
Spain. With the fuzzy approach, however, biogeographical patterns that look different from each other using
a crisp perspective tend to converge. Dissimilarities in
the comparison of SRch maps between Iberia, Spain,
and Portugal disappear almost completely when FSR ch
are compared. This suggests that patterns that could
be derived from data sets showing differences (e.g.,
nonidentical extents, different data sources, different
lattices, different scale, or different taxonomic criteria),
or even from different classification algorithms, could
be more similar, and thus more consistent, using the
fuzzy logic than using a crisp approach.
S UPPLEMENTARY M ATERIAL
Supplementary material, including data files and/or
online-only appendices, can be found at http://www.
sysbio.oxfordjournals.org/.
F UNDING
This work was made possible thanks to the Ministerio
de Educación y Ciencia, Spain (CGL2006-09567/BOS);
the Ministerio de Ciencia e Innovaci ón, Spain, and European Regional Development Fund, European Union
(CGL2009-11316); and the Consejerı́a de Innovación,
Ciencia y Empresa, Junta de Andalucı́a, Spain (P05RNM-00935).
A CKNOWLEDGMENTS
We thank Ramón Hidalgo for his work to develop the
software for the exposed methodology, Dr. Dave Roberts
for his useful comments and his help in the mathematic
notation, and Dr. Adrian Paterson, Prof. John Birks, and
anonymous referees for their valuable remarks about a
previous version of the manuscript.
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