Invariance in species-abundance distributions

15 Pages·2009·0.35 MB·English

TheorEcol(2009)2:89–103
DOI10.1007/s12080-008-0031-3
ORIGINAL PAPER
Invariance in species-abundance distributions
Arnošt L. Šizling&David Storch&Jiří Reif&
Kevin J. Gaston
Received:11April2008/Accepted:24October2008/Publishedonline:4December2008
#SpringerScience+BusinessMediaB.V.2008
Abstract Many attempts to explain the species-abundance such a distribution. Both the log-normal and our area-and-
distribution(SAD)assumethatithasauniversalfunctional taxon invariant distribution fitted data well. However, the
form which applies to most assemblages. However, if such log-normal distributions of two adjoined sub-assemblages
a form does exist, then it has to be invariant under changes cannot be composed into a log-normal distribution for the
in the area of the study plot (the addition of neighboring resulting assemblage, and the SAD composed from two
areas or subdivision of the original area) and changes in log-normal distributions fits the SAD for the assemblage
taxonomic composition (the addition of sister taxa or poorly in comparison to the area-and-taxon invariant
subdivision to subtaxa). We developed a theory for such distribution. Observed abundance patterns therefore reveal
anarea-and-taxoninvariant SADandderiveda formula for area-and-taxon invariant properties absent in log-normal
distributions, suggesting that multiplicative models gener-
ating log-normal-like SADs (including the power-fraction
Electronicsupplementarymaterial Theonlineversionofthisarticle model) cannot be universally valid, as they necessarily
(doi:10.1007/s12080-008-0031-3)containssupplementarymaterial,
apply only to particular scales and taxa.
whichisavailabletoauthorizedusers.
:
A.L.Šizling(*) K.J.Gaston Keywords Scaleinvariance.Areainvariance.
BiodiversityandMacroecologyGroup,
Taxoninvariance.Populationsizes.
DepartmentofAnimalandPlantSciences,
UniversityofSheffield, Frequencydistributions.Relativeabundances
SheffieldS102TN,UK
e-mail:[email protected]
e-mail:[email protected]
Introduction
K.J.Gaston
e-mail:[email protected]
The relative frequency of the abundances, a, of species in
:
A.L.Šizling D.Storch samples, assemblages and communities has long intrigued
CenterforTheoreticalStudy, ecologists. Perhaps foremost this has been manifested in the
CharlesUniversityinPragueandtheAcademyofSciencesofthe
search for a common descriptor, f(a), of species-abundance
CzechRepublic,
distributions (SADs). Indeed, a wide range of models have
Jilská1,
11000Praha1,CzechRepublic been fitted to different data sets, including distributions
derived variously from considerations of simple statistics,
D.Storch
e-mail:[email protected] population dynamics, niche partitioning, or fractal geometry
: (e.g.Motomura1932;Fisheretal.1943;Preston1948,1981;
D.Storch J.Reif
Magurran 1988; Magnussen and Boyle 1995; Engen and
DepartmentofEcology,FacultyofScience,CharlesUniversity,
Viničná7, Lande 1996; Sichel 1997; Tokeshi 1999; Dewdney 2000;
12844Praha2,CzechRepublic Hubbell 2001; Green et al. 2003; Magurran and Henderson
2003; Harte et al. 2005; Pueyo et al. 2007; for reviews see
J.Reif
e-mail:[email protected] May 1975; Marquet et al. 2003; McGill et al. 2007).
90 TheorEcol(2009)2:89–103
Few would disagree that some basic characteristics are shouldbecharacterizedbythesamedistributionfunctionas
common to virtually all real SADs. These include that they describes the two sub-assemblages, albeit with different
are markedly right-skewed (i.e. that the largest number of parameters (Pueyo 2006). We call this property of SADs
species occurs in one of the smallest abundance classes). area-and-taxon invariance (Šizling and Storch 2007, for
Beyondthis,however,thereissurprisinglylittleconsensus. broader concept see Storch and Šizling 2008), since such a
Variousauthorschampiontheimportanceofdifferentbroad universal distribution is invariant under changes in area of
classes of models to describe SADs (see Marquet et al. the study plot and taxonomic composition (i.e. changes in
2003; McGill et al. 2007), of particular different distribu- thesetofspeciesincluded;notethattaxoninvariancerefers
tions within those classes, and of different methods of here to species in different taxonomic groups, not to the
assessing the fit of models. They also place different abundances of those taxonomic groups, e.g. not a genera-
emphases on issues of sampling and thus on the relation- abundance distribution). Area-and-taxon invariances there-
ship between the underlying probability distribution and fore occur when the same functional form that fits species
observed SADs. Nonetheless, identifying an appropriate dataoverasmallareaoralowerleveltaxonomicgroupcan
function characterizing observed SADs has generally been fit data overa larger area (comprising multiple small areas)
considered as crucial for understanding mechanisms gener- and at a higher taxonomic level (comprising species from
ating relative species abundances. several lower taxa).
One of the reasons for the lack of clarity as to which So far, area-and-taxon invariances have been implicit in
functions provide the best models of SADs is that tests have the assumption of a general form of the SAD. If the
focusedalmostexclusivelysimplyonfittingoneormoresuch functional form of SADs is universal, then it should not
functions or model results to one or more observed SADs, change with enlarging or reducing the size of the study
whatever the approach used (but see Kempton and Taylor area.Similarly,itshouldnotchangeifwebroadenthesetof
1974).Thisisproblematicbecausetheoutcomesofsuchtests speciesstudiedbyaddingasistertaxon,orifweuseonlya
areespeciallysensitivetothefitinthetailsofthedistributions subtaxon(i.e.oneparticularorderinstead ofthewholeofa
andthesetailsareusuallybiasedinobservationaldata(Fisher class). Consequently, all models trying to explain or derive
et al. 1943; Preston 1948; Williamson and Gaston 2005; Yin SADs (e.g. broken stick, power fraction, or neutral
etal.2005).Thetailformedbytherarestspeciesisbiasedby communitydynamics)shouldassumeeitherarea-and-taxon
the difficulties associated with finding such species. In invariance or some basic level of description to which the
consequence it has been shown that, for example, small model applies and from which the other levels (and other
samples from log-normal and log-normal-like distributions functional forms of the SAD) can be mathematically
tend to obey log-series distributions (Golicher et al. 2006). derived. For instance, if we assume that a particular model
The tail of an SAD formed by the commonest species can generating the SAD concerns species population abundan-
alsobebiasedbysampling,althoughtheeffectsareprobably ces across whole biogeographic units (continents), and the
less systemic, and dependent on the particulars of the model is not area-and-taxon invariant, then the functional
methodology used (e.g. whether effectively there are upper formsforsmallerareasaredifferentfromthatproducedbythe
bounds above which abundances cannot be differentiated). model,andhavetobesecondarilyderived.Inotherwords,if
The influence of biases in the tails of SADs on the weassumethatthereisauniversalmechanismgeneratingthe
statistical fit of distributions is particularly important given SAD, then it must either lead to an area-and-taxon invariant
that discussion as to the relative merits of different distribution,oritmustapplyonlyatsomefundamentallevel
distributions has come often to focus on rather narrow of description defined by a particular area and taxon level.
differences in such fit (e.g. McGill 2003). Indeed, it has Bothcaseshaveinterestingimplications.Searchingforarea-
increasingly been recognized that other testable properties and-taxoninvarianceisthusnotsimplyamathematicalnicety,
need to be explored to help identify those distributions that as it concerns the question of whether there is a privileged
most properly describe SADs (Wilson 1993; McGill et al. spatialscalethatdeterminesecologicalpatternsatotherscales
2007). One such property is the consistency of the orwhetherallscalesaremoreorlessequivalent.
description of the SAD in its spatial and taxonomic Here we develop a theory concerning the invariance of
complexity: if the observed data are correctly described SADs under variation in area and taxonomic composition.
by some distribution function, then the SAD of an Furthermore, we introduce a new test of abundance
assemblage (e.g. defined spatially, taxonomically, ecologi- distributions based on their behavior when changing area
cally, or in other terms) comprising two sub-assemblages, and/or taxonomic scope, and apply it to the log-normal
each of them described by the distribution function, should SAD which has been widely championed as a null
bewelldescribedbythestatisticalcompositionofthesetwo hypothesis for the form of SADs (Preston 1948; for
distributionfunctions.Moreover,ifthedistributionfunction possible mechanisms see Whittaker 1970; May 1975;
describing the SAD is universal, the composed distribution Engen and Lande 1996; Engen 2001; May et al. 2007)
TheorEcol(2009)2:89–103 91
and to a new functional form which fulfils the condition of SADs are sufficiently flexible to fit well if properly—and
an area-and-taxon invariant distribution. We will show that independently—parameterized.However, theparametersof
although the log-normal distribution provides a good fit to the composite SAD depend on the component SADs, so
the observed SADs at different spatial scales, the compo- that the comparison between fits of the composite and
sition oftwolog-normal SADs neitherfitsmergeddata nor component distributions helps identify the appropriate
leads to another log-normal SAD, i.e. it does not reveal analytical form of the SAD much better than a simple
consistency nor exhibit area-and-taxon invariance. In independent fitting.
contrast, our new distribution retains its good fit to the The consistency does not necessarily imply area-and-
data regardless of changing area. Although we obviously taxon invariance; even a non-invariant SAD could in
cannot prove that our functional form of an area-and-taxon principle be spatially and taxonomically consistent if the
invariant SAD is the most appropriate description of composite SAD (in such a case differing in its functional
observed SADs, unlike the log-normal SAD it is universal form from the component SADs) fits the data well.
and consistent when changing area and/or taxonomic However, in such a case this functional form obviously
composition. We argue that no mechanism proposed as cannot beuniversal, asitchanges with scalesandtaxa, and
the explanation of the log-normal SAD can be universal we never know a priori which is the fundamental scale (or
across scales and taxa. fundamental taxon level) at which the mechanism that
Wealsoofferatoolforallofthecomputationspresented produces a given SAD should apply. A truly universal
here and thus for explorations of any available dataset (see functional form of the SAD must be area-and-taxon
http://www.cts.cuni.cz/wiki/ecology:start). invariant, i.e. its functional form does not change after the
composition from the distributions characterizing sub-
assemblages, although it naturally changes its parameters.
Theory
Composition of two SADs of adjoined sub-assemblages
RequirementofspatialandtaxonomicconsistencyofSADs
It is obviously not a problem to determine the empirical
Observed SADs can be fitted equally well using many SAD (i.e. the cumulative distribution function constructed
functional forms. However, a good fit is not the only over data) of a given assemblage from the sets of
indication of appropriateness of a given functional form. A abundances of its sub-assemblages—it is a simple matter
stronger requirement for an appropriate functional form of of summing the abundances of each species across these
the SAD is consistency of its fit with data when changing sub-assemblages. A greater challenge is if we assume that
theobservedarea and/ortaxonomic scope. Assume thatwe theSADforanassemblagehasaparticularfunctionalform
have two SADs, f and f , characterizing assemblages on (log-normal, geometric series etc.) and we seek to deter-
1 2
someadjoinedareas,describedbythesamefunctionalform mine what is the resulting functional form and whether the
(e.g. if λ≠σ then fðaÞ¼le(cid:2)la and fðaÞ¼se(cid:2)sa are two original functional formisretainedwhen changingthearea
functions representing one functional form). We call these by addition of other assemblages, or, conversely, by
functional forms consistent if the empirical SAD (i.e. the dividing the area into smaller plots. Equally, one might
observedpattern)forthewholeassemblageofthetwoareas pose the question in terms of the addition of other taxa or
taken together is fitted well by the SAD whose functional the splitting of a given taxon into subtaxa.
form, including parameters, can be generated by the Let us start with merging two subtaxa (regardless of
mathematical composition (hereafter composite distribu- whether these represent two monophyletic sister clades or
tion) of f and f (hereafter component distributions). If the otherwise delimited, mutually exclusive sets of species).
1 2
composite distribution for the whole assemblage, on the This situation is simpler than merging two neighboring
otherhand,fitstheempiricalSADconsiderablyworsethan areas, as subtaxa by definition do not share any species.
thefunctionalformsforindividualsub-assemblages,thenf The functional form of the SAD that results by composing
1
and f cannot represent appropriate functional forms two SADs for respective subtaxa is
2
reflecting an underlying mechanism even if their fit to the
sub-assemblages is perfect. f1[2ðaÞ¼p1f1ðaÞþp2f2ðaÞ; ð1Þ
Importantly, the SAD may not be spatially and taxo- where as are abundances, fs are probability density
nomically consistent even if the composite distribution can functions, and π is a proportion of species in the focal
i
be fitted well by the same functional form, but indepen- taxon i (p1þp2 ¼1).
dentlyofthecompositionofSADsofthesub-assemblages, The case of merging two adjoining areas is more
i.e. with parameters tuned according to the empirical complex, as this requires two operations. One is the same
distribution. The reason is that many functional forms of as in the previous case, i.e. addition of the SADs of species
92 TheorEcol(2009)2:89–103
which occur in only one sub-assemblage. The second Area-and-taxon invariant SADs
operationcomprisesconvoluting(splicingtogether)theSADs
of species which are common to both sub-assemblages. The If there is no fundamental scale (sensu area) which
result(i.e.theformoftheresultingSAD)ofthefirstoperation determines abundance patterns at other scales, the SAD is
depends on relative species turnover between the sub- necessarily area and/or taxon invariant. Now we explain
assemblages, as the effect of this operation depends on the why the area invariance and taxon invariance are not
proportion of species which are unique to just one sub- separable from each other and derive an area-and-taxon
assemblage.Theresultofthesecondoperation(belowmarked invariant SAD.
by * for correlated-convolution, hereafter c-convolution) Taxon invariance refers to the situation in which the
c
depends on the correlation between species abundances in functionalformoftheSADdoesnotchangewhenmerging
the two sub-assemblages (Table 1). If the abundances are two subtaxa. A taxon invariant distribution function can be
perfectlycorrelated(i.e.theyareproportionaltoeachother), found when taking Eq. 1 as the functional equation (i.e.
the composition of the respective distributions comprises equation whose solution is a function) and finding its
simple proportional enlargement of species abundances, solution. Apparently, there are two solutions. (1) If both f
1
whereas if they are only poorly correlated (or uncorrelated), and f are identical (including parameters), then the
2
the c-convolution is much more complex (see Fig. 1 and composed function is obviously again the function f
1
Appendix I). More formally, the entire composition of the (≡f ). The solution is therefore any function if its
2
two distributions represents the proportional sum (i.e. the parametersdonotvarybetweentheassemblages.Thiscase
linear combination) of particular SADs seems to be biologically irrelevant, since taxa commonly
varyintheiraverageabundancesandthustheycannotshare
fplot1[plot2 ¼p10 fspinplot1exclusivelyþp01 fspinplot2exclusively identicalSADs.(2)Ifthefunctionsdifferinparameters,the
þp f (cid:3) f
11 commonspinplot1 c commonspinplot2 solution is any function which can be expressed as a sum
ð2Þ (linear combination) of several functional forms (i.e.
PN
where the πs are the proportional sizes of the respective fðaÞ¼ cifiðaÞ; where N is a free parameter which may
groups and π11 is equivalent to Jaccard index (J) of vary beit¼w1een areas). The only condition imposed on the
assemblagesimilarity(numberofspeciesincommondivided sumisfiniteintegrability(i.e.theintegralbetweenzeroand
by total species number; Table 1). infinity must be finite and greater than zero) and positivity
Table1 Definitionsofmeasuresused
K–Sdistance Kolmogorov–Smirnovstatistics,i.e.themaximumdistancebetweentwocumulativedistribution
functions;itrangesbetween0and1;units:proportions
Consistency Difference(cid:5)between(cid:2)meanK–Sdistanc(cid:3)e(cid:4)forparticulardi(cid:6)stributionsandthedistanceforthosecomposed
plusone consD¼f KS þKS 2(cid:2)KS þ1 ;K–Sdistanceisthedistancebetweendataand
plot1 plot2 comp
assumeddistribution;ifthecompositedistributionfitsworsethancomponentdistributionsthe
consistencyisbelowone;anabsolutelyconsistentmodelhasconsistencyequalto1;ifthecomposite
distributionfitsdatabetterthanthecomponents,theconsistencyfallsabove1;itrangesbetween0and
2;units:proportionalvalues
J Jaccardindexofsimilarity,i.e.proportionofthespeciesfoundintwoplotsthatarecommontoboth;
units:[%]
Correlation Thecorrelationquantifiesasituationinwhichtheabundancesofspeciesinagivenplotarerelatedto
theabundancesofthesamespeciesintheotherplot;inthemodel,thedependencebetween
abundancesisimposedbytwoconstraints—i.e.twoincreasinglines(Fig.1b)withinwhichthe
abundancesareindepend(cid:5)entofeachother.Correlati(cid:6)oniscalculatedasthecomplementofanangle
betweentheconstraints correlationD¼fp=2(cid:2)angle ;thebiggeritis,themoretheabundances
constraineachother;itrangesbetween0andπ/2;units:[rad]
Areainvariantfunctionalform(i.e. IffananalyticalformuladescribingaSADdoesnotchangewhencomposingtwoformulas
model)ofSAD characterizingtwoSADsoftwoplots,wesaythatthisformula(i.e.themodelofSAD)isarea
invariant.Theparametersoftheformulamayvaryafterthecomposition
Taxoninvariantfunctionalform(i.e. IffananalyticalformuladescribingSADdoesnotchangewhencomposingtwoformulascharacterizing
model)ofSAD twoSADsoftwotaxa,wesaythatthisformula(i.e.modelofSAD)istaxoninvariant.Theparameters
oftheformulamayvaryafterthecomposition
Fundamentalareaortaxon IffSADisnotareaand/ornottaxoninvariant,itimpliesanexistenceofanareaor/andataxon,whose
SADdetermineSADsofotherareasand/orothertaxa.Suchareaandtaxonarecalledfundamentalin
thetext
TheorEcol(2009)2:89–103 93
a) ƒFig.1 Bivariateplotsofabundances,a anda ,fortwodisjunctplots
1 2
in cases of a uncorrelated, b correlated and c observed abundances.
Both full and open circles represent individual species. The species
that occupy only one plot were excluded. a All the combinations of
abundancesgivingtheabundancea,ifplotsarejoined,followaline
(full line, full circles); b if abundances are correlated, the points are
constrainedbytwoincreasinglines(dotted);abundancesforplot1,a ,
1
arethusboundedby a and a whereσsareslopesofthelines;
cabundancesobservesdmaxoþn1thetrsamninsþe1ctweremostlyweaklycorrelated
witheachotherandfollowedtheassumedpattern.Theabundancesof
thespeciesthatoccupyonlyoneoftheplotsareshowninthisgraph
subtaxa comprises two sets of distinct species, whereas
composing SADs from two adjoining areas comprises both
merging two sets of distinct species and c-convolution of
b)
SADs of common species, area invariance implies also
taxon invariance, provided that some species occur only in
one sub-assemblage. (Note that taxon here necessarily
refers to any set of species, not only to a monophyletic
species clade, i.e. this invariance comprises all possible
taxonomic systems.) In such a case we can speak about
area-and-taxon invariant SADs, and hereafter we will deal
with this universal type of invariance. This invariance
would be obvious if all species were shared by both areas
and if abundances were fully correlated, so that the
convolution would comprise simple rescaling of SADs for
the respective assemblages (see above). However, the truly
(i.e. non-trivially) area-and-taxon invariant SAD is charac-
terized by the robustness against changes in species
turnover (i.e. against the variation in πs) and changes in
c)
the level of correlation of abundances between the sub-
60 assemblages.ThenaturalconditionimposedontheSADby
theareainvarianceisthateverySADmustattaintheorigin
(i.e.zeroabundancehaszeroprobabilitydensity;McGillet
3 al. 2007). The reason is that nonzero probability density at
3
ot zero abundance would imply that we know how many
pl
or species are missing from a given area — but this is
es f 30 apparently arbitrary and relative to the definition of the
c
n whole species pool.
a
d
n Anarea-and-taxoninvariantfunctionasdefinedaboveis
u
b
a a proportional sum of diffonential distributions introduced
by Preston (1981) (the name is based on ‘the difference
between two exponentials’). We will therefore call it the
0 multi-diffonentialdistribution(seealsothemulti-exponential
0 30 60
abundances for plot 32 distribution in Šizling and Storch 2007), and it is expressed
as
XN (cid:2) (cid:3)
(fðaÞ(cid:4)0 for all a); for a similar idea in economics see fðaÞ¼ ci e(cid:2)Aia(cid:2)e(cid:2)aia ; ð3Þ
i¼1
Mandelbrot (1963).
Area invariance characterizes SADs whose functional where a is abundance, the parameters A and α are positive
i i
form does not change when joining two areas, i.e. when it (0<A <a), and c is a real number (see Fig. 2 for
i i i
is transferred from small to large areas by the composition meaning of the parameters). For the rationale and the proof
of the distributions of smaller areas. Since merging two that it obeys the requirement of area-and-taxon invariance
94 TheorEcol(2009)2:89–103
a) b)
2
A=4 5.0
sity A=3 α=5 sity α=50 A=1
n n
e e
d d
bility 1 A=2 bility 2.5 α=10
a a
b b
o A=1 o
pr pr α=5
α=1.0001
0 0.0
0 1 2 0 1 2
abundance abundance
Fig.2 Shapesofadditivetermsofthemulti-diffonentialdistribution term,becauseitsvaluedetermineshowcloseistheentiredistribution
for different combinations of A and α parameters. Each term has a tothefocalterm,whichcanthenbecalledthedominantterm.Sumof
shapesimilartoalog-normalorlog-series(highαinb)distribution, all dominances across all terms is one. Full lines in subplot a show
dependingonproportionofAandαparameters(Eq.3).Thetermsare variation in A (=1, 2, 3, and 4), holding α=5, while they show
shown after normalization (i.e. multiplying by a normalization variation in α (=1.0001, 5, 10, and 50), holding A=1 in subplot b.
constant (¼aA=ða(cid:2)AÞ) to have the area below the curve equal to NotethatA<αbydefinition.Dashedlinesshowtheentiredistribution
one, so that the parameter c (Eq. 3) is replaced by dðaA=ða(cid:2)AÞÞ (proportional sum of the terms) in the case when all the terms are
whereδisarealnumberwhichcanbecalleddominanceofthefocal equallydominantandthusequallycontributetotheentiredistribution
(Eq.2;wheremodelofc-convolution,* ,isdefinedbyEq.5 thebreedingseasons(April–June)of2004and2005.Points
c
in Appendix I) see Appendix II. were separatedby between 300 and 500m, suchthat study
Themulti-diffonentialdistributionhasapotentiallylarge plots of diameters 150 m at each point were approximately
numberoftermsandconsequentlyparameters,whichcould adjacent.Themaximumrecordednumbersofindividualsof
be interpreted as a disadvantage in comparison to simpler eachspeciesateachpointfromthefivevisitswereusedfor
functional forms. However, this is actually a necessary analyses to obtain reliable abundance estimates (see Storch
property of any area invariant distribution, since species et al. 2002).
spatial turnover between sub-assemblages implies that the
functional form is taxon invariant and thus actually is Tests
composed from a certain number of other functions, i.e.
component SADs (see Eq. 1). Only a few additive terms We performed two kinds of tests. Both focused on area-
might, however, be dominant (Fig. 2). related variation of the SAD, because taxon-invariance
inevitably follows from area-invariance, as explained
above. First, we simply evaluated the fit of the multi-
diffonential and the log-normal distributions to the ob-
Methods servedSADsfordifferentareas(infactlengthsoftransects,
see below). We will call this the test of the fit between the
Data modeled and observed shape of the SAD. It consisted of
calculatingtheKolmogorov–Smirnov(K–S)distance(max-
To test the area-and-taxon invariance of SADs we needed imumdifferencebetweencumulativedistributionfunctions;
detailed abundancedata ofadjoined species assemblagesat Table 1, Fig. 3c, and S1) between the observed SAD and
several spatial scales. Such data are not easily available, so the fitted distributions for abundance data for each sample
we performed a multiscale bird survey across the Czech of50 consecutive points,of 100consecutive points, andso
Republic. The data comprise the abundances of 144 bird on up to750 points. All possible sets ofa given numberof
species (excluding Larus ridibundus and Riparia riparia, consecutive points were used. The log-normal and multi-
i.e. the species breeding in large colonies) at each of 768 diffonential distributions vary in number of parameters,
points along a linear East-West transect in south Bohemia thus we also compare them using an information criterion
and Moravia. Birds were mapped by the point count (AIC; Akaike 1974), in the case of multi-diffonential
method (Bibby et al. 1992) within the distance limited by distributions varying numbers of terms between 1–20
150maroundeachpointduringfiveearlymorningvisitsin (log-normal distribution has two parameters).
TheorEcol(2009)2:89–103 95
Fig.3 SADasobserved
a)
(squaresinrankplots;black
barsinPreston-stylehistogram),
andmodelledbylog-normal 5000
(dashedlineinrankplots;white
barsinPreston-stylehistogram)
e 500
andmulti-diffonential(fullline c
n
inrankplots;dashedbarsin a
d
Preston’splot)approachesfora n
u 50
samplesofthefirst300points ab
(1–300)andbnext300points
(301–600)ofthetransect.Sub- 5
plotcshowsthemergeddata
andcomposeddistributions.K–
SistheKolmogorov–Smirnov
0.0 0.5 1.0
distance,whichismaximum
rank
distanceinthedirectionofthe
axisofproportionalrankinthis b)
case;itcanbecalculatedinthis
waybecausetheproportional 5000
rankplotisaninvertedcumula-
tivedistributionfunction—the
proportionalrankestimatesthe e 500
c
cumulativeprobability.Therank an
plotsforthelog-normaland nd
multi-diffonentialapproaches bu 50
a
wereconstructedusing15,000
simulations
5
0.0 0.5 1.0
rank
c)
5000
e 500
c K-S
n
a
d
n
u 50
b
a
5
0.0 0.5 1.0
rank
The weakness of the test of fit is that fit depends on the tion of abundances and species turnover obviously affects
number of free parameters of a given distribution, and on consistency of the SAD, as we have shown, we evaluated
the fitting procedure—and both these properties can the extent to which the consistency across scales (areas) is
naturally differ among different functional forms of SADs. dependent on the correlation between species abundances
Moreover, the fit of a distribution can be affected by in two given sub-assemblages, and on species turnover
sampling issues, as the functional form of the sampled between them.
distribution can differ from the underlying distribution The test of consistency thus evaluated the deterioration
(Green and Plotkin 2007). For this reason we focused on of the fit of SADs which were mathematically composed
the second type of test, which evaluates consistency of the from component SADs of two sub-assemblages (see
functional form of SADs across scales (areas) as explained Fig. 3), and the sensitivity of this deterioration on the
above; hereafter the test of consistency. Since the correla- variation in species turnover between sub-assemblages
96 TheorEcol(2009)2:89–103
and the correlation between abundances in each sub- underlying distribution with confidence of 95%. Although
assemblage. This evaluation was performed as follows the multi-diffonential distribution reveals good fit even in
(see also SI): the case of two additive terms (SIII), we used 10 terms for
all simulations.
1. The SADs for each species assemblage were fitted
using both log-normal and multi-diffonential distribu- Numerical simulations
tions. The species assemblages comprised non-over-
lapping sections of the point transect, starting with 20 Distributions of pairs of adjacent plots were composed
points, then 40, 60,... 180 neighboring points. numerically. This allowed us to use the same method for
2. Component SADs were constructed using the fitted both the distributions and thus to make the results
SADsofeachpairofcompositeassemblages,i.e.SADs comparable with each other. We preferred a method based
of two neighboring sections on the transect. The onrandomsamples,eachfromonecomponentdistribution.
composite assemblages thus consisted of 40, 80, The composition of fitted distributions comprised three
120,... 360 sample points. steps: (1) generating abundances for species common to
3. Kolmogorov–Smirnovdistanceswerecalculatedforthe bothplots,(2)generatingabundancesforspeciesoccupying
comparison between empirical and fitted (component) only the first plot, and (3) generating abundances for
SADs for both sub-assemblages. We call these dis- species occupying only the second plot (see SI). All
tances KS and KS . numbers of generated abundances were proportional to the
plot1 plot2
4. Kolmogorov–Smirnov distance was calculated for the numbers in the respective groups (π , π , and π ), and
11 10 01
comparison between the composite SAD (composed together they comprise 15,000 values. The abundance for
from SADs of the given pair of sub-assemblages) and species common to both plots was generated as the sum of
the empirical SAD for the respective assemblage. We the two abundances drawn from the respective fitted
call this distance KS . distributions; however, the two abundances were selected
comp
5. Consistency was calculated as a co(cid:2)ntrast between d(cid:3)is(cid:4)- so that the empirical observed correlation of species
tances KS and KS , namely as KS þKS abundances between the respective plots was retained (the
plot comp plot1 plot2
2(cid:2)KS þ1. Consistency is thus a measure of the pairs of abundances of respective species from adjacent
comp
“success” of the composition of the two distributions in subplots must lie between two increasing constraints; see
termsofthefitofthecompositedistributiontothemerged Fig. 1b, Eq. 5 in Appendix I, and SI.f, and o–q). This
data: a value of 1 indicates the same fit of the composite approach to the c-convolution is only one of the possible
distribution as the fit of the component distributions, definitions and should properly be called a constrained-
lower values indicate poorer fit of the composite convolution. We preferred it for its simplicity. The
distribution, whereas values higher than 1 indicate a constraints were extracted from the data for each compo-
better fit of the composite distribution in comparison to sition separately.
the component distributions.
Both the K–S distance and composed distributions were Results
calculated numerically in all cases. The K–S distance was
computed as a distance between the empirical cumulative Both the distributions can be accepted at the p=0.05 level
distribution function for data and 15,000 abundances and rejected at p=0.01 level for the test of the fit using a
randomly drawn from the fitted or composed distribution. test for K–S statistics (DKW inequality; pffi0:02),
The log-normal distribution was fitted by calculating mean although the multi-diffonential distribution apparently fit
andvariancefromlogarithmicallytransformeddata,whereas the data better (Fig. 4). There were also no dramatic
forthefittingofthemulti-diffonentialdistributionaspecific changes in the fit between different sized areas (i.e. plots
algorithm was developed (see SII). Drawing from the log- differing in number of census points). However, the mean
normaldistributionwasbasedonthestandardalgorithm,as K–Sforagivenarea(N=14)wascorrelatedwiththesizeof
Exp(sum of 1,000 random values drawn from the uniform the area (although with marginal significance for a log-
distribution(0,1)rescaledtoN(μ,σ)),andfordrawingfrom normal distribution, pffi0:052, r=−0.52, whereas for a
the multi-diffonential distribution the cumulative distribu- multi-diffonential distribution p<10−5, r=−0.98), both the
tion function was used (Eq. 7 in Appendix II, and SI). The distributionsfittingbetterinlargerareas(inagreementwith
number of 15,000 abundances was chosen according to the Preston 1948; Green and Plotkin 2007). The slightly but
Dvoretzky–Kiefer–Wolfowitz(DKW)inequality(Wasserman systematicallybetterfitofthemulti-diffonentialdistribution
2004; wikipedia .../wiki/Dvoretzky_Kiefer-Wolfowitz_ (Fig. 4) could in principle be caused by the fact that this
inequality) not to deviate more than about 1% from the distributionhasmorefreeparameters(Fig.2).However,the
TheorEcol(2009)2:89–103 97
0.2
below one (i.e. reveal inconsistency) for the log-normal
log-normal model distribution, 74 of 104 cases fell above one (i.e. reveal
multi-diffonential consistency) for the multi-diffonential distribution (for
details including significances see Table 2).
e The inconsistency of the log-normal distribution and
c
an overall consistency of the multi-diffonential distribution
dist 0.1 werealsorevealedbythedependenceofconsistencyonthe
S
K-
a)
0
0 120 240 1.0
y
c
length [Km] n
e
st
Fig. 4 Box (50%) and whiskers (95%) of the K–S distances (data- si
n
fitteddistributions)forplotsofvariousareasmeasuredasthelengthof co 0.9
transect in km. Medians of distances for log-normal (black squares)
andmulti-diffonential(whitesquares)distributionsareshownaswell
as the outliers (full and empty circles, respectively). The dotted line log-normal
shows the limit (p<0.05) for rejecting the fit (DKW test for K–S multi-diffonential
distance)
0.8
6 30 60
length [Km]
AICidentifiedallmulti-diffonentialdistributionswith1–20
terms as providing a better fit than the log-normal (for b)
details see Supplement: SIV).
Our main aim, however, was not to test the shape of the
distribution, but its properties, as only the properties might 1.0
y
indicate possible underlying mechanisms. Since many c
n
underlying mechanisms can produce very similar shapes, ste
we tested both the models for consistency. The test of nsi
o
consistency revealed remarkable inconsistency for the log- c
0.9
normal distribution (Fig. 5) and consequently for all of its
underlying processes. The consistency of the multi-diffo-
nential distribution was much higher and could not be log-normal
rejected. This was apparent when comparing the numbers multi-diffonential
0.8
of cases which fell below and above the level of absolute 0.0 0.3 0.6
consistency(consistency=1).Whileallofthe104casesfell
correlation [rad]
c)
„
Fig.5 Testofconsistencyforlog-normal(blacksquares)andmulti-
diffonential (white squares) distributions. Consistency refers to the
difference between K–S distances of component and composed
distributions. Consistency of one (dotted lines) refers to the full
1.0
consistency between composed and fitted distributions when adding
y
assemblagesofnewplots.aConsistencyagainstareameasuredasthe c
n
lengthofthetransectinkm.Boxesandwhiskersshowminimumand e
st
maximum estimates of 95% CIs for expected and observed values, si
n
respectively (t-distribution); circles show the outliers. b The depen- o
denceofconsistencyonthecorrelationofspeciesabundancesbetween c 0.9
two adjacent plots. The full increasing line shows the regression for
the log-normal distribution, the dashed constant line shows the
regression for the multi-diffonential distribution. Note that the point
of absolute correlation is at p=2ffi1:57. c The dependence of
consistency on Jaccard index (J). The full increasing line shows the 0.8
50 70 90
regressionforthelog-normaldistribution,thedasheddecreasingline
showstheregressionforthemulti-diffonentialdistribution Jaccard [%]
98 TheorEcol(2009)2:89–103
Table2 Testsofconsistencyoflog-normalandmulti-diffonentialSADs
Area[nofpoints] Log-normal Multi-diffonential
Belowone Aboveone pvalue Belowone Aboveone pvalue
20 38 0 3.6–10−12 11 27 0.99
40 19 0 1.2–10−6 3 16 0.99
60 12 0 2.5–10−4 3 9 0.98
80 9 0 0.002 5 4 0.5
100 7 0 0.008 2 5 0.94
120 6 0 0.015 3 3 0.66
140 5 0 0.03 1 4 0.97
160 4 0 0.06 1 3 0.94
180 4 0 0.06 1 3 0.94
Overall 104 0 5–10−32 30 74 0.999
Numbersofpairsofadjacentplotsthatrevealinconsistency(columns‘belowone’;consistency<1)andhighconsistency(columns‘aboveone’;
consistency>1),forthelog-normalandmulti-diffonentialdistributions.ThesignificPnancepistheprobabilitythatlessorequalnumbersfallabove
onebychanceassumingequalprobabilitytobeinbothgroups,calculatedasp¼ CN0:5i (Nisthetotalnumberofpointsandnisnumberof
i
pointsthatfellaboveone;Cisthecombinationnumber) i¼0
correlation between abundances in neighboring plots and the observed SAD is likely to be area-and-taxon invariant.
on species turnover (based on the knowledge that all Moreover, the log-normal model of SAD was shown to be
possible distributions reveal consistency in the case of inconsistent when enlarging area, as the composition of two
strongly correlated abundances and low species turnover; log-normal SADs does not fit the SAD of the joined sub-
see the Theory section). The log-normal distribution assemblagesforanyofthesizesofareatested.IfSADswere
revealed low consistency regardless of the level of log-normalforsomearea,thenweshouldsee(inFig.5a)high
correlation between abundances (p≈0.15, r≈0.14, N= consistencywhenjoiningplotsofthatarearegardlessofany
104), indicating that the range of measured correlations lack of area invariance (i.e. regardless of whether the
did not approach the level at which it would reveal composite distribution is also log-normal). Our finding thus
consistency due to the similarity of component SADs. highlights the failure of the log-normal distribution as a
However, the consistency of log-normal SADs increased consistentdescriptorofspeciesabundancepatterns,although
with similarity of assemblage composition as measured by the log-normal SAD may still serve as a useful approxima-
the Jaccard index (p≈0.08, r≈0.17, N=104) (Fig. 5b,c). tion for a particular assemblage for some practical purposes.
The consistency of the multi-diffonential distribution was This has important implications for the mechanisms
invariably high and independent of the correlation of possibly producing observed SADs. The inconsistency
abundances (p≈0.79, r≈0.03, N=104), although it slightly means that no underlying process producing log-normal
decreased with assemblage similarity (p≈0.05, r≈−0.19, SADs (for reviews see May 1975; Sugihara 1980; Engen
N=104). The plot actually showed better agreement with 2001; May et al. 2007) can be considered as universal and
data after composition (i.e. consistency higher than 1) than satisfactory (see also Williamson and Gaston 2005).
before when species turnover between assemblages was Whereas log-normal-like distributions emerge due to
high (andwhentheeffectofrare species shouldrather lead multiplicative processes (May 1975), the processes gener-
to inconsistency). Conversely to the multi-diffonential ating overall SADs are naturally at least partially additive.
distribution, the log-normal distribution not only failed in ThisisbecauseeverySAD canbeconsideredascomposed
producing the same distributional form after the composi- from SADs of sub-assemblages occupying only part of the
tion, but the distribution composed from two log-normal whole area, which share several species. This also means
distributionswhichfittedsub-assemblageSADsitselffitted that the models based on niche division (the broken stick
poorly to the merged data at all tested scales. and power-fraction models, see Tokeshi 1999), are equally
inappropriate as all of these models are implicitly based on
purely multiplicative processes (Williamson and Gaston
Discussion 2005). It would still be possible that multiplicative
processes could determine SADs at the largest scales (i.e.
We have shown that the functional form of the multi- continental or global), whereas SADs at local scales would
diffonential distribution is much more consistent with ob- emergebysamplingtheregionalSAD,buteveninthiscase
served abundance data than is the log-normal, implying that the log-normal SAD would not be an appropriate charac-