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Jou nal o Biogeog aphy. 2020;47:72–86.
wileyonlinelib a y.com/jou nal/jbi
Recei ed: 20 Janua y 2019
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Re ised: 8 Augus 2019
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Accep ed: 9 Augus 2019
DOI: 10.1111/jbi.13697
RESEARCH PAPER
Species–a ea ela ionships in con inuous ege a ion: E idence
om Palaea c ic g asslands
Jü gen Dengle 1,2,3 | Thomas J. Ma hews4,5 | Manuel J. S einbaue 6 |
Sebas ian Wol um7,8 | S e en Boch9 | Alessand o Chia ucci10 | Timo Con adi1 |
Iwona Dembicz11,12 | Co ado Ma cenò13,14 | I zia Ga cía‐Mijangos13 |
A kadiusz Nowak12,15 | Da id S o ch16,17 | We ne Ul ich18 |
Juan An onio Campos13 | Lau a Cancellie i19 | Ma a Ca boni20 |
Giampie o Ciasche i21 | Pie e De F enne22 | Ji i Dolezal23,24 | Ch is ian Dolnik25 |
F anz Essl26 | Edy Fan ina o27 | Go edo Filibeck19 | John‐A id G y nes28 |
Ricca do Gua ino29 | Behlül Güle 30 | Monika Janišo á31 |
Ewelina Klichowska12,32 | Łukasz Kozub11 | Anna Kuzemko33 |
Michael Man hey34 | Anne Mime 35,3 | Ali eza Naqinezhad36 |
Ch is ian Pede sen37 | Robe K. Pee 38 | Vincen Pellissie 35 |
Remigiusz Pielech39 | Gio anna Po enza40 | Leona do Rosa i40 |
Massimo Te zi41 | O solya Valkó42 | Denys Vynoku o 33 | Hannah Whi e43 |
Manuela Winkle 44,45 | Idoia Biu un13
1Plan Ecology G oup, Bay eu h Cen e o Ecology and En i onmen al Resea ch (BayCEER), Uni e si y o Bay eu h, Bay eu h, Ge many
2Vege a ion Ecology G oup, Ins i u e o Na u al Resou ce Sciences (IUNR), Zu ich Uni e si y o Applied Sciences (ZHAW), Wädenswil, Swi ze land
3Ge man Cen e o In eg a i e Biodi e si y Resea ch (iDi ) Halle‐Jena‐Leipzig, Leipzig, Ge many
4GEES (School o Geog aphy, Ea h and En i onmen al Sciences) and Bi mingham Ins i u e o Fo es Resea ch, Uni e si y o Bi mingham, Bi mingham, UK
5CE3C – Cen e o Ecology, E olu ion and En i onmen al Changes/Azo ean Biodi e si y G oup, Uni . dos Aço es, Aço es, Po ugal
6GeoZen um No dbaye n, Depa men o Geog aphy and Geosciences, F ied ich‐Alexande Uni e si y E langen‐Nü nbe g (FAU), E langen, Ge many
7Chai o O ganic Ag icul u e and Ag onomy, Li e Science Cen e Weihens ephan, Technische Uni e si ä München, F eising, Ge many
8Ins i u e o O ganic Fa ming, Soil and Resou ce Mangemen (IAB), Ba a ian S a e Resea ch Cen e o Ag icul u e (L L), F eising, Ge many
9Resea ch Uni Biodi e si y & Conse a ion Biology, WSL Swiss Fede al Resea ch Ins i u e, Bi mensdo , Swi ze land
10Depa men o Biological, Geological and En i onmen al Sciences (BiGeA), Uni e si y o Bologna, Bologna, I aly
11Depa men o Plan Ecology and En i onmen al Conse a ion, Ins i u e o Bo any, Uni e si y o Wa saw, Wa saw, Poland
12Bo anical Ga den Cen e o Biological Di e si y Conse a ion in Powsin, Polish Academy o Sciences, Wa saw, Poland
13Depa men o Plan Biology and Ecology, Uni e si y o he Basque Coun y UPV/EHU, Bilbao, Spain
14UMIB, CSIC‐Uni e si y o O iedo, Mie es, Spain
15Ins i u e o Biology, Uni e si y o Opole, Opole, Poland
16Cen e o Theo e ical S udy, Cha les Uni e si y, P aha 1, Czech Republic
17Depa men o Ecology, Facul y o Science, Cha les Uni e si y, P aha 2, Czech Republic
18Depa men o Ecology and Biogeog aphy, Facul y o Biological and Ve e ina y Sciences, Nicolaus Cope nicus Uni e si y, To un, Poland
19Depa men o Ag icul u al and Fo es y Sciences (DAFNE), Uni e si y o Tuscia, Vi e bo, I aly
20Depa men o Biological Sciences, Uni e si y o To on o ‐Sca bo ough, To on o, Canada
21Bo anical O ice, Majella Na ional Pa k, Sulmona, I aly
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and ep oduc ion in any medium,
p o ided he o iginal wo k is p ope ly ci ed.
© 2019 The Au ho s. Jou nal o Biogeog aphy published by John Wiley & Sons L d
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73
DENGLER E aL.
22Fo es & Na u e Lab, Ghen Uni e si y, Gon ode, Belgium
23Ins i u e o Bo any, Czech Academy o Sciences, P ůhonice, Czech Republic
24Depa men o Bo any, Facul y o Science, Uni e si y o Sou h Bohemia, České Budějo ice, Czech Republic
25Ecology Cen e Kiel, Kiel Uni e si y, Kiel, Ge many
26Di ision o Conse a ion Biology, Vege a ion and Landscape Ecology, Depa men o Bo any and Biodi e si y Resea ch, Uni e si y Vienna, Vienna, Aus ia
27Depa men o En i onmen al Sciences, In o ma ics and S a is ics, Ca' Fosca i Uni e si y o Venice, Venice, I aly
28Depa men o Biological Sciences, Uni e si y o Be gen, Be gen, No way
29Dep . STEBICEF, Uni e si y o Pale mo, Pale mo, I aly
30Biology Educa ion, Facul y o Educa ion, Dokuz Eylul Uni e si y, Buca, İzmi , Tu key
31Ins i u e o Bo any, Plan Science and Biodi e si y Cen e , Slo ak Academy o Sciences, Banská Bys ica, Slo akia
32Ins i u e o Bo any, Jagiellonian Uni e si y, K aków, Poland
33Depa men o Geobo any and Ecology, M.G. Kholodny Ins i u e o Bo any, Na ional Academy o Sciences o Uk aine, Kyi , Uk aine
34Ins i u e o Bo any and Landscape Ecology, G ei swald Uni e si y, G ei swald, Ge many
35Compu a ional Landscape Ecology, Helmhol z Cen e o En i onmen al Resea ch, Leipzig, Ge many
36Depa men o Biology, Facul y o Basic Sciences, Uni e si y o Mazanda an, Babolsa , I an
37Depa men o Landscape Moni o ing, No wegian Ins i u e o Bioeconomy Resea ch, Ås, No way
38Depa men o Biology, Uni e si y o No h Ca olina, Chapel Hill, NC, Uni ed S a es
39Depa men o Fo es Biodi e si y, Facul y o Fo es y, Uni e si y o Ag icul u e in K aków, K aków, Poland
40School o Ag icul u al, Fo es , Food and En i onmen al Sciences, Uni e si y o Basilica a, Po enza, I aly
41Ins i u e o Biosciences and Bio esou ces (IBBR), I alian Na ional Council o Resea ch (CNR), Ba i, I aly
42MTA‐DE Lendüle Seed Ecology Resea ch G oup, Deb ecen, Hunga y
43School o Biology and En i onmen al Science, Ea h Ins i u e, Uni e si y College Dublin, Dublin 4, I eland
44GLORIA Co‐o dina ion, Depa men o In eg a i e Biology and Biodi e si y Resea ch, Uni e si y o Na u al Resou ces and Li e Sciences Vienna (BOKU), Vienna, Aus ia
45GLORIA Co‐o dina ion, Ins i u e o In e disciplina y Moun ain Resea ch, Aus ian Academy o Sciences, Vienna, Aus ia
Co espondence
Jü gen Dengle , Plan Ecology G oup,
Bay eu h Cen e o Ecology and
En i onmen al Resea ch (BayCEER),
Uni e si y o Bay eu h, Uni e si ä ss .
30, 95447 Bay eu h, Ge many; Vege a ion
Ecology G oup, Ins i u e o Na u al
Resou ce Sciences (IUNR), Zu ich Uni e si y
o Applied Sciences (ZHAW), G üen als .
14, 8820 Wädenswil, Swi ze land; Ge man
Cen e o In eg a i e Biodi e si y Resea ch
(iDi ) Halle‐Jena‐Leipzig, Deu sche Pla z
5e, 04103 Leipzig, Ge many.
Email: jue gen.dengle @uni‐bay eu h.de
Funding in o ma ion
S a e Fund o Fundamen al Resea ch o
Uk aine, G an /Awa d Numbe : Ф83/53427;
Eusko Jau la i za, G an /Awa d Numbe :
IT936‐16; Slo enská Akadémia Vied,
G an /Awa d Numbe : VEGA 02/0095/19;
Na odowe Cen um Nauki, G an /Awa d
Numbe : 2017/27/B/NZ8/00316 and
DEC‐2013/09/N/NZ8/03234; Cen e
o In e na ional Scien i ic S udies and
Collabo a ion (CISSC), G an /Awa d
Numbe : NA; Eu asian D y G assland
G oup (EDGG) and he In e na ional
Associa ion o Vege a ion Science (IAVS),
G an /Awa d Numbe : NA; G an o á
Agen u a České Republiky, G an /Awa d
Numbe : GA 17‐19376S; MIUR ini ia i e
“Depa men o excellence”, G an /Awa d
Numbe : Law 232/2016; Ba a ian Resea ch
Alliance, G an /Awa d Numbe : BayIn An_
UBT_2017_58; Bay eu h Cen e o Ecology
and En i onmen al Resea ch (BayCEER),
G an /Awa d Numbe : NA
Abs ac
Aim: Species–a ea ela ionships (SARs) a e undamen al scaling laws in ecology al‐
hough hei shape is s ill dispu ed. A la ge a eas, powe laws bes ep esen SARs.
Ye , i emains unclea whe he SARs ollow o he shapes a ine spa ial g ains in
con inuous ege a ion. We asked which unc ion desc ibes SARs bes a small g ains
and explo ed how sampling me hodology o he en i onmen in luence SAR shape.
Loca ion: Palaea c ic g asslands and o he non‐ o es ed habi a s.
Taxa: Vascula plan s, b yophy es and lichens.
Me hods: We used he G assPlo da abase, con aining s anda dized ege a ion‐plo
da a om ascula plan s, b yophy es and lichens spanning a wide ange o g ass‐
land ypes h oughou he Palaea c ic and including 2,057 nes ed‐plo se ies wi h a
leas se en g ain sizes anging om 1 cm2 o 1,024 m2. Using nonlinea eg ession,
we assessed he app op ia eness o di e en SAR unc ions (powe , powe quad‐
a ic, powe b eakpoin , loga i hmic, Michaelis–Men en). Based on AICc, we es ed
whe he he anking o unc ions di e ed among axonomic g oups, me hodological
se ings, biomes o ege a ion ypes.
Resul s: The powe unc ion was he mos sui able unc ion ac oss he s udied axo‐
nomic g oups. The supe io i y o his unc ion inc eased om lichens o b yophy es
o ascula plan s o all h ee axonomic g oups oge he . The sampling me hod was
highly in luen ial as oo ed p esence sampling dec eased he pe o mance o he
powe unc ion. By con as , biome and ege a ion ype had p ac ically no in luence
on he supe io i y o he powe law.
Main conclusions: We conclude ha SARs o sessile o ganisms a smalle spa ial
g ains a e bes app oxima ed by a powe unc ion. This coincides wi h se e al o he
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DENGLER E aL.
1 | INTRODUCTION
Species–a ea ela ionships (SARs) ep esen one o he mos un‐
damen al laws in ecology (Law on, 1999; Lomolino, 2000). Since
he ea ly s udies by A henius (1921) and Gleason (1922) hey
ha e a ac ed conside able a en ion (e.g. Conno & McCoy, 1979;
D aka e, Lennon, & Hilleb and, 2006; Lomolino, 2001; Rosenzweig,
1995; T ian is, Guilhaumon, & Whi ake , 2012). SARs a e o g ea
heo e ical in e es as di e en heo ies o island biogeog aphy
(e.g. MacA hu & Wilson, 1967), species abundance dis ibu ions
(e.g. Pueyo, 2006; Šizling & S o ch, 2004) and neu al models (e.g.
Hubbell, 2001) p edic di e en shapes o SARs, wi h he impli‐
ca ion ha obse ed SARs can be deployed o es such heo ies.
Fu he mo e, i is in e es ing o es how axonomic g oup, scale,
me hodological se ings and ecosys em o geog aphic con ex in lu‐
ence he ela i e pe o mance o SAR unc ions and hei pa ame‐
e s (e.g. Chia ucci, Viciani, Win e , & Diekmann, 2006; C awley &
Ha al, 2001; D aka e e al., 2006; S o ch, E ans, & Gas on, 2005). In
addi ion, SARs allow he a ea e ec o be con olled in assessmen s
o ecological d i e s o biodi e si y (e.g. P ice, 2004; Whi ake ,
Willis, & Field, 2001). SARs also allow ex apola ion o species ich‐
ness o la ge a eas ha canno be su eyed wi h easonable e o
(e.g. Kunin e al., 2018; Plo kin e al., 2000; Ul ich, 2005; Wilson,
Pee , Dengle , & Pä el, 2012). In addi ion, SARs allow s anda d‐
iza ion o ichness eco ds om se e al di e en ly sized uni s o a
common g ain size, he eby acili a ing scale‐independen di e si y
compa isons and isualiza ions (e.g. Kie e al., 2005) and he iden‐
i ica ion o biodi e si y ho spo s (e.g. Fa o ini, 2007). Finally, he
slope pa ame e s o ce ain ypes o SARs a e sui able measu es o
be a‐di e si y (DeMalach, Saiz, Zaady, & Maes e, 2019; Ju asinski,
Re ze , & Beie kuhnlein, 2009).
His o ically, s udies o SARs ha e la gely been es ic ed o
wo unc ions, (a) he powe unc ion (o en called he powe law;
A henius, 1921; P es on, 1962) and (b) he loga i hmic unc ion
(some imes e oneously e med he ‘exponen ial’ unc ion; Gleason,
1922). This was mainly because he i o hese wo unc ions was
easily explo ed using leas squa es linea eg ession echniques.
A compa ison o a b oade se o unc ions became possible wi h
he ad en o nonlinea eg ession echniques (e.g. Dengle , 2009;
Fla he , 1996; Guilhaumon, Gimenez, Gas on, & Mouillo , 2008;
S iles & Scheine , 2007). In ecen yea s, a wide a ay o di e en
unc ion ypes has been p oposed and es ed (Dengle , 2009, 2010;
Tjø e, 2003, 2009). Consequen ly, se e al comp ehensi e s udies
ha e been conduc ed on he i o di e en unc ions and pa ame‐
e s o island SARs as well as o he b oadscale SARs. T ian is e al.
(2012) compa ed 20 di e en models o 601 ue island da ase s
a ound he wo ld and ound s ong suppo o he powe unc ion
o e all. Ma hews, Guilhaumon, T ian is, Bo egaa d, and Whi ake
(2016) ex ended his s udy o 182 habi a islands, wi h a simila ind‐
ing. In a u he s ep, hey es ed how ecological con ex a ec s he
slope pa ame e o he powe unc ion, and hey ound sys ema ic
di e ences be ween island ypes and spa ial scales, bu no be‐
ween majo axa.
While knowledge o unc ions and pa ame e s o island SARs has
been b oadly consolida ed du ing he las decade, compa able empi ‐
ical e idence on small‐g ain SARs in con inuous habi a s is s ill lack‐
ing ( o heo y see S o ch, 2016; Williamson, 2003). Wi h con inuous
habi a o ege a ion, we e e o si ua ions whe e he sampling uni s
do no ha e a na u al bo de such as islands o habi a islands, bu
a e delimi ed by he esea che . The in luen ial s udy o C awley and
Ha al (2001) on how biodi e si y depends on scale in con inuous
ege a ion, o example, a p io i only conside ed he powe unc ion.
Some egional s udies ha e ound a p e alence o he powe unc‐
ion using mul imodel in e ence, bu we e es ic ed o less han 20
da ase s (e.g. Dengle , 2009; Dengle & Boch, 2008). Fu he mo e,
Rosindell and Co nell (2007) ob ained powe unc ion SARs om a
spa ially explici ecological d i model (Hubbell, 2001) wi hin a ho‐
mogeneous g id model assuming skewed dispe sal ke nels. By con‐
as , he e is a belie ha he loga i hmic unc ion should be mo e
sui able a small spa ial scales (Gleason, 1922; an de Maa el, 1997).
Sa u a ed unc ions (i.e. unc ions wi h a ho izon al uppe asymp o e)
a e also o en assumed o ep esen SARs in con inuous ege a ion
weöö, inspi ed by he s ill widesp ead, bu lawed (see Ba kman,
1989) concep o so‐called ‘minimal a eas’ (e.g. Muelle ‐Dombois &
Ellenbe g, 1974), which was assumed o be he scale a which spe‐
cies ichness is sa u a ed o a gi en communi y. Addi ional con usion
comp ehensi e s udies o SARs a di e en g ain sizes and o di e en axa, hus
suppo ing he gene al app op ia eness o he powe unc ion o modelling species
di e si y o e a wide ange o g ain sizes. The poo pe o mance o he Michaelis–
Men en unc ion demons a es ha ichness wi hin plan communi ies gene ally
does no app oach any sa u a ion, hus calling in o ques ion he concep o minimal
a ea.
KEYWORDS
loga i hmic unc ion, Michaelis–Men en unc ion, minimal a ea, nes ed‐plo sampling,
nonlinea eg ession, Palaea c ic g assland, plan biodi e si y, powe law, scaling law, species–
a ea ela ionship (SAR)
Handling Edi o : Holge K e
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DENGLER E aL.
a ound small‐g ain SARs was caused when con ounding di e en
sampling schemes wi h SARs in he s ic sense (i.e. hose o iginally
conside ed by A henius, 1921, o P es on, 1962). Fo example, S iles
and Scheine (2007) and DeMalach e al. (2019) epo ed ha he
logis ic unc ion (a sa u a ed unc ion) and he loga i hmic unc‐
ion, espec i ely, pe o med much be e han he powe unc ion.
Howe e , hey had analysed species accumula ion cu es, me ging
non‐con iguous sample uni s (also called species–sampling ela‐
ionships, SSRs; see Dengle , 2009; F idley, Pee , an de Maa el, &
Willems, 2006), and no SARs in he s ic sense. In conclusion, his
si ua ion calls o a comp ehensi e, mul imodel in e ence analysis o
small‐g ain SARs in con inuous ege a ion, compa able o he analy‐
ses o T ian is e al. (2012) and Ma hews e al. (2016) o island SARs.
As he Palaea c ic biogeog aphic ealm comp ises mo e han one
hi d o he wo lds’ ice‐ ee e es ial su ace and spans a wide ange
o clima ic and opog aphic g adien s, i ha bou s a high numbe o
ege a ion ypes and conside able biodi e si y (Rounse ell, Fische ,
To e‐Ma in Rando, & Made , 2018). A ound 22% o he Palaea c ic
is composed o a ious g assland ypes (Tö ök & Dengle , 2018),
some o hem being he wo ld eco d holde s o small‐g ain as‐
cula plan di e si y (Wilson e al., 2012). A la ge p opo ion o he
Palaea c ic g asslands a e p ima y g asslands such as s eppes and
a c ic‐alpine g asslands. E en in egions whe e he po en ial eg‐
e a ion is o es , na u al g asslands occu in azonal and ex azonal
condi ions. Mo eo e , ag icul u al ac i i ies and pas o alism long
p esen in he Palaea c ic has esul ed in he c ea ion o seconda y
g asslands dependen on human land use ha p e en s succession
owa ds sh ublands o o es s (Tö ök & Dengle , 2018). The co e ‐
age o majo ecological g adien s and he high di e si y o ege a‐
ion ypes ac oss se e al biogeog aphic egions highligh Palaea c ic
g asslands as an excellen model sys em o s udy small‐g ain SARs
and how hey a e a ec ed by di e en ac o s.
He e, we used mo e han 2,000 nes ed‐plo se ies om he
G assPlo da abase (Dengle e al., 2018), om a wide ange o
g assland ypes ac oss six biomes, o pe o m a comp ehensi e anal‐
ysis o small‐g ain (1 cm2–1,024 m2) SARs in con inuous ege a ion
o ascula plan s, b yophy es and lichens. Speci ically, we aimed o
add ess he ollowing ques ions using he Palaea c ic g asslands as
an example:
1. Which unc ion is mos app op ia e o desc ibe small‐g ain
SARs?
2. Does he pe o mance o he di e en unc ions depend on ac‐
o s such as sampling me hod, axonomic g oup, biogeog aphic
se ing o ege a ion ype?
2 | MATERIALS AND METHODS
2.1 | Vege a ion‐plo da a
We used plo da a om he collabo a i e ege a ion‐plo da abase
G assPlo (Dengle e al., 2018; h p://b.link/g ass plo ), which is
egis e ed in he Global Index o Vege a ion‐Plo Da abases (GIVD;
Dengle e al., 2011) as EU‐00‐003. G assPlo collec s ege a ion‐
plo da a (bo h ichness and composi ion) oge he wi h me hodo‐
logical, en i onmen al and s uc u al in o ma ion om g asslands as
well as o he plan communi ies domina ed by he bs, dwa ‐sh ubs
o c yp ogams om he Palaea c ic biogeog aphic ealm ( o de‐
limi a ion see Figu e S1.1). Requi emen s o inclusion a e ha he
plo s (sampling uni s) we e p ecisely delimi ed in he ield and ca e‐
ully sampled wi h he aim o achie ing comple e species lis s. One
s eng h o G assPlo is he nume ous mul i‐scale da ase s de i ed
om a di e si y o nes ed‐plo sampling schemes (e.g. Dengle e al.,
2016) o a eas om 1 cm2 o 1,024 m2 (schemes o he h ee main
ypes o sampling designs in Figu e S2.1).
We e ie ed all nes ed‐plo se ies con ained in G assPlo ( .1.27
on 4 Janua y 2019) ha comp ised a leas se en di e en g ain sizes
(see o e iew o he 69 da ase s wi h hese da a in Table S1.4). In o al,
he e we e 2,057 se ies wi h ascula plan in o ma ion (Figu e 1), o
which 757 also con ained b yophy e da a and 780 lichen da a (Figu e
S1.2). The plo s we e dis ibu ed o e 26 di e en coun ies om
34.9° o 68.9°N, om 9.1°W o 161.8°E and co e ed an al i udinal
g adien om 0 o 4,387 m a.s.l. (Figu e 1, Figu e S1.2). In o al, he
nes ed‐plo se ies consis ed o 139,265 indi idual subplo s wi h ich‐
ness da a, o en wi h se e al eplica es pe g ain size. Fu he cha ac‐
e is ics o he used da ase s a e p o ided in Appendix S1.
Fo hose nes ed‐plo s se ies wi h mo e han one subplo o a
ce ain g ain size, we a e aged ichness alues ac oss subplo s and
s o ed he in o ma ion on how many subplo s he a e age was based
on. Thus, we ob ained one single ichness alue pe each g ain size
wi hin each nes ed‐plo se ies, i possible, o ou di e en axonomic
g oups (1 – comple e e icolous mac oscopic ege a ion; 2 – ascula
plan s; 3 – e icolous b yophy es; 4 – e icolous lichens). We also e‐
co ded whe he plo s we e sampled wi h he shoo p esence o wi h
he oo ed p esence me hod ( o e minology, see Dengle , 2008).
2.2 | SAR modelling
F om he nume ous di e en unc ions p oposed o modelling SARs
(Dengle , 2009; Tjø e, 2003), we selec ed h ee main unc ions ha
ha e speci ically been sugges ed and used o SAR modelling in con‐
inuous ege a ion (DeMalach e al., 2019; Dengle & Boch, 2008):
he powe unc ion, he loga i hmic unc ion (o en e oneously
e med he exponen ial unc ion) and inally he Michaelis–Men en
unc ion as a simple wo‐pa ame e example o a SAR wi h sa u a‐
ion (i.e. an uppe h eshold o ichness). To accoun o possible
‘scale dependence’ o he SAR, we added wo a ian s o he powe
unc ion ha allow o exponen s o change wi h a ea: he ‘quad a ic
powe unc ion’ wi h a con inuous change o he exponen , and he
‘b eakpoin powe unc ion’ wi h an ab up change o he exponen
a a ce ain g ain (e.g. Dengle , 2010). The i e unc ions we e se‐
lec ed o ep esen undamen ally di e en shapes (Dengle , 2008;
see also Figu e S3.1) as well as di e en complexi ies (numbe o
i ed pa ame e s; Table 1).
We i ed all i e unc ions o bo h species ichness S (S‐space;
‘linea space’) and o log S (log S‐space; ‘loga i hmic space’) as
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DENGLER E aL.
dependen a iables using nonlinea eg ession (Table 1). Bo h ap‐
p oaches a e alid, ha e been used in he li e a u e, and ha e di ‐
e en s eng hs and limi a ions (see Dengle , 2009). Due o he
di e en ea men o he e o s uc u e, he pa ame e es ima es
in he wo spaces usually sligh ly de ia e. Gene ally, i ing in S‐space
gi es mo e weigh o good i a la ge g ain sizes, whe eas i ing in log
S‐space gi es mo e weigh o good i a small‐g ain sizes. Mo eo e ,
i ing in log S ypically educes he e oscedas ici y in he esiduals.
As i ing in log S‐space is no possible i some subplo s ha e
S = 0 (excluding such cases is no ecommended; Dengle , 2010;
Williams, 1996), we add essed his issue as ollows. Fi ing nes ed‐
plo se ies in he op imal case means ha he ichness alue o
he smalle g ain sizes is ep esen a i e o he whole a ea o he
la ges plo , which could be achie ed by ull essella ion o i s a ea
and a e aging he ichness alues o all esul ing subplo s. In such
op imal sampling, e iden ly he mean ichness alue o any smalle
FIGURE 1 Densi y and spa ial
dis ibu ion o he 2,057 nes ed‐plo
se ies in he Palaea c ic biogeog aphic
ealm ha we e analysed in his s udy.
The colou scale indica es he numbe
o a ailable se ies pe 10,000‐km2 g id
cell. The map uses he Eu ope Lambe
Con o mal Conic p ojec ion [Colou igu e
can be iewed a wileyonlinelib a y.com]
1
2−4
5−9
10−19
20−49
50−99
100−199
200−499
TABLE 1 The i e unc ion ypes used in his s udy o model species–a ea ela ionships (SARs). All unc ions we e i ed bo h in S‐space
and in log S‐space. The ollowing no a ions a e used: S = mean species ichness; A = a ea/m2; log = log10. The k i ed pa ame e s (excep he
a iance) a e e med c, z, z1, z2, b0, b1 and T
Func ion name Abb e ia ion kFo mula in S‐space Fo mula in log S‐space Meaning o pa ame e s
Powe powSAR 2S = c A^zlog S = log c + z log A c = ichness a uni a ea (1 m2);
z = s eepness pa ame e (exponen in
S‐space o slope in log S‐space)
Powe quad a ic powQSAR 3S = 10^(log c + z1 log A + z2
(log A)^2)
log S = log c + z1 log A + z2
(log A)^2
c = ichness a uni a ea (1 m2);
z1 = s eepness pa ame e ;
z2 = change o s eepness wi h in‐
c easing a ea
Powe b eakpoin b eakSAR2 4S = 10^[log c + (log A < log
T) (z1 log A) + (log A ≥ log
T) (z1 log T + z2 (log A – log
T))]
log S = log c + (log A < log T)
(z1 log A) + (log A ≥ log T) (z1
log T + z2 (log A – log T))
c = ichness a uni a ea (1 m2);
T = b eakpoin (a ea a which he
s eepness changes); z1 = s eepness
pa ame e o A > T; z2 = s eepness
pa ame e o A ≥ T
Loga i hmic logSAR 2S = b0 + b1 log Alog S = log (b0 + b1 log A)b0 = in e cep (in S‐space); b1 = s eep‐
ness pa ame e
Michaelis–Men en mmSAR 2S = b0 A/(b1 + A) log S = log (b0 A/(b1 + A)) b0 = sa u a ion alue (modelled
maximum ichness); b1 = s eepness
pa ame e
No e: The logical exp essions (log A < log T) and (log A ≥ log T) e u n 1 i hey a e ue and 0 i hey a e alse.
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77
DENGLER E aL.
g ain size would be >0 i he e was a leas one species in he la ges
plo . Howe e , in mos cases, he nes ed‐plo sampling schemes in
G assPlo eco ded only one o ew eplica es o smalle plo s. In
such cases, he eco ded (a e age) ichness alue may be S = 0, while
he ue a e age (calcula ed om a la ge numbe o plo s) would be
posi i e. As ichness can ake only posi i e alues, an obse ed ich‐
ness o 0 based on a single plo is a biased es ima e as i ep esen s
he ange o [0, 0.5), while an obse ed ichness based on a single
plo o 1 is an unbiased alue o he ange o [0.5, 1.5). The e o e,
we eplaced 0 wi h 0.25, ha is, he mean o he lowe and uppe
bo de o he ange o alues o which i s ands. Likewise, i an ob‐
se ed mean ichness o 0 was based on n eplica es, we assigned a
mean ichness o 0.25/n.
The i e models we e i ed in R (Ve sion 3.5.1; R Co e Team,
2018) using a combina ion o linea and nonlinea eg ession. Fi ing
in S‐space always employed nonlinea eg ession, and op imiza‐
ion used he ‘mle2’ unc ion in he ‘bbmle’ R package (Bolke & R
Co e Team, 2017). As he op imiza ion algo i hm was sensi i e o
he s a ing pa ame e alues p o ided, a b u e‐ o ce app oach was
used o ind pa ame e alues ha maximized he likelihood, o a
gi en model. Fo each model (e.g. powe , b eakpoin ), a g id o mul i‐
ple di e en s a ing pa ame e alues was c ea ed. The size o his
g id depended on he model, wi h he mo e complex models ha ing a
la ge numbe o po en ial s a ing pa ame e alues. Model op imi‐
za ion was hen unde aken mul iple imes using he 'mle2' unc ion
and he s a ing pa ame e alues om each ow in he g id. Finally,
he s a ing pa ame e alues ha esul ed in he model i wi h la g‐
es maximum likelihood we e chosen. The AICc and R2
adj. alues o
he model i op imized using hese s a ing pa ame e alues we e
hen calcula ed. We conside AICc and R2
adj. as adequa e measu es
o he ela i e app op ia eness/supe io i y o he compa ed SAR
unc ions despi e he non‐independence o he da a poin s in ou
nes ed‐plo da a. In Appendix S4 (R codes in Appendix S5 and S6),
we sampled om i ual landscapes whe e he shape o he SARs is
known o es whe he nes ed‐plo sampling in oduces biases in he
model selec ion using AICc and R2
adj. The ‘ ue shape’ o he SARs in
hese i ual landscapes was de e mined by a e aging he esul s o
se e al andom non‐nes ed plo se ies o di e en g ain sizes. The
esul s show ha he model anking ob ained by nes ed‐plo sam‐
pling is close o he ue pa e n and ac ually depic s, on a e age,
he ue pa e n be e han SARs cons uc ed om a se ies o single
non‐nes ed plo s in he same landscape would do. Acco dingly, we
conside ou app oach as alid.
In a small numbe o cases, he e we e mul iple op imized model
i s (i.e. wi h di e en pa ame e es ima es) wi h iden ical (maxi‐
mum) likelihood alues; he e, we simply selec ed one se o pa ame‐
e alues a andom. Following s anda d s a is ical con en ion, he
a iance was always conside ed as an addi ional pa ame e when
calcula ing AICc. Thus, o example, he powe model was consid‐
e ed o ha e h ee pa ame e s when calcula ing AICc. Fo he powe
b eakpoin model, a u he model‐ i ing s ep was implemen ed. In
ce ain cases, he bes ‐selec ed powe b eakpoin model using he
a o emen ioned app oach con ained a z‐ alue ha was g ea e han
1 o less han 0. This z‐ alue was hen ixed a ei he 0 o 1 (de‐
pending on which o hese alues i was ini ially closes o) and he
model i ing p ocess epea ed. I bo h o iginal z‐ alues we e ou o
bounds, his addi ional s ep was no unde aken. Fo a gi en model
and plo se ies, he abo e model i ing p ocess was epea ed ac oss
all ou axonomic g oups.
Fo he log S‐space analyses, he loga i hmic, Michaelis–Men en
and b eakpoin powe unc ions we e i ed using he nonlinea i ‐
ing p ocedu e ou lined abo e, whe eas he powe model and he
quad a ic powe unc ion we e i ed using linea eg ession and he
s anda d 'lm' unc ion in R. The o e all model i ing p ocess was el‐
a i ely compu a ionally demanding and ook app oxima ely 48 h on
a 24‐co e compu e clus e (100 GB RAM). Due o he b u e‐ o ce
app oach, we achie ed con e gence o all models o all axa in all
da ase s in he log S‐space, and a negligible amoun o non‐con e ‐
gence in he S‐space (maximum 4% o lichens, bu 0% o comple e
ege a ion). The R code used o un he analyses is a ailable as
Appendix S7.
2.3 | Ranking and compa ison o he SAR unc ions
We anked model pe o mance in i e ways. Fi s , we coun ed o
how many nes ed‐plo se ies a ce ain unc ion pe o med bes
among all compa ed unc ions, using model selec ion based on
AICc (Bu nham & Ande son, 2002). Second, o each unc ion we
calcula ed he Akaike weigh s based on AICc in each nes ed‐plo se‐
ies. Akaike weigh s can be in e p e ed as he p obabili y ha he
unc ion i is he bes model o he obse ed da a, gi en he se o
i e candida e models (Johnson & Omland, 2004). Thi d, o each
unc ion by nes ed‐plo se ies combina ion we calcula ed Δi, ha
is, he di e ence in AICc o he pa icula unc ion compa ed o
he espec i e bes pe o ming unc ion (‘del a AICc’). Fou h, we
anked models using R2
adj., which was calcula ed using he o mula:
1 – (1 – R2) (n – 1)/k, whe e R2 is he s anda d R2, n is he numbe
o da a poin s and k is he esidual deg ees o eedom. Fi h, we
de e mined he bes pe o ming unc ion based on he Bayesian
In o ma ion C i e ion (BIC) as he e is no clea ag eemen on he
supe io i y o AIC/AICc e sus BIC (Bu nham & Ande son, 2002;
Johnson & Omland, 2004). The i e compa isons we e unde aken
only in cases whe e ou i ing p ocedu e yielded a esul o all i e
models. No e ha model compa isons a e es ic ed o each ‘space’,
ha is, measu es o goodness o i o in o ma ion con en (e.g.
R2
adj, AICc) canno be compa ed be ween S‐space and log S‐space
(Dengle , 2009).
As sampling me hodology has been epea edly sugges ed o in‐
luence he shape o SARs (Dengle , 2008; Williamson, 2003), we
es ed o an e ec o some key sampling me hod aspec s using
ANOVAs and linea eg essions: (a) shoo e sus oo ed sampling
o plan s (bo h me hods a e widesp ead; see Dengle , 2008); (b)
whe he he ichness o smalle g ain sizes was a e aged om se ‐
e al eplica ed subplo s o no ; and (c) numbe o g ain sizes in a se‐
ies (dis ibu ion o he di e en me hodological choices and o he
da a in Table S1.1 and Figu e S1.4). Fu he mo e, we es ed whe he
78
|
DENGLER E aL.
he pe o mance o he unc ions depended on (d) axonomic g oup
( ascula plan s, b yophy es, lichens), (e) biome (B uelheide e al.,
2019; based on Schul z, 2005), ( ) ege a ion ype o (g) ichness in
he la ges plo o he se ies (see Figu e S1.1, Tables S1.2 and S1.3).
3 | RESULTS
3.1 | Gene al sui abili y o he compa ed unc ions
Gi en he wide ange o ege a ion ypes s udied, he species–a ea
cu es also a ied widely (Figu es S8.1 and S8.2). Fo all axonomic
g oups and i espec i e o S‐space e sus log S‐space, he powe
unc ion was by a he bes model when using AICc as a model
selec ion c i e ion (Figu e 2). Fo he ichness o he comple e eg‐
e a ion, i was he bes model in 68.1% o all cases in S‐space, wi h
alues sligh ly d opping om ascula plan s (57.8%) o b yophy es
(56.0%) o lichens (49.5%). The supe io i y o he model was e en
sligh ly highe in he log S‐space han in he S‐space. Fo he com‐
ple e ege a ion, he second bes model, hough clea ly in e io ,
was he quad a ic powe unc ion (in bo h spaces), while he loga‐
i hmic unc ion was second bes o ascula plan s in bo h spaces
as well as o b yophy es and lichens in log S‐space. The Michaelis–
Men en sa u a ion unc ion gene ally pe o med poo ly, bu was
he second bes model o b yophy es and lichens in S‐space.
When conside ing BIC ins ead o AICc (Figu e S8.3), he anking
o unc ions changed. The b eakpoin powe unc ion pe o med
bes ollowed by he ‘no mal’ powe unc ion and he quad a ic
powe unc ion, while he loga i hmic unc ion and he Michaelis–
Men en unc ion had negligible suppo . The o e all suppo o
he h ee a ian s o he powe unc ion combined inc eased om
c. 60%–90% in case o AICc o c. 90%–95% in case o BIC.
When conside ing no only he bes model, bu also he ela i e
pe o mance o all i e models ia Akaike weigh s (Figu es S8.4 and
S8.5) o del a AICc (Figu e S8.6), he esul s emained quali a i ely
simila , bu he supe io i y o he powe unc ion was e en clea e ,
wi h he mean Akaike weigh eaching as high as 71.1% in he case
o he comple e ege a ion in log S‐space. Based on R2
adj., ha is,
conside ing only he i , no he complexi y, he h ee powe unc‐
ions pe o med e y well (mos ly abo e 0.95 o all axa and ascu‐
la plan s, and mos ly abo e 0.85 o b yophy es and lichens), while
he pe o mance o he loga i hmic and Michaelis–Men en unc ions
was subs an ially wo se (Figu e S8.7). Owing o he one o wo ad‐
di ional pa ame e s, he quad a ic and b eakpoin powe unc ion,
espec i ely, had a sligh ly be e i han he no mal powe unc ion.
The esul ing pa ame e es ima es o all i e models and hei de‐
sc ip i e s a is ics a e p o ided in Appendix S9. He e, we summa ize
only he esul s o he powe unc ion pa ame e es ima es, as i was
clea ly he o e all bes model. In pa icula , we ocus on a ew pa ame‐
e s ha a e pa icula ly ele an o in e p e a ion. The slope pa am‐
e e (z‐ alue) o he o e all bes pe o ming unc ion (powe unc ion)
in S‐space was 0.20 ± 0.05 (mean ± s anda d de ia ion) o all axa,
wi h sligh a ia ion among he h ee axonomic g oups ( ascula
plan s: 0.26 ± 0.11; b yophy es: 0.19 ± 0.12; lichens: 0.28 ± 0.14). In log
S‐space, he alues showed a simila pa e n wi h li le de ia ion in ab‐
solu e alues om S‐space (Table S9.1). The z2 es ima e o he quad a ic
powe unc ion was signi ican ly nega i e ( ‐ es wi h 0 mean as null
model) o all axa (p < .001; mean: −0.017 ± 0.047, median: −0.012),
wi h simila ends o ascula plan s (p = .09; mean: −0.061 ± 1.634,
median: −0.019), b yophy es (p < .001; mean: −0.105 ± 0.548, median:
−0.009) and lichens (p = .07; mean: −0.844 ± 8.204, median: −0.041)
(Table S9.2).
3.2 | Fac o s in luencing unc ion pe o mance
The ela i e pe o mance o he i e models was s ongly in lu‐
enced by se e al me hodological ac o s: (a) oo ed sampling
d as ically dec eased he ela i e pe o mance o he powe unc‐
ion compa ed o shoo sampling (Figu e 3), while he quad a ic
powe and b eakpoin powe models pe o med ela i ely be e
(Figu e S8.8). (b) Likewise, in nes ed‐plo se ies whe e he smalles
plo s we e no eplica ed and a e aged, he ela i e pe o mance
o he powe unc ion was much wo se han when an a e aging
had aken place (Figu e 4, Figu e S8.9). (c) The numbe o included
g ain sizes (no necessa ily co ela ing wi h he g ain size ange)
also dec eased he supe io i y o he no mal powe unc ion, while
he wo o he a ian s o he powe unc ion inc eased in ela i e
pe o mance, and oge he all h ee a ian s o he powe unc‐
ion we e e en mo e supe io when mo e g ain sizes we e sampled
(Figu e S8.10).
Biome had ha dly any in luence on he supe io i y o he
powe unc ion i espec i e o axonomic g oup (Figu e 5). Only
o ascula plan s he ela i e pe o mance o he powe unc‐
ion sligh ly was wo se in he ‘D y opics and sub opics’ and in
he ‘Sub opics wi h win e ain’ han in he o he ou biomes.
FIGURE 2 Model pe o mance in compa ison o he i e
unc ion ypes: powe (powSAR), powe quad a ic (powQSAR),
powe b eakpoin (b eakSAR), loga i hmic (logSAR) and Michaelis–
Men en (mmSAR), exp essed as ac ion o cases whe e a gi en
model pe o med bes based on AICc. The compa isons we e un
o he comple e e icolous mac oscopic ege a ion (all species),
ascula plan s, e icolous b yophy es and e icolous lichens, and
bo h in S‐space and log S‐space [Colou igu e can be iewed a
wileyonlinelib a y.com]
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79
DENGLER E aL.
Likewise, he 18 di e en majo ege a ion ypes ha dly showed
any di e ence in he supe io i y o he powe unc ion; he ew
signi ican di e ences in he ANOVA we e mos ly ela ed o
ypes wi h only e y ew eplica es (indica ing ha his migh
jus be a andom de ia ion and no a p ope y o he espec i e
ype) (Figu e S8.11). Howe e , one ege a ion cha ac e is ic had
a signi ican in luence on he ela i e pe o mance o unc ions,
a leas in ascula plan s: he ela i e pe o mance o he powe
unc ion s ongly inc eased wi h he numbe o species in he big‐
ges plo (Figu e S8.12).
4 | DISCUSSION
4.1 | The na u e o he species–a ea ela ionship
We ound s ong suppo o he powe unc ion SAR a small‐g ain
sizes in con inuous ege a ion using mo e han 2,000 nes ed‐plo s
o e la ge ecological, geog aphical and di e si y g adien s o h ee
majo axa and when ocusing on he comple e ege a ion. Using
AICc and R2 as measu es, he ‘no mal’ powe unc ion was on a ‐
e age he bes model. Using BIC, he b eakpoin powe unc ion
p e ailed, and he quad a ic powe unc ion had a simila le el o
suppo o he no mal powe unc ion. This di e ence is no as on‐
ishing as BIC penalizes complexi y o a unc ion di e en ly han
AICc, bu ac ually less s ongly o small sample sizes, which migh
lead o o e i ing. I basing he conclusions on BIC, he e migh be
some scale dependence o he SAR, ha is, a mino change o he
exponen z wi h g ain size (see also C awley & Ha al, 2001). I all
h ee a ian s o he powe unc ion a e conside ed join ly, hei
p e alence as he bes model inc eased om c. 60%–90% based on
AICc o c. 90%–95% based on BIC. Wi h ou simula ion (Appendix
S4), we could u he demons a e ha his esul was no caused
by he non‐independence o he nes ed plo s, bu ha his sampling
app oach, i a all, migh e en sligh ly unde es ima e he supe io i y
o he no mal powe unc ion.
The gene al supe io i y o he powe unc ion was la gely
una ec ed by axonomic g oup, biome o ege a ion ype. This
inding is in line wi h p e ious egional s udies analysing small
subse s o he cu en da abase (Dengle , 2009; Dengle & Boch,
2008; F idley e al., 2006). Al hough we es ic ed ou compa ‐
isons o p agma ic easons o a smalle se o unc ions, which
s ill p o ides a good ep esen a ion o he o e all ange o pos‐
sible SARs, ou indings a e consis en wi h hose o T ian is e
al. (2012) and Ma hews e al. (2016) o ue islands and habi a
FIGURE 3 Di e ences in model pe o mance o he powe unc ion exp essed as AICc weigh s be ween he wo undamen al ways o
eco ding plan s, oo ed p esence and shoo p esence. ‘Roo ed p esence’ coun s species in he poin whe e hey a e a ached o he g ound
i espec i e whe he hey ha e oo s in he ana omic sense o no . The displayed alues a e o he S‐space ( esul s in log S‐space we e
consis en ) [Colou igu e can be iewed a wileyonlinelib a y.com]
FIGURE 4 Di e ences in model
pe o mance o he powe unc ion
exp essed as AICc weigh s be ween
sampling schemes whe e smalle g ain
sizes we e eplica ed and hei ichness
a e aged and cases wi h only one
subplo pe g ain size (non‐a e aged).
The displayed alues a e o he S‐space
( esul s in log S‐space we e consis en )
[Colou igu e can be iewed a
wileyonlinelib a y.com]
80
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DENGLER E aL.
islands, despi e he e y di e en s udy sys ems and scales.
This sugges s ha , in spi e o he commonly accep ed no ion
ha con as ing ac o s in luence species di e si y a di e en
spa ial scales (B own & Pee , 2003; Field e al., 2009; Shmida &
Wilson, 1985; Sie e e al., 2012), he esul ing SARs a e as on‐
ishingly simila o e many o de s o magni ude (see also Wilson
e al., 2012) and ac oss axa and ecological condi ions. Al hough
he powe unc ion has been epea edly c i icized (e.g. Pan,
Zhang, Wang, & Zhu, 2016; S iles & Scheine , 2007), ou s udy
suppo s he idea ha i is indeed he one model ha , a leas o
plan communi ies, can be uni e sally applied (no e ha e en in
hose cases whe e i was no he bes model, i pe o med e y
well; see Figu e S8.7). In con as , o he models a e sui able, a
bes , in only a ew speci ic cases.
This poses he ques ion o why a single unc ion (bu wi h
a ying pa ame e s, see nex subsec ion) can be sui able ac oss
so many di e en si ua ions. In ac , powe law SAR‐like ela ion‐
ships a e a om es ic ed o species di e si y e sus a ea, bu
can likewise be ound in o he na u al phenomena, such as species
equency e sus body size, o body size e sus a ea (Sou hwood,
May, & Sugiha a, 2006), o e en in comple ely di e en ealms o
science and e e yday li e (Nekola & B own, 2007; bu see S ump
& Po e , 2012, o a c i ical iew). A gene al inding om hese
di e en disciplines is ha powe unc ions mos o en esul om
non‐equilib ium condi ions (Mi zenmache , 2012) o skewed unde ‐
lying dis ibu ions (e.g. Rosindell & Co nell, 2007). Powe law ela‐
ionships a e likely he consequence o complex dynamical sys ems,
no necessa ily o speci ic ecological mechanis ic p ocesses (Nekola
& B own, 2007), e en i he slopes o he powe law SARs migh
well be e ec ed by such p ocesses. In his espec , i is in e es ing
o compa e he ela i e pe o mance o he powe unc ion ac oss
axonomic g oups. Pe o mance was highes o all species g oups
combined, ollowed by ascula plan s, b yophy es and lichens,
which co esponds o he mean species ichness o each g oup.
Mo eo e , in ascula plan s ( he g oups wi h he bigges da ase ),
we ound a s ong inc ease in he supe io i y o he powe unc‐
ion wi h he ichness in he bigges plo . I seems ha he mo e
elemen s (he e: species) wi h sligh ly di e en p ope ies (e.g. e‐
quencies, habi a p e e ences, sizes) a e in ol ed, he mo e closely
powe unc ions a e app oached.
In addi ion, ou indings sugges ha he e is likely no sa u a‐
ion in SARs in con inuous ege a ion as ou sa u a ion unc ion
(Michaelis–Men en) pe o med on a e age much wo se han he
unc ions wi hou sa u a ion ac oss a wide a ay o di e en eco‐
logical condi ions. We belie e ha e en he ew indi idual da ase s
whe e he Michaelis–Men en unc ion appea ed o be supe io a e
likely a e ac s o insu icien eplica ion a he smalle g ain sizes.
As Dengle and Boch (2008) ha e shown, he ela i e pe o mance
o he powe unc ion e sus sa u a ion unc ions imp o es when
he eplica ion o smalle subplo s is inc eased and hus he cal‐
cula ed a e age ichness is close o he ue mean ichness. This
is in line wi h ou inding o bes i s o he Michaelis–Men en
unc ion o b yophy es and lichens in S‐space. As hese g oups
o en ha e ew species in g asslands, in many cases none o he
smalle subplo s (ac oss se e al g ain sizes) con ained any species,
esul ing in a eco ded ichness o 0, despi e he ac ha he ue
a e age mus be highe and inc ease wi h g ain size (see Me hods).
We hus ecommend ha he concep o ‘minimal a ea’ (which
only has a meaning i sa u a ion exis s), ha has been p esen ed in
nume ous ex books o ege a ion science (e.g. Ba bou , Bu k, &
Pi s, 1999; Ken , 2012; Muelle ‐Dombois & Ellenbe g, 1974) o
o e a cen u y, should be comple ely abandoned, as has al eady
FIGURE 5 Compa ison o model pe o mance o he powe unc ion exp essed as AICc weigh s be ween he six biomes ep esen ed in
he s udy. The displayed alues a e o he S‐space ( esul s in log S‐space we e consis en ) [Colou igu e can be iewed a wileyonlinelib a y.
com]
