Jordi Bascomptea,*,Bartolo Luqueb,Joseolarreab,Lucaslacasab

Aprobabilistic modelofreservedesign

Jordi Bascomptea,*,Bartolo Luqueb,JoseOlarreab,LucasLacasab

aIntegrativeEcologyGroup,Estacio´nBiolo´gicadeDon˜ana,CSIC Apdo.1056,E-41080Sevilla,Spain

bDepartamentodeMatema´ticaAplicadayEstadı´stica ETSI Aerona´uticos,UniversidadPolite´cnicadeMadrid Plaza Cardenal Cisneros3, Madrid28040,Spain

Abstract

Wedevelopaprobabilistic approachtooptimum reservedesignbasedonthespecies–arearelationship. Specifically,wefocusonthe distributionofareasamong asetofreservesmaximizingbiodiversity. Webeginbypresenting analyticsolutions fortheneutral casein whichallspecieshavethesamecolonization probability.Theoptimum sizedistributionisdetermined bythelocal-to-regionalspecies richness ratio k. There isa critical kt ratio defined by the number of reservesraised to the scaling exponent of the species–area relationship. Belowkt,auniform area distributionacross reservesmaximizes biodiversity. Beyond kt,biodiversity ismaximized by allocating acertain areatoonereserveanduniformly allocating theremaining areatotheother reserves.Weproceedbynumerically exploringtherobustness ofouranalyticresultswhendeparting fromtheneutralassumptionofidenticalcolonization probabilities across species.

r2007ElsevierLtd.Allrightsreserved.

Keywords:Marine reserves;Islandbiogeography; Species–area;Power-laws;Nestedness; Colonization;Extinction

1. Introduction

Thetheory ofislandbiogeography predicts thenumber ofspeciesinanisland asabalance betweencolonization andextinction events(MacArthurandWilson,1967).The number ofspeciess(hereafter biodiversity) ofanislandof areaA canbedescribedbythefollowingpower-law relationship:

s¼cAz, (1)

wherecisafittedconstant andthescalingexponent zhas valuesintherange0.2–0.4 (Williamson,1988). Several explanations fortheabovespecies–arearelationship have beenproposed, includingspecies abundancedistributions (May,1975),populationdynamics(Hubbell,2001),andthe interplaybetweenaskewedspeciesabundancedistribution and intraspecificspatialaggregation(GarcıaMartınand Goldenfeld, 2006).The smallrange ofempirical z-values has recently been derived from the specific form of the

*Correspondingauthor.

E-mailaddress:(J.Bascompte).

canonicallognormal species abundancedistribution (Southwoodetal.,2006), whichservedtounifythe species–arearelationship withtwootherpowerlaws in ecology:speciesfrequency versusspecieslength, and maximalbodysizeversusarea(Southwoodetal.,2006).

Thetheoryofislandbiogeography hasbeenusedto generate simple rules of thumb in conservationbiology. Oneclassicalexampleistheproblem ofchoosing between one largeor two smallreserves.Higgsand Usher (1980) used the species–area relationship and elegantly showed thattheanswerdependsonthespeciesoverlap,thatis,the fraction of common species contained in both smaller reserves.Thus,itis bettertohavetworeservesforlow overlaps,whereasonereservemaximizesbiodiversityifthe overlapislargerthan aspecificthreshold.

Herewe extendtheoneversustworeservesapproach (HiggsandUsher, 1980)forthecaseofmultiplereserves. Given asetofrreserves,weaskthefollowingquestions: (i) what is the size distribution among these reserves that maximizes biodiversity? and (ii)how doesthis solu- tion depend on the total protected area and regional diversity?

Ouranalytic approximationassumesneutrality. MacArthurand Wilson (1967)assumed that all species areequivalent inthesenseofhaving thesameextinction andcolonization rates(see alsoHubbell, 2001foran importantgeneralization attheindividual level).However, research in island biogeographysincethe decade of the

1980shasunequivocally shownthat speciesaredistributed non-randomlyacrossreserves.Specifically,duetodifferent colonization (and/or extinction) rates, some species are more widespread than others. The observed pattern is nested, in which species inhabiting small reserves form perfectsubsetsofthespecies inhabiting largerreserves (Darlington,1957;Patterson,1987;Atmar and Patterson,

1993;Cook and Quinn, 1998;Fischer and Lindenmayer,

2002).Toassesstowhatextendthesenon-randompatterns ofspeciesdistributionaffectouranalyticresults,weendup byanalyzing numerically an extension ofour model. We thusask:(iii)howrobustareouranalyticresultswhennon- neutral, species-specific colonization rates are incorpo- rated?Ouranalyticalapproach differsfromalternative approaches inreservedesignsuchassite-selectionalgo- rithms(Nicholsonetal.,2006;CabezaandMoilanen, 2003; Arponen etal.,2005;Halpern etal.,2006;Wilsonetal.,

2006)thatanalyzerealsystemsandpredicttheoptimumset ofreservesgivensomefinitebudget.Ourpaperpresentsan idealizedsystemthat, although necessarilysimplistic,itis abletopredict general, robust rulesofthumb based ona few ubiquitousgenerallawssuchasthespecies–area relationship.

2. Maximizingbiodiversity:tworeserves

Let us start by illustrating the case of two reserves. AlthoughthisreproducesHiggsandUsher(1980),itwillbe important forourgeneralization torreservesinthenext section.HiggsandUsher(1980) assumedafixedarea distributionbetweenboth reservesandderivedthecritical speciesoverlapdictatingwhetheritis bettertohavealarge

Inarealisticscenariotherearespecies withhigh colonization rates(theseoneswill likelyappear inboth reserves),and specieswithlowcolonization rates (wewill hardlysee anyofthese).Letusassumethefollowing probabilitydistributionofreservecolonization acrossthen speciesinthepool:

PðxÞ/x—g, (4)

withx¼1;2;...;n.

Noticethattheaboveprobability distributionwould produceanestedpattern asfoundin islandbiogeography (Darlington,1957;Patterson,1987;Atmar and Patterson,

1993). For example, only the species with the highest

colonization probabilitywould be found in the far dis- tant reserve, while this and the other species would be found in the closest reserve. That is, species in remote reservesformwell-definedsubsetsofthespecies foundin closereserves.

To be able to derive analytical results, we start by assumingthateveryspecieshasthesamecolonization rate. This corresponds to the limiting case g¼0, that is, a uniform colonization probabilitydistribution.Thisneutral scenariowillprovidetheminimumoverlapbetweenspecies inthe two reserves. In the last section wewillrelax this neutral assumption.

Let’s take a number s1 of different species randomly fromthenspeciespooltooccupythefirstreserve.For the second reserve we must choose randomly s2 different speciesfrom thepool.Wecannowimaginethat thepool hasbeendividedintwourns:thefirstwiths1 speciesand thesecond withn—s1 different species.Wewillcompute theprobabilityqmthat,aftertakings2random species,mof themwereactually present inthefirsturn. qm isthus the probability ofhavinganoverlapofmcommonspecies betweenthetworeserves.

The s2 speciesgroup willbe constituted by mspecies fromtheurnwiths1 speciesands2—mfromtheurnwith n—s1 species.Thereare s1 different, evenpossibilitiesof

reserveortwosmallones.Ourapproachinhereisslightly

choosing

mspeciesfrom thefirsturn. Similarly,thereare

different:weassumethatwe havetworeserves(rinthe followingsection) and areableto tune thearea distribu- tion.That is,havinginmindthat thetotal areaAsatisfies A¼A1þA2, we can determine to our convenience p

satisfyingA1¼pAandA2¼ð1—pÞA.Letusassumethat

n—s1 different,evenwaysofchoosings mspeciesfrom

s2—m

thesecondurn.Havinginmindthatevery choiceis independent, that weassume a uniform probability distributionofcolonization,and that thetotal number of

nistheregionalnumberofspecies(i.e.,thetotalnumberof

choicesisn

2

,the probability qm ofhaving mcommon

speciesinthenearby continent). Eachoneofthesespecies

hasaprobability ofcolonizing anyoftheabove reserves. Thenumber ofspeciess1 inreserve1willbe:

speciesisgivenbythehypergeometric distribution:

s1 n—s1

s1¼cAz¼cAzpz,(2)

andsimilarly,thesecondreservewillhosts2 speciesgiven

qm¼

ms m

n ,(5)

s2

by

s2¼cAz¼cAzð1—pÞz.(3) The problem is then to calculate the value of p

maximizing biodiversity, i.e.,the total number ofspecies

where,ifs2Xs1,m¼0;1;2;...;s1;andifs2ps1,

m¼0;1;2;...;s2.

Themeanspeciesoverlapbetweenbothreservesis deter-

minedbythemeanofthehypergeometric distribution:

s1s2

inboth reserves.

hqi¼

,(6)

n

1.6

1.5

1.4

1.3

1.2

1.1

1

0.50.60.70.80.91

p

1.02

1.01

1

0.99

0.98

k=kc=0.94655

0.50.60.70.80.91

p

Fig.1. RelativebiodiversityBðp;kÞversusrelativereservesizepbetweentworeservesfordifferentvaluesofthelocal-to-regionalspeciesrichnessratiok: (a)k¼0:1;0:2;...;0:9fromtop tobottom; and (b)k¼0:91;0:93;0:95;0:96;0:97,and0:98fromtop tobottom.Dots represent numerical simulations (averageover100realizations, wheretheregionalpoolisn¼10000species),andlinesdepictthetheoretical equation (9).(a)Bðp;kÞ41Indicatingthatitis alwaysbetter to choose tworeservesto maximizebiodiversity. (b)When k4kc 0:947,choosing oneor tworeserveswilldepend on p:forlowto moderate valuesofp,Bðp;kÞo1indicating that thebestoption isnowchoosingonlyonereserve.Notealsothat forvaluesofkbelowkt 0:862,two identical reserves(p¼0:5)givesthemaximum biodiversity forallk,but beyond thisthreshold, p¼0:5changes from amaximum toaminimum of biodiversity. Thissituation canbeeasilyunderstoodbylookingatFig.3;z¼0:3.

Weareinterestedinmaximizingbiodiversity. Therefore, weneed to maximize the following function (Higgs and Usher, 1980):

s1s2

1

0.99

Fðp;s;nÞ¼s1þs2—hqi ¼s1þs2—

.(7)

n

0.98

ONLYONE RESERVE

Takingintoaccount thespecies–arearelationship (1,2,3), biodiversity isgivenby

2

0.97

MAXIMIZES BIODIVERSITY

Fðp;s;nÞ¼s½pz þð1—pÞz]—s

pzð1—pÞz.(8)

0.96

Letusdefinetheratiok¼s=n,whereoncemoresisthe number ofspeciessupportedbyasinglereserveoftotal area A(1),and nistheregional speciespool. kisthus a local-to-regionalspecies richnessratio;smallk-values indicaterichcontinents, diversetaxons,and/or asmall protected area.IfwenowdivideEq.(8)bys,we candefine

0.95

0.94

0.93

TWO RESERVES MAXIMIZES BIODIVERSITY

anindexofrelativebiodiversity Bðp;kÞ:

0.50.60.70.80.91

Fðp;kÞz

zzzp

Bðp;kÞ

s¼p

þð1—pÞ

—kpð1—pÞ,(9)

Fig. 2. TheisoclineBðp;kÞ¼1inthespacep—kseparates theregions

Thesolution Bðp;kÞ¼1definesacriticallineinsucha

way that for Bðp;kÞ41, having two small reserves maximizesbiodiversity, whereasifBðp;kÞo1, having only one reserve isthe best option. Note that, aslong asthe speciespoolnislargerthans,0ok¼s=np1soasafactof symmetry,weonlyhavetoconsiderthesituation0:5ppp1. Thebehavior ofBðp;kÞforseveralvaluesofkisplotted

inFig.1.Hereafter weassumewithout lackofgenerality z¼0:3.Note that for values of k between 0:1and 0:9 (Fig. 1a),therelative biodiversity Bðp;kÞ isalwayslarger than1. Thismeansthatregardlessofthereservesize distributionp,itis alwaysbettertohavetwosmallreserves than abigone.

Abovesomecriticalvaluekc¼0:94655...,choosingone ortworeservesdependsstronglyonthesizedistributionp

wheretheoptimal choiceinorder tomaximizebiodiversity iseitherone reserveortworeserves.

(see Fig. 1b). For low p-values, one reserve is better ðBðp;kÞo1Þ,butafteralargeenoughp-value,tworeserves maximize biodiversity as before ðBðp;kÞ41Þ. kc can be derivedeasilybysolvingBðp;kÞjp¼1=2 ¼1.

Theabove results aresummarized inFig. 2,wherethe

isocline Bðp;kÞ¼1isplotted in the space p—k. Points ðp;kÞ belowthecriticallineindicatesituations inwhichtwo reservesmaximizebiodiversity.

k doesnotonlydeterminewhetheroneortworeserves maximize biodiversity through the critical kc value explored above. Within thedomain oftworeserves,there

is another criticalkvalue(kt)thatdeterminestheoptimum sizeallocation between thetwo reserves.Note inFig. 1a that foreveryvalueofkp0:8,relativebiodiversityreaches itsmaximum whenp¼0:5,that is,fortworeservesofthe samesize.However,forkX0:91,p¼0:5 stillrepresentsan extremaof thebiodiversityindex,buthaschangedfrom maximumtominimum(Fig.1b). Themaximumrelative biodiversity isnow associated to higher values ofp.All theseconclusionscanbederivedindetailfromtheextrema analysis of Bðp;kÞ. In order to find directional extrema

ðp;kÞ* of Bðp;kÞ, we fixk. This converts Bðp;kÞ into a

parametricfunction ofk,sayBk.Wethensolve:

reservesofthesamesize) maximizesthefunction. This extrema turns into a minimum above kt, and a new maximumappearswithp40:5 (favoringanasymmetric distributionofreserves).

3. Generalization torreserves

Theproblem canbegeneralizedfrom tworeservestoa genericnumber r.Theargument isasfollows:

First, suppose again that we can determine the area distribution of the reserves, so that the area of the ith

reserve willbe Ai¼pA (with A¼Pm

qB p

qp¼0.(10)

i

eachreservei,wehave

i¼1piA). Then, for

si¼cAzpz¼spz,(13)

Afirstsolution ofthisequation isp¼0:58k.Now we tacklethesecondderivative,whichgivesinformationboth

i

whereagain,s

i

¼cAz.

on the function’s convexity and on the nature of the extrema. Now wecanevaluate forwhichvalueofk¼kt, the size allocation p¼0:5 changes from maximum to minimum. That is

q2Bk

If we have only one reserve, the function that we

should maximize willbe, trivially, the constant function F1¼s1.Wehavejustseenthatinthecaseoftworeserves, wecoulddividethepoolintwourns, onewiths1 species and the other one with n—s1. This fact leads us to maximize the function F2¼Fðp;s;nÞ¼s1þs2—ðs1s2=nÞ

p¼1=2

¼0.(11)

giving the total number of different species in both

reserves.

Thesolution tothisequation is

kt ¼ð1—zÞ2z, (12)

thatinourcase(z¼0:3)iskt ’0:862.Thisisthethreshold that distinguishes the domain where p¼0:5 represents eitheramaximum oraminimumofbiodiversity. InFig.3 werepresent theextrema ðp;kÞ* ofBðp;kÞ.Wecanclearly

Inthecaseofthreereserves,wecanrepeattheprocessof dividingthepoolintwourns:nowthefirsturnwillcontain F2differentspeciesandtheotheronen—F2.Reasoning as before, we would obtain a new function F3¼F2þ s3—ðF2s3=nÞ. We can generalize for r reserves through thefollowingrecurrence equation:

observe the extrema bifurcation: under kt, p¼0:5(two

Fr ¼Fr

1þs

Fr—1sr

—

.(14)

n

0.75

Itiseasytodemonstratebyinduction that

(rsi )

0.7

Fr ¼n 1—Y1

i¼1n

.(15)

0.65

Using the species–area relationship (1),defining again k¼s=n and dividing it by s, we find a generalized expressionfortherelativebiodiversity:

Fr1(r)

0.6

Brðfpig;kÞ

s ¼k 1—

i¼1

1—kpi

.(16)

0.55

0.5

Thus,theproblemnowbecomesasearchof thearea distribution fpig that maximizes Brðfpig;kÞ. This corre- sponds tominimizingthefollowingfunction:

r

GrðfxigÞ¼Y1—xzÞ,(17)

0.850.8550.860.8650.870.8750.88

ði

i¼1

k

Fig. 3. Extrema ðk;pÞ oftherelative biodiversity function Bðp;kÞ. Note thatatthethreshold kt 0:862anextremabifurcation takesplace.Below thisthreshold, p¼0:5isamaximumofBðp;kÞ.Aboveit,p¼0:5converts into aminimum ofBðp;kÞ, and anewmaximum ofBðp;kÞ appears for p40:5.Thismaximum stronglydependsonp.

fori¼1;...;r,wherewehavedefinedthevariablesxi such that xi ¼k1=zp.

WeusetheLagrange multipliersmethodtoperform this task. As long as the logarithmic operator is a mono- tonically increasing function, the minimum of Gr will coincide with the minimum of logðGrÞ. Applying this

transformation:

r

logGrðfxigÞ¼Xlogð1—xzÞ,(18)

i¼

willbethefunction tominimize.Note that

and jH2j40. Solving this set of inequalities, we find againtheexpectedsolution kokt ¼ð1—zÞ2z.

Inthegeneralcaser42,weproceedasfollows:

Thefirstcondition ishiio0,whichissatisfiedtrivially8r. The second condition is that the determinant of the Hessian changes from positive to negative at somevalue

rrr

.That is,weneedto find k

that satisfiesjHj¼0.To

Xxi ¼Xk1=zp

¼k1=zXp

¼k1=z,(19)kttr

i¼1

i

i¼1

i

i¼1

solvethedeterminantofanr-order matrix isingenerala

tough problem. However, due to the fact that the

sothat wecanwritetheLagrangianassociated to(18)as

r"r#

determinant is an algebraic invariant, we just have to diagonalize the Hessian, and ask when any eigenvalue

L¼Xlogð1—xzÞ—lXxi—k1=z

.(20)

becomes null. As a fact of symmetry, we find that the

i¼1

i¼1

Hessianhasthefollowingshape:

Solvingthesystemandundoingthechangestoxi weget01

z—1

l¼kpz—1

¼ ¼

z—1

i

kpz—1

¼ ¼

z—1

r

kpz—1

.(21)

BC BC

Hr ¼BC

Atrivialsolution istheuniform distribution:

1

p1¼ ¼pi¼ ¼pr ¼r.(22)

Asecondsolution is

p1¼p,

1—p

BC

B babC

b ba

whichisacirculant matrix rxrwithreigenvalues:

l1 ¼a—b withmultiplicity sðl1Þ¼r—1,

l2 ¼aþðj—1Þb withmultiplicity sðl2Þ¼1.

pi¼r

;i¼2;...;r.ð23Þ

—

Hence, jHrj¼0provides two solutions depending on

Notethatwhenr¼2wegetourpreviousresults.Infact,

ifwefixr¼2in(21),weget Eq. (10)as expected (the solution ofLagrange multipliers givesustheextrema).

Again, if we set r42, we have that the uniform distribution(22)actsasamaximum until acriticalvalue

whether a¼bora¼ð1—rÞb.

The first possibility givesus a mathematical solution with kt41, which has no physical meaning. The second possibilitygivesustherelation:

rzðz—1Þ

kt isreached, from which it acts as a minimum, letting

distribution(23)actasthemaximum.

kt ¼

zð2—rÞ—1

,(27)

From nowonwewillfocusontheuniform case,where (22) maximizesbiodiversity.StartingfromEq.(16) and assuming auniform distribution(22)ofreservesizes,the relativebiodiversity willbe

1 k r

whichisongoodagreement withthecaser¼2andisthe generalsolution oftheproblem. Wecanconclude that in the case of r reserves, the size distribution p¼1=r maximizes biodiversity as long as the local-to-regional

speciesrichnessratio kislowerthan thecriticalvaluekt.

Brðp;kÞ¼k 1— 1—rz

.(24)

Beyondthisthreshold andasafactofconsistency,thesize distribution that will maximize biodiversity will be the

As in the case of r¼2, we have to check that this

distributionmaximizesbiodiversityuntilsomethreshold kt (thatis,thatthisdistribution,beinganextremaofBrðp;kÞ, changesfrommaximumtominimum).Forthis,wehaveto solve Hr ¼ðhijÞrxr, the Hessian of Brðp;kÞ, fixing k and assumingp¼1=r.Thus,thediagonal termsoftheHessian willbe

other extremefound in(23).

4. Relaxingtheneutralassumption

Up to here wehave assumed neutrality, i.e., that all species have the same colonization probability. This allowed analytic tractability.In order to seehow robust

zðz—1Þ

1 r—1

previous results areinthe faceofrelaxing neutrality, we

hii¼

rz—21—rz

a,(25)

willnowpresentnumericalresultsforthegeneralcasewith amorerealisticcolonization probability distribution.

andthenon-diagonalterms

Finding an analytical expression of the distribution

—z2k

hij¼

1 r—2

b.(26)

overlap similartoEq. (5)isadifficult problem whenthe colonization probability distributionisnolongeruniform,

r2z—21—rz

Notethatwhenwesetr¼2,theconditions underwhich

p¼1=rrepresents amaximum ofbiodiversity are h11o0

butapowerlaw(Eq.(4)).However,weareonlyinterested

inthemeanofthatdistribution,i.e.,themeanoverlap.We canassume,forafixedk,thefollowingansatzforthemean

1.03
1.025
1.02
1.015
1.01
1.005 / ecology(Southwoodetal.,2006), namelyspeciesfrequency versusspecieslength,andmaximumbodysizeversusarea. Here we add to this work by showing yet another relationship ofthe species–area exponent z.Interestingly
enough, thecriticalvaluekt separating thetwooptimum
reserve size allocation is determined by the number of
reservesraised to the power-law exponent ofthe specie-
s–arearelationship (seeEq.(27)).Thisconnection between
identicalvariablessetsupthepossibilityofextendingsome
ofthe current findings inthe context ofother ecological
laws.For example, the commonly observed value ofthe
exponent zisrelated totheunderlying lognormal species
abundancedistribution (Southwoodet al., 2006;Garcıa
Martın andGoldenfeld, 2006),andthusonecouldexplore
howspeciesabundancedistributionsmayaffectoptimum
1
0.5 / 0.6 / 0.7 / 0.8 / 0.9 / 1 / reserve design. Exponent zalso depends on habitat and
scale(Garcıa Martın andGoldenfeld, 2006),sodespitethe
P / spatially implicit assumptions ofour model, such details

Fig.4.SimilartoFig.1 butforrreservesanddifferentcolonization probability distributionsdescribed by values of gin Eq. (4). Squares represent Monte Carlo simulations and linesrepresent the ansatz (28). k¼0:9,and from top to bottom,g¼0(correspondingto the uniform probability distribution),0.1,0.25,0.5,and0.75.Asnoted,departing from neutrality ðg¼0Þdoesnotaffectlargelytheanalyticsolution.

ofthat distribution:

s1s2g

couldbeincorporatedthrough z.

Ouranalyticalsolutionsdependonlyon the underlying species–area relationship,which although seemsto be a good descriptor of real distributions if: (i) individuals clusterinspaceand(ii)ifabundancedistributionissimilar toPreston’slognormal, itisindependent onspecificdetails oftheseproperties (Garcıa Martın andGoldenfeld, 2006). This suggests that our approachisalso independenton details.

hqi¼

PðgÞ2— ,(28)

Thenumerical solutions intheprevious sectionallowus

torelaxtheneutrality assumption.Ouranalytic resultsare

wherePðgÞisapolynomial whosecoefficientswillhaveto

beestimatedthrough fitting.InFig.4we comparesome numericalresultswiththisansatzforthecasek¼0:9.Note thattheagreementis quitegood.Wefindasthebestfitting forPðgÞasecondorderpolynomial ofthefollowingshape: PðgÞ 1:0þ0:7gþ0:41g2. Unfortunately, we have not found a general simple ansatz so that this polynomial mustbefittedforeachvalueofk.

Thenumerical results shown inFig. 4clearlyillustrate thatforvaluesofgo1,thespecies-specific colonization probabilities reducerelativebiodiversity bylessthan 3%.

5. Discussion

Wehave developed a probabilistic framework to optimum reservedesign.Itdictatestheoptimum size allocation amongasetofrreserves.Wehavefoundthat a simple variable k depending on the area allocated to reservesand the regional speciesrichness isakey determinantofthebestsizedistribution.Forhighregional species richnessandlowreserveareas,auniformarea distributionmaximizesbiodiversity. For lowregional species richnessandhighreserveareas,theoptimum size allocation consists of allocating a certain area to one reserve and uniformly distributing the remaining area among theremaining reserves.

Recentresearchhaslinkedthe species–arearelationship with two other independently derived power laws in

robustformoderatedepartures fromneutrality. Thisimplies thatspecific complexitiesinthecolonization ratesacross species wouldprobably affectonlyquantitatively butnot qualitativelyouranalyticresults.Thissuggest thevalueof simple,yet generalanalyticpredictions,whichdespitetheir simplicitycanbeusedtoprovidegeneralrulesofthumb.

Acknowledgments

This work was funded by the European Heads of Research Councils, the European Science Foundation, and the EC Sixth Framework Programme through a EURYI(EuropeanYoung Investigator) Award (to J.B.), bytheSpanish Ministry ofEducationandScience(Grant REN2003-04774 to J.B., and Grant FIS2006-08607 to B.L.-L.L.), and by Universidad Politecnica de Madrid (Grant UPM AY05/10921toJ.O.-B.L.).

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