255
SEGREGATION AND SYMMETRY IN TWO-SPECIES
POPULATIONS AS STUDIED BY NEAREST-NEIGHBOUR
RELATIONSHIPS
BY E. C. PIELOU
WestSummerland,
BritishColumbia
I. INTRODUCTION
Much workhas been done in recentyearson thespatialpatternsof naturalpopulations
of plants,and it has been customaryto studyone species only at a time. Suppose,
however,an investigator
wereconcernedwitha populationcontainingtwo co-dominant
species. Then, besides determining
the densityand spatial patternof each of the codominantspecies consideredseparatelyit would also be interesting
to know how the
individualsof one species were arrangedwithrespectto the individualsof the other
species.The two speciescould be describedas 'unsegregated'if any individualplant of
one of the species was as likelyto be foundgrowingclose to membersof the other
speciesas to membersof its own species.Conversely,
thetwo speciescould be described
as 'segregated'iftheyhad a tendency,
howeverslight,to occurin one-speciesclumps,so
thatany individualwas morelikelyto be foundnear membersof its own speciesthan
near membersof the other.
In studying
therelationship
betweentwo speciesgrowingin thesame area it has been
usual to samplethe area withquadratsand thendetermine
whetherthenumberofjoint
occurrencesof the two species in the same quadrat was significantly
greater(or less))
thanchanceexpectation;if so, thetwo speciesweresaid to be positively(or negatively)
associated. However, as Greig-Smith(1957, Ch. 4) has shown, the resultobtained
inevitablydependson the size of quadrat used. It is thusmeaninglessto assertthat a
pair of speciesis, say,positivelyassociated,unlessat thesame timeone specifiesat what
scale this positive association manifestsitself.Nevertheless,
the degreeto whichthe
individualsin a two-speciespopulationare, colloquiallyspeaking,'mingledtogether'is
clearlyan intrinsicpropertyof the whole population,and one independentof scale.
it shouldbe possibleto measurethisproperty
Furthermore,
in sucha waythattheresult
obtainedis independentof any arbitrarycharacteristic
(such as quadrat size), of the
samplingmethod.It is forthispurposethatthe conceptof segregation,
as describedin
thepreviousparagraph,is proposed.The smallerthedegreeof 'mingled-togetherness'
of
a pair of species,the greaterwillbe theirdegreeof segregation.
A populationof two species,A and B, may exhibittwo entirelydifferent
typesof
pattern,as shownin Fig. 1.
One typeof pattern,the 'unsegregated'type,is shownin Figs. la and lb. In Fig. la
the two species are presentas two co-extensiverandom populations.In Fig. lb the
populationas a whole is clumped,but withineveryclumpthe individualsof speciesA
and the individualsof speciesB are presentin the same proportionsand are arranged
of each other.
independently
Figs. lc, 1d, le and If show examplesof the second,or 'segregated',typeof pattern.
and symmetry
Segregation
256
In Fig. 1c a fullysegregatedpopulationis shown.The plants are aggregated,each
clump containsindividualsof one species only,and the clumpsare entirelyseparate
fromone another.
Figs. Id, le and If show threeexamplesof partialsegregation.Fig. Id shows two
clumps of an aggregatedpopulation with one clump containingmore membersof
speciesA thanof speciesB and theothermoremembersof speciesB thanof speciesA.
In Fig. le a clump of A's and a clumpof B's partlyoverlapeach other.In Fig. If a
randompopulationof speciesA occupiesthesame area as an aggregatedpopulationof
speciesB.
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(c): Fullysegregated.
(a) and(b): Unsegregated.
populations
fortwo-species
Fig. 1. Sixpossiblepatterns
(d), (e) and(f): Partlysegregated.
It is easy to visualizeotherpatternscharacterizedby partialsegregation.To list all
themin thefieldis probably
conceivablepossibilitiesand devisemethodsof recognizing
perse is easilydetectedby an investigationof the
an impossibletask but segregation
relationshipsof the populationmembers.As will be shown below,
nearest-neighbour
one maycalculatetheexpectedproportion
assumingthetwo speciesto be unsegregated,
for
of individualsof speciesA havingotherA's as theirnearestneighbours;and similarly
the
Then if the two speciesare partlysegregated,
the threeotherpossiblerelationships.
numberof individualshavingnearestneighboursof the same speciesas themselveswill
exceedexpectation,whilethe numberof individualsof one specieshavinga memberof
is total,
theotheras nearestneighbourwillbe less thanexpected.Whenthesegregation
none of the individualsof speciesA will have an individualof speciesB as its nearest
neighbourand vice versa.
its possible
of segregation,
thedetectionand measurement
Beforeconsideringfurther
causes willbe discussed.
E. C. PIELOU
257
II. CAUSES OF SEGREGATION
Two speciesmaybe segregatedeitherbecausethemembersof one or bothofthemoccur
of thehabitat.
in familialclumps,or as a resultof heterogeneity
or fromthecrowdingofa
reproduction,
Familialclumpingmayresultfromvegetative
membersof
large numberof seedlingsaround theirparent.If, owingto competition,
the otherspeciesare excludedfromsuch clumps,the two specieswillbe segregated.
by threedistinctmechanisms:
of thehabitatcan cause segregation
Heterogeneity
1. Both speciesmaybe aggregatedbecause of spatialvariationin habitatfactors,but
the factorswhose patchinesscontrolsthe patternof one of the speciesmay be quite
the other.Or onlyone of the speciesmay be
unconnectedwiththe factorscontrolling
theotherbeingat random.Patternslikethose
aggregatedbecause of habitatvariability,
in Fig. Ic, le or l would thenresult.
factor,
2. If two aggregatedspeciesowed theirpatchinessto the same environmental
wouldoccur.For example,
thensegregation
butifthetwodiffered
in theirrequirements,
thedensitiesof bothmightdependon soil moisturebutiftheoptimaforthetwo species
differed,
the patchesin whichone species attaineda high densitywould,notcoincide
withthehighdensitypatchesof theother.It could thenbe said thateach specieshad its
own nichewithinthe area.
3. If two species had identicalrequirements
but one was able to toleratea greater
departurefromoptimumconditionsthan the other,then one would expect theless
tolerantspeciesto be confinedto smallpatchesof groundonly,whilethemoretolerant
specieswould be morewidespreadin its occurrence(as in Fig. If). As Skellam (1951)
growin a heterogeneous
has shown,iftwo annualspecieshavingthesame requirements
and
the otherin abilityto
in
reproductive
capacity
habitat,withone speciesexcelling
thenthe formerspecieswould tend to be confined
establishitselfagainstcompetition,
to the poorerhabitatpatchesand the latterto the betterpatches.If, on the contrary,
as in Fig. lb, one would have to
two aggregatedspecieswerecompletelyunsegregated
but also theirtolerances,
requirements,
conclude not only that theirenvironmental
wereidentical,and thattheywerecompeting,so to speak,on equal terms.
III. THE DETECTION AND MEASUREMENT OF SEGREGATION
randompopulations.Let
Suppose two speciesof plant,A and B, occur as co-extensive
so that
the proportionsof plantsbelongingto speciesA and B be a and b respectively,
a+b=
1.
and
Then: Pr {a plant chosen at random is of species A} = a
of
a
chosen
is
Pr {the nearestneighbourof randomly
speciesAI = a
plant
A
an
as itsnearest
{
is
an
and
has
A
a
Pr randomlychosenplant
Therefore,
} JPAA= a2
neighbour
LikewisePAB = PBA= ab and PBB= b2
refersto thespeciesoftherandomlychosenplant,and
wherethefirstletterin each suffix
the secondto the speciesof its nearestneighbour.
In two-speciespopulationsin whichthe speciesare not at random,however,it does
not necessarilyfollowthatPAB = PBA. In any populationof plants some individuals
maybe so placed thattheyare notthenearestneighbourof anyotherindividual;others
willserveas nearestneighboursto 1, 2, ..., 5 individuals.(From geometricalconsiderationsit is easyto see thatit is impossibleforanyindividualto serveas nearestneighbour
to more than fiveotherswhen,as always,the distancesare measuredfromcentreto
258
Segregationand symmetry
centreof the plantsconcerned.)Thus, as has been shownby Clark & Evans (1955), the
proportions,qr of individualsthat are nearestneighboursto r (r = 0, 1, ..., 5) other
5
individualsforma definite
distribution
and q, = 1. Now, in a two-speciespopulation
r=O
likethatdescribedabove,we have,as before,Pr { a treepickedat randomis an A } a
But unlesstheproportionof A's servingas nearestneighbourto r otherindividuals(of
fortheB's, it is no longer
eitherspecies)is aqr forall possiblevaluesof r, and similarly
truethatthe probabilityis a thatthe nearestneighbourof a randomlychosenplantis
an A. For supposethoseindividualseach servingas nearestneighbourto two or more
of species B, and converselythat those
other individuals,say, were predominantly
of species
servingas nearestneighboursto 0 or 1 otherindividualswerepredominantly
A. Then, denotingby a' the probabilitythata plantpickedat randomhas an A as its
nearestneighbour,and by b' theprobability
thatit has a B as its nearestneighbour,we
a' a and b' >b (a'+b' = 1)
have
So
ab' >a'b
PBA
PAB=
Suppose, then,we scrutinizeeverymemberof a two-speciespopulationcontaininga
totalof N plantsof both species.If we determine
the speciesof each individualand the
willfallintofour
relationships
speciesof its nearestneighbour,the N nearest-neighbour
categories:AA, AB, BA, BB, wherethefirstletterin each pair denotesthespeciesofthe
base treeand the second the speciesof its nearestneighbour.Denotingtheirobserved
frequencies
byfAA, fAB, fBA andfBB respectively
(Ef = N), thenumbersof A's and B's
in thepopulationare givenby:
Na = fAA+.fAB and Nb =fBA +fBB
whilethe numbersof timesA's and B's occur as neighboursare givenby:
Na =fAA+fBA
and Nb' ==fAB+fBB
It is now possibleto comparetheobservedand expectedproportionsof each of thefour
table thus:
typesof nearest-neighbour
relationship
by meansof a 2 x 2 contingency
Base plant
Totals
SpeciesA
SpeciesB
f SpeciesA
Nearestneighbour
SpeciesB
fAA
fA B
fBA
fBB
Naf
Nb'
N
Nb
Totals
Na
Applicationof the x2 testin the usual way (see, e.g. Fisher& Yates 1953) will reveal
are morefrequent,
and AB and BA
whetherAA and BB nearest-neighbour
relationships
in thepopulationbeingstudied,thanwould
lessfrequent
nearest-neighbour
relationships
be expectedifthetwospecieswereunsegregated.
Iftheobserveddeparturefromexpectation is significant
at the probabilitylevel chosen,the conclusionfollowsthat the two
speciesmaybe regardedas partlysegregatedfromeach other.
In applyingthistestto a naturalpopulationit is no morelaboriousto scrutinizeevery
In order
memberofthepopulationthanto selecta randomsampleofthemforscrutiny.
to selectsuch a randomsampleit would be necessaryto label each individualand then
findit again laterifit happenedto be one of the ones randomlychosen; so it is simpler
to recordthespeciesof each plantand of itsnearestneighbourin thefirstplace. Whicha tract
evermethodis used it is necessaryto defineas thepopulationbeinginvestigated
259
E. C. PIELOU
smallenoughforit to be feasibleto examineeverymemberof the two specieswithinit.
it seemsdesirableto have some
segregation
For populationsof two speciesexhibiting
measureof theirdegreeof segregation.Since,,if the two species are segregated,the
relationshipswill be less than the
observednumberof AB and BA nearest-neighbour
numberexpectedin an unsegregatedpopulationcontainingthe same numberof A's
a statisticS, suchthat
ofsegregation',
define,as 'coefficient
and,B's, one mayconveniently
relationships
1 Obseryednumberof AB and BA nearest-neighbour
relationships
Expectednumberof AB and BA nearest-neighbour
S
--
1 JAB+fBA
N(a'b+b'a)
population.S willequal unitywhenthereare
ClearlyS willbe zero in an unsegregated
relationships;for this to occur not only mustthe
no AB nor BA nearest-neighbour
far apart
theymustalso be sufficiently
clumpsof A's and of B's be non-overlapping;
(as in Fig. ic) forno memberof an A-clumpto have a memberof a B-clumpas its
nearestneighbour,and vice versa.
It is also possiblefor S to take negativevalues. Suppose a populationconsistedof
N widelyspaced groups,withan individualof speciesA at thecentreof each groupand
n individualsof species B around each centralA. The populationwould thenconsist
that withineach groupthe
of N A-individualsand nN B-individuals.Suppose further
B's wereso spaced thattheywere all nearerto the centralA than to each other.(As
remarkedabove, thisis onlypossibleifn<5.) We shouldthenhave:
fAB =N,5fBA
=
nN,fAA =fBB
=
0
a = b' = 1/(n+l) and b a' = n/(n+1)
Then
S = 1-(nI+ 1)2/(n2+1)
So
S attainsitsminimumvalue of -1 whenn = 1. So S =-1 willoccuronlywhenthe
numbersof A's and of B's in the populationare equal and the populationconsists
ofisolatedAB pairs.The totalrangeof S is therefore
from-1 to + 1. Negative
entirely
values of S would be expectedto occur when one of the specieswas parasiticon the
other,forinstance,or in a populationof breedinganimalsin whichtheA's weremales
and the B's females.
In testingforsignificance
the difference
betweentwo values of S obtainedfromtwo
different
comparingtwo 2 x 2 tables,one fromeach site.This may
sitesone is, in effect,
be done by combiningthe two tables into one 2 x 2 x 2 table and testingfor 'second
orderinteraction'by the methoddescribedby Snedecor(1948).
IV. SYMMETRICAL AND UNSYMMETRICAL POPULATIONS
It seemsof value to distinguish
betweenthosepopulationsin whichthe numberof AB
fromthe numberof BA
relationshipsdoes not differsignificantly
nearest-neighbour
and those in whichit does. It is convenientto call a
relationships,
nearest-neighbour
two-speciespopulation 'symmetrical'when fAB = fBA and 'unsymmetrical'when
fAB
JfBA.
exceedsfBA. Then it followsthatthe B's
Suppose,forexample,thatfAB significantly
occur proportionately
moreoftenas nearestneighboursthan do theA's, havingregard
the population;per contra,the A's are
to the proportionsof B's and A's constituting
moreisolated.This mayresultfromtwo distinctcauses.
predominantly
On the one hand, it is obvious that (measuringfromcentreto centre)a large plant
Segregationand symmetry
260
will have less chance than a small one of servingas nearestneighbourto any other
individualin thepopulation.So iftheindividualsof speciesA, say,are of greatermean
size than those of species B, the B's will functionas nearestneighboursmore often
thantheA's and we shallhavefAB>fBA.
evenwhentheindividualsof
On the otherhand,a populationmaybe unsymmetrical
both speciesare of negligiblesize (i.e. whenit may be assumedthattheindividualsare
not competingwithone anotherforgroundspace). Then,iftheirpatternis like thatin
Fig. If, withthe A's widelyspaced and the B's in compactclumps,it will frequently
neighboura memberof the nearestB-clump,
happenthatan A willhave as its -nearest
whilethisB will have as its nearestneighbouranothermemberof the same clump.It
will then be foundthatfAB>fBA. In this case, as in that describedin the preceding
paragraph,themeanarea of groundper individualis again greaterfortheA's thanfor
the B's. But in thiscase, althoughthe mean area availableto each plantis greaterfor
theA's thanfortheB's it is assumedthattherootsystemof each plantdoes notoccupy
the wholeof the area availableto it.
is to count
The quickestwayto determine
whetheror not a populationis symmetrical
relationshipas describedabove, and testforsignithe fourtypesof nearest-neighbour
ficancethe difference
betweenfABandfBA. However,in a populationin whichthe two
greatly
or in whichthedensityof one of thespeciesdiffers
speciesare highlysegregated,
fromthat of the other,thesenumbersmay be small. A more laborioustest,but one
is to examineeach memberof the populationand recordits
usingmoreinformation,
speciesand the number(whichmustbe 0, 1, ..., 5) of otherindividualsof both species
forwhichit functionsas nearestneighbour.
members,a single
From anypopulation,ifwe disregardthe speciesof its constituent
frequencytable may be obtained showingthe observedproportionsof individuals
This has
functioning
as nearestneighbourto 0, 1, ..., 5 otherindividualsrespectively.
randomone by
been done forthreenatural,aggregatedpopulationsand fora synthetic
consistsoftwospecies,
Clark & Evans(1955). If,now,thepopulationunderinvestigation
A and B, the observationsmay be sortedinto two sets of proportions,one set, say
of A's servingas nearestneighbourto 0, 1, ..., 5
a5, showingtheproportions
a%, al,
otherindividualsrespectively;
and the otherset fo, Pi, ... P5, showingthe proportions
in the sensedefinedabove
of B's so functioning.
Onlyif the populationis symmetrical
will the expectedproportionsof A's and B's servingas nearestneighboursto r other
individualsbe thesame forall valuesof r. So one maytestby x2 whetheror notthetwo
fromeach otherand if theydo the population
sets of proportionsdiffersignificantly
may be assumedto be unsymmetrical.
Applicationsof boththe above testswillbe illustratedin PartVII of thispaper.
...,
V. SEGREGATION AND SYMMETRY
It is clear thatunsymmetrical
ones, may be eithersegrepopulations,like symmetrical
To takean extreme
case, supposethetwospeciesin thepopulation
gatedor unsegregated.
consistedof large,widelyspaced trees(theA's) witha densegroundcoveramongthem
of smallherbs(the B's). As beforelet the numbersof A's and of B's in the population
be Na and Nb respectively
(a+b= 1). Owing to the large size of everyA-individual
that all inter-plantdistances are measured fromthe centresof
(and remembering
individuals),it is assumedthatnone servesas nearestneighbourto any otherindividual
of eitherspecies.
261
E. C. PIELOU
Nb
ThenfAA =fBA =,
fAB -Na,
fBB=
WhenceS= 0
Onlyifsomeofthetreesgrewin denseclumpsamongwhichno lowerstoryof B's could
establishitselfwould segregationoccur. This exampleis givento show that,although
one will usuallybe concernedwithpairs of specieswhichdo not differgreatlyin size,
the concept of segregation,as definedin termsof nearest-neighbour
relationships,
is applicablewithoutrestriction
whateverthe relativesizes of thetwo speciesconcerned
and howeverunsymmetrical
(as in the example)the populationmay be.
VI. SEGREGATION AND AGGREGATION
If, of two speciesgrowingin an area, one is aggregatedor clumped,and the otherat
random(as in Fig. if), thentheymustbe segregated.But ifbothspeciesare aggregated
it does not necessarilyfollowthattheywill be segregatedfromeach other;even when
the two speciesdifferfromeach otherin theirdegreeof aggregationit is stillpossible
if theyare segregated,
it does not followthat
forthemto be unsegregated.
Conversely,
in theirdegreeof aggregation.
theymustdiffer
00
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0*
0
0
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00
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0
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00
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00
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0
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0.0
0.*
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c
d
betweensegregation
and aggregation.
Fig. 2. Interrelationships
and equallyaggregated.
(a): Unsegregated
and unequallyaggregated.
(b): Unsegregated
and equallyaggregated.
(c): Segregated
and unequallyaggregated.
(d): Segregated
Suppose we measurethe amountof aggregationby the index a (Pielou 1959) where
a = CroD(D = Densityand co= mean of the squaresof distancesfromrandompoints
to theirnearestplant). Then, giventwo species A and B, havingindexes aA and ai
all fourof the followingconditionsare possible:
respectively,
(a) S = 0, aA= aB
(b) S = 0, aA:# aB
(c) S>0,
(d) S>0,
aA
=
aB
aA A aB
are shownin Fig. 2. Condition(a) willoccurwhen
Examples,labelledcorrespondingly,
of each
thedensitiesof the two speciesare equal and both are arrangedindependently
otherin commonclumps.Condition(b) willoccurwhenthepatternis likethatin (a) but
the densitiesof the two speciesare unequal. So long as the relativeproportionsof A's
0. But now aA = ITWADA
and B's are the same in everyclumpwe shall have S
A
262
Segregationand symmetry
aB because in thiscase DA #/DB. Thus althoughthe A-clumpsand the Bwithineveryclump(so that
are
and the two speciesunsegregated
co-extensive
clumps
valuesof a.
densitiesof thetwo speciesgiveriseto different
S = 0), the different
Conditions(c) and (d) willoccurwhentheclumpsof A's are distinctor partlydistinct
fromthe clumpsof B's. Obviouslythiscan happen regardlessof whetheraA = aB or
aA = aB. In Figs. 2c and 2d the extremecase whereS = 1 is shown.
TWcBDB=
VII. A POPULATION OF PINUS PONDEROSA AND
PSEUDOTSUGA MENZIESII
madeon it
(1) Descriptionof thepopulationand theobservations
may
of the way in whichthe conceptsof segregationand symmetry
As an illustration
betweentwo speciesgrowingin the same area, the
help to explaintheinterrelationships
resultsobtained from an actual population with two co-dominantspecies will be
described.
The populationinvestigatedconsistedof a square tract of 5625 sq. m withinan
extensivearea of open woodland of PinusponderosaLaws. and Pseudotsugamenziesii
(Mirb.) Franco (P. taxifolia (Poir.) Britton).No otherspeciesof treewas present.Pinus
ponderosa,which can endureextremedroughtbut whose seedlingsare intolerantof
shade (Haig, Davis & Weidman 1941), is a commontree at low altitudesin the dry
Okanagan valleyof BritishColumbia. In areas too dryforany othertreeto growit
and
oftenformspurestands.Pseudotsugamenziesiihas seedlingsthatare shorter-rooted
than those of Pinusponderosa(McMinn 1952) and cannot therefore
slower-growing
establishitselfat thelowest,driestsites.But,higherup thehillsides,whereprecipitation
forPseudolsuga
is greaterand wintersnow lies longer,the soil moistureis sufficient
menziesiiseedlingsto survive,and open woodlandof thesetwo speciesis found.At still
higheraltitudeswhereamplemoistureallowsthegrowthof denseforestsofP. menziesii
and otherspecies,the shade is too dense forPinusponderosa.Mixed woodland of P.
ponderosaand Pseudotsugamenziesiithus occupies a wide transitionzone betweenthe
aridvalleyfloor(at an altitudeof 343 m) and theluxuriantforestsof themountainsides.
300 m above thevalleyfloor.
The populationstudiedwas at an altitudeof approximately
The tractwithinwhichobservationsweremade contained228 treesin all, 160 indivithe
duals of P. menziesiiand 68 individualsof Pinusponderosa.Hereafter,forbrevity,
treeswill be referred
to simplyas Douglas firsand pines. The veryfew trees of less
than 2 m heightwereignored.In connectionwithobservationsto be describedbelow,
at a heightof 1-5 m fromthe ground,was
the trunkcircumference
in centimetres,
are shownin Fig. 3. For the Douglas
measuredforeverytree,and thesize distributions
firsthe trunkcircumferences
rangedfrom4 to 156 cm witha mean of 31-1cm; forthe
pines therangewas from6 to 155 cm witha mean of 42 0 cm.
It was obviouson inspectionthatthetreeswereaggregated;as wellas distinctclumps
of trees,therewerealso isolatedindividualsof both speciesin the spaces betweenthe
fromeach otherin theirdegrees
whetherthetwo speciesdiffered
clumps.To determine
a (Pielou 1959) was foundforeach. To
of aggregation,
the indexof non-randomness,
do this100randompointswereplacedin thearea bymeansof randomco-ordinates;the
co-ordinatesweretakenfroma randomnumberstable and the pointsthenlocated by
pacing, along a compass bearing,fromthe boundariesof the area. From each point
thedistancerD,to thecentreofthenearestDouglas fir,and thedistancerp,to thecentre
and denotingthe
of the nearestpine, were measured.Then, puttingco= (Xr2)/100,
E. C. PIELOU
263
numberofDouglas firsand thenumberofpinesperunitarea byDD and Dp respectively,
it was foundthat:
forthe Douglas firsaD = TCODDD = 4079
and forthepinesap = -rcipDp= 1P984.
When a sampleof 100 point-to-plant
distancesis measured,a value of a greaterthan
1 234 indicatessignificant
at the 1% level,in the directionof aggreganon-randomness,
tion (Pielou 1959). One maytherefore
assumethatboth specieswereaggregated.Comparingthe two values of a by means of a t-test,it was foundthat t = 2-976with 198
moreaggregated
degreesoffreedom.This impliesthattheDouglas firsweresignificantly
thanthe pines.
30.
Pines
?20.
a).
m10L
C:-
0-
06
08
10
12
14
16
Log10 (Circumference in cm)
18
2-0
2-2
Fig. 3. Sizedistribution
fortheDouglasfirsandpinesinthe
populationinvestigated.
Each treein the populationwas nextexaminedin turnand for each the following
factswererecorded:
(1) Its speciesand the speciesof its nearestneighbour,
(2) thenumberof othertrees(of bothspecies)forwhichit servedas nearestneighbour,
and
(3) its trunkcircumference
in centimetres
and thatof its nearestneighbour,at 1P5m
above theground,and the distance(centreto centre)betweenthem.
The data providedby (1) wereused to determine
whetheror not theDouglas firsand
the pinesweresegregated.From (2) it was possibleto testwhetherthe populationwas
or unsymmetrical
in the sensedefinedin PartIV.
symmetrical
The measurements
in (3) weremade in orderto ascertainwhetherthe spatialpattern
of the treeswas, at least in part,determinedby competitionamong them.As I have
shownin an earlierpaper(Pielou 1960)a populationmaybe aggregatedbecause clumps
of small individualsoccupy the gaps among isolated large individuals.When this
mechanismis operatingthe distancebetweenany individualand its nearestneighbour
mustbe positivelycorrelatedwiththe sumof some measureof the sizes(e.g. the sumof
thetrunkcircumferences)
of the two treesbetweenwhichdistancewas measured.
264
Segregationand symmetry
The resultsobtainedforthe threesetsof observationslistedabove willbe considered
in turn.
(2) The testfor segregation
As describedin Part III, one may testforsegregationbetweenthe Douglas firsand
the pinesby meansof a 2 x 2 table. The resultsare shownin Table 1. Whetherto use a
one-tailor a two-tailtestmustbe decidedin the lightof whatseemsreasonablein any
givencase. Thus,ifone takesas nullhypothesis
thatthepopulationis unsegregated
and
as alternative
hypothesisthatany segregationoccurringwillbe positive,a one-tailtest
is indicated.If, however,it is consideredthata populationis as likelyto be negatively
as positivelysegregated,
thena two-tailtestis needed.For the Douglas firsand pines
thereseemedno reasonto expectnegativesegregationso a one-tailtestwas used. The
value of x2 (withYates's correction)was 22 022 and thisis significant
at the05 % level
as foundfromFisher& Yates's (1953) Table VIII.
The coefficient
of segregation
is S = 1-
38+23 = 0 318.
52-2+ 37-2
Table 1. The testfor segregation
in thenaturalpopulation
Base tree
rDouglas fir
Nearest neighbour
Pine
Douglas fir
137
(122-8)
23
L
(372)
160
Totals
Pine
38
(52-2)
Totals
175
30
53
(158)
68
228
The upper figuresare the observed frequencies. The lower figures,in
brackets,are the expectedfrequenciesin an unsegregatedpopulation.
XI (withYates'scorrection)= 22 022
P(X2/1) <
0?005
(3) The testsfor symmetry
Denotingby/hp thenumberof Douglas firshavingpinesas theirnearestneighbours,
and byfPD the numberof pineshavingDouglas firsas theirnearestneighbours,
it will
be seenfromTable 1 that:
fDp =
23 andfpD 38
The factthat thesefrequenciesare unequal suggeststhat the populationmay be unsymmetrical
in the sense definedin Part IV, withthe pines showinga tendencyto be
relatively
moreisolatedthantheDouglas firs.One maytestforsignificance
thedifference
betweenthese two frequenciesby calculatingX2 (withYates's correction),obtaining:
2
(IfPD-fDP I-1)2
fPD +>fDP
(38-23-1)2/61 = 3 213.
Withone degreeof freedomthe probabilityof gettinga value of X2greaterthan thisis
0073. So this test alone would not justifya conclusionthat the population was
unsymmetrical.
265
E. C. PIELOU
In applyingthistestthe nearest-neighbour
relationshipsof only 61 of the 228 trees
could be considered.A bettertest,therefore,
consistsin comparingtheobserveddistributions, for each of the two species, of the numberof individualsservingas nearest
neighbour to r other trees (r = 0, 1, ..., 5). This comparison is shown in Table 2. None
of the treeswas foundto serveas nearestneighbourto as manyas fouror fiveother
trees,and in calculatingx2thevaluesforr = 2 and r = 3 werepooled so thatno expected
frequency
shouldbe less thanfive.The numberof degreesof freedomwas therefore
two
and the testshowsthatthe populationwas significantly
unsymmetrical
at the 2 % level
withthe pines showinga greatertendencyto isolationthan the Douglas firs.Thus the
smallerindex of non-randomness,
a, for the pines than for the Douglas firs,cannot
have resultedwhollyfromthe factthatthe pines wereless dense; theymustalso have
been moreevenlyspaced thanthe Douglas firs.
Table 2. The testfor symmetry
in thenaturalpopulation
The numberoftreesforwhicheachtreeservedas nearestneighbour
A
-5
Totals
2
3
0
1
4
5
160
fDouglasfirs
39
78
32
11
0
0
Species
dPines
SPecie~
L
Totals
(47 7)
29
(20 3)
68
(73 7)
27
(31-3)
105
(29 5)
10
(12-5)
42
(9 1)
2
(3-9)
13
(0)
0
(0)
0
(0)
0
(0)
0
68
228
The upperfiguresare the observedfrequencies.
The lowerfigures,
in brackets,are the expected
frequencies
assuming
thepopulationto be symmetrical.
= 0019.
x2= 7-874,P(X2/2)
Poolingthevaluesin thelastfourcolumns,
distancesand treesizes
(4) The correlations
betweeninter-tree
In an earlierpaper it was shown(Pielou 1960) thatin a pure stand of P. ponderosa,
the distancebetweeneverytreeand its nearestneighbourwas significantly
correlated
withthe sum of the trunkcircumferences
of the pair of treesconcerned;and further,
that the correlationprovidesevidencethatthe patternof the populationwas, at least
partly,due to the effectsof inter-tree
competitionwithsmall,youngtreesconfinedto
weremade
clumpsat somedistancefromlarge,old ones.The samesetsofmeasurements
on everytree in the two-speciespopulationnow being consideredand the resulting
scatterdiagramis shownin Fig. 4. Althoughmeasurements
weremade at all 228 trees,
only 161 pointsappear in the scatterdiagram.This is because 134 of thenearest-neighbourrelationships
werereflexive,
thatis, thetreeconcernedwas thenearestneighbourof
its own nearestneighbour.For these 67 reflexivepairs a singlepoint only has been
marked.The remaining
made at treesthatdid not belong
94 pointsshowmeasurements
to reflexive
pairs; in thesecases each treecontributes
once in its capacityas a base tree;
as well,any of themmayalso contribute
not at all, or else several(up to five)timesas a
nearest-neighbour
tree. Distinctsymbolshave been used to show the three typesof
pairs: a Douglas firwitha Douglas firas nearestneighbour;a pine witha pine; and a
memberof one specieswitha memberof the other.
From the diagramas a whole it is obvious that the two variates were positively
the patternof the
inter-tree
correlated,and that therefore
competitionwas affecting
population.But if such competitionwere the only cause of clumping,withthe seeds
Segregationand symmetry
266
of both speciesbeingscatteredat randomand withthe habitathomogeneous,thenthe
two specieswould not have been segregatedfromeach other.The factthattheywere
segregatedshowsthathabitatpatchinessor familialclumping(or both) mustalso have
thepatternof at least one of the species.
been factorsin determining
If each of the species were confinedto its own nicheswithinthe area, one would
expectcompactclumpsoftrees,whentheyoccurred,to containindividualsofone species
only,membersof the otherbeingexcluded.Had thisbeen so, 'mixed'pairs(a pine with
a Douglas firas nearestneighbouror vice versa) would have occurredonly when a
isolatedpine,say,had as itsnearestneighboureitheran isolatedDouglas
comparatively
fir,or a memberof a Douglas firclump; and mutatismutandis.In that case the mean
distanceformixedpairswouldhave been greaterthanforpairsin whichboth
inter-tree
3
E
2
)oDouqlas
fir- douglas fir
0
0
* Pine-pine
0
0
*Douglas fir- pine
0
o
3
C
~~~~0
~~~ 0
?O
?o
0 oO
)
0
~~0*00
0-4-
01
0-3-
uJQ
E
0Q
-
E 0]
V)
~~o
02
o
03 0
0
*
0
o0cb000
00
0
0
00
0
OG0
0
0
000
E
0
~~~~~~~~~~~~00
Q)
o
e
00e
0
Go 0
0
0
0
0
0000
0
0
0
0
0
0o
10
0
00
000
0
3 4 5
2
I
0
034065
Distance frombase tree to neorest neighbour;metres.
10
forthethreetypesof
distanceand sumof trunkcircumferences
Fig. 4. The relationbetweeninter-tree
are logarithmic.
pairsin thepopulationstudied.Bothco-ordinates
simplyto commembersbelongedto the same species('self' pairs). It is not permissible
typesof pair,however,since,
distancesforthedifferent
pare the meansof theinter-tree
as Fig. 4 shows,thesedistancesare correlatedwiththe sizes of the treesformingeach
distancesweregreaterformixedpairsthanforselfpairs,thepoints
pair. But ifinter-tree
in the scatterdiagrampertainingto mixedpairs shouldhave appearedto the rightof
thosepertaining
to selfpairs.Thisis clearlynotso and providesevidence,albeitnegative,
thatnichespecificity
did not accountforthe observedsegregation.
wereoperating,
The same arguments
wouldlead one to expectthatifnichespecificity
reflexive
pairs of individuals(in whicheach of the two treesof the pair was the nearest
of two pines,or of two Douglas
neighbourof the other)would consistpredominantly
was testedbymeansofa
firs,and onlyrarelyof a pineand a Douglas fir.Thispossibility
in thetable are almostidenticalwith
2 x 2 table (see Table 3). The observedfrequencies
theexpectedfrequencieson theassumptionthatmixedpairsare as likelyas selfpairsto
be reflexive.
This providesadditionalevidencethattherewas no nichespecificity.
267
E. C. PIELOU
and typeofpair
Table 3. The relationbetweenreflexiveness
Reflexive
paits
Non-reflexive
pairs
Totals
Selfpairs Mixedpairs
36
98
(35 85)
(98 15)
25
69
(25415)
(68 85)
61
167
Totals
134
94
228
X2= 0-002,P(X2/1)>095
of thepair(base
A selfpairis one in whichbothmembers
treeand nearestneighbour)belongto the same species.A
belongto thetwo
mixedpairis onein whichthetwomembers
different
species.
(5) Discussionoffieldresults
The resultsof the observationsmade on the naturalpopulationlead to thefollowing
conclusions:
moreso thanthepines.
theDouglas firssignificantly
1. Both specieswereaggregated,
2. The Douglas firsand thepineswerepartlysegregatedfromeach other.
in that the pines showed a greatertendency
3. The populationwas unsymmetrical
isolated.
than did the Douglas firsto be relatively
4. Therewas no evidencethattheobservedsegregation
resultedfromnichespecificity,
so it remainsto considerwhatothercause could accountforthe segregation.
thereare threeotherpossiblecausesfor
As shownin PartII, excludingnichespecificity,
segregation:familialclumping;responseof the two speciesto unrelatedhabitatfactors
whichwerevariableover the area; and the possibilitythat,thoughboth specieswere
affected
by the same habitatfactor(or factors),one was moretolerantthan the other.
of Douglas firin the New Forest,foundthat
Jones(1945) studyingthe regeneration
but thatthe clumpsweremostdensefarfrom
seedlingsof thisspecieswereaggregated,
couldnothaveresultedfromclumpingof offspring
thesourceofseed; so theaggregation
arounda parent.He showed,moreover,that therewas a definiterelationshipbetween
the densityof the seedlingsand the natureof thelitter,so thatthiswas a clear case of
of habitat.AlthoughI have been unable to
aggregationresultingfromheterogeneity
of similarstudieson P. ponderosa,it seemsunlikelythat
findanyreportsin theliterature
familialclumpingwould occur in thisspecieseither.Many of the individualtreeswere
would not be expectedto occurin distinct
tall and closelyspaced so thattheiroffspring
clumps.
It is possiblethatamongthe youngtrees,the densityof the Douglas firsvarieswith
the natureof thelitter,and the densityof thepineswiththepatternat groundlevel of
light and shade, for, as Haig, Davis & Weidman (1941) have shown,P. ponderosa
Thus segregationbetweenyoungtrees
seedlingsneed lightfor successfulgermination.
of thesetwo speciescould resultfromthefactthattheirdensitiesdependedon unrelated
habitatfactors.It is not clear,however,to what extentthiseffectwould persistas the
treesgrewolder.
Lastly,thereis the possibilitythat whileboth species have the same requirements,
from
resultsfromthefactthattheDouglas firsare less tolerantofdepartures
segregation
confinedto smallerpatchesof ground,thanthemore
optimumconditionsand therefore
268
Segregationand symmetry
tolerantpines.It is knownthatthe pines are more droughtresistantthanthe Douglas
firs.This explanationwould also accountforthe factsthatthe Douglas firsweremuch
moreaggregatedthanthepinesand thatthepopulationwas unsymmetrical.
It shouldbe emphasizedthatthe conclusionsarrivedat above are intendedto apply
onlyto the actual populationstudied.Theyhave been describedto show how observations on the 'segregation'and 'symmetry'
of two-speciespopulationsmay throwlight
on the causes underlying
populationpatterns.A studyof numeroustractswithinthe
mixed forestof pines and Douglas firswould be needed to ascertainwhetherthe
conclusionsdrawnweregenerallyapplicablein thisforest.
VIII. SUMMARY
Whenthespatialpatternsof two speciesin a community
besides
are beinginvestigated,
determining
the spatialpatternof each specieswithrespectto the ground,it is also of
value to determine
the spatialarrangement
of the membersof one specieswithrespect
to themembersof the other.
Thus two speciesmaybe describedas 'segregated'if an individualof eitherspeciesis
morelikelyto be foundamongmembersofthesame species,thanamongmembersofthe
otherspecies.Conversely,
if neither
the two speciesmaybe describedas 'unsegregated'
shows a tendencyto occur as one-speciesclumps.A two-speciespopulationmay be
segregatedor unsegregatedregardlessof whetheror not the populationas a whole is
clumped.
Segregationmay be detectedand measuredby a studyof the nearest-neighbour
relationshipsof the membersof the population.By examiningeach individualof a twospeciespopulationin turnand determining
its speciesand thatof its nearestneighbour,
one may draw up a 2 x 2 table showingthe observedfrequenciesof the fourpossible
types of nearest-neighbour
relationship.These may be comparedwith the expected
frequencies
in an unsegregated
populationby a X2test.
A coefficient
ofsegregation
fora two-speciespopulationis definedin termsof nearestneighbourrelationships.
'Negativesegregation'is also possible; thisoccurswhenan individualof one of the
two speciesis morelikelyto be foundnearindividualsoftheother,thannearindividuals
of its own species.
Regardlessof its degreeof segregation,
a two-speciespopulationmay also be 'symmetrical'or 'unsymmetrical'.
the membersof the two speciesare
If it is symmetrical,
equally likelyto functionas nearestneighboursto otherpopulationmembers.In an
unsymmetrical
population,individualsof one of the speciestend to be more isolated
than those of the otherand thus serveless oftenas nearestneighbours.Two testsfor
are described.
symmetry
Field resultsfroma mixed stand of Douglas firand ponderosa pine are used to
illustratetheconceptsdeveloped.
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Science,
in biologicalpopulations.
Clark, P. J. & Evans, F. C. (1955). On someaspectsofspatialpattern
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Fisher,R. A. & Yates, F. (1953). Statistical Tables for Biological, Agriculturaland Medical Research.
4thEd. Oliver& Boyd,London.
London.
Greig-Smith,P. (1957). QuantitativePlant Ecology. Butterworths,
whitepinetype.
in thewestern
Haig, I. T., Davis, K. P. & Weidman,R. H. (1941). Naturalregeneration
Tech. Bull. U.S. Dep. Agric.767.
E. C. PIELOU
269
Jones,E. W. (1945).Theregeneration
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in thenorthern
McMinn,R. G. (1952).The roleofsoil drought
Rocky
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(Received1 March 1960)
D
;.E.