Taylor’s Power Law: Order and Pattern in Nature

629 Pages·2019·26.424 MB·English

by  R.A.J. Taylor

Taylor’s Power Law
Taylor’s Power Law
Order and Pattern in Nature
R. A. J. Taylor
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Preface
InAutumn1964,myfamilytookaroadtripfromManhattan,Kansas,toWin-
nipeg, Manitoba. Sitting in the back seat of our Oldsmobile, I listened to my
parentsdiscussinganequationmyfatherwastryingtounderstand.Itwasappar-
entlyimportantforpracticalreasonsandperhapsothers,butwasenigmatic.It
didn’tseemtofitintothescienceofthetime.Fiveyearslater,Isatinalecture
theateratImperialCollege,London,listeningtoProfessorT.R.E.(Dick)South-
wood lecturing on the analysis of spatial distribution of organisms and its
importanceinunderstandingpopulationchange.Hetootalkedofanenigmatic
equationthat had multiple practical applications and an important function in
interpretingspatialdistribution,butwhoseoriginswereelusive.Fouryearslater
at Silwood Park, Professors Dick Southwood and Michael Way gave me the
freedomtoinvestigatewhatwasbythenknownasTaylor’spowerlaw.Ilearned
then that practically any rule governing organization in space iterated in time
wouldresultinaconvincingpowerlawrelatingspatialvariancetopopulation
density.OneruleIused,basedonthepotentialenergyequationofphysics(the
Δ-model),wasparticularlysuccessfulinthis,butwashighlydependentonini-
tialconditions.Thesameinputparametersappliedtodifferentstartingpatterns
produceddifferentpowerlaws,apparentlyatrandom.TheΔ-modelhasdescrip-
tiveandinterpretivevalue,asfieldbiologistshaveshown,butlittlepredictive
value.Onepropertyitdidhave,whichmanyothermodelsfailedtodisplay,was
itsabilitytogeneratepowerlawswithslopessteeperthanvarianceproportional
to the mean squared. As far as Roy Taylor and I were aware in 1977, this
variance-mean relationship was unique to ecology. In the last 20 years, the
powerlawhas beenfound toapplytodata indisciplinesasdiverse asastron-
omy,geology,meteorology,criminology,sociology,economics,andcomputer
science.Thus,itisnotuniquetoecologyandmayhavenothingtodowithecol-
ogyperse.Thisbookgrewoutofmycontinuinginterestinthisenigmaticbut
ubiquitousequationandthedesiretounderstanditsorigins.Myapproachisnot
a modeling approach, although models are important andare notignored, but
thatofbasicbiology:therubricofcomparativeanatomy.Thebookisorganized
in three parts: an introductory section deals with background and methods; a
survey section describes both biological and non-biological examples; and a
concludingsectioncoverspracticaluses,models,andinterpretation.Themid-
dlesectionisintendedtodothreethings:tocoveraswideataxonomicrangeas
possible,toshowthefullrangeofpowerlawslopes,andtohighlightsimilarities
xvii
xviii Preface
and differences between the biological and non-biological examples. Only in
thepenultimatechapterdoIattemptasynthesis.ElsewhereIhavetriedtoreport
accurately and to refrain from criticism. Those cited can judge whether I was
successful.
SpecialthanksgotomyteachersatImperialCollege,NadiaWaloff,George
Murdie, Michael Way, and Dick Southwood, and for the many conversations
withmyfather,L.R.Taylor.Inadditiontomymentors,allalasnowgone,Iwant
tothankmydaughterKarawhodesignedthebookcover,andthefollowingpeo-
plewhoprovideddataorcheckedmyuseoftheirdata:JohnCardina,Malcolm
Elliott,DanGrear,LarryMadden,AdelineMurthy,ArnePeters,WaynePolley,
Peter Smits, and Xiangming Xu. Malcolm Elliott, Cathy Herms, Dan Herms,
ParwinderGrewal,SamMa,LarryMadden,AdelineMurthy,JoePerry,Wayne
Polley,KaraTaylor,andRobertTaylorreadandcritiquedparts.Severalcoau-
thorsalsocontributeddataanddiscussion:AndyChapple,DickLindquist,Mike
McManus,SunnyPark,CharliePitts,LesShipp,IanWoiwod,andthelateBen
Stinner. To all, I offer my sincere thanks for their assistance, support, and
friendship.
Thisbookisdedicatedtomyparents,JeanandRoyTaylor,andtomyfam-
ily,wifeKate,daughterKara,andsonRobertwhonotonlyputupwithmewhile
developingthisbook,butreadalotofittoo.However,oftheinevitableerrors,I
alone claim responsibility.
Wooster, Ohio
January2019
Chapter 1
Introduction
Thattigershuntaloneandwolvesinpacksarewell-knownattributesofthese
twopredatorymammals.Thelikelihoodoffindingevenonetigerthesedaysis
distressinglylow,butevenacenturyagowhentheyweremuchmoreabundant
throughoutsoutheastAsiaandSiberia,itwouldhavebeenraretofindoneadult
tigerinthecompanyofanothernothercub.Bycomparison,itwasandstillis
rare to encounter a lone wolf. These species’ spatial behavior is easily recog-
nized as being different; but how different, and how does one measure it?
I chose these two species to exemplify extremes in spatial behaviors because
they are familiar archetypes. But their characteristic distributions and abun-
dancesarenodifferentinkindtothoseofthemyriadotherspecies,microbes,
plants,andanimalsvertebrateandinvertebrate,withwhichwesharethisplanet.
Thisbookisprimarilyaboutthespatialdistributionsofpopulationsofliving
things, and how their distributions are measured. It also touches on nonliving
entitiesthatappeartobehavemathematicallyinasimilarfashion.Itisconceiv-
ablethatthesamerulesthatgovern,oratleastdescribe,livingorganisms’dis-
tributions in space and time also apply to nonliving things. If this is the case,
then the mathematical construct that is central to the theme of this work may
havedeepermeaningthanthemathematico-statisticalcuriosityseemedtohave
55years agowhenit was first described andnamed.
Taylor’ power law (TPL; Lincoln and Boxshall, 1982) was referred to by
thatnameinT.R.E(Dick)Southwood’sEcologicalMethodswithParticular
ReferencetotheStudyofInsectPopulations,publishedin1966.AsfarasIhave
beenabletoascertain,thisistheearliestreferencetotheeponymouspowerlaw
relating sample variance and mean population published in Nature by
L.R.(Roy)Taylor(Taylor,1961,hereafterreferredtoasLRT61).RoyTaylor’s
ownbookonecologicalexperimentspublishedtheyearfollowingSouthwood’s
references the relationship butdoes notname it (Lewis and Taylor, 1967).
Taylor’s power law states that sample variance and mean population are
relatedby a simple power law:
V¼aMb (1.1a)
or
LogðVÞ¼A+blogðMÞ (1.1b)
Taylor’sPowerLaw.https://doi.org/10.1016/B978-0-12-810987-8.00001-X
©2019ElsevierInc.Allrightsreserved. 1
2 Chapter 1 Introduction
whereVisthevarianceofasetofsamplesandMistheaveragevalueofthatset;
A¼log(a) and b are parameters estimated from a collection of variances and
meansgeneratedfrommultiplesetsofsamplestakenatdifferenttimesand/or
placesandextendingoverasbroadarangeofmeansaspossible.Ihavechosen
touseitalicstorepresentcalculatedestimatesofvarianceandmeanastheseare
generallythebasisforpresentationanddiscussionofaggregationasmeasuredby
TPL.Wherepopulationortheoreticalvalues,asopposedtoestimates,aredis-
cussed,normaltextforvariablesandGreeklettersforparameterswillbeused:
V¼αMβ: (1.2)
TPL data are usuallyanalyzed by taking logarithms of the mean and vari-
ances. Common logs, log (∙), will normally be used, represented by log(∙).
10
Natural logs, loge(∙)when used will bedenotedby ln(∙).
TPLismathematicallysimpleanditsparametersprovidesimplemetricsfor
describingthedegreeofaggregationoforganismsandotherentitiesthatrange
from regular to random to highly clumped. But its genesis and scope are not
simple.Indeedmuchpaperhasbeenconsumedtryingtoaccountforthem.This
book is intended to collect together the thoughts and suggestions of the hun-
dredswhohavewrittenandspeculatedaboutTPL.Therearetwomainobjec-
tives: to review as wide a taxonomic range of examples and nonbiological
examples of TPL and to examine the various models developed to account
for it. There are also counter examples where the power law does not apply
whenonemightexpectitto.Theexceptionstothelawmayprovetobefarmore
illuminatingthan the thousands of exemplars.
Earlyon,inLRT61andSouthwood(1966)forexample,itwasproposedthat
aislargelyasample-sizeparameterwhilebisapopulation-specific,orpossibly
species-specific,descriptionofaggregation.Intheinterveningyears,thesesim-
pledistinctionshavebeenfoundtooversimplifytheinterpretationofaandb.In
fact,theyareoftencorrelatedmakingthissimpledichotomyuntenable.Com-
plicatingmatters,thereareactuallythreevariantsofthepowerlawdepending
onhowthesamplesweretakenandhowtheyareprocessed.Thesevariantstell
different storiesthat collectively shed new lighton TPL.
It is no accident that the relationshipbetween varianceand mean was dis-
covered by an agricultural and ecological entomologist. With the possible
exceptionofplanktonandotheraquaticinvertebrates,themostextensivedata
of population density and distribution are of insects, mostly those of eco-
nomic—agricultural,veterinary,ormedical—significance.Muchofstatistical
theoryoriginatedwiththesedisciplines.RonaldFisher,thearchitectofexper-
imentaldesign,wasemployedatRothamstedExperimentalStationinHarpen-
den, England, to analyze long-term data of agricultural experiments (Fisher,
1935). While at Rothamsted, Fisher also developed methods for ecological
research, including the analysis of diversity with C.B. Williams (Fisher
etal.,1943)aswellasspatialdistribution(BlissandFisher,1953).Fisher’stra-
dition continues at Rothamsted (now Rothamsted Research) and includes the
Introduction Chapter 1 3
workofRoy Taylorandcolleagues. While Fisher’scontributions were essen-
tially theoretical, their value lies in their application to applied entomology,
plant pathology, agronomy, horticulture, agriculture and genetics, as well as
population andcommunityecology.
ThefundamentalfeatureofTPListhedensitydependenceofthefrequency
distribution of abundance. Locally, the frequency distribution of population
numberperunitareamaybedescribedbythePoisson,negativebinomial,Ney-
man’s type A, or one of dozens of other statistical distributions. But at larger
scale, we generallyfind that many statisticaldistributions are needed tochar-
acterizeabundance.Theastonishingthingisthatifweplotthevarianceagainst
the mean for each of the empirical frequency distributions, we find they are
relatedbyastraightlineonlogarithmicscales.Fig.1.1Aoffourdistributions
ofEuropeancornborer(Ostrinia(¼Pyrausta)nubilalis)suggestsacontinuum
inshapeoffrequencydistributioninwhichthemodeshiftstotherightandthe
right-handtailgetslongerwithincreasingmeandensity(McGuireetal.,1957in
LRT,1965).Thisplotsuggestsasmoothchangeinfrequencydistributionwith
density,butaplotofOoencyrtuskuvanaeparasitismofgypsymoth(Lymantria
dispar)eggs(BrownandCameron,1982)showsoverlapsbetweenthebestfit
distributions as density increases(Fig. 1.1B).
Clearly,thedistributionandabundanceoforganismsareintimatelyrelated,
andtogetherdefineapopulation’ssizeandstructureinspace.TPLimpliesthat
theshapeofthefrequencydistributionofnumberspersampleisitselfdensity
dependent.Furthermore,transitionaldistributionsmaynotbewelldescribedby
any knownstatistical frequencydistributionas is evident inFig.1.1A.
InhisbookPatternsintheBalanceofNature,Williams(1964)makesthe
casethatthestatisticalpatternsofpopulationsandcommunitiesofpopulations
are the numerical and geographic manifestation of a natural balance. He sug-
gested that there is a connection between the frequency distribution of abun-
dance of species in a community and the spatial distribution of individuals in
that community. Specifically, he suggested thatalthoughthe numerical abun-
danceofaspeciesorothergroupinginacommunityisalwaysinastateofflux,
it is possible that the overall statistical pattern remains more or less constant.
ThehighdegreeofrepeatabilityexhibitedbyTPLarguesforasimilarnatural
pattern.ItiswiththatinmindthatIparaphrasedWilliamsinthesubtitleOrder
and Pattern inNature.
LRT61(Fig.1.2)isoneofthemostwidelycitedpapersinecologyandagri-
culture.AsofDecember2018,TheWebofScience(ThomsonReuters,2018)
listsnearly2000citations(Fig.1.3)with40–50newpapersperyear.Thereare
manymorethatrefertoTPLbutcitederivativeorlaterpublications.Therecent
growthisdueinlargeparttothediscoveryofvariance-meanpowerlawsindis-
ciplines as diverse as economics, physics, computer science, genetics, and
molecularbiology.Thisdiversitywashighlightedbyamajorreviewof“fluc-
tuation scaling,” as the variance-mean power law is known in physics (Eisler
et al., 2008).
4
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FIG.1.1 (A)ThreeoffourfieldsamplesofEuropeancornborerfromMcGuireetal.(1957)arewellfitbydifferentfrequencydistribution;thefourthisnotfitbya
distribution.Themean-variancepairsalllieonalinewithequationV¼1.5M1.25.(B)ThefrequencydistributionoftheeggparasiticwaspOoencyrtuskuvanaeonor
neargypsymotheggmasseschangesasthewasp’sdensityincreases:Theoriginaldatawereexpressedaswaspspereggmass.ThefittedlineisV¼2.31M1.27,
r2¼0.94.((A)FromLRT(1965).CourtesyoftheCouncilforInternationalCongressesofEntomology.(B)AdaptedfromBrownandCameron(1982).)