Philosophicalapproachto populationmodeling
Overthepastcentury,populationmodelshaveevolvedfrombeingsimplycleverwaysof plottingareasonablecurveofabundanceversustimetobeingatoolbothtoimprove understandingofpopulationdynamicsandtomakepredictionsforthepurposeof management.Thatunderstandinghasbeengainedbyexamininghowthemechanisms embodiedintheequationsleadtotheparticularpatternofabundanceovertimeand space.Thisfocusontheconnectionbetweenthemodelstructure(theformofthe equations)andemergentpopulationpropertiescontainedinthemodelsolutions(what kindofpopulationdynamicsdotheequationsproduce?)hasallowedbroaderandmore usefulemploymentofpopulationmodels,inboththescientificendeavorofunderstandingpopulationdynamicsandintheapplicationofpopulationdynamicstopractical problems.Thischapterdescribeshowtheapproachtopopulationmodelinghasevolved overthepastseveraldecadesandthenatureofitscurrentstate(seeBox1.1foravocabulary ofconceptualtermsthatwewillusethroughoutthebook).
Webeginourdescriptionofmodelingphilosophyinthischapterwithasomewhat historicaldigressiontogivethereaderasenseofthewaythatecologistsbegantoapproach populationmodelinginthemidtwentiethcentury,andhowthatapproachhasevolved (Section1.1).Oneremnantofthatmeanderinghasenduredasausefulconcept:the differencebetween strategic and tactical populationmodels.
Nextwedescribetheformallogicunderlyingboththescientificandthepracticaluseof models(Section1.2).Thislogicprovidesusefulwaysofstructuringhowoneusesresults frompopulationmodelsinthecontextofone’sgoalsandavailabledata.Thissectionalso servesasareminderthatusingmodelstoachievegreaterscientificunderstandingand usingthemforpredictionsinpracticalapplicationareactuallybasedondifferentlogical foundations.Theprincipaloutcomeofthissectionisthefactthatforbothscientific progressandpracticalapplications,weneedmodelsthatprovidetheopportunityto compareexplicitlythestructureofthemodeltotherealworld,eitherempiricallyor,at least,conceptually.Werefertothatcharacteristicas realism,anditconnotesaqualityof testabilityorobservabilityinmodels.Thischaracteristicguidestheorganizationofmodel descriptionsthroughoutthebook.
Nextweturntothebasicresearchonsystemtheoryduringthe1970sforaformal definitionofthe state ofasystemtoprovideuswiththatrealism(Section1.3).Whilethis topicmaysoundabitesoteric,itprovidesthekeyrationalefordevelopmentofmodelsto achievescientificunderstandingandeffectivemanagement.Thebasicideaunderpinning thisbookisthatadescriptionofthestateofasystematanytimeinstantneedstoinclude alloftheinformationnecessarytoprojectuniquelythestateofthatsystematthenext
PopulationDynamicsforConservation.LouisW.Botsford,J.WilsonWhite,andAlanHastings, OxfordUniversityPress(2019).©LouisW.Botsford,J.WilsonWhite,andAlanHastings. DOI:10.1093/oso/9780198758365.001.0001
timeinstant.Thisconceptformsthebasisforunderstandingthedifferentrolesofmodels withagestructure,sizestructure,stagestructure,andspatialstructure,thebackboneof thisbook.
Thenextsection(Section1.4)exploresthelimitationsinourabilitytodetecttheactual structureandstateofanecologicalsystemintherealworld.Theselimitationsarecast intermsofthedifferentkindsofuncertaintyweencounter.Althoughwedonotinclude estimationofpopulationparametersinthisbook,ourdevelopmentofpopulationmodels, ouranalysisofthem,andourexamplesofapplications,areheavilyinfluencedbythe relativeuncertaintyinvariouscomponents.Basically,weseekwaystoformulatemodels thatallowourconclusionstodependascloselyaspossibleontheparametersandconcepts wecanknowwell,andweidentifytheimportantaspectswewouldliketoknowbetter,i.e. themostcriticalremaininguncertainties.Theroleofuncertainties,ofstochasticversus deterministicapproaches,willbeanotherimportantthemeofthisbook.
Nextweturntoadescriptionofhowtheanalysisandstudyofpopulationsisrelated tootherdisciplinesthatfocusondifferent levelsofintegration inbiology(Section1.5). Differentdisciplinesfocusondifferentlevelsofintegration.Researchatthedifferentlevels ofintegrationemploysdifferentapproaches,andtheyleadtodifferentemergentproperties,witheachleveltendingtoexplainthenexthigherlevel.Forexample,population dynamicsdependsonthesurvivalandreproductiveratesofindividualorganisms.Also, lowerlevelsofintegrationareusuallymoresensitivetosmallertemporalandspatialscales ofenvironmentalvariability.Thesedifferencesareimportantbecausepopulationmodels canservetoscale-upindividual-levelobservationstothepopulationscale.
Weendthischapterwithasynthesisofcurrentthinkingonthesetopicsasreflectedin recentpublications,asof2019.Theseillustratehowtheconversationintheecological literatureregardinghowbesttoformulateandusepopulationmodelscontinuesto beactive.
1.1Simplicityversuscomplexity,andfourcharacteristicsofmodels
Anobviousquestionaddressedinanymodelingactivityishowcomplexshouldthemodel be?Simplermodelsarepreferredbecausetheyareeasiertoanalyzeandtounderstand.On theotherhand,simplemodelsarenotusefuliftheyomitkeyaspectsthatareessential tothedynamics.Thebruteforcemethodofmakingsuretoavoidthiserrorofomission istodevelopamodelofsufficientcomplexitytocompletelyreplicatetheactualsystem beingmodeled.However,thiselusivegoalisnotnecessarilydesirable,especiallywhenthe operationofsomemechanismsorcomponentsincludedarepoorlyunderstood.
Twodifferentapproachesforreachingamodeloftheappropriatecomplexityare possible.Onecouldstartwiththesimplestpossiblemodel,suchasdeterministicexponentialgrowth,andaddfeaturestothemodelthatwouldallowbetterunderstanding. Alternatively,onecouldstartwithacomplexdescriptionthatincludesallfeaturesone couldenvision,andthenmakesimplifyingassumptions.Eachapproachhasitsadvantages anddisadvantages,aswillemergelater.
Thetopicofmodelcomplexityforecologicalmodelsiscurrentlyanareaofactive discussioninpopulationscienceandmanagement.Forexample,nationalandinternationalpanels(e.g.thePewOceansCommission(PewOceansCommission,2003)and theUSCommissiononOceanPolicy(USCommissiononOceanPolicy,2004))have recommendedthatresourcemanagementtransitionfromusingsingle-speciespopulation modelstomodelsthatincludeallcomponentsoftheecosystem,includingfoodweb interactions,thevariablephysicalenvironment,andsocioeconomicfactors(i.e.ecosystem
basedmanagement,EBM;Pikitchetal.,2004).However,theseadditionalfactorsareoften poorlyunderstood.Achallengingquestion,therefore,iswhetheritisbettertoaddpoorly understoodstructuretothemanagementmodel;i.e.willitactuallyimprovemanagement (Botsfordetal.,1997)?Thisremainsavexingproblemthatinfluenceshowpopulation dynamicswillbedescribedinthesemodels(Collieetal.,2014).
Animportantstepinthehistoryofpopulationmodelingwasmathematicalecologist RichardLevins’(1966)analysisofthequestionofsimplicityversuscomplexity.Hepointed outtheproblemswithmodelsofhighcomplexity:(a)therearetoomanyparametersto estimate;(b)theequationsarenotsolvableanalyticallyandwouldexceedthecapacityof thefastestcomputers(in1966),and(c)theresultingexpressionsaresocomplexastobe meaningless.Theseproblemsarestillpresentfortyyearslater,exceptthatcomputational limitationscontinuetoshrinkwithtime.
Levins(1966)declaredthatitwas“ofcoursedesirabletoworkwithmanageablemodels whichmaximized generality, realism,and precision towardtheoverlappingbutnotidentical goalsofunderstanding,predicting,andmodifyingnature” [emphasisours].Hefurther proposedthatwecouldnotachieveallthreequalitiesinasinglemodel,butratherthat onlytwooutofthreecouldbeachievedinanyspecificcase.Wedonotquibblewith whythesethreecharacteristicsshouldbetheoneschosen,norwiththebasisforthe statementthatonecanachievetwo,butnotthree.However,Levinsdidnotdefinethe threecharacteristics.Wewill.Thesimplestdefinitionof generality isthecharacteristic ofapplyingtoallpossibleexamples. Realism isopentoseveraldifferentdefinitions, butitisreasonablyclearthatwhatLevinsmeantisthecharacteristicintroducedinthe introductiontothischapter,thatofallowingfordirectcomparisontotherealworld.For example,amodelthatincludedthemortalityratesandreproductiveratesofindividuals atallageswouldbeconsideredrealistic,butamodelthatincludeda(presumed)simple summaryoftheireffects,suchaspopulationcarryingcapacity,wouldbelessrealistic.We canobservetheamountofreproductionandtheratesofdeathofindividuals,butcarrying capacityisanemergentpropertyofamodelsolutionandwouldrequirelonger-term observationstodetermine.Notethatthisdefinitionofrealismincludesnorelationshipto truth,asthemorecommon,everydaydefinitionofrealismimplies.Inthecontextofthe truth,Levins’definitionofrealismcanbeinterpretedasmeaningthatanelement’struth intherealworldistestable,notthattheelementisatruedepictionofreality.Levins’ definitionof precision isthestandardstatisticaldefinition,thoughheprobablyactually meantaccuracy,i.e.notjustconsistentlysimilaranswers,butconsistentlythecorrect answer(seeBox1.1).
Box1.1 DEFINITIONSRELATEDTOMODELINGPHILOSOPHY
Generality—strictlyspeaking,thequalityofastatementapplyingtoallcases.
Realism—havingthesamestructuralformasarealobject.
Precision—thequalityofastatisticalestimatehavinganarrowdistributionoferrorabouta point,butnotnecessarilyaboutthetruevalue.
Accuracy—thequalityofastatisticalestimatehavinganarrowdistributionoferrorabout thetruevalue.
Holism—includingalloftherelevantfactors
Strategicmodels—modelsdevisedtoanswerverygeneralquestionsaboutpopulation behavior,withlittleattentiontoaccuratelyportrayingaspecificsituation. Tacticalmodels—modelsdevisedtoanswerspecificquestionsaboutrealsituationsforthe purposeofmakingprojectionsonwhichmanagementwillbebased.
AccordingtoLevins’two-out-of-threerule,somemodelssacrificegeneralityforthe sakeofachievingrealismandprecision.Levinsofferedthemodelsusedinfisheries managementasanexample.Thosemodelsincludeindividualgrowthrates,reproductive rates,andmortalityrates,whicharecomparabletotherealworld,andvaluesofcatchcan beprojectedprecisely,butthemodelersdonotintendforasinglemodeltodescribeall fishedpopulations.
LevinsofferedVolterra’s(1926)predator–preysystemasanexampleofthesecond category:modelsthatsacrificerealismforgeneralityandprecision.Thismodelomitsthe timelagsofteninvolvedinpredator–preycyclesandtheeffectsofaspecies’population densityonitsbirthanddeathrates.Instead,themodelusesphenomenologicalmodel parameterssuchascarryingcapacity.Althoughthismodelparametercannotbedirectly linkedtoanobservableecologicalinteractioninthefield(hencethelackofrealism), themodelnonethelessrepresentstheessentialcharacteristicsofgeneralpredatory–prey dynamics,andtheresultscanbestatedveryprecisely.
Theremainingclassofmodelsinthetwo-out-threeschemesacrificesprecisionfor thesakeofrealismandgenerality.Levins’exampleforthiscasewasthesetofsimple biogeographicalmodelsdescribingbroadlatitudinalpatterns.Forexample,Bergmann’s rulestatesthatanimalbodysizewillbelargerincolderclimates(Bergmann,1847;Meiri andDayan,2003).Thesemodelsaregeneralinthattheyapplytomanyspecies,andthey possessthequalityofrealismsincetheycaneasilybecomparedtoobservations,butthey donotyieldprecisevalues,rathertheyaremerelyqualitativepredictionsintheformof inequalities.
ThevalueofLevins’contributionwasthatheacknowledgedthatmodelscanbequite differentinfunction,andthatdifferentkindsofmodelsmaybebestfordifferentuses. Thistopichascontinuedtoevolveuptothepresent.
ThenotedsystemsecologistC.S.(“Buzz”)Holling(Holling,1968)assessedLevins’ (1966)viewandaddedafourthcharacteristicofmodels: holism,thequalityofincluding allrelevantfactors.Asnotedinthesecondparagraphofthissection,requiringthatall relevantfactorsbeincludedinamodelisaquestionablestepwhenthereisonlyapoor understandingofthedynamicbehavioroftheproposedadditions.Themodernadvicein favorofincludingallspeciesofanecosysteminecosystemmodelsisanexampleofthe questforgreaterholism.However,asnotedpreviously,thereisnotaclearadvantageto addingapreyspeciestoamodelwhentheconsumptionratesandtheconsequencesof consumptionarenotwellknown(Collieetal.,2014),andapredatorthatconsumesmany differentpreyspeciescanberepresentedjustaswellbyamodelwithoutpreyexplicitly included(Murdochetal.,2002).
AnassessmentofLevin’s(1966)viewsbyRobertMay(May,1973),aphysicist-turnedmathematicalecologist,concludedthatmostecologicalmodelswereneithergeneral, realistic,norprecise,andthatthecentralissuedidnotinvolvethosethreequalities,but rathershouldfocusontherelativeadvantagesofsimple,generalmodelsversuscomplex, specificmodels.Mayheldthatbothsimpleandcomplexmodelshadtheirplace,withthe formerbeingusefulfordescriptionsofthegeneraltrendsinlargecomplexsystems,while thelatterwereusefulindealingwithspecificaspectsofpartsofsuchasystem,aswould occurinamanagementsituation.Additionalfactorsthatwouldinfluencemodelcomplexity includetheamountofdataavailable:indata-poorsituationsthereislessjustificationfor complexmodels(aswellasaneedformoredata,topermittheuseofmorecomplexmodels). ThesetwonotionsweresimilartowhatHolling(1973)hadreferredtoas strategic and tactical approachestomodeling,respectively.Thesetwotermsarenowcommonlyusedto differentiatebetweenthesetwodifferentkindsofmodelingactivities(Box1.1).
Anillustrativeexampleofthedifferencebetweenstrategicandtacticalapproaches isthedevelopmentofpopulationmodelstodesignmarinereserves,followingtheir increaseinpopularityasamanagementtoolinthe1990s.Astheideaofmanaging marineresourcesbycreatingareasofnofishing(i.e.marinereserves)begantogain prominenceinthe1990s(Botsfordetal.,1997;Murrayetal.,1999),itbecamenecessaryto developpopulationmodelstoevaluatehowmarinemetapopulationswouldpersistwhen distributedoverheterogeneoushabitatswithfishingpermittedinsomelocationsbutnot inothers.Amarinemetapopulationisanumberofseparatesubpopulationsdistributed overspace,linkedbyadispersinglarvalstage(Roughgardenetal.,1988;Botsfordetal., 1994;KritzerandSale,2004;Chapter9).Initially,ecologistsusedsimplified,strategic modelstounderstandtheeffectsofmarinereservesonfishpopulations.Theyaddressed broadquestionssuchashowdoesmanagementwithreservescomparetoconventional managementintermsoffisheryyield(e.g.HollandandBrazee,1996;Mangel,1998; HastingsandBotsford,1999;Hart,2006;WhiteandKendall,2007)?Theanswersall indicatedthedualnatureofconventionalmanagementandmanagementbyreserves:a certaincatchwaspossible,anditcouldbeachievedbyarangeofpairsofvaluesoffishing mortalityratesandfractionsofthecoastlineinreserves.Anotherimportantquestion waswhichspatialconfigurationsofreservesconnectedbyadispersinglarvalstagewould supportpersistentfishpopulations(e.g.Botsfordetal.,2001;Gainesetal.,2003;Kaplan, 2006,Whiteetal.,2010a)?Thegeneralanswerswerethat(a)singlereserveswouldsupport persistentpopulationsofspecieswithdispersaldistancesshorterthanthewidthofthe reserve,and(b)combinationsofmanyreservesthatcoveredacertainfractionofthe coastlinecouldsupportspecieswitharangeofdispersaldistances.Thesestrategicmodels typicallydescribedpopulationsalongidealized,linearcoastlineswithlogisticdynamics orsimpleagestructure.Theyprovidedearlyguidancewhendecisionswereneededtobe madeinthefaceoflimiteddataandknowledge,andtheyalsolaterprovidedacheckon whethertheeventual,moredetailedtacticalmodelsmadesense.
Whenmanagementagenciesactuallybegantoimplementmarinereservesearlyinthe twenty-firstcentury,thedecision-makingprocessrequiredtheabilitytocomparethecosts andbenefitstospecificfishspeciesofspecificproposedMPAsatspecificlocations,hence tacticalmodelingwasrequired.Themodelsbecamemuchmorerealistic,includingage structuredmodelswithdensity-dependentrecruitment,linkedtolifehistorydatafrom severalspecies,andusingdataonhabitatdistributionandoceancurrentsalongreal coastlines.Thesemodelsprojectedtherelativeamountsofbiomassandfisherycatch expectedatspecificlocationsafterreserveimplementations(e.g.Kaplanetal.,2006; Pelletieretal.,2008;Kaplanetal.,2009;Whiteetal.,2010c,2013b;Hopfetal.,2015; reviewedbyPelletierandMahevas,2005;Whiteetal.,2011).
Fortunately,theresultsofsimulationsoftheserealistic,tacticalmodelsturnedouttobe consistentwiththecharacteristicspredictedbytheearlierstrategicmodels.Forexample, whenthefishingratealreadymanagedaspeciesatthemaximumsustainedyield,adding areservewouldonlycausetheyieldtodecline,acharacteristicofthedualnatureof reserveandconventionalfisherymanagementidentifiedearlierinthestrategicmodeling. Moreover,thestrategicmodelingoftenprovidedvaluableinterpretationsoftheresults oftacticalmodeling,thatmaynothavebeenappreciatedotherwise.Thecharacteristics ofpersistenceandyieldinreservesmentionedherearedevelopedmorethoroughlyin Chapters9and11.
WhatusefulinformationcanwetakefromtheseearlymusingsbyLevins,Holling, andMay?IdeallywewouldwantallthreeofLevins’characteristics,butrealismseems mostimportant.Forscientificprogressandusefulpracticalapplications,weneedtobe
abletoconnectourresultstotherealworld,henceweneedtousemodelcomponents thatcanbeobservedandmeasured.Itwillbeadvantageoustobegeneral,butweneed notalwaysbe;sometimesourinterestwillbefocusedonspecificsituations.Insome situationswewillneedtobeprecise,e.g.whenmakingprojectionsoftheconsequencesof proposedmanagement,butinothersituationsacomparativedescriptionofresults(e.g. thisparametervalueleadstogreaterpopulationstabilitythanalowervalue)willstillbe valuable.
Obviouslywewillsometimesbeconcernedwithabroadrangeofinstancesandissues, andwillbefollowingastrategicapproach,whileinotherswewillbeconcernedwith specificcases,andwillbefollowingatacticalapproach.Anessentialquestionconcerns thetruedegreeofgeneralityinvolvedinstrategicmodels.Modelsthatyieldmoregeneral resultsbyapplyingtoagreaternumberofspecificinstancesoftendosobecausetheyapply lesswelltoanyspecificinstance.Thistradeoffisillustratedbytheoldjokeaboutthedrunk seenlookingforhislostringoutsideofSam’sBar.Whenaskedwhatheislookingfor,he repliesthatheislookingfortheringhelostwhilecomingoutofJoe’sBar.Whenitis pointedoutthatJoe’sBarisseveralblocksdownthestreet,thedrunkreplies,“Iknow,but thelightisbetterherenearSam’sBar.”
Thekeytomakinggooduseofbothstrategicandtacticalapproachesistochoosesimple strategicmodelsthatpossessenoughrealismintheircriticalaspectsthattheirgeneral resultsareapparentinthespecificresultsoftacticalmodelimplementations.Theresults describedaboveforstrategicandtacticalmodelsofreservesareanexample(discussed furtherinChapter11).
1.2Logicalbasisforpopulationmodeling
Wecanalsogaininsightsintotheappropriateuseofpopulationmodelsfromaformal descriptionofthelogicalbasesfortheiruse.Inthisbooktheplannedusesofmodelscan beviewedastwofold:(1) scientific usetoimproveunderstandingofthemechanismsof populationdynamics,and(2) practical usetoprovideprojectionsformanagement.These dependondifferentlogicalbases.Wereviewthoseherewiththecaveatthattheseareonly sketchesofthebasicideas;adequatetodiscernthenatureofthetypeofmodelneededfor aparticularusage,butnotnecessarilyexpertviewsofthecurrentstatusofthephilosophy ofscience.Alsonotethatthesetwousesofpopulationmodelsdonotcoveralloftheir uses.Forexample,anotheruseofmodelsissimplyasa pedagogicaltool toillustratethe behaviorofpopulationswithoutgreatconcernforactualmechanisms.Wewillnotbe concernedwithsuchuseofmodelsinthisbook,althoughsomeofthemodelsinChapter 2areoftenusedinthatway.
1.2.1 Deductivereasoningandthescientificusesofmodeling
Wecharacterizethescientificuseofmodelsintermsofhowwemightanswerspecific questionsregardingthecausesofobservedpopulationphenomena.Forexample,thereis oftenconcernoverwhycertainpopulationshavedeclinedovertime,orhaveincreasedto dramaticallyhighabundance,orhaveexhibitedcyclicbehavior.Inalmostallinstances, therearemultipleproposedcausesfortheobservedbehavior.Takingcyclicbehavioras anexample,hypotheticalfactorspotentiallyresponsiblefortheobservedcycleswould include(a)anover-compensatorystock-recruitmentrelationship(i.e.acasewhererecruitmentactuallydecreasesasstockincreases,forlargeenoughstocklevels),(b)acyclic
environmentalvariable,(c)apredator/preyrelationship,andsoforth.Werefertothese ashypotheses,anddescribeaprogramforusingpopulationmodelstoevaluatethese hypothesesascausesoftheobservedcycling,decline,orincrease.
Thebasicdeductiveapproachtotestinghypothesesinpopulationmodelingisto incorporateeachhypothesisinapopulationmodelofthespeciesofinterest,andrun asimulationorsolvethemodeltoseewhetheritsoutput(e.g.thepatternofabundance overtime)producestheobservedbehavior.Wethenevaluatethemodelanditsderived consequencesasaconditionalargumentfromdeductivelogic(Box1.2).
Box1.2 CONDITIONALARGUMENTS
Conditionalargumentsfromdeductivelogicareargumentsofthefollowingform,withtwo premises(P1andP2)andaconclusion(C):
P1:If a then b
P2: a true(or b true,or a false,or b false)
C: b true(or a true,or b false,or a false,respectively).
Here“a”iscalledthe antecedent and“b”iscalledthe consequent.Ofthefourpossible outcomesofP2andC,twoarevalidargumentsandtwoareinvalid.Thevalid argumentsare(1):
P2: a true
C: b true,
whichiscalledconfirmingtheantecedent,and(2):
P2: b nottrue
C: a nottrue, whichiscalleddenyingtheconsequent.
Thelogicallyinvalidargumentsare(1):
P2: b true
C: a true, whichiscalledconfirmingtheconsequent,and(2):
P2: a nottrue
C: b nottrue,
whichiscalleddenyingtheantecedent.Notethattheargumentsarenamedafterthe statementinP2,whichcontainstheinformationavailableonwhichwecanbasethe conclusion.
Inusingmodelsandtheseconditionalargumentsfromdeductivelogic,weobviouslywant tomakeuseofthevalidargumentsandavoidusingtheinvalidarguments(Box1.2).We willdescribethisapproachformallyusingtheterminologyinBox1.2,andthenshowan exampleofitsapplicationtocyclicpopulationdynamicsinDungenesscrabs.Thebasic argumentbeginswiththeinitialpremiseP1,
If [MODEL] then [MODELOUTPUT],
wheretheMODELcontainsoneofthehypotheticalexplanationsoftheobservedphenomenonwhosecauseisinquestion;itis“true”ifitisthecorrectexplanation.MODEL OUTPUTisthepatternofpopulationdynamicsproducedbythemodel;itis“true”if itmatchestheobservednaturalphenomenon.Ifthemodeloutputdoesnotmatchthe behaviorinquestion,thenthesecondpremiseP2willbe“[MODELOUTPUT] nottrue”
(whichcorrespondsto“P2: b nottrue”inBox1.2),whereasifthemodeloutputdoesmatch thebehaviorinquestion,thesecondpremisewillbe“[MODELOUTPUT] true”(which correspondsto“P2: b true”).WecanseefromBox1.2thatif b isnottrue(i.e.themodel outputdoesnotmatchtheobservedpopulationbehavior),wecandrawtheconclusion(C) that a isnottruebyapplyingthevalidargumentofdenyingtheconsequent.Thiswould meanthatMODELisnottrue,whichwetaketomeanthatthehypothesizedmechanism inthatmodelisnotthecauseoftheobservedbehavior(presumingtheotherassumptions ofthemodelaretrue).If,ontheotherhand, b istrue,thecorrespondingargumentwould beconfirmingtheconsequent,aninvalidargument.If b istrue,wedonothaveavalid argumentavailable,hencewecannotdrawaconclusionaboutthetruthoftheMODEL. Theseresultsfromdeductivelogichavebeenusedtoformulateaprogramforresearch thatdependsonstrongresultsthatrejecthypotheses,whileavoidingweakresultsthat confirmhypotheses(Popper,1959;Platt,1964).Itconsistsofsequentiallytestingall proposedhypotheticalcauses.Ifthehypothesisisrejected(i.e.modeloutputdoesnot matchtheempiricalpopulationobservation),thenextstepistotestthesub-hypotheses, i.e.theassumptionsofthemodelotherthanthehypothesisbeingtested.Thisstepis taken(andrepeated)becauseoftheimportanceofthestepofrejectingahypothesis.If, ontheotherhandthehypothesisisnotrejected,theprogramsimplymovesontothe nexthypotheticalcauseofthephenomenonatissue.
Researchthatusedage-structuredmodelstostudythecauseofthecyclesinDungeness crabpopulationsinnorthernCaliforniainthelate1970s(Fig.1.1)provideanexampleof theuseofthisapproachininterpretationofpopulationmodels.Atthattime,researchon whetherthecyclescouldbecausedbyover-compensatorydensity-dependentrecruitment
Fig.1.1 ThecatchofDungenesscrabinCaliforniasince1950.Thispatternofcyclicvariability wasanalyzedinthelate1970stodetermineitscause.Catchisareasonablycloseapproximation tomaleabundanceinthisspeciesbecauseitisamale-onlyfisherywithhighlevelsofharvest ofmalesgreaterthanaminimumsizelimit.DatafromtheNOAAFisheriesInformationSystem https://www.fisheries.noaa.gov/national/commercial-fishing/fisheries-information-system-program
focusedontwohypotheticalmechanisms:(a)cannibalismofoldercrabsonnewrecruits tothebenthichabitat,whichhadbeenshowntooccur;and(b)densitydependencein fecundity,whichwasproposedtoexistbuthadnotbeendocumented.Tworesearch groupsconcernedwithdeterminingthecauseofthecyclesbothshowedthatinagestructuredmodelswithdensity-dependentrecruitment,cyclescouldoccurwithperiod roughlytwicethemeanageofadultcrabs.Basedonthatresult,BotsfordandWickham (1978)notedthatcannibalismofoldercrabsonyoungcrabscouldcausecycles,butthat theirperiodwouldbelessthantheperiodoftheobservedcycles(i.e.10y).However, McKelveyetal.(1980)interpretedthesameresultfromasimilar,age-structuredmodelas evidencetorejectcannibalismasthecause,anddeclareddensity-dependentfecundityto bethecause,becauseamodelwithdensity-dependentfecundityproducedcyclesofthe observedperiod.
Thedifferencesinthetwoapproacheslayintheirconclusionsandrecommendednext steps.Bothgroupsrejectedcannibalismasmodeled,butBotsfordandWickham(1978) recommendedfurthertestingofthesub-hypotheses(essentiallytheassumptionsoftheir modelthatcannibalismratewouldbeproportionaltoindividualmetabolicdemand), beforerejectingcannibalismasapossiblemechanism.McKelveyetal.(1980),ontheother hand,rejectedcannibalismasapossiblecause,andacceptedfecundityasthecause(thus confirmingtheconsequent).Theyrecommendedaprogramofstudytoshowthatdensity dependenceoffecundityexisted.BotsfordandWickham(1978)focusedonthepotential forastrongresult,i.e.possiblybeingabletorejectthecannibalismhypothesisbyfurther studyoftheactualagedependenceofcannibalism.McKelveyetal.(1980),ontheother hand,rejectedcannibalismandfocusedonfurtherstudyofadifferenthypothesisbased onthedeductivelyinvalidargumentofconfirmingtheconsequent.Theirrecommended firststepwouldbetoshowthatfecunditywasdensitydependent.Thequestionofthe causeofthecycleswentbeyondscientificcuriositysincethepracticalimplicationofthe differenthypothesesforfisherymanagementdiffered.Ifthefemaleswereresponsiblefor thecycles,fishingwouldnotbeaffectingthecyclicbehavior.However,ifmaleswere involvedinthedensitydependence,asincannibalism,truncationoftheagestructure ofmalesbyfishingwouldhaveadestabilizingeffectonthepopulation(moreonthisin Chapter7).
AsweshallseeinSection1.2.2,thelogicfollowedbyMcKelvey,etal.(1980),while invalidindeductivelogic(confirmingtheconsequent),isnotunlikethatfollowedinan acceptableinductiveargument.Inessencetheirconclusionswerebasedonchoosingthe modelthatcurrentlyprovidedthebestfit,acommonapproachinappliedecology.
1.2.2 Inductivereasoningandpracticalapplicationsofmodeling
Inductiveargumentsconsistofconclusionsdrawnfromanumberofobservations.The basicformofaninductiveargumentisthatoutof n trialsacertainpropositionhasbeen observedtobetrue Z percentofthetime,thereforewecanconcludethatitistrue Z percent ofthetime.Thisisreferredtoasanargumentbyenumeration(Salmon,1973).Inductive argumentsarenotdescribedasbeing“valid”or“invalid,”rathertheyarereferredto asbeingrelatively“stronger”or“weaker.”Anargumentbyenumerationisstrongerfor greater n,andconverselyweakerforlower n.
Inrepresentingpopulationdynamics,wemakeuseofaspecificformofanargument byenumeration,referredtoasanargumentbyanalogy.Theformofthatargumentis:if anobject, A,isthesameasanotherobject, B,incharacteristic1,incharacteristic2,in characteristic3,andsoon,uptocharacteristic n,thenonecanconcludethat A and B will
bethesameintermsofanewcharacteristicnotyetcompared.Thisargumentbecomes strongerasthenumberofcomparisonsincreases.Inthecaseofpopulationdynamics,the argumentcouldbethatapopulationmodelissimilartothepopulationintermsofthe patternofsurvivalversusage,fecundityversusage,individualgrowthversusage,andpast abundanceversustime;thereforeonecanconcludethatthemodelwillbesimilartomodel behaviorpredictedforthefuture.Again,thisargumentbyanalogyalsobecomesstronger asthenumberofcomparisonsincreases.Doingthisformallywithstatisticalapproaches andinformationcriteriarequiresstochasticmodels(butoftenlessformalapproachesare usefulandappropriate).
Stockassessmentsmadeforthepurposeofmanagingfisheriesareagoodexampleof turningtoargumentsbyinductionforthepurposesofmanagement.Examplesofthese canbeseenatwebsitesfortheregionalfederalfisherymanagementcouncilsintheUSA (e.g.thePacificFisheryManagementCouncil(PFMC)forthewestcoastofthecontiguous USA),andothersimilarsiteselsewherearoundtheworld.Thesearetypicallybasedona modelfittoseveraltypesofdataatonce:dataonagestructureofthecatch,afishery independentsurveyofabundance,andexistinginformationongrowthversusagefit previouslytosize-at-agedata.
1.2.3 Consequencesofdeductiveandinductivelogicforpopulationdynamics
Thequestionofwhyweneedtwodifferentkindsoflogic(inductiveanddeductive),with twodifferentsetsofseeminglyconflictingrules,mayhaveoccurredtothereader,justaswe imagineitdidtotheearlyGreekstudentsoflogicseveralmillenniaago.Deductivelogicis veryconservative,leadingusonlytoanumberofpotentialcausesofphenomenathathave beentestedandfoundnottobeactualcauses;itwithholdsjudgementonphenomena thathavenotbeenrejected,andneverconfirmsthataparticularcauseiscorrect(this wouldbeconfirmingtheconsequent).Thisdoesnotprovidetheinformationneededfor practicalapplicationstomanagement.Tomanagepopulationsinthefuture,weneedto knowwhattheactualcausesofthedynamicbehaviorare,andthatrequirestheapproach ofinductivelogic.
Theseparateusesofinductiveanddeductivelogichavealonghistoryinecology ingeneral(e.g.Dayton,1973).Also,Caswelletal.(1972)longagoseparatedmodels intothosedevelopedforthepurposesofbetterunderstandingandthosedevelopedfor prediction.Hewasaddressingthe“problemofvalidation,”whichhastodowithwhat onecanconcludefrompositiveoutcomesofmodelpredictions.Asstatedpreviously,these outcomesmeansomethingininductivelogic,butmuchlessindeductivelogic.
Muchmorecouldbesaidaboutinductiveanddeductivelogic,butherewedescribe themonlybriefly,solelytoassesstheirrequirementsofmodels(seeSalmon,1973and Chapter7ofFord,2000formoreinformation).Wesimplyconcludeforourpurposeshere thatwhetheryouareemployingdeductivelogicinascientificuseofmodelsorinductive logicinapracticalapplicationofmodels,youwillbemoreeffectiveifthemodelsexplicitly containtheactualobservablemechanismsinthepopulations.Thiswouldbesatisfied bythemodelshavingahighdegreeofwhatiscalled“realism”inthetrichotomyof Levins(1966).Intheuseofdeductivelogic,oneultimatelyneedsspecifichypothetical mechanismsthatcanbetested.Intheuseofinductivelogic,theargumentsimproveas thenumberofcomparisonsincreases,andrealismisbydefinitioncomparabilitywith reality.
1.3Thestateofasystem
Aformalapproachtodecidinghowtoconstructpopulationmodelscanbefoundin theearlydevelopmentsofsystemscience(Caswelletal.,1972;Zadeh,1973;Caswell 2001).Inthe1960s,scientistswerebeginningtograpplewithcomplexsystemsthrough mathematicalmodelsandcomputersolutionsforthosemodelsinaneffortcalledsystem science.Asystemwas“acollectionofobjects,eachbehavinginsuchawayastomaintain behavioralconsistencywithitsenvironment”(Caswelletal.,1972).Systemscience requiredaformal,consistentmethodforconstructingmodels,whichwasachievedby carefullydefiningthe state ofasystem.Thebasicideawasthatthebehaviorofasystem woulddependnotjustonthecurrentstimulusfromtheenvironment,butalsoonits history,i.e.howithadrespondedtopastenvironments.Systemscientistsrepresented theeffectofthepastcompletelybyexpressingthecurrentstateofthesystem.Inorder todescribehowasystemwouldrespondtoanexternalstimulusfromtheenvironment, thestateandthestimulus–response–staterelationshiphadtosatisfyseveralconditions. Themostilluminatingconditionforourpurposeswasthatthecombinationofthe statevariable,thecurrentstimulusfromtheenvironment,andthestimulus–response relationshiphadto uniquely determinetheresponseofthesystem(i.e.thestateofthe systematthenexttimeinstant;seeCaswell(2001)foramoreextensivediscussion).The ideathatthestateofthesystematonetimepointshouldcompletelydeterminethestate ofthesystematthenexttimepointisanalogoustothe Markovian propertyofcertain stochasticmodels(Box1.3).
Box1.3 MARKOVIANPROCESSES
Oneimportantpropertyofaproperlydefinedstatevariableisthataddinginformation aboutthepaststateofthesystemprovidesnomoreinformationaboutthefuturethanthe knowledgeofthecurrentstate.Thisisthesamepropertysharedbyanimportantclassof stochasticmodelscalledMarkovprocesses.OnecommonexampleofaMarkovprocessisa Markovchain,whichisarandomsequenceofevents(suchasthepositionofaparticleina fluidexhibitingBrownianmotion)inwhichthepositionatanypointintimedependsonlyon thepositionattheprevioustime.Thatis,predictionsofthefuturestateofthesystembased onthecurrentstatearenotchangedbyaddinginformationaboutpaststates.Markovchains arecommonlyencounteredinstatistics,andinChapter8wediscusstheiruseinstochastic populationmodels.Mostofthemodelswediscussinthisbookarenotstochastic,butthe statevariablesnonethelesshappentosharethissamepropertyofbeingMarkovian,and ofcontainingalloftheinformationattime t neededtopredictthestateofthesystemat time t + 1.
Thissimpleconceptofstatemaysoundobviousandtrivial,butitisverypowerful.In asenseitdeterminestheorganizationoftherestofthisbook.Glancingatthetableof contents,youwillseethatwebegininChapter2withsimplemodelsthatrepresenta populationintermsofthetotalnumberofindividuals(i.e.abundance, N),thenwemove quicklytoaddingagestructure,andrepresentingthepopulationintermsofthenumber
ateachage.Thereasonfordoingsoisthatweshowthatrepresentingapopulationby itscurrentabundance N isnotsufficientforustopredict,uniquely,howmanythere willbenextyear.Thatdepends(atleast)ontheagestructureofthepopulation,i.e.how manyofthe N wereyoungerthanageofmaturity,andhowmanywereolderandhence reproductive.
Whilewedevelopafairlycomprehensiveviewofpopulationdynamicsusingagestructuredmodels,weultimatelydecidedthatwehadtomoveontomodelsthatadd representationofpopulationstateintermsofsizestructure,i.e.thenumberofindividuals ateachageandsize.Thisisbecauseformanytaxa,reproductivematuritydependsonsize, notonage.Ifallindividualsgrewalongthesameplotofsizeversusage,wewouldnot needamodelwithbothageandsize,butcoulduseeitheranage-structuredorasizestructuredmodel.However,manytaxaexhibitplasticgrowth,wherebythegrowthatany timecoulddependonpopulationdensityortheenvironment(e.g.temperatureorfood). Also,formanyspecies,individualsatacertainagedonotallhavethesamesize.
1.3.1 Modelsofi-statesandp-states
Theconceptofstatecanapplytoeitheranindividualorapopulation(Metzand Diekmann,1986).Fromanindividualperspective,the i-statevariablesarethethingsyou needtoknowaboutanindividualtobeabletouniquelypredictwhatitsstatewillbein thenexttimeinstantanditsresponseintermsofreproductionandmortality.Thesocalled i-statescouldincludetheindividual’sage,size,reproductivestatus,levelofenergy reserves,andsoforth.
Fromapopulationperspective,oneofthewaysofformulatinga p-state,i.e.adescription ofthestateofapopulation,istodescribethe p-stateasthenumberofindividualsateach i-stateinthepopulation.Thestimulus–responsefunctionisthenalawofmassactionthat describesthe“flow”ofindividualsthroughthespacedefinedbythe i-statevariables.The numberateach i-stateattime t isdeterminedbythenumbertherepreviouslythatdied,the numberreproducedtothatstate,andthenumbersthat“grew”or“traveled”toandfrom thatstate.Oneoftheconditionsnecessaryforthis i-statedistributiontobeasufficient descriptionofthe p-stateisthatindividualswiththesame i-staterespondidenticallyto theenvironmentandeachother.Asecondnecessaryconditionisthatthepopulation output,theproductionofnewindividualsandtheir i-states,canbecalculatedfromthe distributionoverthe i-state.Together,thesetwoconditionsaresufficienttosatisfythe definitionofstate(MetzandDiekmann,1986).Onemightnoticethattheyarealsoan exampleoftheMarkovianpropertyweintroducedearlier(Box1.3).
Thesedefinitionswereenvisionedinadeterministiccontext.Theuniquepredictionof thefuturewouldobviouslynotbepossibleifoutcomeswerestochastic.Whenpopulationsbecomelocallysmall,theybecomeinherentlystochasticbecauseofdemographic stochasticity(seenextparagraph).Thismeansthattheapproachofusingthedistribution over i-statesasthestatevariablewillonlyworkifthepopulationabundanceremainshigh enoughatallpointsinthe i-state.Highenoughheremeanssuchthattheoutcomesof mortalityand/orbehavioralinteractionsdonothavetobeexpressedintermsofdiscrete numbersofindividuals.
Thisconditioncanbeunderstoodbyexplainingdemographicstochasticityasanexample,aphenomenonwewillrefertoinChapter8onrandomvariabilityinpopulations, butwhichweintroduceheretounderstandtheconsequencesofpopulationsbeing locallysmall.Supposewehaveapopulationinwhich,onaverage,ninetypercentofthe individualssurviveeachyear.Therearetwodifferentwaysthatwecouldrepresentthat
inamodel.Onewouldbesimplytomultiplythenumberatthebeginningoftheyearby 0.9toobtainthenumberattheendoftheyear.Toseehowthatwouldwork,assumewe startwithfifteenindividuals,asanexample.Multiplyingthatby0.9givesus13.5,which isproblematic,sincetherealpopulationwillnothavefractionsofindividuals.Wecan getaroundthisdeparturefromrealitybyfollowingadifferentproceduretoobtainthe numbersurviving:foreachindividualinthepopulation,eachyearweconceptuallyflip acointhatcomesupheadsninetypercentofthetimeandtailstenpercentofthetime. Headsmeansthattheindividualsurvives,tailsmeanstheydie.Theresultofthiswill beanintegereachyear,thusgettingaroundtheproblemofendingupwithfractionsof individuals.However,nowofcoursetheoutcomewillnotbethesameforeverysequence offifteencoinflips.Thuswehaveintroducedakindofrandomnessthatisactuallypresent inapopulation.Fortunatelythiskindofrandomnessiswellstudied,andthedistribution ofoutcomesisabinomialdistribution(Box1.4).
Box1.4 THEBINOMIALDISTRIBUTION
Thebinomialdistributionisusefulinunderstandingdemographicstochasticityandin decidinghowmanysimulationrunstomakewhentryingtocomputeaprobabilityofan outcomesuchasextinction.
ThebinomialdistributiondescribestheoutcomeofanumberofBernoullitrials.A Bernoullitrialistheprocessofconductingarandomexperimentthathasoneoftwopossible outcomes.Themostcommonexamplesareflippingacoin,ordrawingamarbleoutofa bagthatcontainsredandwhitemarbles.Inthelattercase,theprobabilityofdrawingared marblecouldrangebetween0and1,dependingontheratioofredtowhitemarblesinthe bag.Wewillusecoinflippingasourexample,butimagineacoindoesnotnecessarilyhave aprobabilityof0.5ofbeingheadsand0.5ofbeingtails,butratherhasaprobability p of beingheads(likethebagofmarbles, p couldbeanyvaluebetween0and1)andprobability 1 p ofbeingtails.
Imagineyouflipthecoinseveraltimesinarow.Theprobabilityofobtaining k headsin n flipsofthecoinis p(k) = (n k ) pk(1 p)(n k),where (n k ) isthebinomialcoefficient,whichis thenumberofwaysthatasequenceof n flipswith k headscanoccur: (n k ) = n!/ [k! (n k)!]
Theimportantcharacteristicsforourpurposesarethemeanandvarianceofthisdistribution. Themean,orwhatiscalled expectedvalue instatistics,ofthisdistributionis np,whichmakes intuitivesense(thenumberofflipstimestheprobabilityofgettingheadsoneachflip),and thevarianceis np(1 p).Inbothoftheexamplesforwhichwewillusethis,wewillbe interestedintherelativeamountofvariability,thestandarddeviationdividedbythemean, whichiscalledthecoefficientofvariation(CV).
Thismeansthattheamountofdemographicstochasticity(i.e.variabilityintheoutcomeof theBernoullitrials)declinesinproportiontothesquarerootofthenumberofindividualsin thepopulation.Thissimpleresultisalsousefulinotherways;forexample,whenestimating aprobabilityofextinctionbyrepeatedrandomsimulations,therelativestandarderrorof thatestimatealsodeclinesinproportiontothesquarerootofthenumberofsimulations (Harrisetal.,1987).
UsingtheformulasdescribedinBox1.4,wecanseethattherelativeamountofvariability duetothisbinomialprocess(survival)declineswiththesquarerootof n.Thus,aslong aswestartwithahighenoughnumber,theerrorincurredbysimplymultiplyingthe
numberofindividualsby0.9issmall.Forexample,withfifteenindividualsthecoefficient ofvariationwouldbe0.0775,whileifwehad150individualsitwouldbe0.0245,andwith 1500individualsitwouldbemerely0.0077.
Demographicstochasticityisanimportantsourceofvariabilityinpopulationdynamics. Theexamplewejustusedisdemographicstochasticityinsurvival,butitisnotdifficultto imaginethatthiskindofrandomnesscanariseinprocessesotherthansurvival,suchas thesexratioofamother’soffspring,orspecifickindsofbehavioralinteractionsbetween individuals.Demographicstochasticityisthekeyreasonwhythe p-statedescriptionin termsofthedistributionofabundanceateach i-statebreaksdownwhenpopulation numbersatany i-statearesmall.Itisimportanttoemphasize,forexample,thatina populationwithverylargetotalnumbers,butsmallnumbersofreproducingindividuals, stochasticitywillbeimportanteventhoughtotalnumbersarelarge.
1.3.2 Individualbasedmodels(IBM)
Inthelate1980sandearly1990s,mathematicalecologistsbegantodevelopanalternative toformingarepresentationofa p-stateasadistributionover i-states.Theybeganto realizethatdistribution-based p-stateswerenotconsistentwithtwoofthebasictenetsof populationbiology:(1)thatindividualsareinherentlyheterogeneous,andareinherently differentinmorewaysthancanbedescribedwithafew i-statevariables,and(2)thatin someofthemostimportantpopulationsituations,oneofthecausesofproblemsisthat populationsbecomelocallysmall.Oneexampleofthelatterislowspatialdensity,but otherswerelessobvious,e.g.behavioralinteractionsamongindividuals.Characteristics (1)and(2)areofcoursenotindependent,beingtwowaysofstatingthesamething:adding moredimensionstoamulti-dimensionalspacemakesindividualdensitiesperunitspace lower,bydefinition.Inotherwords,addingmore i-statesinordertoimproverealismwill necessarilyreducethenumberofindividualsineach i-statecategory,andthusreducethe precisionof p-staterepresentationsof i-statedistributions.
Somemathematicalecologistsdecidedtoavoidformulatingadistribution-based p-state model,andinsteadtowritecomputerprogramsthatjustkepttrackofthe i-statesofall individualswithinapopulationdirectly(Hustonetal.,1988;DeAngelisandGross,1992; Judson,1994;GrimmandRailsback,2005;RailsbackandGrimm,2011).Thismeantthat insteadofanalyzinghowadistributionover i-stateschangedwithtime,insteadthey numericallycomputedhowthe i-stateofeachindividualchanged,thenaddedthemup topresentresults.Thesemodelsweretermed individual-basedmodels (IBM),whichisabit ofamisnomersincethe p-statedistributionmodelswerealsoultimatelyindividualbased. Thesearenowalsocalled agent-basedmodels (RailsbackandGrimm,2011).Wedonot discussIBMsfurtherinthisbookbecausetheytendtodependonbruteforcecomputations ratherthanrequiringmathematicaldescriptionsofpopulationdynamics.Thefactthat computerscontinuetobecomefasterandIBMsrequirelittlebackgroundinmatrixalgebra orpartialdifferentialequationshasmadethempopular,andotherbooksaddressthem directly(GrimmandRailsback,2005;RailsbackandGrimm,2011).
Populationviabilityanalyses(PVA)ofthered-cockadedwoodpecker(Picoidesborealis) providesagoodexampleofusingIBMsinordertoaccuratelyrepresentpopulation dynamicsatlowabundance.Inthe1990s,initialPVAanalysesforthisspecieswere accomplishedwithdeterministicandstochasticstage-structuredmodels(Heppelletal., 1994;Maguireetal.,1995).However,thesemodelscouldnotadequatelyrepresentthe breedingbehaviorofredcockadedwoodpeckers.Someredcockadedwoodpeckersdisperse toacquirebreedingpositions,butothersremainontheirnatalterritoryasnon-breeding
helpers(Waltersetal.,1988).Thesehelpersconstituteapoolofreplacementbreederswho canreplacebreedersthatdie.Thishasabufferingeffectonbreedermortalitythatreduces thevariabilityinreproductioninawaythatrequiredanIBMtorepresent.Subsequent analysesshowedthatwhileprobabilitiesofextinctionincreasedwithareductionin breedingcolonies,greaterclumpingofterritoriesamelioratedthatbymaintaininglarger potentialreplacementpoolsofhelpers(Letcheretal.,1998;Waltersetal.,2002).
ThefactthatIBMsoragent-basedmodelsareprimarilysimulationbasedlimitstheeasy applicationofanalyticalsolutions,andthepotentialeasyorelegantinterpretationof thosesolutions.Italsolimitsopportunitiestomakeuseofvariousmathematicalmethods suchasstabilityanalysisandoptimization.However,simulationapproachescanbea usefulcheckontheformulationofanalyticmodels—ifthebehavioroftheanalyticmodel andthesimulationapproacharesimilaroverarangeofparameters,thenonehasmuch greaterconfidenceinusingthesimpleranalyticmodel.Additionally,MetzandDiekmann (1986)emphasizedthattheirapproachbasedon i-statesand p-stateswasessentiallyan analyticdescriptionofanIBMapproach.However,the i-state, p-stateapproachisnota panaceaasmodelsbasedonthisapproachcanquicklybecomecomplex.Yet,theapproach hasbeenshowntobepowerfulforunderstandingstructuredpopulations(deRoosand Persson,2013;andtheexampleswegiveintherestofthisbook).
1.4Uncertaintyandpopulationmodels
Inthisbookwedonotincludethetopicofestimatingpopulationparametersor populationstatesfromdata,hencewedonotdirectlyaddressaspectsofrandomness associatedwithestimation.Nonethelesswewillneedtorefertothedifferentkindsof variabilityoruncertaintyinpopulationmodeling.Thethreecategoriesofuncertaintyin populationmodelsare:(1)processerror,(2)measurement(orobservation)error,and(3) structuralerror.Thefirst,processerror,isvariabilitythatwedecidenottoaccountforwith explicitcausalmechanismsinapopulationmodel;rather,wesimplytreatitasnoisein thepopulationdynamics.Thedramaticvariabilityinrecruitmenttomostfishpopulations isagoodexample.Wecouldconceivablydescribethemultipleinteractingphysicaland biologicalprocessesthatleadtothatvariability,e.g.Caselleetal.(2010),buttheincrease inexplanatorypowerwouldlikelynotbeworththehugeeffort,sowesimplytreatitas “noise.”
Measurementerror(oralternatively“observationerror”)resultsfromtheinherent imprecisioninthemethodsweusetoestimatepopulationratesorcumulativestates suchasabundanceorbiomass.Mostobservationsofpopulationsincludeacombination ofmeasurementandprocesserror(Fig.1.2),andamajorchallengeistoseparatethem (deValpineandHastings,2002).Wetrytoseparatethembecauseprocesserrorwill affectdynamics,whereasmeasurementerrordoesnot.Forexample,populationvariability (e.g.year-to-yearchangesinthemortalityrate)influencesthelevelofriskofextinction, butobservationerror(e.g.slightlyoverestimatingthenumberofanimalsoneyearand underestimatingthenext)doesnot(directly).Therefore,onewouldnotwanttouse thetotalvariabilityinanabundancetimeseriestocalculaterisk,becausethatwould overestimatetheprocesserror.
Structuralerrorreferstotheeffectsofusingthewrongmodelformulation.Thesequence ofchaptersinthisbookfromChapter3toChapter9showseffortstodeterminethe i-statesthatleadtoavaliddefinitionofthestateofapopulation,hencetheyareefforts toreducestructuralerrors.Anotherfamiliarefforttoreducestructuralerrorswouldbe
Fig.1.2 Aschematicillustrationofprocesserrorandmeasurementerror.Attime t,theobserveddata, NO(t),differsfromtheactualstate, NA(t),becauseofmeasurementerror.Theactualstateisexpected tomovealongthedashedlinebutdoesnot,ratheritmovestoahighervaluebecauseithasbeen perturbedbyprocesserror(e.g.environmentalnoise).Again,at t + 1,theactualstatediffersfromthe observedstatebythemeasurementerror(whichhasadifferentvalueat t + 1thanat t).
thecalltomodelwholeecosystemsinsteadofindividualpopulations(ecosystem-based management;Pikitchetal.,2004;Fulton,2010).
Ofthethreetypesofuncertainty,thisbookwilldealwithhowprocesserrorand structuralerroraffectpopulationdynamics.Withregardtoprocesserror,wehave mentionedpreviouslyhowthepresenceofdemographicstochasticityinfluencesthe conceptualfoundationforourchoiceof i-statevariables.Wewillalsoaccountforasecond sourceofprocesserror:thelargeamountofenvironmentalvariabilityinpopulation vitalrates,especiallyinyoungerstages.Forexample,marinefishpopulationsexhibit order-of-magnitudevariabilityinrecruitment.Neartheendofthebookwewillseethat agestructureofpopulationscausesthemtobemoresensitivetocertainfrequenciesof variabilityinenvironmentaldrivers(suchastemperatureorrainfall)thanothers.Sincethe frequencycontentofenvironmentalvariabilityhaschangedinthepast(e.g.Cobbetal., 2003),andwilllikelychangewithachangingclimateinthefuture(e.g.Timmermann etal.,1999),wewillbeinterestedinhowpopulationsrespondtosuchchanges.
Whilewewillnotdealwithpopulationestimationinthisbook,observationerrorwill influenceourefforts.Forexample,wewillseeinChapters4,10,and11thatoneofthe criticalparametersforpopulationpersistenceisonethatishighlyuncertainandextremely difficulttoestimate.Asaconsequence,weareforcedtopursuemodelingavenuesthatseek tominimizedependenceonknowingthatparameter.
Aspreviouslynoted,structuralerrorplaysafundamentalroleinthisbook,atleast conceptually.Itguidesourchoicesamongthedifferentage,size,andspatialmodels throughoutthebook,togetherwithconsiderationofmeasurementuncertainty.The essentialtensionbetweenthesetwotypesofuncertaintyasmodelcomplexityvariesis illustratedinFig.1.3.Simplemodels(Chapter2),forexample,donotpossesstheage structurenecessarytorepresentthebehaviorofmostpopulationsfaithfully,sowouldbe ontheleftsideofthatfigure.Inthisbook,wegenerallyincreasemodelcomplexity,thus reducingstructuralerror.However,asthisfigureindicates,insodoing,weaccumulate agreaternumberofparameterswhosevaluesneedtobedetermined,thusnominally increasingmeasurementuncertainty.Thiseffectbecomesincreasinglyimportantasone goesfromdescribingpopulationbehaviortoattemptingtodescribethebehaviorof
Fig.1.3 Aschematicviewoftherelativeamountsofstructuralandparameter(measurement) uncertaintyasmodelcomplexityincreasesfromlefttoright.
wholecommunitiesorecosystems(Plagányietal.,2014;Collieetal.,2014).Notethat communitymodelershaveevaluatedtheeffectsoflumpingcomponentsincommunities (referredtoasaggregationerror,Gardneretal.,1982),butweknowofnosimilarattempts inpopulationdynamics.
1.5Levelsofintegrationinecology
Mostreaderswillhavebeenintroducedtothedifferentlevelsofecologicalintegration, perhapsevenasearlyasintheirsecondaryeducation(Fig.1.4).Inthisbookwewill obviouslybefocusedonthepopulationlevel,andinthischapterwehavespentconsiderabletimedescribingsomerelationshipsbetweenthepopulationandtheindividual levels.Thoserelationshipsareaconsequenceofthemoregeneralcharacteristicofthe levelsofecologicalintegration:theexplanationofbehavioratonelevelwillbefound intheratesatthenextlowerlevel.Thatis,populationdynamicsaredeterminedbythe combinationofindividualreproductive,mortality,growth,andmovementrates.Bythe sametoken,ecosystemdynamicswillbedeterminedbythegrowthratesofpopulations andtheirinteractions.
AsecondimportantcharacteristicofthelevelsofintegrationinFig.1.4isthatboth temporalandspatialscalesofvariabilitygenerallyincreasewithincreasinglevelsof integration.Thismeans,forexample,thatwewillgenerallybebetterabletostudy individualsoversufficienttimeandspacetounderstandtheirprocesses,thanweareable tostudypopulationsorecosystems.Recentdevelopmentsinoceanacidificationillustrate thisidea.Sincethe1990s,whenitbegantobeobviousthatincreasingCO2 levelsinthe atmospherewouldchangethebicarbonatechemistryintheocean,therebydecreasingpH, therehavebeenmanystudiesofchangesinindividualsurvivalandgrowthratesbrought aboutbylowerpH(seeKroekeretal.(2010)forareview),butveryfewstudiesofthe effectsofpHonmarinepopulationsorcommunities.Thestudiesattheindividuallevel providelittledirectionformanagement,sothereisagrowingappreciationoftheneedfor modelingstudiesto“scaleup”effectsattheindividualleveltotheirconsequencesatthe populationandecosystemlevels,e.g.LeQuesneandPinnegar(2012).
