Interval Analysis and its Applications to Optimization in 區(qū)間分析及其在優(yōu)化中的應(yīng)用

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IntervalAnalysisanditsApplicationstoOptimizationinBehaviouralEcology

by

JustinTung

CS490IndependentResearchReport

Instructor:DavidSchwartz

Date:December19,2001

TableofContents

Abstract 4

Introduction 5

IntervalAnalysis 5

BasicsandNotation 5

UncertaintyandApproximatingValues 5

IntervalArithmeticandFunctions 6

ForagingTheory 7

BasicsofForagingModels 7

SimplisticAnalyticForagingModel 8

TheOptimalResidenceTime 11

2.ResearchProblemandMethods 12

Motivation 12

ProblemswithFixedPointOptimizationinForagingModels 12

IntervalAnalysisasUncertaintyinMethod 13

ResearchProblem 13

Software 14

Methodology 14

Fixed-PointAnalysis 14

GeneralMethod 14

Algorithm:BisectionMethod 15

IntervalAnalysis 16

GeneralMethod 16

Algorithm:IntervalNewton’sMethod 16

VariationandConstraintsonParameters 17

3.NumericalAnalysisofModel 18

FixedPointAnalysis 18

GraphicalAnalysis 18

OptimizationandAnalysis 23

IntervalAnalysis 26

TrueSolutionsandIntervalOptimization 26

StabilityAnalysis 29

4.ConclusionsandFutureExploration 30

ResultsofNumericalStudy 30

ComparisonofFixedPointandIntervalRoots 30

ApplicationstoForagingModel 30

FutureExploration 32

Bibliography 33

Abstract

IntervalAnalysisisameansofrepresentinguncertaintybyreplacingsingle(fixed-point)valueswithintervals.Inthisproject,intervalanalysisisappliedtoaforagingmodelinbehaviouralecology.Themodeldescribesanindividualforaginginacollectionofcontinuouslyrenewingresourcepatches.Thismodelisusedtodeterminetheoptimalresidencetimeoftheforagerinaresourcepatchassumingtheforagerwantstomaximizeitsrateofresourceintake.Beforeapplyingintervalanalysis,fixed-point(non-interval)optimizationwillbedonetoserveasabasis.Certainparametersinthemodelwillthenbereplacedwithintervalsandinterval-basedoptimizationconducted.Acomparisonoftheintervalandfixed-pointresultswillbedoneaswellasanalysisofparameterintervalsandtheirconstraints,rootapproximations,andapplicationstothemodel.

Chapter1:Introduction

1.1IntervalAnalysis

1.1.1BasicsandNotation

ThispaperwillexplainonlythebasicsofIntervalAnalysis(IA)neededtounderstandthetopicscoveredandassumessomepriorknowledgeofIAandMatlab(see2.1.4regardingMatlab).ForaformalmathematicalintroductionandindepthcoverageofconceptsseeSchwartz(1999)orMoore(1966)listedinthereferences.Intervalanalysiswasinitiallydevelopedinthelate1960’stoboundcomputationalerroranditisadeterministicwayofrepresentinguncertaintyinvaluesbyreplacinganumberwitharangeofvalues(Schwartz17).Fixed-pointanalysisissimplyanalysisusingnon-intervalvalueswherethereisnouncertaintyinthevalues.Asaresult,IAuncertaintyconceptscanbeusedtomodelvaryingbiologicalparametersintheecologicalmodeltobeexploredinsection1.2andalsotoframefixed-pointresults.

IA’smathematicaldefinitionsandnotationsareextendedfromsettheoryandorderednumericalsetscalledintervals(Schwartz30).Thispaperconsidersclosedintervalanalysiswiththefollowingdefinitionsofaninterval(usingMatlabupperbound,lowerboundstylenotation):

inf(x)–denotestheinfinum,orlowerboundofx

sup(x)–denotesthesupremumorupperboundofx

1.1.2UncertaintyandApproximatingValues

Thereareaseveralusefulquantitiesrelatedtotheconceptoftheinterval:size,radius,andmidpoint.Thesize(orthickness)ofanintervalindicatestheuncertaintyinavalueandisspecifiedasawidth:w(x)=sup(x)-inf(x)(Schwartz32-33).Intervalswithzerothicknessarecrispintervalswhereasnon-crispintervalssaidtobethick.Theconceptsofradiusandmidpointareusefulindescribingintervalsaswellasconstructingthem.Theradiusandmidpointaredefinedas(Schwartz33):

rad(x)=w(x)/2

mid(x)=(sup(x)+inf(x))/2

Toconstructanewinterval,onewayistouseanoriginalvalue,whichisavaluethatsuppliesthemidpointpointofanewinterval.Then,acertainradius(uncertainty)canbeaddedtoandsubtractedfromtheoriginalvaluetoobtainanewinterval(Schwartz35).Similarly,themidpointcanalsoserveasanapproximationtoavaluewithanerrorofplusorminustheradius.Usingthesedefinitions,thepercentageuncertaintyinamidpointvaluewouldbe:p=100*rad(x)/mid(x)(Schwartz148).

1.1.3IntervalArithmeticandFunctions

Theresultsandpropertiesofintervalarithmeticwillbeomittedforthissection;however,IrecommendreferringtoSchwartz(1999)tounderstandthebasicsofintervalarithmetic.Thefundamentalprinciplesinintervaloperationsareindependenceandextremes.Independencemeansnumericalvaluesvaryindependentlybetweenintervalsandextremesmeansintervaloperationsgeneratethewidestpossibleboundsgiventherangesofvalues(Schwartz37-38).Interval-valuedfunctionsfollowfromintervalarithmeticofwhichtherearetwotypes:intervalextensionsandunitedextensions(ortruesolutionsets).Intervalextensionsarefunctionswhereintervalarithmeticisappliedtocalculateresults.Unitedextensionsaremorecomputationallyintensiveandinvolvecalculatingfixed-pointresultswithallpossiblecombinationsofvariableintervalendpoints.Thedisadvantageoftheintervalextensionsisthattheycanoverexpandthetruesolutionsetsofafunction(Schwartz45-49).Thisqualityofintervalextensionsisunfortunatesincebothtypesofextensionsguaranteecontainmentofallpossiblenumericalresultsofthefunctiongiventheinputs.Also,bothextensionssatisfyapropertycalledinclusionmonotonicity(giveninputs,anextensiongeneratesthewidestpossiblebounds),whichissimilartotheextremesprincipleofintervalarithmetic(Schwartz56).

1.2ForagingTheory

1.2.1BasicsofForagingModels

Foragingmodelsingeneralstudytwobasicproblemsofaforager:whichfood/preyitemstoconsumeandwhentoleaveanareacontainingfood(aresourcepatch).Thispaperwillconcentrateonthelatterasanoptimizationproblem.Beforegoingintodetailsofthemodel,itisimportanttounderstandtheframeworkofforagingmodels.StephensandKrebspointoutthatforagingandoptimalitymodelshavethreemaincomponents,decision,currency,andconstraintassumptions(5).Decisionassumptionsdeterminewhichproblems(orchoices)oftheforageraretobeanalyzedandthesechoicesareusuallyexpressedasvariables.Theoptimizationproblemcomesfromassumingbehaviourandevolutionarymechanismsoptimizetheoutcomeofaforager’schoices.Currencyassumptionsprovidemeansofevaluatingchoices.Thesechoicesusuallyinvolvemaximization,minimization,orstabilityofasituation.Choiceevaluationisembodiedinthecurrencyfunction(arealvaluedfunction),whichtakesthedecisionvariablesandevaluatestheiroutcomeintoasinglevalue.Constraintassumptionsarelimitationstothemodelandrelatedecisionvariableswiththecurrency.Limitationscanbegeneralizedto2types,extrinsic(environmentlimitsonanimal)andintrinsic(animal’sownlimitations).Also,therearethreegeneralconstraintassumptions(alsoassumedbythemodelinsection1.2.2)forconventionalforagingmodels:

1)Exclusivityofsearchandexploitation–thepredatorcanonlyconsumeorsearchforpatches/preyandnotperformbothactionsarethesametime

2)SequentialPoissonencounters–items(preyorpatches)areencounteredoneatatimeandthereisaconstantprobabilityregardingprey/patchmeetingsinashorttimeperiod

3)Completeinformation–theforagerbehavesasifitknowstherulesofthemodel

Thesethreeconceptsofdecision,currency,andconstraintprovidemeansofoptimizationgivenchoices,howtodeterminetheirsuccess,andlimitations(StephensandKrebs6-11).

1.2.2SimplisticAnalyticForagingModel

Giventhestructureofaforagingmodel,itiseasytoframeamodelexaminingtheforagingofasingleanimaloveracollectionofdistinctresourcepatches.Ihavetakenthemodelalongwithitsdecision,currency,andconstraintassumptionsfromWilson(2000)soderivationsofthemodel’sequations,itsorigins,andananalysisandextensionsofthemodelinCcanbefoundinhisbook.Therulesoftheforagerinthemodelarethattheanimalstaysforafixedtimebeforemovingtoanewresourcepatch,timeisdiscrete,andpatchresourcevalues(biomass,energy,etc)growlogistically.Thedecisionassumptionliesinthedeterminationofthefixedtimevalue.Thecurrencyfunctionallowsustooptimizetheanimal’ssituationgiventhisfixedtimeandalsoallowsustoapplyconstraintstothemodel(Wilson152).Despiteitsname,thesimplisticanalyticforagingmodelactuallycharacterizesforagingsimulationresultswellusingthemodel’sspecifications.Hereisalistofparameterstakeninbythemodel:

Toderivethemodel,wecanstartanalyzingtheresourcesideassumingthattheithpatchwithouttheforagerconsuminggrowslogistically:

Wheretistimeandriistheamountofresourcesintheithpatch.Ifaforagerentersagivenpatchf,resourcedynamicscanbemodeledasfollows:

Inthefthpatch,theconsumerdecreasestherateofgrowthbyafactorrelatedtobeta,theconsumingrate.Tomodeloverallpatchgrowth,averagepatchgrowthforN-1identicalpatchesisaddedtothepatchgrowth(ordecay)ofthefthpatch:

Toachieveanequilibriumresourcedensityr*,wesetdr/dt=0andsolveforryielding:

AquickanalysisofthelimitasNapproachesinfinityshowsthattheforager’seffectisinsignificantatequilibriumsincethetermcontainingbetagoestozeroandr*goestoKasexpected(Wilson153-154).Theresourcesideprovidesuswithanenvironmentandextrinsicconstraintsthatwillaffectthecurrencyfunctionwhichliesontheforagersideofthemodel.Keytoresourceexploitationmodelsisthegainfunction,g(t).Thegainfunctionspecifiestheamountconsumedfromaresourcepatchgiventimet.Assumingtheforagerlandsonaresourcepatchalwaysinequilibriumr*,g(t)isthetimeintegralofitsinstantaneousconsumptionrateminusitsmetaboliccosts.Metaboliccostsrepresentintrinsicconstraintssincetheanimalmust“pay”thesecostswhenforaging.

Thextandxmconstraintsactasintrinsiclimitationsonthemodelsincethetravelingcostspreventstheforagerfrommovingquicklyfrompatchtopatchandskimmingresources,whilethemetaboliccostcausestheforagertogatherresourcesforthreatofdeath.Inordertoevaluateg(t),werequireananalyticalsolutiontorf,whichmeasurestheresourcesinthepatchtheforagerisin.Solvingthefirstorderdifferentialequationfromtheresourcesidederivationsforrfusingseparationofvariablesyields:

Thensubstitutingthisequationintog(t)andsolvingtheintegralweget:

Usingthisgainfunction,thecrucialequationfromtheforagerperspective,thenetforagingratefunctioncanbecalculatedas:

R(t)isthecurrencyfunctionforourmodelsinceitisthebasisofchoiceevaluationandoptimizationforthemodel(Wilson154-155).Italsocombinesthedecisionvariablesandconstraintsintoonevalueandwillbeabasisforgraphicalanalysislateron.

1.2.3TheOptimalResidenceTime

Assumingbehaviouralandevolutionarymechanismsdriveforagerstooptimizethetimespentoneachpatch.Thisassumptionimpliesthattheywillstaylongenoughtooptimizetherateofresourceconsumption,r(t).Mathematically,thischoiceimpliesthemaximizationofr(t):

wheret*iscalledtheoptimalresidencetime.Forratemaximization,r’’(t*)<0mustalsobechecked;however,assumingr(t*)isatamaximum,weobtaintheequation:

whichrelatesr(t)tothederivativeofthegainfunction.Thesecondfunctionaboveisthefunctiontobeusedforrootfinding.Essentially,theoptimaltimetoleaveapatchiswhentheexpectedrateofresourcereturndecreasestotheaveragerateofreturnofanewpatch.Thisresult,whichisastatementofthemarginalvaluetheoremforforaging,isareasonableestimationofanimalbehaviourconsideringaforager’sdesiretomaximizeonitsresourceintake(Wilson156-157).Oneprobleminoptimizationproblemsisfiguringoutwhetheranoptimalpointexistsandinthiscase,wemustbesurer’’(t*)<0andbesuresuchat*exists.Unfortunately,duetothecomplexityofg(t)isitnotpossibletosolveforanexplicitsolutionoft*sotojustifyexistenceofasolution,weturntotheoreticaljustificationsand,laterinsection3.1.1,graphicalmeans.Duetotheconditionsspecifiedinforagingmodels,gainfunctionsare“well-defined,continuous,deterministic,andnegativelyaccelerated”functions(StephensandKrebs25-26).Thisoutcomeresultsfromassumingpatchresourcesaresufficient(i.e.theequilibriumresourceissufficientlylarge)enoughthat,whentheforagerentersapatch,r(t)willreachamaximumandthendecreaseuntilthegainfunctionreachesanasymptoticmaximumwhenfurthertimespentintheresourcepatchdoesnotyieldsignificantlymoregaininresources.Thereasonforthegainfunction’spropertiesistheassumptionthatpatchescontainafinitenumberofresourcesandforagingdepletesthem.Thisassumption;however,reliesontheassumptionsaboutresidencetime,foragingrates,andresourcepatchdynamicsingeneral(StephensandKrebs25-26).Duetothesimplenatureofthemodel,thesegainfunctionpropertiesholdsor(t)doesreachamaximumatt*andr’’(*t)<0.

Chapter2:ResearchProblemsandMethods

2.1Motivation

2.1.1ProblemswithFixedPointOptimizationinForagingModels

StephensandKrebs(1986)discussvariouslimitationsandcriticismsofbehaviouralecologyoptimalitymodels.Onecriticismhastodowith“staticversusdynamic”modelingsincebasicforagingmodelsoftendonottaketheanimal’sstateintoaccount(i.e.whetherananimalisstarvingorfullyrestedandfed)(StephensandKrebs34).Also,aproblemthatoccursduringtestingphasesofamodeliswhenitbreaksdown.Atthatpoint,theecologistmustre-analyzethemodeltofindwhatisincorrect,oftencheckingconstraints(StephensandKrebs208)

2.1.2IntervalAnalysisasanUncertaintyMethod

IAcanaddressbutnotcompletelysolvetheproblemsstatedabove.OneofthestrengthsofIAisitsabilitytoevaluateawholerangeofvaluesinonecalculationthatwouldtakeaninfinitenumberoffixed-pointcalculationstoproduce.Asaresult,IAcouldsimulatethepresenceofmultiplestatesofananimaland/oritsenvironmentbyplacinguncertaintyinthemodel’sparameters.Thismethodprovideseasydeterministicimplementationofmultiplestatemodelsandproducesrangesofvaluesforevaluation.Thismethodpartiallyaddressesthesecondproblemofwhenamodelbreaksdown.Amodelcouldbreakdownduetoincorrectassumptionsaboutconstantforagerorenvironmentstates.AnotherapplicationofIAtotestingisthatIAisnumericallysuperiorwhenitcomestotestingdifferentacceptableuncertaintiesinvaluescouldhelpidentifyproblemsinthemodelorunrealisticassumptions.

2.1.3ResearchProblem

ThepurposeofthispaperistointroduceIAmethodstothesimplisticanalyticforagingmodelandcalculateintervaloptimalresidencetimesforintervalparameters.Atsametime,solvingthefixedpointoptimalresidencetimewillprovideframeworkfromwhichtoanalyzetheintervalresults.TheoptimizationwillbedoneforpatchsizesN=3,5,10,and20.Theparameterswithuncertaintywillbe:

Theseuncertaintiescouldbestrengthenedwithfieldworkstudies;however,forsimplicitytheyaredeterminedapriori.Aftercomputingintervaloptimalresidencetimes,theintervalsandtheirfixed-pointapproximationswillbecomparedtothefixed-pointoptimaltimestocomparealgorithmsandtoanalyzethefunctions’behaviourunderbothmethods.Onamoretheoreticalside,stabilityanalysisofthemodelwillbeconducted.Thisanalysisinvolvesvaryingoneuncertainparameter,whileholdingtheothersconstantuntilthemodelfails.Therefore,stabilityanalysisisusedtoseeperformanceofthemodelunderparameteruncertaintyperturbations.Conditionsforfailurewillbespecifiedinsection.

2.1.4Software

ThelanguagetobeusedisMatlabversion5.3withanadd-ontoolboxcalledINTLABprogrammedbySiegfriedRump(2001).RefertothereferencesfordocumentationontheINTLABtoolbox.Forthispaper,theINTLABtoolboxisusedtoprovideintervaldatastructures,implementationofintervalarithmeticandinterval-valuedfunctions,aswellasbasicfunctionsforradius,midpoint,andintersectionintervalfunctionsinMatlab.

2.2Methodology

2.2.1Fixed-PointAnalysis

GeneralMethod

Rootfunc(t)specifiedbelowisthefunctionwhoserootweareseeking:

Sincerootfunc(t)hasrelativelycheapfunctionevaluationsitisusefultoperforma“graphicalsearch”fortheroot(VanLoan294).Thisprocedureinvolvesplottingthefunctioninthetimeintervalofinterestandexaminingitsroots.Inadditiontothisfunction,duringthefixed-pointanalysis,wewillalsoplotr(t)tosearchfortheexistenceofmaximumsaswellasrootfunc’’(t)toconfirmthatrootfunc’’(t)isindeednegativeintheintervalofinterest.Although,graphicalsearchesarerathertrivial,theyprovidealargeamountofinformationconfirmingtheoreticalconclusionsinsection1.2aswellasenablingapictorialviewoftheobjectiveandrelatedfunctions.Anotheruseoftheplottingoffunctionsbeforeoptimizationistousetheplotstogeneratestartingintervalsforiterationsofrootfindingmethods.Inordertoplotrootfunc,r(t),andr’’(t)itisnecessarytoimplementequationsforg(t),g’(t),andr’’(t).Thedetailsforcalculatingg’(t)andr’’(t)areleftoutsinceg(t)isarathermessyfunction,butthederivativesareimplementedintheMatlabcodeforsection3.1.1.Assumingthepropertiesofthegainfunctiondiscussedin1.2.3,whichwillbeconfirmedinsection3.1.1,itisnecessarytochooseanalgorithmthatwillproducearootgiventheconditionsofrootfunc(t).

Algorithm:BisectionMethod

Sincerootfunc(t)iscontinuousandchangessignwithintheintervalofinterest,thebisectionmethodcanbeused.Thismethodinvolvescalculatingasequenceofsmallerandsmallerintervalsthatbracket(contain)therootofrootfunc(t).Themainalgorithmproceedsasfollows(ifrootfunct(t)=f(t))givenabracketinginterval[a,b]:

assumef(a)f(b)0andletm=(a+b)/2

eitherf(a)f(m)0orf(m)f(b)0

inthefirstcaseweknowtherootisin[a,m]elseitisit[m,b]

Ineithersituation,thesearchintervalishalvedandthisprocesscanbecontinueduntilasmallenoughintervalisobtained.Sincethesearchintervalishalvedwitheachiteration,thebisectionmethodexhibitsO(n)convergence.Theonlytrickypointsaretooptimizethemethodsothatonlyonefunctionevaluationisrequiredperiterationafterthefirstandtoestablishasafeconvergencecriterionsothatthetoleranceintervalisnotsmallerthanthegapinthefloatingpointnumbersbetweenaandb.VanLoanprovidesthecoreofthecodeforthebisectionmethodwithslightmodificationstofittheparametersofthemodel(280).AlthoughthebisectionmethoddoesnotexperienceO(n^2)convergenceliketheNewtonmethod,rootfunc(t)issimpleenoughthatitconvergesquicklyforpracticalpurposes.Also,itissimple(algorithmicallyanddoesn’trequirerootfunc’(t)implementation)andtranslateswellintotheideaofsearchingusingastartinginterval[a,b],whichwewilluselaterintheintervalrootfinding.

2.2.2IntervalAnalysis

GeneralMethod

Whenchangingfromfixed-pointtointervalbasedrootfindingtherearesomeimmediatedifferences.Therootisnolongeracrispintervalsinceiterationsusinganinterval-valuedfunctionproduceintervals.Asaresult,convergencecriterionsandthegeneralmethodsofrootfindingmusthaveasetvaluedapproach.Becausetheoptimalresidencetimeswillinherentlybethickintervalssincetherootfunc(t)isnowanintervalfunctionwithuncertainparameters,convergencewillbesolvedsimplebysettingamaximumnumberofiterations.ThereasonthisconvergenceguaranteesanenclosureoftherootisduetotheIntervalNewtonmethod,whichisbasedonthefixed-pointone.

IntervalNewtonMethod

Fordetailsofthemathandconvergencepropertiesofthealgorithm,refertoKulischetal.(2001).Thisalgorithm,whenfindingrootsoffixed-pointfunctionsexhibitsO(n^2)convergence.LiketheBisectionMethod,itrequiresabracketingintervaltobeginandwitheachiterationgeneratessmallerandsmallerintervals(ifpossible),whichareboundedbyintersectionswithpreviousiterations.Thealgorithmisasfollows:

wherethex’sareintervals,m(x)isthemidpointofatheintervalx,andfisthefunctionwhoserootweseek(Kulischetal.35-36).Thesimilarityofthisalgorithmtothefixed-pointNewtonmethodisthatastartingintervalmustbesuppliedandtheintervalsizeisdecreasedusingaf(x)/f’(x)termduringiterations.Thisalgorithmisalmostassimpleasthebisectionmethodsinceaneasyconvergencecriterionhasbeenspecified;however,theintervalNewtonrequiresimplementationofrootfunc’(t)andrequiresforanevaluatedintervalx.Throughcomputationaltrials,Ihavedecidedtoincreasethecomplexityofthefunctionevaluationsforthef(x)/f’(x)bycomputingitsunitedextensioninsteadofaintervalextension.Sincebothf(x)andf’(x)involvemanyintervalarithmeticcalculationsinanintervalextension,thevaluesareoverexpandedfromtheirtruesolutionsetandincomputingoptimalresidencetimesweareinterestedinfindingtightboundsontherootgivenmodeluncertainties.Asaresult,theintervalNewtonstepiscalculatedbyfindingtheminimumsandmaximumsoff(x)andf’(x)givenallthecombinationsoftheendpointsoftheparametersandthetimeintervalandproducingunitedextensionvaluesforbothquantities.

VariationandConstraintsonParameters

Asoutlinedin2.1.3,percentuncertaintiesinthealpha,beta,xt,andxmparameterswillbeaddedwhenanalyzingthemodelusingintervalrootfinding.Thisvariationofparametersservestoaddresstheproblemswithfixed-pointoptimizationoutlinedin2.1.1.Thepotentialuseforintervalparametersliesinmodeltesting,simulatingarangeofenvironmentandforagerstates,andrelaxationofconstraints.Besidesreplacingparameterswithintervals,wealsowanttoconductthestabilityanalysismentionedin2.1.3.Thisprocessinvolvesincreasinguncertaintiesofagivenparameterwhilekeepingallotherparametersconstantuntilthemodelfails.AfterspecifyingtheintervalNewtonalgorithmwecanconstructafailurecondition.FailureofthemodelcanbeconsideredtooccurwhentheintervalNewtonmethodreturnsoftheinitialintervalsuppliedtotheintervalrootfinderastheoptimaltimeintervalforanyNnumberofpatches(i.e.thealgorithmwasunabletoprovidetighterboundsforoptimalresidencetimethantheinitialinterval).Thisinitialintervalwillbechosentobeasufficientlythickintervalfromfixed-pointanalysis,whichenclosesthetimeintervalofinterest.Asaresult,theintervalwillbetheonechosentoinitiatethebisectionmethod.Thereareotherconditionsthatcouldclassifyfailure;however,fromamodelstandpoint,averythickintervalforanoptimalresidencetimeisnotveryusefulsinceitimpliestoomuchvariabilityinaforager’sbehaviour.Consequently,stabilityanalysisisanumericallyintensiveprocedureinvolvinggradualincreaseduncertaintiesofaparameteruntilmodelfailure.Itisinterestingmorefromatheoreticalstandpointsinceitdescribeslimitationsofthemodelaswellasextremesituationsandtheireffectsonaforager’soptimalresidencetime.

Chapter3:NumericalAnalysisofModel

Inthischapter,IwillincludeMatlabcodeofkeyfunctionsimplementingthegeneralmethodsdiscussedin2.2,supportingfunctions,andalgorithmsastheyareusedinthenumericalanalysisofthemodel.

3.1Fixed-PointAnalysis

3.1.1GraphicalAnalysis

Insection,thegraphsofr(t),rootfunc(t)androotfunc’(t)wereseentobeinformativeforselectinganinitialintervaltobeginthebisectionandintervalNewtonmethods.Also,thesegraphshelpvisualizethebehaviourofthemodelastimepassesaswellasconfirminganoptimalresidencetimeexists.Thefollowingfunctionsarerequiredtoimplementthefunctionsandgraphthem.Also,thefollowingparameterspecificationsfromWilsonareused(155):

functiongain=gain(t,alpha,K,beta,xt,xm,N)

%GAIN(t,alpha,K,beta,xt,xm,N)GainFunction

%

%Generalevaluationoftheg(t)functionwhich

%takesinthevariablesinorder:time,patchgrowth

%rate,patchcarryingcapacity,consumingrate,

%travelcost,metabolicrate,andnumberof

%resourcepatches

rstar=(1-beta/alpha/N)*K;

deltaBA=beta-alpha;

num=(alpha*rstar+...

K*(deltaBA))*exp(deltaBA*t)-...

alpha*rstar;

denom=K*deltaBA;

inLog=num/denom;

gain=(beta*K/alpha)*(log(inLog)-deltaBA*t)-...

(xt+xm*t);

functionintakeRate=r(t,alpha,K,beta,xt,xm,N)

%R(t,alpha,K,beta,xt,xm,N)IntakeRateFunction

%

%Calculatesthenetintakerate.Takesinthevariables

%inorder:time,patchgrowthrate,patchcarrying

%capacity,consumingrate,travelcost,metabolicrate,

%andnumberofresourcepatches

g=gain(t,alpha,K,beta,xt,xm,N);

intakeRate=g./t;

functiongainprime=gainprime(t,alpha,K,beta,xt,xm,N)

%GAINP(t,alpha,K,beta,xt,xm,N)GainFunctionDerivative

%

%Generalevaluationoftheg'(t)function

%

%Takesinthevariablesinorder:time,patchgrowth

%rate,patchcarryingcapacity,consumingrate,

%travelcost,metabolicrate,andnumberof

%resourcepatches

rstar=(1-beta/alpha/N)*K;

deltaBA=beta-alpha;

exppart=(alpha*rstar+K*deltaBA)*exp(deltaBA*t);

num=deltaBA*exppart;

denom=exppart-alpha*rstar;

gainprime=(beta*K/alpha)*(num./denom-deltaBA)-xm;

functionrootfunc=rootfunc(t,alpha,K,beta,xt,xm,N)

%ROOTFUNC(t,alpha,K,beta,xt,xm,N)RootFunction

%

%Thisfunctiontheonewewanttofindtherootof

%Takesinthevariablesinor