Oscillations in a Size-Structured Prey Predator Model

MISSING IMAGE

Material Information

Title:
Oscillations in a Size-Structured Prey Predator Model
Physical Description:
1 online resource (69 p.)
Language:
english
Creator:
Bhattacharya,Souvik
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Martcheva-Drashanska, Maia
Committee Members:
Rao, Murali
Hager, William W
Yan, Liqing
Bolker, Benjamin M

Subjects

Subjects / Keywords:
Mathematics -- Dissertations, Academic -- UF
Genre:
Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
This dissertation introduces a predator-prey model with the prey structured by body size based on reports in the literature that predation rates are prey-size specific. The model is built on the foundation of the one-species physiologically structured models studied earlier. Three types of equilibria are found: extinction, multiple prey-only equilibria and possibly multiple predator-prey coexistence equilibria. The stabilities of the equilibria are investigated. Comparison is made with the underlying ODE Lotka-Volterra model. It turns out that the ODE model can exhibit sustain oscillations if there is an Allee effect in the net reproduction rate, that is the net reproduction rate grows for some range of the prey's population size. In contrast, it is shown that the structured PDE model can exhibit sustain oscillations even if the net reproductive rate is strictly declining. Those occur, however, if reproduction is size specifc and limited to individuals of large enough size. Simulations are presented to support our hypothesis that size-specifc predation can destabilize the predator-prey equilibrium in the PDE model.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Souvik Bhattacharya.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Martcheva-Drashanska, Maia.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2011
System ID:
UFE0043216:00001


This item is only available as the following downloads:


Full Text

PAGE 1

OSCILLATIONSINASIZE-STRUCTUREDPREYPREDATORMODELBySOUVIKBHATTACHARYAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

PAGE 2

c2011SouvikBhattacharya 2

PAGE 3

Tomymotherandfather,whoseconstantsupportandinspirationhasleadmetoachievemygoal 3

PAGE 4

ACKNOWLEDGMENTS IamreallyblessedtobeastudentoftheDepartmentofMathematicsatUniversityofFlorida.Thedepartmentbringsmeclosetomanyfamousprofessorsintheeld,whosecareandguidancehelpedmebecomeacompletemathematician.IdeeplyacknowledgethecareandguidancereceivedfrommyadvisorDrMaiaMartchevawhoseconstantsupporthelpedmeunderstandthedifferentaspectsofMathematicalBiologyandalsotodeveloptheskillrequiredinmodelingmathematicalbiologicalsystems.Iwouldalsoliketotakethisopportunitytothanksallotherprofessorsinourdepartmentwhoseperiodicalsupportisalsonoteworthy.Myendlessgratitudetoeachandeveryoneofthem.Lastbutnottheleast,myunendinggratitudetomymotherandfatherwhosesacrice,unconditionalloveandblessingwasalwayswithme,guidingmealongtheway.IshallalsoliketomentiontheunconditionalloveandsupportfrommybrotherwhichIfoundinallmystepstillthisstage. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 6 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 8 1.1Background ................................... 8 1.1.1Food-webEcology ........................... 8 1.1.2PredominanceofBody-sizeinFood-chainNetwork ......... 10 1.2RelatedMathematicalWork .......................... 10 2THEMODELANDINVESTIGATIONOFTHEEQUILIBRIA ........... 12 2.1TheODEModelanditsAnalysis ....................... 12 2.2ThePDEModel ................................. 15 2.2.1EquilibriaoftheSize-structuredPredator-preyModel ........ 17 2.2.2Prey-onlyEquilibria ........................... 19 2.2.3Predator-preyCoexistenceEquilibria. ................ 24 2.2.4StabilityAnalysis ............................ 27 2.2.5ExtinctionEquilibrium(u=0,P=0) ................. 29 2.2.6Prey-onlyEquilibria ........................... 31 2.2.7StabilityofaCoexistenceEquilibrium ................. 38 3HOPFBIFURCATIONANDOSCILLATIONS ................... 42 3.1-stepFunction,EverythingElseisConstant. ................ 42 3.2-stepFunction&-stepFunction ..................... 45 4NUMERICALANALYSISOFTHEPDEMODEL .................. 51 4.1NumericalScheme ............................... 51 4.2NumericalAnalysis ............................... 51 4.3SimulationsusingtheNumericalDiscretization ............... 62 5DISCUSSION ..................................... 64 REFERENCES ....................................... 67 BIOGRAPHICALSKETCH ................................ 69 5

PAGE 6

LISTOFFIGURES Figure page 1-1Oscillationsinlynx-haresystem.Thenumbersareinthousands ........ 9 2-1Agraphofthefunctions(N)and(N).Eachpointofintersectionofthetwocurvesgivesonesolutiontotheequation(N)=(N),andoneprey-onlyequilibrium. ...................................... 13 2-2AgraphofthefunctionsN(t)andP(t).Thegraphshowsoscillationsinthenumbersofthepredatorandtheprey. ....................... 15 2-3AleeeffectinthecasewhenR0>1.TheequationR(N)=1hasauniquesolution. ....................................... 21 2-4AlleeeffectinthecaseR0<1.EquationR(N)=1canhavetwosolutions(illustratedintheleftgure)ornosolutions(illustratedintherightgure). ... 22 2-5Multiplesubthresholdequilibria.WehaveR(0)<1. ............... 23 4-1Parametricplotinthe(P,dP). ......................... 63 4-2Theleftgureshowsthecycleinthe(N,P)planewithtimeasaparameter.Therightgureshowstheoscillationsinthetotalnumberofpreyandthepredatorasfunctionsoftimewhenthoseoscillationshavestabilized. .......... 63 6

PAGE 7

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyOSCILLATIONSINASIZE-STRUCTUREDPREYPREDATORMODELBySouvikBhattacharyaAugust2011Chair:MaiaMartchevaMajor:MathematicsThisdissertationintroducesapredator-preymodelwiththepreystructuredbybodysizebasedonreportsintheliteraturethatpredationratesareprey-sizespecic.Themodelisbuiltonthefoundationoftheone-speciesphysiologicallystructuredmodelsstudiedearlier.Threetypesofequilibriaarefound:extinction,multipleprey-onlyequilibriaandpossiblymultiplepredator-preycoexistenceequilibria.Thestabilitiesoftheequilibriaareinvestigated.ComparisonismadewiththeunderlyingODELotka-Volterramodel.ItturnsoutthattheODEmodelcanexhibitsustainoscillationsifthereisanAlleeeffectinthenetreproductionrate,thatisthenetreproductionrategrowsforsomerangeoftheprey'spopulationsize.Incontrast,itisshownthatthestructuredPDEmodelcanexhibitsustainoscillationsevenifthenetreproductiverateisstrictlydeclining.Thoseoccur,however,ifreproductionissizespecicandlimitedtoindividualsoflargeenoughsize.Simulationsarepresentedtosupportourhypothesisthatsize-specicpredationcandestabilizethepredator-preyequilibriuminthePDEmodel. 7

PAGE 8

CHAPTER1INTRODUCTION 1.1BackgroundInecologypredationdescribesabiologicalinteractionwhereapredator(anorganismthatishunting)feedsonitsprey(theorganismthatisattacked)[ 15 ].Predator-preyinteractionshavefascinatedmathematicalbiologistsforalongtime.Severallong-termdatasetshavebeencollectedforpredatorandpreyinteractionsinnature.Predator-preyinteractionscanbeonepartinthehugechainornetworkwhereeveryspeciesislinkedtotheotherspeciesinoneormoreways,comprisingaglobalfoodchain. 1.1.1Food-webEcologyAnimportantaspectofecologyistostudythefoodrelations.Differentanimalslivingandpersistingintheenvironmentaresubjectedtothefoodrelations.Oneisdependentontheother.Itwasasearlyas1959whenHutchinsonexplainedthissysteminacoherentfashion.Eatorbeeaten,asHutchinsonputit,wastermedastherawdictumoftheenvironment.In[ 18 ]Hutchinsonexplainedthelinksbetweenfoodchain,naturalselection,effectofsize,effectsofterrestrialplantsandhowtheinterrelationoffoodchainaffectsthesystem.Healsoexplainedhowlimitationofdiversityandnicherequirementsplayahugeroleinit.Oneofthequestionheraisedwaswhetherhigherlevelofdiversityleadstomoreorlessstabilityincommunityecology.Thisissuewaslateraddressedin[ 19 ]and[ 20 ].Ithasbeenalongstandingquestioninecologytoidentifytheinteractionstrengthandstabilityoftheinteractionbetweenpredatorandprey.SalaandGrahamdevisedmethodstoestimatethedistributionofpredatorpreyinteractionstrengthswithinasubtidalherbivorecommunityof45species[ 14 ].Incontrasttothetheoriespresentedearlier,theauthorsprovedthatintermediatesizepredatorsunderrealisticcircumstancescanprovethemselvesasthemosteffectiveconsumers.Article[ 21 ]designedbothobservationalandexperimentalapproachesfor 8

PAGE 9

estimatingtheinteractionstrengthamongthespeciesanddiscussedtheirtiestothetheory.Allthesediscussionsprovethatthereisahugebondamongdifferentorganisms,abondwhichisthecauseforthespeciestoexistinthenaturalenvironment.Oneofthemostwell-knownexamplesinfood-chainecologyistheoneofthedynamicalinteractionsofCanadianlynxandsnowshoeharewiththedatacollectedbytheHudsonbayCompanyinCanadaduringtheperiod1821-1940(seeFigure 1-1 ).Thisexampleisnowdiscussedinmanymathematicalbiologytextbooks[ 1 ].Thesedatasetshavesuggestedthatthepredator-preyinteractionsinnatureoftenpersistintheformofoscillations.Thequestionwhataccountsfortheperiodicityinthepredator-preydynamicshasbeenacentralquestioninmathematicalbiologyformanyyearsleadingtoamultitudeofarticlesdiscussingoscillationsinordinarydifferentialequationmodels[ 11 ]. Figure1-1. Oscillationsinlynx-haresystem.Thenumbersareinthousands 9

PAGE 10

1.1.2PredominanceofBody-sizeinFood-chainNetworkBodysizeplaysabigroleindeterminingthedifferentfactorsinvolvedinpredator-preydynamics.Predatorspreferpreyofcertainrangeinsize.Itisanaturalinstinctoflivingbeingstopreventthemselvesfrombecomingpreytoothers.Hencepreytriestoescapepredationbyevolvingatalargersize.Theauthorin[ 24 ]hascollectedsamplesfrom26differentsitestoprovethatthereisarelationshipwhichexistedbetweenthepreybodysizeandpredation.Hisobservationrevealsthepresenceofnegativecorrelationofpreysizewiththelargestpredatorlengthinthesystemwheretheexperimentwasperformed.Overtheyearsecologistshaveperformeddifferentexperimentstoprovetherelationshipbetweenbodysizeandpredation.Inanothercasestudytheauthorsin[ 23 ]haveshownthatguppiesevolveatalargersizeinordertoavoidpredationatsmallersizes.Furthermore,JenningsandWarr[ 16 ]claimedthatsmallermeanbody-sizeratiosarecharacteristicofmorestablepredator-preyenvironment.WarrenandLawtonsuggestthatifthebodysizesofpredatorsarelargerthanthebodysizesoftheirpreysthenatrophichierarchy(cascademodel)mightexistbasedonbodysize.Holt[ 22 ]wassuccessfulindesigningmodelsbasedoncommunitymodules(suchasapredator-preymodule)whichnotonlyportraytheecologicalinterplayofthespeciesinvolvedbutalsoallowtheintegrationofevolutionaryperspectivethroughmodelparameters.Themainquestionweaddressinthisdissertationis:Areoscillationsinanecologicalsystemwheresize-dependentpredationplaysahugerole? 1.2RelatedMathematicalWorkInthisdissertationwesetforththehypothesisthatpredationandpreyindividuals'bodysizemayberesponsiblefortheoscillationsobservedinthepredator-preyinteractionsinnature.Theinteractionsbetweenthepredatorandthepreyarestronglyinuencedbothbythesizeofthepredatorandthesizeoftheprey[ 13 ].Biologicalliteratureaboundswitharticlesdiscussingtheroleofsizeinpredator-preyinteractionsinavarietyofnaturalsystems[ 12 ].Yet,theroleofsize,asacontinuousvariable,in 10

PAGE 11

thecontextofpredator-preymodels,hasrarelybeendiscussedinthemathematicalbiologyliterature.DeRoosatal.investigatetheroleoffoodavailabilityonthesizeofthepredatorDaphniaandndthatthereiscoexistenceinastableequilibrium,andstablecycles[ 10 ].Inthisdissertationwetaketheoppositeperspective:welookattheimpactofthesizeofthepreyonthepredator-preyinteraction.Oursize-structuredpartialdifferentialequationmodelisbasedonthenon-linearsingle-populationsizestructuredmodelanalyzedin[ 2 ].Ourinvestigationismotivatedbyreportsinthebiologicalliteraturethatpredatorspreferpreyofcertainbodysize,whilethebodysizeofallpreysmayvaryinalargerange[ 3 ].Ithasbeensuggestedthatpredatorstendtoprefermedium-sizepreyastoolargepreymaybetoodifculttohandle,whiletoosmallpreymaybetoochallengingtocatch.However,ifadequaterefugeisavailable,thenthepredationratedeclineswiththesizeoftheprey[ 6 ].Thequestionwhetherpreysizemaybeadestabilizingfactorinthepredator-preyinteractionsseemsanopenandinterestingquestionthatweaddressinthisdissertation.Mathematically,ourresultsparallelmostcloselytheinvestigationofapredator-preymodelwhichaccountsforpreyage-structure[ 4 ].Linds,justaswedo,threetypesofequilibria:anextinctionequilibrium,aprey-onlyequilibrium,andacoexistenceequilibrium,andperformspartialanalysisoftheirstabilities.AmoregeneralmodelinwhichboththepredatorandthepreyarephysiologicallystructuredisintroducedbyLoganetal.[ 8 ](seealso[ 7 ]).However,suchamodelisrathercomplex,andtheauthorsconsideranumberofmoretractablespecialcases.Incontrastwiththeirmodel,whichincludesaHollingIIfunctionalresponse,ourmodelonlyincludesalinearsize-dependentfunctionalresponse.OurreasonforaccountingonlyforlinearfunctionalresponseistoeliminatethepossibilitythattheHollingfunctionalresponse,wellknowntodestabilizepredator-preyinteractions,haddestabilizedthecoexistenceequilibrium.Withlinearfunctionalresponse,andmonotonedecreasingrecruitmentrate,theunderlyingODEmodelofourPDEsystemwillnotexhibitoscillations. 11

PAGE 12

CHAPTER2THEMODELANDINVESTIGATIONOFTHEEQUILIBRIA 2.1TheODEModelanditsAnalysisWerstintroduceanODELotka-Volterramodel.LetN(t)bethepopulationsize,P(t)bethepopulationsizeofthepredator.TheOrdinaryDifferentialEquationisgivenasfollows. N0=((N))]TJ /F5 11.955 Tf 11.95 0 Td[((N))N)]TJ /F5 11.955 Tf 11.96 0 Td[(PNP0=PN)]TJ /F4 11.955 Tf 11.96 0 Td[(dP (2) N(0)=N0,P(0)=P0where(N)isthebirthrateofprey,(N)isthedeathrateoftheprey.Theparametergivesthepredationrate,isthepredatorconversionefciency.Finally,thedeathrateofthepredatorisgivenbyd.ThissystemhasanextinctionequilibriumE0=(0,0)inwhichboththepredatorandthepreypopulationsgoextinct.Theextinctionequilibriumislocallystableif(0)<(0),andunstableotherwise.Fortheremainderofthissectionwewillassumethat(0)>(0)sothatatleastthepreypopulationisviable.Intheabsenceofthepredator,theprey-onlyequilibriaareobtainedassolutionstotheequation (N)=(N).(2)Assume(N)!0asN!1and(N)isnondecreasinginN.Theseassumptionsandthefactthat(0)>(0)implythatthisequationhasatleastonepositivesolutionN.However,equation( 2 )mayhavesolutionsevenif(0)<(0).AssumetheequationhasksolutionsN1,...Nk,allofwhicharesimplesolutionsandorderedinincreasingorder(seeFigure 2-1 ). 12

PAGE 13

Figure2-1. Agraphofthefunctions(N)and(N).Eachpointofintersectionofthetwocurvesgivesonesolutiontotheequation(N)=(N),andoneprey-onlyequilibrium. Eachofthesesolutionsgivesaprey-onlyequilibriumEj=(Nj,0).Eachoftheseequilibriaislocallystableif 0(Nj)<0(Nj)andNj)]TJ /F4 11.955 Tf 11.96 0 Td[(d<0(2)andunstableotherwise.ThesecondinequalitysaysthatthepredatorcannotinvadetheE0jthequilibriumoftheprey.Denethethresholdquantity ^N=d .(2)Clearlyallprey-onlyequilibriaEjsatisfyingNj>^Nareunstable.Inotherwords,iftheprey-onlypopulationislargeenough,itwillsupportthepredatortoexist.Wecallthequantity^Nminimumthresholdpreypopulationsizeforexistenceofthepredator.Sinceequilibriaareallsimple,theinequalitybetweenthederivativesofandchangeswitheachequilibrium.Forinstance,onFigure 2-1 ,therearevesolutions:N1,...,N5.Wehave0(N1)<0(N1),0(N2)>0(N2),etc.Iftheminimumthresholdpreypopulationsizeforexistenceofpredatorislargeenough,thenprey-onlyequilibriaN1,N3,N5inFigure 2-1 willbelocallystable. 13

PAGE 14

System( 2 )hasauniquecoexistenceequilibriumofthepredatorandthepreyE=(^N,^P)where^Nisgivenin( 2 ),and^P=((^N))]TJ /F5 11.955 Tf 11.96 0 Td[((^N))=. Theorem2.1. Thecoexistenceequilibriumislocallystableifandonlyif 0(^N)<0(^N).(2) Proof. TheJacobianofthesystematthecoexistenceequilibriumpointisgivenbythefollowingmatrix,J=0B@(0(^N))]TJ /F5 11.955 Tf 11.95 0 Td[(0(^N))^N)]TJ /F5 11.955 Tf 9.3 0 Td[(^N^P01CAWeobservethatthedeterminantofthematrixJispositive.Wenowfocusonthetraceofthematrix.Thetworootsofthejacobianarenegativeorhavenegativerealpartsifandonlyif0(^N)<0(^N).Thisprovestheconditionforstability.Iftheconditionisnotsatisedthenthetraceofthematrixispositiveandhencehastwopositiverootsorrootswithpositiverealparts.Ineithercaseitisunstable. NotethatthetworootsoftheJacobianaregivenbythefollowingequation, 1,2=(0(^N))]TJ /F5 11.955 Tf 11.96 0 Td[(0(^N))^Nq (0(^N))]TJ /F5 11.955 Tf 11.95 0 Td[(0(^N))2^N2)]TJ /F3 11.955 Tf 11.95 0 Td[(42^N^P 2(2)Thesystemwillhaveaimaginaryrootonlyinthecasewhen0(d )=0(d ).Weshowbelowthatthederivativeoftherealpartoftherootwithrespecttodisgivenby(00(^N))]TJ /F5 11.955 Tf 11.96 0 Td[(00(^N))^N 2 14

PAGE 15

whichisnotequaltozerounless(00(^N)=00(^N)).Wecancertainlychooseafunctionlikethat.HencetheconditionforHopfBifurcationissatisedandwehaveoscillationintheODEmodel.If( 2 )fails,thecoexistenceequilibriummaybecomedestabilizedandsustainedoscillationsarepossible(seeFigure 2-2 ).Hence,thesimplepredator-preymodelin( 2 )iscapableofcomplexbehavior. Figure2-2. AgraphofthefunctionsN(t)andP(t).Thegraphshowsoscillationsinthenumbersofthepredatorandtheprey. 2.2ThePDEModelInthissectionweconsiderasize-structuredpopulationmodelintroducedinthepaperbyCalsinaandSaldana[ 2 ].Themodelisanon-linearrstorderpartialdifferentialequation,equippedwithnonlocalboundaryconditions.Asweareinterestedinthesizespecicpredationeffectsofaspecialistpredatorwearegoingtoextendthatmodeltoincorporateapredatorpopulation.Theresultingextendedmodelisasize-structuredversionofthewell-knownLotka-Volterrapredationmodel.Themodel,asintroducedin[ 2 ],describesthedynamicsofthesize-structuredprey.Itispresentedbelowinnotationconsistentwithourextendedsize-structuredpredator-preymodel: ut+(g(x,N(t))u(x,t))x+(x,N(t))u=0,x2[0,1) (2) g(0,N(t))u(0,t)=Z10(x,N)u(x,t)dx,t>0 (2) u(x,0)=(x),x2[0,1) (2) 15

PAGE 16

wherethetotalpopulationsizeN(t)attimetisgivenby:N(t)=Z10u(x,t)dx.Thefunctionu(x,t)representsthedensityofthepreypopulationofsizexandattimet.InparticularthatmeansthatRbau(x,t)dxrepresentsthenumberofpreyfromsizeatosizebwherebothaandbarepositivenumbers.Here(x,N)representsthenaturalpercapitasize-dependentdeathrate.Furthermore,(x,N)givesthepercapitasize-dependentbirthrate.Themodelassumesthatallbirthsoccurtothesameinitialsizewhichwehaveshiftedtobezero.Bothandarenonnegative,LipschitzcontinuouswithrespecttoxandNfunctions.Furthermore,thebirthanddeathratessatisfy: Assumption1. Thebirthanddeathratesatisfy limN!1(x,N)=0 isaboundedfunctionwithrespecttobothxandN,thatis,supx,N(x,N)=. (X,N)ispositiveasafunctionofxonasetofpositivemeasure. limN!1(x,N)=1. isboundedfrombelow:(x,N) .Thefunctiong(x,N)isthegrowthrate.Weassumeitisacontinuouslydifferentiablefunctionwithrespecttoxandboundedbyg(0).Fortheremainderofthisworkwewillassumethatg(x,N)is,infact,independentofN,thatisg(x,N)=g(x).Furthermore,g>0forallx2[0,1).Theinitialcondition(x)isanon-negativeandintegrablefunction,whichispositiveonasetofpositivemeasure.Inthepresentdissertationweincludetheeffectofaspecialistpredator.Theinclusionofaspecialistpredatorintroducesanadditionalvariableandalsoanadditional 16

PAGE 17

equationinthesystem.Weassumethepredatorfeedsonpreyofspecicsizeanddenotethesize-specicpercapitapredationrateas(x).Thepredationrate(x)isabounded,Lipschitzcontinuousfunctionwithcompactsupport.Weassumethat(x)>0onasetofpositivemeasure.Denoteby =supx(x).Weincorporatetheeffectofpredationaspredator-introducedadditionalmortalityonthepreywhichissize-dependent.Theresultingmodelisgivenasfollows.Thisisthemainmodelthatweconsiderinthisdissertation. ut+(g(x)u(x,t))x=)]TJ /F5 11.955 Tf 9.3 0 Td[((x,N)u)]TJ /F5 11.955 Tf 11.96 0 Td[((x)Pug(0)u(0,t)=Z10(x,N)u(x,t)dx (2) u(x,0)=(x),P(0)=P0P0=PZ10(x)(x)u(x,t)dx)]TJ /F4 11.955 Tf 11.95 0 Td[(dPThetime-dependentfunctionP(t)representsthetotalnumberofpredatorsattimet.Theparameter(x)istheprey-sizedependentpredatorsmetabolicefciencybywhichthebiomassofconsumedpreyistransformedintopredator'sbiomass.Finally,disthedeathrateofpredators.ThetotalnumberofpredatorsPisanonnegativefunction.Predatorsdeathratedandmetabolicefciency(x)arealsononnegative.Thenumberofpredatorsattimet=0isgivenbythenonnegativenumberP0. 2.2.1EquilibriaoftheSize-structuredPredator-preyModelExistenceanduniquenessofsolutionstomodel( 4 )canbeprovedsimilarlyasin[ 2 ].Inthissectionweareinterestedintime-independentsolutions(equilibria)ofthe 17

PAGE 18

model( 4 ).Thesystemfortheequilibriais (g(x)u(x))x=)]TJ /F5 11.955 Tf 9.3 0 Td[((x,N)u)]TJ /F5 11.955 Tf 11.96 0 Td[((x)Pu (2) g(0)u(0)=Z10(x,N)u(x)dx (2) 0=PZ10(x)(x)u(x)dx)]TJ /F4 11.955 Tf 11.96 0 Td[(dP (2) wherePisthetotalpredatorsizeattheequilibrium.TheconstantNrepresentsthetotalpreysizeattheequilibriumandisgivenby:N=Z10u(x)dx.AssumingthatNandParegivenconstants,thedifferentialequationforu( 2 )canbeintegrated u(x)=u(0)g(0) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0((x,N) g(s)+(s) g(s)P)ds (2) =u(0)g(0)(x,P,N)whereforaxedPandNwehaveintroducedthefollowingnotation(x,P,N)=1 g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(x,N) g(s)ds)]TJ /F12 7.97 Tf 6.59 6.42 Td[(Rx0(s) g(s)Pds.Thefunctioncanbeinterpretedastheprobabilityofthepreytosurvivetillsizex.Tondu(0),PandNweplacetheformulaforu(x)intotherenewalequation,theequationforthepredator,andtheequationofthetotalpreysize.Weobtainthefollowingnonlinearsystemofthreeequationsintheunknownsu(0),PandN. g(0)u(0)=u(0)g(0)Z10(x,N)(x,P,N)dx0=Pu(0)g(0)Z10(x)(x)(x,P,N)dx)]TJ /F4 11.955 Tf 11.95 0 Td[(dP (2) N=u(0)g(0)Z10(x,P,N)dx 18

PAGE 19

Anequilibriumsolutionofthesystem( 4 )isgivenbythetriple(u(0),P,N),whereu(0),P,andNareasolutionofthesystem( 2 ).System( 2 )alwayshasthetrivialsolutionwhereu(0)=0,P=0,N=0.ThetripleE0=(0,0,0)givestheextinctionequilibrium.Besidestheextinctionequilibrium,wehavetwotypesofotherequilibria.Thersttypearepredator-freeequilibria,wherethepredatorgoesextinctbutthepreypopulationsizepersists.Thesecondtypeofequilibriaarecoexistenceequilibriawherebothpredatorandpreyarepresent.Weconsiderthefollowingtwocases. 2.2.2Prey-onlyEquilibriaInthiscasewehaveP=0.Hencetheequilibriaherewouldbeoftheform(u(0),0,N).Wehavetondthevaluesofu(0)andN.WithP=0system( 2 )takestheform 1=Z10(x,N) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(x,N) g(s)dsdx (2) N=Z10u(0)g(0) g(x)e)]TJ /F12 7.97 Tf 8 6.43 Td[(Rx0(x,N) g(s)dsdxwhere u(x)=u(0)g(0) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(x,N) g(s)ds.(2)Therstequationinthesystem( 2 )isindependentofthesecondequationanddependsonNbutnotonu(0).Wecanrstsolvetherstequationinsystem( 2 )forN.Thenweobtainu(0)fromthesecondequationinsystem( 2 ).Hence,giventhatweknowN,wegetu(0)=N R10g(0) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(x,N) g(s)dsdx.Therefore,system( 2 )isessentiallyadecouplednon-linearsystem.Wenotethatsinceg(0)>0thedenominatorintheformulaforu(0)isnonzero.Wenowfocusonthenumberofsolutionsoftherstequationin( 2 ). 19

PAGE 20

Wedenethenetreproductionrateasafunctionofthepreypopulationsize: R(N)=Z10(x,N) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(x,N) g(s)dsdx.(2)ThebehaviorofthisfunctionofNdeterminesthemechanismsofgrowthofthepreypopulation.TounderstandbetterthesolutionsoftheequationR(N)=1weassumesometypicaltypesofbirthanddeathratefunctionsandthentrytoexploretheexistenceofequilibria. Assumption2. Assumethebirthanddeathrateshavethefollowingproperties: (x,N)=R00(N)(x). (x,N)=0(N)+m(N)(x). [0(0)])]TJ /F6 7.97 Tf 6.59 0 Td[(1=R10(x) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(s,0) g(s)dsdx. Allfunctionsandconstantsarenonnegative(positive).TheequationforthetotalpopulationsizeR(N)=1takestheform 1=Z10R00(N)(x) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx00(N)+m(N)(x) g(s)dx(2)WehavetoprovethatthereexistssuchNwhichsatisesequation( 2 ).ForN=0wehavebytheAssumptions 2 aboveR(0)=R0.WecallR0intrinsicreproductionnumberofthepreypopulation.Inwhatfollowsweconsiderspecicexamples.Case1:Suppose(s,N)doesnotdependonNandweassumespecicvaluesfor0andm.Inparticular,let0(N)=0,m(N)=1.Then,byAssumption 2 thenetreproductionrateofthepreypopulationtakestheform R(N)=R00(N) 0(0) (2) 20

PAGE 21

Figure2-3. AleeeffectinthecasewhenR0>1.TheequationR(N)=1hasauniquesolution. Wewillassumeaparticularformofthefunction0(N),ormoreprecisely,aparticularformofthenetreproductionrate,andthenwewillshowthatthereexistsNsuchthatR(N)=1.Weconsiderthefollowingspecicformofthenetreproductionrate: R(N)=R0e)]TJ /F6 7.97 Tf 6.59 0 Td[((N)]TJ /F10 7.97 Tf 6.59 0 Td[(a)2 e)]TJ /F10 7.97 Tf 6.59 0 Td[(a2=R0ea2)]TJ /F6 7.97 Tf 6.58 0 Td[((N)]TJ /F10 7.97 Tf 6.58 0 Td[(a)2(2)whereaisapositiveparameter.SinceR0(N)=R0ea2)]TJ /F6 7.97 Tf 6.59 0 Td[((N)]TJ /F10 7.97 Tf 6.58 0 Td[(a)2()]TJ /F3 11.955 Tf 9.3 0 Td[(2(N)]TJ /F4 11.955 Tf 12.18 0 Td[(a)),thederivativeR0(N)=0atN=aonly.AlsoweobservethatN=aisalocalmaximumforthegraphandR(N)!0asN!1.Weconsidertwocases 1. IfR0>1.InthiscasethereexistsonlyoneNsuchthatR(N)=1.Figure 2-3 illustratesthisscenario. 2. IfR0<1.Inthiscasethegrapheitherdoesnotcrosstheliney=1orcrossestwice.Thus,thereexisteither0ortwoNsuchthatR(N)=1.ThegraphinFigure 2-4 showsthat.InthecaseR0<1thecriticalvalueoftheparameterasuchthattheequationR(N)=1transitionsfromhavingtwosolutionstohavingnosolutionsisdenotedbyacr.Thiscriticalvalueoccurswhen,throughmanipulationsona,thegraphofR(N)=R0ea2)]TJ /F6 7.97 Tf 6.58 0 Td[((N)]TJ /F10 7.97 Tf 6.58 0 Td[(a)2touchestheliney=1.InthiscasewehavethatR0(N)=0atN=a. 21

PAGE 22

Figure2-4. AlleeeffectinthecaseR0<1.EquationR(N)=1canhavetwosolutions(illustratedintheleftgure)ornosolutions(illustratedintherightgure). Hencethecriticalvalueofa=acrisgivenbythesolutionofthefollowingequation R(acr)=1(2)Thus,thecriticalvalueoftheparameterais acr=r ln1 R0(2)ThecorrespondingvalueofNobtainedwhena=acrisNcranditisadoublerootoftheequationR(N)=1.AllotherrootsoftheequationR(N)=1,whena6=acraresimpleroots.WenotethatR0<1,wehave1 R0>1andthesquarerootiswelldenedandpositive.TheequilibriathatareobtainedinthecaseR0<1arecalledsubthresholdequilibria.Theaboveexamplecanbeextendedtoallowformorethantwoequilibria.Forinstance,assumea1thereexistseitheroneorthreeNsuchthatR(N)=1.IfR(0)<1,theequationR(N)=1mayhaveno 22

PAGE 23

solutions,twosolutionsorfoursolutions,ifallsolutionsaresimple.WeillustratethislastcaseinFigure 2-5 .WeconcludethateveninthecasewhenthemortalityratedoesnotdependonN,wemayhavemultiplesuperandsubthresholdequilibria. Figure2-5. Multiplesubthresholdequilibria.WehaveR(0)<1. Case2:Inthiscasewe,infact,considerthegeneralcasewhereboththebirthrateandthedeathratemaydependonN.Werecallthatthegeneralnetreproductionratedenedasafunctionofthetotalpreysizeisgivenby( 2 ).EquilibriaofthetotalpreypopulationsizearesolutionsoftheequationR(N)=1.Wedeneinanalogywiththeexampleabove,theintrinsicreproductionnumberofthepreypopulationas R0=R(0)=Z10(x,0) g(x)e)]TJ /F12 7.97 Tf 7.99 6.42 Td[(Rx0(x,0) g(s)dsdx(2)TheassumptionsonthebirthrateanddeathrateasfunctionsofNguaranteethatlimN!1R(N)=0whichstatesthatthenetreproductionrateofthepreypopulationapproacheszeroasthepreypopulationsizegrowstoinnity.Thus,inthecasewhenR0>1,thatisR(0)>1theequationR(N)=1hasatleastonepositivesolutionN.InthecaseR0<1,astheexamplesabovesuggest,theequationR(N)=1mayormaynothavesolutions.WesummarizethendingsinthefollowingTheorem. Theorem2.2. Weconsiderthefollowingtwocases: 23

PAGE 24

1. LetR0>1.Then,thereisatleastonepositiveprey-onlyequilibriumE1=(u1(0),0,N1).IftherearemultiplesolutionstotheequationR(N)=1andtheyareallsimple,thenthereisanoddnumberofthemN1...Nkwherekisodd.Eachofthesesolutionsgivesaprey-onlyequilibriumEj=(uj(0),0,Nj)forj=1,...,k. 2. LetR0<1.Then,theremaybenopositiveprey-onlyequilibrium.IftherearemultiplesolutionstotheequationR(N)=1andtheyareallsimple,thenthereisanevennumberofthemN1...Nkwherekiseven.Eachofthesesolutionsgivesaprey-onlyequilibriumEj=(uj(0),0,Nj)forj=1,...,k.Wenotethattherequirementthatallequilibriaaresimpleisveryimportant.Thegeneralcasewhensomeequilibriacanhavehighermultiplicitiesismuchmorecomplex.However,thisconditioncanfail,andsomesolutionsofR(N)=1canhavehighermultiplicity.Inthiscasetheparametersofthemodelhavetosatisfyadditionalconstrains.Consequently,forveryfewchoicesoftheparameters,equilibriaofhighermultiplicityarepossible.Intheexample,theparametervalueforwhichtherootNhasahighermultiplicity,isonlyacr. 2.2.3Predator-preyCoexistenceEquilibria.Inthiscasewearelookingforequilibria(u(0),P,N)whereP6=0. Assumption3. Assume Thereproductionnumberofthepreypopulationintheabsenceofthepredatorsatises:R0>1. TheequationR(N)=1hasksolutionsN1,...Nk,wherekisodd.Weassumethatallsolutionsaresimple.WedenethepredatorreproductionnumberattheNjprey-onlyequilibrium Rp,j=NjZ10(x)(x)(x,0,Nj) dZ10(x,0,Nj)dx.(2) 24

PAGE 25

Thepredator'sreproductionnumbergivestheabilityofthepredatortoinvadetheNjequilibriumoftheprey.Inparticular,ifRp,j>1,thenthepredatorcaninvadethejthprey-onlyequilibrium. Assumption4. AssumethatthereexistsNj>0suchthat: Thepredator'sreproductionnumberatthepreviousprey-onlyequilibriumRp,j)]TJ /F6 7.97 Tf 6.59 0 Td[(1<1,thatisthepredatorcannotinvadetheNj)]TJ /F6 7.97 Tf 6.59 0 Td[(1thprey-onlyequilibrium; Thepredator'sreproductionnumberRp,j>1,thatis,weassumethatthepredatorcaninvadethejthprey-onlyequilibrium. Assumealso,jisodd.WenotethatifRp,1>1,thentheaboveassumptionwouldbetriviallysatised.Inthecaseofpredator-preycoexistenceequilibria,thenon-linearsystemfortheequilibria( 2 )doesnotdecouple.Fromthesecondequationwemayexpressu(0)andeliminateitfromthesystem.Wehave: u(0)g(0)=d R10(x)(x) g(x)e)]TJ /F12 7.97 Tf 7.99 6.42 Td[(Rx0(x,N) g(s)ds)]TJ /F12 7.97 Tf 6.58 6.42 Td[(Rx0P(s) g(s)dsdx.(2)Replacingu(0)g(0)intheequationforthetotalpreypopulationsizeweobtainthefollowingequationinPandN:N=R10d g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(x,N) g(s)ds)]TJ /F12 7.97 Tf 6.59 6.42 Td[(Rx0P(s) g(s)dsdx R10(x)(x) g(x)e)]TJ /F12 7.97 Tf 7.99 6.42 Td[(Rx0(x,N) g(s)ds)]TJ /F12 7.97 Tf 6.58 6.42 Td[(Rx0P(s) g(s)dsdx.Thisequationcoupledwiththerenewalequationleadstothefollowingnon-linearsystemforthevariablesNandP. Z10(x,N)(x,P,N)dx=1N=Z10d(x,P,N)dx Z10(x)(x)(x,P,N)dx.(2)Thisisanon-linearsysteminN,P.Itdoesnotdecouple.Wearelookingforconditionsthatgiveanon-zeropositivesolutionofthatsystem.Eachpositivesolutionofthe 25

PAGE 26

system( 2 )givesonecoexistenceequilibriumE=(u(0),P,N).Therstequationinsystem( 2 )promptsustodenethenetreproductionrateofthepreyinthepresenceofthepredator.Denotethenetreproductionrateofthepreyinthepresenceofthepredatorby R(N,P)=Z10(x,N)(x,P,N)dx.(2)Therstequationinsystem( 2 )givesR(N,P)=1.WeusetheimplicitfunctiontheoremtosolveforPasafunctionofN.ForeacharbitrarybutxedN,theequationR(N,P)=1asanequationofPonlyhasauniquesolution,which,however,maybepositiveornegative.ThisdenesP=f(N)asacontinuousfunctionforallN0.Moreover,@R(N,f(N)) @P=)]TJ /F11 11.955 Tf 11.29 16.27 Td[(Z10(x,N)Zx0(s) g(s)ds(x,f(N),N)dx<0since(x)>0onasetofpositivemeasure.ThefunctionP=f(N)hasthefollowingproperties: f(Nj)=0,thatisthepredatorequilibriumsizeattheprey-onlyequilibriaiszero. Sinceallprey-onlyequilibriaaresimple,f(N)hasanalternatingsignintheconsecutiveintervals.Since,R0>1thesignsarethefollowing 8>>><>>>:f(N)>0on(0,N1)f(N)<0on(N1,N2)...f(N)>0on(Nk)]TJ /F6 7.97 Tf 6.59 0 Td[(1,Nk)(2)NowwereplaceP=f(N)inthesecondequationof( 2 ).AfterreplacingPthesecondequationin( 2 ),weobtainanequationinNonly.Wecanrearrangethetermsinthatequationtogetthefollowingform. Z10(N(x)(x))]TJ /F4 11.955 Tf 11.95 0 Td[(d)1 g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(x,N) g(s)ds)]TJ /F12 7.97 Tf 6.59 6.42 Td[(Rx0f(N)(s) g(s)dsdx=0(2) 26

PAGE 27

Werecallthatweassumethatthereexistsaprey-onlyequilibrium,Njsuchthatthepredatorcaninvadethisequilibrium,thatisRp,j>1.Weintroducethefollowingnotation.Letthelefthandsideoftheequation( 2 )bedenotedbyF(N): F(N):=Z10(N(x)(x))]TJ /F4 11.955 Tf 11.96 0 Td[(d)1 g(x)e)]TJ /F12 7.97 Tf 7.99 6.42 Td[(Rx0(x,N) g(s)ds)]TJ /F12 7.97 Tf 6.58 6.42 Td[(Rx0f(N)(s) g(s)dsdx(2)ClearlywecanseethatF(0)<0.Thus,ifRp,1>1,wehaveF(N1)>0.Therefore,thereexists^N2(0,N1)suchthatF(^N)=0.Then,thecorrespondingvalueofthepredatorpopulationsizeisgivenby^P=f(^N),andsinceR0>1,thisvalueofthepredatorsizeispositive.InthegeneralcasesinceRp,j)]TJ /F6 7.97 Tf 6.59 0 Td[(1<1,thatimpliesthatF(Nj)]TJ /F6 7.97 Tf 6.59 0 Td[(1)<0.Ontheotherhand,since,Rp,j>1,wehaveF(Nj)>0.Therefore,thereexistan^N2(Nj)]TJ /F6 7.97 Tf 6.58 0 Td[(1,Nj)suchthatF(^N)=0.Atthesametimewehavethat^P=F(^N)and^P>0.Onecanexpressthecorresponding^u(0)fromequation( 2 ).Wesummarizethendingsinthefollowingtheorem. Theorem2.3. LetAssumption 3 andAssumption 4 hold.ThenthereisatleastonecoexistenceequilibriumofthepredatorandthepreyE=(^u(0),^P,^N).Severalremarksareinorder. ConditionsinAssumption 3 andAssumption 4 aresufcientconditionsforacoexistenceequilibriumtoexist.Acoexistenceequilibriummayexistifoneormoreconditionsfail.Forinstance,evenifprey'sintrinsicreproductionnumberR0<1,coexistencemaystilloccur. Inthesize-structuredcase,unlikeODEcase,thecoexistenceequilibriummaynotbeunique.Intuitively,thatmaybethecasesincesize-specicpredationmayaffectsome(saymoreabundant)sizesofaprey-onlyequilibrium,andleadtocoexistence.Sincemorethanoneprey-onlyequilibriumexists,eachofthemcanbepotentiallyperturbedthiswaytoacoexistenceequilibrium,leadingtomultiplecoexistenceequilibria. 2.2.4StabilityAnalysisInthissectionweconsiderthelocalstabilityoftheequilibriaofthemodel( 4 )aroundanequilibriumpoint.Welinearizethemodelaroundageneralequilibrium 27

PAGE 28

E=(u(0),P,N)whereu(0)correspondstou(x)=u(0)g(0)(x,P,N).Weintroducethefollowingperturbations u(x,t)=u(x)+(x,t)P(t)=P+(t) (2) N(t)=N+n(t).ThelastequalityholdssinceN(t)=Z10u(x,t)dx=Z10(u(x)+(x,t))dx=N+n(t).Itisclearfromtheabovecomputationsthat n(t)=Z10(x,t)dx.(2)Sincethebirthrate(x,N)andthedeathrate(x,N)arenon-linearfunctionsofthetotalpopulationsize,weexpandthemaroundtheequilibriumpointas (x,N)=(x,N+n(t)) (2) =(x,N)+n(t)0(x,N)+h.o.t (2) (x,N)=(x,N+n(t)) (2) =(x,N)+n(t)0(x,N)+h.o.t (2) where,'h.o.t'intheequationaboverepresentsthehigherordertermsi.ethetermsthatinvolveproductsofperturbations.Sinceweconsiderthelocalstabilityaroundtheequilibriumpointweneglectthehigherorderterms.Hencethelinearizedequationsof 28

PAGE 29

themodel( 4 )reducetothefollowingsystem: t+(g(x)(x,t))x=)]TJ /F5 11.955 Tf 9.29 0 Td[((x,N))]TJ /F5 11.955 Tf 11.95 0 Td[(0(x,N)u(x)n(t))]TJ /F5 11.955 Tf 11.95 0 Td[((x)P)]TJ /F5 11.955 Tf 11.95 0 Td[((x)u(x)g(0)(0,t)=Z10(x,N)(x,t)dx+n(t)Z100(x,N)u(x)dx0=PZ10(x)(x)(x,t)dx+Z10(x)(x)u(x)dx)]TJ /F4 11.955 Tf 11.95 0 Td[(d(2)wheren(t)isgivenby( 2 ).Weusethelinearizationsabovetoinvestigatethestabilityofeachtypeofequilibria:extinction,prey-only,andcoexistenceequilibria. 2.2.5ExtinctionEquilibrium(u=0,P=0)Hereweconsidertheequilibriumwherethereisnopredatororpreyinthemodel.Weperturbtheextinctionequilibriumwithasmallvalueandthenobservethebehaviorinthelongrun.Thesystemfortheperturbationsabovetakestheform t+(g(x)(x,t))x=)]TJ /F5 11.955 Tf 9.3 0 Td[(0(x)(x,t)g(0)(0,t)=Z100(x)(x,t)dx0=)]TJ /F4 11.955 Tf 9.3 0 Td[(d(2)wherewehaveusedthefollowingnotation:0(x)=(x,0)and0(x)=(x,0).Tondthestabilityoftheequilibriumpointweinvestigatetheeigenvaluesofthelinearizedoperatorbysetting(x,t)=et (x)and(t)=et .Hencewehavethefollowingeigenvalueproblemforthestabilityoftheextinctionequilibrium: (x)+(g(x) (x))x=)]TJ /F5 11.955 Tf 9.3 0 Td[(0(x) (x)g(0) (0)=Z100(x) (x)dx =)]TJ /F4 11.955 Tf 9.3 0 Td[(d (2)Clearly,=)]TJ /F4 11.955 Tf 9.3 0 Td[(disoneoftheeigenvalueswhichisnegative.Further,weassume6=)]TJ /F4 11.955 Tf 9.3 0 Td[(dsothat =0.Tondtheremainingeigenvalueswelookfornonzerosolutionofthersttwoequations.Inparticular,wesolvetherstequation: (x)=g(0) (0) g(x)e)]TJ /F12 7.97 Tf 7.99 6.42 Td[(Rx00()+ g()d(2) 29

PAGE 30

andreplaceitinthesecondequation,obtainingthefollowingcharacteristicequation Z100(x) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx00()+ g()ddx=1.(2)WedenotebyG()=Z100(x) g(x)e)]TJ /F12 7.97 Tf 7.99 6.42 Td[(Rx00()+ g()d.Hence,thecharacteristicequationbecomesG()=1.Forreal,G()isdecreasing.Furthermore,itisclearfromthedenitionofG()thatas!1,G()!0.Furthermore,G(0)=R0.Wehavethefollowingtwocases: 1. IfR0>1:InthiscaseG(0)>1.Therefore,thenthereexistspositiverealsolution>0totheequationG()=1.Hence,theextinctionequilibriumisunstable. 2. IfR0<1:InthiscaseG(0)<1,thenbecauseofthemonotonicityofG(),theequationG()=1hasauniquerealsolution<0.WenowshowthatallothersolutionstothecharacteristicequationG()=1,whicharecomplex,havenegativerealpart.Indeed,forcomplexwith<0wehavejG()jG(<)R0<1.Hence,theredonotexistanynon-negativerealsolutions,orsolutionswithnon-negativerealpartofforthecharacteristicequationG()=1.Wemayconcludethattheextinctionequilibriumislocallyasymptoticallystable.WesummarizethesendingsinthefollowingTheorem. Theorem2.4. IftheintrinsicreproductionnumberofthepreypopulationR0<1,thentheextinctionequilibriumE0=(0,0,0)islocallyasymptoticallystable.IftheintrinsicreproductionnumberofthepreyR0>1,thentheextinctionequilibriumisunstable.WenotethatwecannotestablishglobalstabilityoftheextinctionequilibriuminthecaseR0<1becauseofthepresenceofsubthresholdequilibria(seeFigure 2-4 ). 30

PAGE 31

2.2.6Prey-onlyEquilibriaTheequilibriainthiscasearegivenbyEj=(uj(x),0,Nj)whereNj=R10uj(x)dx.Forarbitraryprey-onlyequilibriumE=(u(x),0,N)thelinearizationoftheoriginalsize-structuredmodel( 4 )isobtainedfromthelinearizedsystem( 2 ). t+(g(x)(x,t))x=)]TJ /F5 11.955 Tf 9.3 0 Td[((x,N)(x,t))-221()]TJ /F5 11.955 Tf 21.25 0 Td[(0(x,N)un(t)g(0)(x,t)=Z10((x,N)(x,t)+0(x,N)un(t))dx0(t)=(t)Z10(x)(x)udx)]TJ /F4 11.955 Tf 11.95 0 Td[(d(2)whereu(x)ifgivenby( 2 ).Wecanintegratethelastequationinthissystemtoobtain (t)=0e(A)]TJ /F10 7.97 Tf 6.59 0 Td[(d)t(2)whereA=R10(x)(x)u(x)dx.Werecallthatforthejthprey-onlyequilibriumthepredatorinvasionnumberisgivenbyRp,j=NjZ10(x)(x)(x,0,Nj)dx dZ10(x,0,Nj)dxWenotethatthedenitionforAandformulaforu(0)givenby( 2 )implythatRp,j=A=d.Therefore,ifthepredatorcaninvadetheprey-onlyequilibriumRp,j>1,thenA>d,andthejthprey-onlyequilibriumisunstable.Wesummarizethatinfollowingtheorem. Theorem2.5. Thejthprey-onlyequilibriumEj=(uj(0),0,Nj)isunstableifRp,j>1. Proof. IfRp,j>1thenthatimpliesA>dandasaresultfromthesolution(t)giveninequation( 2 )wehavethat(t)!1ast!1.Hencetheequilibriumisunstable. Intheremainderofthissectionweconsiderprey-onlyequilibriaEjwhichcannotbeinvadedbythepredator,thatisRp,j<1.Wewillagaindropthesubscriptj.To 31

PAGE 32

obtaintheeigenvalueproblemforthelinearizedoperator,wearelookingforasolutionof( 2 )thathastheform(x,t)=etz(x)and(t)=et.Fromthedenitionofn(t)wecangetthatn(t)=Z10(x,t)dx=etZ10z(x)dx=n0etIfwesubstitutetheabovequantitiesintheequations( 2 )thenwecanobtainthefollowingeigenvalueproblem: z(x)+(g(x)z(x))x=)]TJ /F5 11.955 Tf 9.3 0 Td[((x,N)z(x))]TJ /F5 11.955 Tf 11.95 0 Td[(0(x,N)n0u(x))]TJ /F5 11.955 Tf 11.96 0 Td[((x)u(x)g(0)z(0)=Z10(x,N)z(x)dx+n0Z100(x,N)dx=A)]TJ /F5 11.955 Tf 11.96 0 Td[(d(2)Intheabovesystemwearelookingforanon-trivialsolution(z(x),).Option1: Let=A)]TJ /F4 11.955 Tf 12.5 0 Td[(d.Then<0.Furthermore,thelastequationissatisedforeverywhere6=0.Wemaychooseasanonzerosolutionofequations( 2 )(z(x),),wherez(x)isthesolutionofthersttwoequationswiththegiven,chosen.Suchasolutionz(x)existsif=A)]TJ /F4 11.955 Tf 12.09 0 Td[(disnotaneigenvalueofthersttwoequationsofthesystem( 2 )with=0.Option2: 6=A)]TJ /F4 11.955 Tf 12.13 0 Td[(d.Weneedtohave=0.Then=(A)]TJ /F4 11.955 Tf 12.13 0 Td[(d).Wemayhaveanon-zeroeigenvector,iftheremainingtwoequationshaveanon-zerosolutionz(x).Theremainingeigenvaluesaresolutionsofthefollowingsystem. z(x)+(g(x)z(x))x=)]TJ /F5 11.955 Tf 9.29 0 Td[((x,N)z(x))]TJ /F5 11.955 Tf 11.96 0 Td[(0(x,N)un0 (2) g(0)z(0)=Z10(x,N)z(x)dx+n0Z100(x,N)udx (2) Herez(x)canbepositiveornegative.Wecall!(x)=0(x,N)un0.Subcase1: Hereweassume(x,N)=(x)thatisdoesnotdependonNsothatwehave0(x,N)=0andasaresult!(x)=0.Thisreducesthesystemtothefollowing 32

PAGE 33

setofequations. z(x)+(g(x)z(x))x=)]TJ /F5 11.955 Tf 9.29 0 Td[((x)z(x) (2) g(0)z(0)=Z10(x,N)z(x)dx+n0Z100(x,N)udx (2) whichcanbeeasilysolvedtoobtaing(x)z(x)=g(0)z(0)e)]TJ /F12 7.97 Tf 8 6.42 Td[(R10+(s) g(s)ds.Substitutingthisinthesecondequationabovewehavethefollowingcharacteristicequation 1=Z10(x,N) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0+(s) g(s)dsdx+Z100(x,N)udxZ101 g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0+(s) g(s)dsdx (2) Wedene G()=Z10(x,N) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0+(s) g(s)dsdx+FZ101 g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0+(s) g(s)dsdx(2)wheretheconstantFisgivenbyF=Z100(x,N)udx.Furthermore,wedenoteby K()=Z10(x,N) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0+(s) g(s)dsdx,L()=Z101 g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0+(s) g(s)dsdx.(2)Hence,thefunctionG()takestheformG()=K()+FL().Clearlyfor=0wehave G(0)=K(0)+FL(0)=R(N)+FL(0)=1+FL(0) (2) sinceNisanequilibriumtotalpopulationandisasolutiontotheequationR(N)=1.ItisclearfromtheformofG()thatforreal,as!1,G()!0.HencethecharacteristicequationG()=1hasapositiverealsolutionifG(0)>1.Apositiverealsolutiontothecharacteristicsequationimpliesthattheprey-onlyequilibriumEjisunstable.SinceG(0)>1ifandonlyifF>0,thenF>0isaconditionthatimplies 33

PAGE 34

instabilityofthesystem,evenif=A)]TJ /F4 11.955 Tf 11.99 0 Td[(d<0.Wesummarizethisresultinthefollowinglemma. Lemma1. Assume(x,N)=(x).IfF>0,thenthejthprey-onlyequilibriumEjisunstable.Concerningstabilityoftheprey-onlyequilibriawehavethefollowinglemma. Lemma2. AssumeF<0,thebirthrate(x,N)=(N).Furthermore,assumethatfortheprey-onlyequilibriumwithtotalpopulationsizeNthefollowinginequalityholds:)]TJ /F4 11.955 Tf 9.3 0 Td[(F<2(N).ThenthecharacteristicequationG()=1hasonlyrootswithnegativerealpart. Proof. WeconsiderthecharacteristicequationG()=1,whereG()isgivenby( 2 ).TheassumptionthatF<0impliesthatG(0)<1.Assumethereexistsaroot=a+ibofG()=1wherea0.Thatimpliesj(G()j=1.WehaveG()=Z10(N) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0+(s) g(s)dsdx+FZ101 g(x)e)]TJ /F12 7.97 Tf 7.99 6.42 Td[(Rx0+(s) g(s)dsdx=((N)+F)Z101 g(x)e)]TJ /F12 7.97 Tf 7.99 6.42 Td[(Rx0+(s) g(s)dsdxTakingabsolutevaluesandusingthefactthatG()=1,wehave1=jG()j=j((N)+F)Z101 g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0+(s) g(s)dsdxjj((N)+F)jZ101 g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0Re()+(s) g(s)dsdx
PAGE 35

Thustherecannotexistsarootwhichhasanon-negativerealpartwhenG(0)<1.Thus,ourclaimisproved. Acoupleofremarksareinorder. TheconditiononthesignofFisrelatedtothenetreproductionrateR(N).Inparticular,inthecasewhenisnotafunctionofN,therateofchangeofthepreypopulationsnetreproductionrateattheequilibriumtotalpreypopulationsizeNisgivenbyR0(N)=Z100(x,N) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(s) g(s)dsdx.Hence,F=u(0)g(0)R0(N),thatisFhasthesignoftherateofchangeofthenetreproductiverateattheprey-onlyequilibrium.Soweestablishedthatifthenetreproductionrateisincreasingattheequilibrium,theequilibriumisunstable.Ifthenetreproductionrateisdecreasing,theequilibriummaybestablesubjecttoadditionalassumptions. Asweshowbelow,ifisconstant,andF<0,equilibriumislocallyasymptoticallystablewithgeneralbirthrate(x,N). InthecasewhenF<0,ifthebirthandthedeathratedependonx,thenoscillationsmaybepossible.Subcase2: Nowweassume(x,N)=(N),thatis,dependsonNonly.Further,assumethat0(N)0.System( 2 )-( 2 )takestheform z(x)+(g(x)z(x))x=)]TJ /F5 11.955 Tf 9.3 0 Td[((N)z(x))]TJ /F5 11.955 Tf 11.96 0 Td[(0(N)un0g(0)z(0)=Z10(x,N)z(x)dx+Z100(x,N)udxn0(2)Werecallthatu(x)=u(0)g(0) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(N) g(s)ds.Wesolvethedifferentialequationforz(x)treatingn0asgiventoobtain z(x)=g(0)z(0) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(N)+ g(s)ds)]TJ /F5 11.955 Tf 13.15 8.08 Td[(0(N)u(0)g(0)n0 24e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(N) g(s)ds g(x))]TJ /F4 11.955 Tf 13.15 8.08 Td[(e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(N)+ g(s)ds g(x)35. (2) Fromdenitionofn0=R10z(x)dxandusingtheformulaforz(x)wecanexpressn0intermsofg(0)z(0): n0=g(0)z(0) +(N)+0(N)g(0)u(0) (N). (2) 35

PAGE 36

Wesubstitutez(x)andn0backinthesecondequationof( 2 ).Weobtainthefollowingcharacteristicequation: 1=Z10(x,N) g(x)e)]TJ /F12 7.97 Tf 8 6.43 Td[(Rx0(N)+ g(s)dsdx"1+0(N)u(0)g(0) (+(N)+0(N)g(0)u(0) (N))# (2) )]TJ /F5 11.955 Tf 42.69 8.09 Td[(0(N)u(0)g(0) (+(N)+0(N)g(0)u(0) (N))+F +(N)+0(N)g(0)u(0) (N) (2) whereF=R100(x,N)u(x)dx.DenotingbyG()theexpressionontherighthandsideoftheaboveequation,wecanwritethecharacteristicequationasG()=1.Theeigenvaluesofthesystem( 2 )aretherealandcomplexsolutionsofthatequation.Asbefore,itcanbeseenthatforrealG()!0as!1.G(0)isdenedasthelimitofthefunctionG()as!0.Hence,G(0)=1+F (N)+0(N)g(0)u(0) (N))]TJ /F16 7.97 Tf 23.87 5.48 Td[(0(N)u(0)g(0) ((N)+0(N)g(0)u(0) (N))Z10(x,N) g(x)Zx01 g(s)dse)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(N) g(s)dsdxTherelationshipofG(0)withoneisdeterminedbythesignoftheexpressionF)]TJ /F5 11.955 Tf -421.36 -23.91 Td[(0(N)g(0)u(0)R10(x,N)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(N) g(s)ds(Rx0ds g(s))dx.WehavethatG(0)>1ifandonlyif F)]TJ /F5 11.955 Tf 11.96 0 Td[(0(N)g(0)u(0)Z10(x,N)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0(N) g(s)dsZx0ds g(s)dx>0.(2)Inthiscasebythesimilarargumentasbeforewecansaythattheequilibriumisunstable.WenowshowthatthesignoftheexpressionaboveisdeterminedbytherateofchangeofthenetreproductionrateattheequilibriumtotalpreypopulationsizeN.Inparticular,wehaveR0(N)=Z100(x,N)e)]TJ /F16 7.97 Tf 6.59 0 Td[((N)Rx01 g(s)dsdx)]TJ /F5 11.955 Tf 9.3 0 Td[(0(N)Z10(x,N) g(x)Zx01 g(s)dse)]TJ /F16 7.97 Tf 6.58 0 Td[((N)Rx01 g(s)dsdx 36

PAGE 37

Thus,wendagainthatifthenetreproductionrateoftheprypopulationisincreasingthroughtheequilibrium,theprey-onlyequilibriumisunstable.Wesummarizethatinthefollowinglemma: Lemma3. Assume(x,N)=(N)and0(N)0.IfR0(Nj)>0,thentheprey-onlyequilibriumEjisunstable.Stabilityofanequilibriuminthiscaseisgivenbythefollowinglemma. Lemma4. Assume(x,N)=(N)and0(N)0.IfF<0therecannotexistanyrootofthecharacteristicequationG()=1with<()0. Proof. Assume<()0.WerewritethecharacteristicsequationG()=1inthefollowingform. "1+0(N)u(0)g(0) (+(N)+0(N)g(0)u(0) (N))#==Z10(x,N) g(x)e)]TJ /F12 7.97 Tf 7.99 6.42 Td[(Rx0(N)+ g(s)dsdx"1+0(N)u(0)g(0) (+(N)+0(N)g(0)u(0) (N))# (2) +F +(N)+0(N)g(0)u(0) (N)Simplifyingtheexpressionwehave Z10(x,N) g(x)e)]TJ /F12 7.97 Tf 7.99 6.42 Td[(Rx0(N)+ g(s)dsdx=1)]TJ /F4 11.955 Tf 121.85 8.09 Td[(F (+(N)+0(N)g(0)u(0) (N))+0(N)g(0)u(0) (2) Theabsolutevalueofleftsideofthepreviousequationislessorequaltoone.Wecanshowthattheabsolutevalueofrightsideoftheequationisgreaterthanone.Wecantreattherightsideoftheequationas1+k 2+r+s.Itcanbeshownthattherealpartofthisexpressionisstrictlygreaterthanone.Henceabsolutevalueisgreaterthanone,whichleadstoacontradiction.Thisshowsthattheredonotexistanyrootswithnon-negativerealpartforthecharacteristicequationG()=1whenFisnegative. 37

PAGE 38

2.2.7StabilityofaCoexistenceEquilibriumTheequilibriuminthiscaseisrepresentedby(u(x),P,N),whereN=R10u(x)dx.Wewillinvestigatethelocalstabilityofthecoexistenceequilibriumstartingfromthelinearizedsystem( 2 ).FirstwenoticethatfromtheequationfortheequilibriumwehaveZ10(x)(x)u(x)dx=d.Thatsimpliesthethirdequationin( 2 ).Wearelookingforasolutionoftheform(x,t)=et(x),(t)=et,n(t)=net.Wesubstitutethisformofsolutionintoequations( 2 )toget, (g(x)(x))x=)]TJ /F5 11.955 Tf 9.29 0 Td[((x))]TJ /F5 11.955 Tf 11.95 0 Td[((x))]TJ /F5 11.955 Tf 11.96 0 Td[(0u(x)n)]TJ /F5 11.955 Tf 11.96 0 Td[((x)P(x))]TJ /F5 11.955 Tf 11.96 0 Td[((x)ug(0)(0)=Z10(x,N)(x)dx+nZ100(x,N)udx (2) =PZ10(x)(x)(x)dxwheren=Z10(x)dx.Inwhatfollowsinthissubsectionweconsidertwocases:constantpredationandsize-specicpredation.Case1:Constantpredation:Wemakethefollowingsimplifyingassumptions Assumption5. Assume isafunctionofNonly,thatis,(x,N)=(N); (x)isconstant:(x)=; (x)isconstant:(x)=.Intheprevioussectionweshowedthatwiththeaboveassumptionon,andanyfunction,theprey-onlyequilibriumislocallyasymptoticallystable.Inthissectionwewillseethat,evenifpredationisconstantwithrespecttosize,ithastheabilitytodestabilizethepredator-preycoexistenceequilibrium. 38

PAGE 39

Withtheaboveassumptions,fromequations( 2 )wehave: (g(x)(x))x+++P g(x)g(x)(x)=)]TJ /F3 11.955 Tf 9.29 0 Td[((0(N)n+)u(x)g(0)(0)=Z10(x,N)(x)dx+Fn (2) =Pn Integratingtherstequationinthesystemaboveweobtainthefollowingformula: g(x)(x)=g(0)(0)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0++P g(s)ds (2) )]TJ /F4 11.955 Tf 9.3 0 Td[(u(0)g(0)(0n+)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0++P g(s)dsZx0eRx0 g()d g(s)ds (2) Thisleadstothefollowingformoftheequations(x)=g(0)(0)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0++P g(s)ds g(x))]TJ /F4 11.955 Tf 13.15 8.09 Td[(u(0)g(0)(0n+) g(x)e)]TJ /F12 7.97 Tf 8 6.43 Td[(Rx0++P g(s)ds[eRx0 g(s)ds)]TJ /F3 11.955 Tf 11.95 0 Td[(1] =g(0)(0)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0++P g(s)ds g(x))]TJ /F4 11.955 Tf 13.15 8.09 Td[(u(0)g(0)(0n+) [e)]TJ /F12 7.97 Tf 7.99 6.42 Td[(Rx0+P g(s)ds g(x))]TJ /F4 11.955 Tf 13.15 8.09 Td[(e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0++P g(s)ds g(x)]Weknowthatn=R10(x)dx.Integrating(x)wehave n=g(0)(0) ++P)]TJ /F4 11.955 Tf 25.97 8.09 Td[(u(0)g(0)(0n+) (+P)(++P) (2) Solvingfornfrom( 2 )wehaven=g(0)(0) 2+L+K=g(0)(0)P 2+L+Kwherewehaveintroducedthefollowingnotation:K=u(0)g(0)2P +PL=+P+u(0)g(0)0 +P 39

PAGE 40

Substitutingthevalueofandnwehavethefollowingformof(x). (x)=g(0)(0)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0++P g(s)ds g(x))]TJ /F4 11.955 Tf 10.49 8.09 Td[(u(0)g(0)g(0)(0) (2+L+K)[0+P ][e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0+P g(s)ds g(x))]TJ /F4 11.955 Tf 13.15 8.09 Td[(e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0++P g(s)ds g(x)] (2) Substitutingthisinthesecondequationof( 2 )weobtainthecharacteristicequationG()=1wherewehavethefollowingformofG(): G()=Z10(x,N) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0++P g(s)ds1+u(0)g(0) 2+L+K(0+2P ))]TJ ET BT /F1 11.955 Tf 433.45 -231.48 Td[((2) u(0)g(0) 2+L+K(0+2P )+F 2+L+KWeobserveherethatG(0)hasthefollowingform: G(0)=1)]TJ /F4 11.955 Tf 13.15 8.09 Td[(u(0)g(0)2P KZ10(x,N) g(x)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0+P g(s)dsZx0ds g(s)dx (2) Thisshowsthatwealwayshave:G(0)<1.Wecanrewritethecharacteristicequationinthefollowingform: 1)]TJ /F4 11.955 Tf 127.39 8.09 Td[(F2 3+L2+(K+u(0)g(0)0)+2Pu(0)g(0)= (2) Z10(x,N) g(x)e)]TJ /F12 7.97 Tf 7.99 6.42 Td[(Rx0++P g(s)dsdx (2) Stabilityofthecoexistenceequilibriumcanbeestablishedinthefollowingspecialcase: Lemma5. AssumeF<0,andthesize-specicbirthratehastheform(x,N)=(N)e)]TJ /F12 7.97 Tf 8 6.42 Td[(Rx0 g(s)dswhereisagivennon-negativeparameter.Thenthepredator-preycoexistenceequilib-riumislocallyasymptoticallystable. 40

PAGE 41

Proof. Thedenominatoroftheleft-handsideinthecharacteristicequationcanberewrittenintheform: 3+L2+(K+u(0)g(0)0)+2Pu(0)g(0)=(++P)[2+Q+R](2)whereQ=u(0)g(0)0 +PR=u(0)g(0)2P +PWerewritethecharacteristicequationintheformF2 3+L2+(K+u(0)g(0)0)+2Pu(0)g(0)=1)]TJ /F5 11.955 Tf 39.59 8.08 Td[((N) +++PTheequationsfortheequilibriumimply++P=(N)Hence,thecharacteristicequationsimpliestoF 3+L2+(K+u(0)g(0)0)+2Pu(0)g(0)=1 +++Pwhichrewrittenasacubicequationbecomes3+A2+B+C=0wherethecoefcientsaregivenby A=Q+(+P))]TJ /F4 11.955 Tf 11.96 0 Td[(F (2) B=(+P)Q+R)]TJ /F4 11.955 Tf 11.95 0 Td[(F(++P) (2) C=(+P)R (2) Since,F<0,itfollowsthatA>0,B>0,andC>0.Furthermore,itisnothardtoseethatAB>C.Thus,Routh-Hurwitzcriteriaimplythattherootsarenegativeorhavenegativerealparts.Oscillationsinthiscasedonotoccur. 41

PAGE 42

CHAPTER3HOPFBIFURCATIONANDOSCILLATIONS 3.1-stepFunction,EverythingElseisConstant. Theorem3.1. AssumeF<0,isaconstantthatdoesnotdependonN,andthesize-specicbirthratehasthefollowingform(x,N)=(N)[A,1)(x)where[A,1)isthecharacteristicfunctionoftheinterval[A,1),thatis(x)=1,ifx>Aandzeroelsewhere.Assumefurtherthatthefollowinginequalityholds: u(0)g(0)2P +P<2(+P)()]TJ /F4 11.955 Tf 9.29 0 Td[(F).(3)Then,HopfbifurcationoccursforsomevalueofA,andthesystemexhibitssustainedoscillations.Note:Wenotethatinequality( 3 )canberewrittenintheform1 22P +Pe(+P)A<)]TJ /F5 11.955 Tf 9.29 0 Td[(0(N)whichcanbeobtainedfromwritingFintermsoftheparameters.ThisinequalityimpliesthattheoscillationsoccurifthepredatorsizePatequilibriumandpredatorpredationrateParesufcientlysmallrelativetothegrowthrateofthepreyattheequilibrium)]TJ /F5 11.955 Tf 9.3 0 Td[(0(N).Inotherwords,foranequilibriumforwhichprey'snetreproductiverateisdecreasing(0(N)<0),predator'sabundanceshouldbelowforoscillationstooccur. Proof. Throughoutthisproofwewillconsiderthespecialcasewheng(x)=1.Withthisformofthebirthrate,wecanintegratetheintegralintheright-handsideofthecharacteristicequation( 2 ).Withtheassumptionthat0=0,andthedenominatorintheform( 2 ),thecharacteristicequationbecomes: 1)]TJ /F4 11.955 Tf 62.06 8.08 Td[(F2 (++P)(2+R)=(N) ++Pe)]TJ /F6 7.97 Tf 6.58 0 Td[((++P)A(3) 42

PAGE 43

Wenotethattheequationfortheequilibriaimpliesthat+P=(N)e)]TJ /F6 7.97 Tf 6.59 0 Td[((+P)A.Hence,thecharacteristicequationsimpliesto +)]TJ /F4 11.955 Tf 20.6 8.08 Td[(F2 2+R=e)]TJ /F16 7.97 Tf 6.58 0 Td[(A(3)where=+P.Lemma 5 impliesthatforA=0(with=0)theaboveequationhasonlyrootswithnegativerealpartsandthecoexistenceequilibriumislocallystable.AsarststeptoestablishingHopfbifurcationforsomeA0>0,welookforpurelyimaginarysolutionsofthesimpliedcharacteristicequation( 3 ).Set=i!.Theequation( 3 )becomesi!++F!2 R)]TJ /F5 11.955 Tf 11.96 0 Td[(!2=e)]TJ /F10 7.97 Tf 6.58 0 Td[(i!ASeparatingtherealandimaginarypartintheaboveequation,weobtainthat!shouldsatisfythefollowingsystem:+F!2 R)]TJ /F5 11.955 Tf 11.96 0 Td[(!2=cos(!A)!=)]TJ /F5 11.955 Tf 9.3 0 Td[(sin(!A)Weeliminatethetrigonometricfunctionsbysquaringbothsidesoftheeachequationabove,andaddingtheequations.Thus,!shouldsatisfythefollowingequation +F!2 R)]TJ /F5 11.955 Tf 11.95 0 Td[(!22+!2=2.(3)Weset!2=ztoobtain +Fz R)]TJ /F4 11.955 Tf 11.95 0 Td[(z2+z=2.(3)Rewritingtheaboveequationasapolynomialequationinzweobtainz[z2)]TJ /F3 11.955 Tf 11.95 0 Td[((2F)]TJ /F4 11.955 Tf 11.95 0 Td[(F2+2R)z+(R2+2RF)]=0. 43

PAGE 44

Assumption( 3 )guaranteesthattheaboveequationhasthreerealroots:anegativeone,zero,andapositiveone.Letz0=!20bethepositiveroot.Then!0=p z0.Tocompletethebifurcationanalysis,wechoosethematurationsizeAasabifurcationparameter.Weviewthesolutionsofthecharacteristicequation( 3 )asfunctionsoftheparameterA,namely(A)=(A)+i!(A).ForsomevalueA0wehave(A0)=0,and!(A0)=!0.Weneedtoshowthattherootscrosstheimaginaryaxiswithnon-zerospeed,thatisweneedtoshowthatd<(A) dAjA=A0>0.Toseethislastinequality,wedifferentiatethecharacteristicequation( 3 )withrespecttothebifurcationparameterAtoobtain:1+Ae)]TJ /F16 7.97 Tf 6.59 0 Td[(A)]TJ /F3 11.955 Tf 23.57 8.09 Td[(2FR (2+R)2d dA=)]TJ /F5 11.955 Tf 9.3 0 Td[(e)]TJ /F16 7.97 Tf 6.58 0 Td[(A.Tosimplifythecomputation,welookattheinverseofd(A) dA:d dA)]TJ /F6 7.97 Tf 6.58 0 Td[(1=1)]TJ /F6 7.97 Tf 18.97 4.71 Td[(2FR (2+R)2 )]TJ /F5 11.955 Tf 9.3 0 Td[(e)]TJ /F16 7.97 Tf 6.59 0 Td[(A)]TJ /F4 11.955 Tf 13.15 8.08 Td[(A (3)=1)]TJ /F6 7.97 Tf 18.96 4.71 Td[(2FR (2+R)2 )]TJ /F5 11.955 Tf 9.3 0 Td[()]TJ /F5 11.955 Tf 5.48 -9.68 Td[(+)]TJ /F10 7.97 Tf 16.55 4.71 Td[(F2 2+R)]TJ /F4 11.955 Tf 13.15 8.08 Td[(A (3)Weset=i!0andrationalizethedenominator:d dA)]TJ /F6 7.97 Tf 6.59 0 Td[(1j=i!0=1)]TJ /F6 7.97 Tf 19.47 4.88 Td[(2FRi!0 ()]TJ /F16 7.97 Tf 6.58 0 Td[(!20+R)2 )]TJ /F4 11.955 Tf 9.3 0 Td[(i!0i!0++F!20 )]TJ /F16 7.97 Tf 6.59 0 Td[(!20+R)]TJ /F4 11.955 Tf 17.17 8.09 Td[(A i!0 (3)=h1)]TJ /F6 7.97 Tf 19.46 4.88 Td[(2FRi!0 ()]TJ /F16 7.97 Tf 6.59 0 Td[(!20+R)2i)]TJ /F4 11.955 Tf 9.29 0 Td[(i!0++F!20 )]TJ /F16 7.97 Tf 6.59 0 Td[(!20+R )]TJ /F4 11.955 Tf 9.3 0 Td[(i!0!20++F!20 )]TJ /F16 7.97 Tf 6.59 0 Td[(!20+R2+Ai !0Wetaketherealpartoftheexpressionintheright-handsideandusing( 3 )weobtain:
PAGE 45

TheexpressionontherighthandsideisnotautomaticallypositivesinceF<0.Toseeitspositivity,werecallthat!20=z0,andz0istherightmostsolutionoftheequationh(z)=2whereh(z)=+Fz R)]TJ /F4 11.955 Tf 11.95 0 Td[(z2+z.Weobservethat,sincez0isthepositiverootoftheequation,z0=R+2F)]TJ /F4 11.955 Tf 11.95 0 Td[(F2+p (2F)]TJ /F4 11.955 Tf 11.95 0 Td[(F2)2)]TJ /F3 11.955 Tf 11.95 0 Td[(4RF2 20,thatis 2FR (R)]TJ /F4 11.955 Tf 11.95 0 Td[(z0)2+Fz0 R)]TJ /F4 11.955 Tf 11.95 0 Td[(z0+1>0.(3)Thelastinequalityimpliesthatd< dA)]TJ /F6 7.97 Tf 6.58 0 Td[(1jA=A0>0.Thiscompletestheproof. 3.2-stepFunction&-stepFunctionForthissectionwemakethefollowingassumptions: Assumption6. Assume (x,N)=isaconstant; 45

PAGE 46

(x)isgivenby(x)=(00xA (x)isconstant:(x)=. Theorem3.2. Supposeassumption 6 holds.Alsoweassumeg(x)=1and0(N)=0.Asbeforewetakepreybirthrateasaseparablefunction(x,N)=(N)0(x)with0(x)=[A,1).ThisconditionguaranteesthepresenceofsustainedoscillationsinthepresentPDEmodel.Note:Theassumptionsinthetheorem,modelapredatorwhichfeedsselectivelyonlyonlargerreproductivepreysizes. Proof. Westartagainfromsystem( 2 ).Solvingthedifferentialequationweobtain (x)=(0)(;x,P))]TJ /F5 11.955 Tf 11.95 0 Td[(u(0)(;x,P)(x)1 [ex)]TJ /F4 11.955 Tf 11.95 0 Td[(eA].(3)ComputingtheintegralofwehaveZ1A(x)dx=(0)e)]TJ /F6 7.97 Tf 6.58 0 Td[((+)A1 ++P)]TJ /F5 11.955 Tf 13.15 8.08 Td[(u(0) e)]TJ /F16 7.97 Tf 6.59 0 Td[(A1 +P)]TJ /F3 11.955 Tf 46.41 8.08 Td[(1 +mu+P=(0)e)]TJ /F6 7.97 Tf 6.58 0 Td[((+)A1 ++P)-222()]TJ /F5 11.955 Tf 56.02 8.09 Td[(d (++P)wherewehaveusedtheequilibriumequationu(0)e)]TJ /F16 7.97 Tf 6.58 0 Td[(A=d(+P).Substitutingintheequationforin( 2 )weobtainthefollowingformulafor: =(0)Pe)]TJ /F6 7.97 Tf 6.58 0 Td[((+)A 2+L+K(3)whereL=+PandK=Pd.Theintegralsofaregivenby: Z1A(x)dx=(0)e)]TJ /F6 7.97 Tf 6.58 0 Td[((+)A 2+L+KZ10(x)dx=(0) +)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(e)]TJ /F6 7.97 Tf 6.58 0 Td[((+)A(3) 46

PAGE 47

Substitutingintheequationfor(0),andcanceling(0)weobtainthefollowingcharac-teristicequation: 1=(N)e)]TJ /F6 7.97 Tf 6.59 0 Td[((+)A 2+L+K+F1 +)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(e)]TJ /F6 7.97 Tf 6.58 0 Td[((+)A+e)]TJ /F6 7.97 Tf 6.58 0 Td[((+)A 2+L+K.(3)Theabovecharacteristicequationsimpliesto(+)]TJ /F4 11.955 Tf 11.96 0 Td[(F)(2+L+K)=[(N)(+))]TJ /F4 11.955 Tf 11.95 0 Td[(FP(+d)]e)]TJ /F6 7.97 Tf 6.59 0 Td[((+)A.Usingthecorrespondingequationfortheequilibria: e)]TJ /F16 7.97 Tf 6.58 0 Td[(A(N)=+P(3)thecharacteristicequationtakestheform: (+)]TJ /F4 11.955 Tf 11.96 0 Td[(F)(2+L+K)=L[(+))]TJ /F4 11.955 Tf 11.96 0 Td[(VP)(+d)]e)]TJ /F16 7.97 Tf 6.58 0 Td[(A(3)whereV=F=(N).Furthermore,thecharacteristicequationcanberewrittenas: 3+a12+a2+a3=[T12+T2+T3]e)]TJ /F16 7.97 Tf 6.58 0 Td[(A(3)where a1=)]TJ /F4 11.955 Tf 11.96 0 Td[(F+L (3) a2=L()]TJ /F4 11.955 Tf 11.96 0 Td[(F)+K (3) a3=()]TJ /F4 11.955 Tf 11.96 0 Td[(F)K (3) and T1=LT2=L()]TJ /F4 11.955 Tf 11.96 0 Td[(VP) (3) T3=LVK 47

PAGE 48

InordertoderivetheHopfBifurcationinthismodelweassume=i!isarootof 3 .Thisimpliesthat!hastosatisfythefollowingequation. )]TJ /F4 11.955 Tf 11.96 0 Td[(i!3)]TJ /F4 11.955 Tf 11.96 0 Td[(a1!2+a2!i+a3=iT2!cos(!)+(T3)]TJ /F4 11.955 Tf 11.96 0 Td[(T1!2)cos(!)+ (3) T2!sin(!))]TJ /F4 11.955 Tf 11.96 0 Td[(i(T3)]TJ /F4 11.955 Tf 11.96 0 Td[(T1!2)sin(!) (3) Weseparatetherealandimaginarypartstoobtainthefollowingsetsofequations. a2!)]TJ /F5 11.955 Tf 11.96 0 Td[(!3=T2!cos(!A))]TJ /F3 11.955 Tf 11.96 0 Td[((T3)]TJ /F4 11.955 Tf 11.95 0 Td[(T1!2)sin(!A) (3) a3)]TJ /F4 11.955 Tf 11.95 0 Td[(a1!2=(T3)]TJ /F4 11.955 Tf 11.95 0 Td[(T1!2)cos(!)+T2!sin(!) (3) Fromtheabovetwoequationsweeliminatesin(!A)andcos(!A)bysquaringandaddingthefunctiontoobtainthefollowingequation. !6+(a21)]TJ /F3 11.955 Tf 11.96 0 Td[(2a2)]TJ /F4 11.955 Tf 11.96 0 Td[(T21)!4+(a22)]TJ /F3 11.955 Tf 11.95 0 Td[(2a1a3+2T1T3)]TJ /F4 11.955 Tf 11.96 0 Td[(T22)!2+a23)]TJ /F4 11.955 Tf 11.96 0 Td[(T23=0(3)Sincethisequationlacksanyoddpowersof!wecanusethesubstitutionz=!2.Thisreduces( 3 )toathirdorderequationinzgivenasfollows. h(z)=z3+z2+Z+#=0(3)wherethecoefcientsaregivenasfollows. =a21)]TJ /F3 11.955 Tf 11.95 0 Td[(2a2)]TJ /F4 11.955 Tf 11.95 0 Td[(T21 (3) =a22)]TJ /F3 11.955 Tf 11.96 0 Td[(2a1a3+2T1T3)]TJ /F4 11.955 Tf 11.95 0 Td[(T22 (3) #=a23)]TJ /F4 11.955 Tf 11.95 0 Td[(T23 (3) 48

PAGE 49

If#<0i.e.a230withoutlossofgenerality.)Thusfrom( 3 )and( 3 )wecaninferthat j=1 !0arccos((a1T1)]TJ /F4 11.955 Tf 11.95 0 Td[(T2)!40+(a2T2)]TJ /F4 11.955 Tf 11.96 0 Td[(a3T1)]TJ /F4 11.955 Tf 11.96 0 Td[(a1T3)!20+a3T3 T22!20+(T3)]TJ /F4 11.955 Tf 11.95 0 Td[(T1!20)2)+2j !0,j=0,1,2,....(3)InordertoprovetheHopfBifurcationwestatethefollowingTheoremandlemma. Theorem3.3. Letusassume!0isthelargestpositiverootof( 3 ).Theni!(A0)=i!0isasimplerootof( 3 )and(A)+i!(A)isdifferentiablewithrespecttoAinaneighborhoodofA=A0.Itiseasytoobservethati!0isasimplerootandisanalytic.UsingtheanalyticversionofImplicitFunctionTheorem(A)+i!(A)isdenedandanalyticinaneighborhoodofA0. Lemma6. Letx1,x2,x3betherootsoftheequation g(x)=x3+x2+x+#=0,(<0)(3)andx3isthelargestpositivesimpleroot,then,dg(x) dxjx=x3>0.ToestablishtheHopfBifurcationatA=A0weneedtoshowthatdRe(A) dAjA=A0>0.Differentiating( 3 )withrespecttoAwegetthefollowingequation, (32+2a1+a2)d dA=[)]TJ /F5 11.955 Tf 9.29 0 Td[(e)]TJ /F16 7.97 Tf 6.59 0 Td[(A(T12+T2+T3)+e)]TJ /F16 7.97 Tf 6.59 0 Td[(A(2T1+T2)]d dA)]TJ ET BT /F1 11.955 Tf 433.45 -616.13 Td[((3) e)]TJ /F16 7.97 Tf 6.58 0 Td[(A(T12+T2+T3) (3) 49

PAGE 50

Thisleadsto (d dA))]TJ /F6 7.97 Tf 6.59 0 Td[(1=32+2a1+a2+Ae)]TJ /F16 7.97 Tf 6.59 0 Td[(A(T12+T2+T3))]TJ /F4 11.955 Tf 11.95 0 Td[(e)]TJ /F16 7.97 Tf 6.58 0 Td[(A(2T1+T2) )]TJ /F5 11.955 Tf 9.3 0 Td[(e)]TJ /F16 7.97 Tf 6.59 0 Td[(A(T12+T2+T3) (3) =32+2a1+a2 )]TJ /F5 11.955 Tf 9.3 0 Td[(e)]TJ /F16 7.97 Tf 6.59 0 Td[(A(T12+T2+T3)+2T1+T2 )]TJ /F5 11.955 Tf 9.3 0 Td[(e)]TJ /F16 7.97 Tf 6.59 0 Td[(A(T12+T2+T3))]TJ /F4 11.955 Tf 13.15 8.09 Td[(A (3) =23+a12)]TJ /F4 11.955 Tf 11.96 0 Td[(a3 )]TJ /F5 11.955 Tf 9.3 0 Td[(2(3+a12+a2+a3)+T12)]TJ /F4 11.955 Tf 11.96 0 Td[(T3 2(T12+T2+T3))]TJ /F4 11.955 Tf 13.15 8.08 Td[(A (3) Thuswehave, SignfRe dAg=i!0=SignfRe(d dA))]TJ /F6 7.97 Tf 6.59 0 Td[(1g=i!0 (3) =SignfRe[23+a12)]TJ /F4 11.955 Tf 11.96 0 Td[(a3 )]TJ /F5 11.955 Tf 9.3 0 Td[(2(3+a12+a2+a3)]=i!0+Re[T12)]TJ /F4 11.955 Tf 11.96 0 Td[(T3 2(T12+T2+T3)]g (3) =SignfRe[)]TJ /F3 11.955 Tf 9.3 0 Td[(2!30i)]TJ /F4 11.955 Tf 11.95 0 Td[(a1!20)]TJ /F4 11.955 Tf 11.95 0 Td[(a3 !20()]TJ /F5 11.955 Tf 9.3 0 Td[(!30i)]TJ /F4 11.955 Tf 11.95 0 Td[(a1!20+a2!0i+a3)]+Re[)]TJ /F4 11.955 Tf 9.3 0 Td[(T1!20)]TJ /F4 11.955 Tf 11.95 0 Td[(T3 )]TJ /F5 11.955 Tf 9.3 0 Td[(!20()]TJ /F4 11.955 Tf 9.3 0 Td[(T1!20+T2!0i+T3)]g (3) =Signf2!60+(a21)]TJ /F3 11.955 Tf 11.96 0 Td[(2a2)!40)]TJ /F4 11.955 Tf 11.96 0 Td[(a23 !20[(a1!20)]TJ /F4 11.955 Tf 11.96 0 Td[(a3)2+(!30)]TJ /F4 11.955 Tf 11.96 0 Td[(a2!0)2]+T23)]TJ /F4 11.955 Tf 11.96 0 Td[(T21!40 !20(T3)]TJ /F4 11.955 Tf 11.95 0 Td[(T1!0)2+T22!20g (3) =Signf2!60+(a21)]TJ /F3 11.955 Tf 11.96 0 Td[(2a2)]TJ /F4 11.955 Tf 11.96 0 Td[(T21)!40+T23)]TJ /F4 11.955 Tf 11.96 0 Td[(a23 !20[(a1!20)]TJ /F4 11.955 Tf 11.96 0 Td[(a3)2+(!30)]TJ /F4 11.955 Tf 11.96 0 Td[(a2!0)2]g (3) Signf3!40+2(a21)]TJ /F3 11.955 Tf 11.95 0 Td[(2a2)]TJ /F4 11.955 Tf 11.96 0 Td[(T21)!20+(a22)]TJ /F3 11.955 Tf 11.95 0 Td[(2a1a3+2T1T3)]TJ /F4 11.955 Tf 11.96 0 Td[(T22) (a1!20)]TJ /F4 11.955 Tf 11.96 0 Td[(a3)2+(!30)]TJ /F4 11.955 Tf 11.96 0 Td[(a2!0)2g (3) Sinceh(z)=z3+Z2+z+#,Wecanconcludethat, dh(z) dz=3z2+2z+=3z2+2(a21)]TJ /F3 11.955 Tf 11.95 0 Td[(2a2)]TJ /F4 11.955 Tf 11.96 0 Td[(T21)z+(a22)]TJ /F3 11.955 Tf 11.95 0 Td[(2a1a3+2T1T3)]TJ /F4 11.955 Tf 11.96 0 Td[(T22)(3)As!0isthelargestpositivesimplerootof( 3 )fromtheprecedingLemmawehave,dh(z) dzjz=!0>0.Hence dRe() dA=dh(!0) dz (a1!20)]TJ /F4 11.955 Tf 11.95 0 Td[(a3)2+(!30)]TJ /F4 11.955 Tf 11.96 0 Td[(a2!0)2>0(3)ThisprovestheexistenceofHopfBifurcationinthePDEmodelinthiscasewhenthepredatorfeedpreferentiallyonlargersizes. 50

PAGE 51

CHAPTER4NUMERICALANALYSISOFTHEPDEMODEL 4.1NumericalSchemeLetusconsiderthemodelwithasconstant. ut+(g(x)u(x,t))x=)]TJ /F5 11.955 Tf 9.3 0 Td[((x,N)u)]TJ /F5 11.955 Tf 11.96 0 Td[((x)Pug(0)u(0,t)=Z10(x,N)u(x,t)dx (4) u(x,0)=(x),P(0)=P0 (4) P0=PZ10(x)u(x,t)dx)]TJ /F4 11.955 Tf 11.95 0 Td[(dP (4) Thenumericalschemeisgivenby ^ujk+1)]TJ /F3 11.955 Tf 13.61 0 Td[(^ujk t+gj^ujk+1)]TJ /F4 11.955 Tf 11.95 0 Td[(gj)]TJ /F6 7.97 Tf 6.59 0 Td[(1^uj)]TJ /F6 7.97 Tf 6.59 0 Td[(1k+1 x+kj^ujk+1+jPk^ujk+1=0(4)Weusethefollowingnotationintheanalysis.kj=u(xj,tk))]TJ /F3 11.955 Tf 13.61 0 Td[(^ujk,k=Pk)]TJ /F3 11.955 Tf 13.24 2.65 Td[(^Pk,k=Nk)]TJ /F3 11.955 Tf 13.52 2.65 Td[(^NkWetakethemaximaltimetobeTandmaximalsizetobeA.Wediscretizethexvariablewithstepxasxj=x0+jx,j=1,2,...,mandthetimevariabletwithsteptastk=kt,k=1,2,..,n. 4.2NumericalAnalysis Theorem4.1. Ifthefunctions(x,N),(x),,(x,N)areboundedandiftherateofchangeofandwithrespecttoNisalsorestrictedwithinproperbounds,thenthenumericalschemeofthePDEmodelisconvergingoftheordert. Proof. Weconsidermodel 4 .Toshowtheconvergenceofthenumericalscheme,weintroducethefollowingdiscretizationofthemodel. 51

PAGE 52

u(xj)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)g(xj)]TJ /F6 7.97 Tf 6.59 0 Td[(1)=u(xj)]TJ /F3 11.955 Tf 11.95 0 Td[(x,t)g(xj)]TJ /F3 11.955 Tf 11.96 0 Td[(x)=u(xj,t)g(xj))]TJ /F5 11.955 Tf 16.48 8.08 Td[(@ @x(g(xj)u(xj,t))x+@2 @x2(g()u(,t))(x)2 2wherexj)]TJ /F6 7.97 Tf 6.59 0 Td[(1xj.Solvingfor@ @x(g(xj)u(xj,t))givesthefollowingform @ @x(g(xj)u(xj,t))=u(xj,t)g(xj))]TJ /F4 11.955 Tf 11.96 0 Td[(u(xj)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)g(xj)]TJ /F6 7.97 Tf 6.58 0 Td[(1) x+ (4) @2 @x2(g()u(,t))(x) 2 (4) Wereplacetin( 4 )bytk+1toobtainthefollowingequation. @ @x(g(xj)u(xj,tk+1))=u(xj,tk+1)g(xj))]TJ /F4 11.955 Tf 11.96 0 Td[(u(xj)]TJ /F6 7.97 Tf 6.58 0 Td[(1,tk+1)g(xj)]TJ /F6 7.97 Tf 6.59 0 Td[(1) x+ (4) @2 @x2(g()u(,tk+1))(x) 2 (4) Nowweexpandu(x,t)intermsofthetimevariable. u(x,tk)=u(x,tk+1)]TJ /F3 11.955 Tf 11.96 0 Td[(t) (4) =u(x,tk+1))]TJ /F5 11.955 Tf 15.87 8.09 Td[(@ @tu(x,tk+1)t+@2 @t2u(x,)(t)2 2 (4) whereisdifferentfromtheonebeforeandsatisestk<
PAGE 53

@ @tu(x,tk+1)=u(x,tk+1))]TJ /F4 11.955 Tf 11.96 0 Td[(u(x,tk) t+@2 @t2u(x,)(t) 2(4)Combining( 4 )and( 4 )wehavethefollowingsetsofequation. @ @tu(x,tk+1)+@ @x(g(xj)u(xj,tk+1))= (4) u(x,tk+1))]TJ /F4 11.955 Tf 11.96 0 Td[(u(x,tk) t+u(xj,tk+1)g(xj))]TJ /F4 11.955 Tf 11.95 0 Td[(u(xj)]TJ /F6 7.97 Tf 6.59 0 Td[(1,tk+1)g(xj)]TJ /F6 7.97 Tf 6.58 0 Td[(1) x+ (4) @2 @t2u(x,)(t) 2+ (4) @2 @x2(g()u(,tk+1))(x) 2 (4) Wenowexpandthetermscontaining(x)and(x),(xj)P(tk+1)u(xj,tk+1)=(xj)u(xj,tk+1)P(tk+t)=(xj)u(xj,tk+1)[P(tk)+tP0()]=(xj)u(xj,tk+1)P(tk)+(xj)u(xj,tk+1)P0()t(xj,N(tk+1))u(xj,tk+1)=u(xj,tk+1)(xj,N(tk)+N0()t)=u(xj,tk+1)[(xj,N(tk))+N0()@(xj,) @Nt]Substitutingbackallthederivedtermsaboveintheoriginalequationweobtainthefollowingequation. u(x,tk+1))]TJ /F4 11.955 Tf 11.96 0 Td[(u(x,tk) t+u(xj,tk+1)g(xj)u(xj)]TJ /F6 7.97 Tf 6.59 0 Td[(1,tk+1)g(xj)]TJ /F6 7.97 Tf 6.58 0 Td[(1) x (4) +(xj,N(tk+1))u(xj,tk+1)+(xj)P(tk+1)u(xj,tk+1)= (4) )]TJ /F4 11.955 Tf 9.3 0 Td[(u(xj,tk+1)N0()@(xj,) @Nt)]TJ /F5 11.955 Tf 11.95 0 Td[((xj)u(xj,tk+1)P0()t)]TJ /F5 11.955 Tf 15.88 8.09 Td[(@2 @t2u(x,)(t) 2+ (4) @2 @x2(g()u(,tk+1))(x) 2 (4) 53

PAGE 54

Thenumericalschemeisgivenby ^ujk+1)]TJ /F3 11.955 Tf 13.61 0 Td[(^ujk t+gj^ujk+1)]TJ /F4 11.955 Tf 11.95 0 Td[(gj)]TJ /F6 7.97 Tf 6.58 0 Td[(1^uk+1j)]TJ /F6 7.97 Tf 6.58 0 Td[(1 x+kj^ujk+1+jPk^ujk+1=0(4)Takingthedifferencebetween( 4 )and( 4 )weobtainu(xj,tk+1))]TJ /F3 11.955 Tf 13.61 0 Td[(^ujk+1)]TJ /F3 11.955 Tf 11.95 0 Td[((u(xj,tk))]TJ /F3 11.955 Tf 13.61 0 Td[(^ujk t+gj(u(xj,tk+1))]TJ /F3 11.955 Tf 13.61 0 Td[(^ujk+1))]TJ /F4 11.955 Tf 11.95 0 Td[(gj)]TJ /F6 7.97 Tf 6.59 0 Td[(1(u(xj)]TJ /F6 7.97 Tf 6.59 0 Td[(1,tk+1))]TJ /F3 11.955 Tf 19.12 0 Td[(^uj)]TJ /F6 7.97 Tf 6.59 0 Td[(1k+1) x+((xj,N(tk+1))u(xj,tk+1))]TJ /F5 11.955 Tf 11.95 0 Td[(kj^ujk+1)+((xj)P(tk+1)u(xj,tk+1))]TJ /F5 11.955 Tf 11.95 0 Td[(jPk^ujk+1)=)]TJ /F4 11.955 Tf 9.29 0 Td[(u(xj,tk+1)N0()@(xj,) @Nt)]TJ /F5 11.955 Tf 11.95 0 Td[((xj)u(xj,tk+1)P0()t)]TJ /F5 11.955 Tf 13.22 8.09 Td[(@2 @t2u(x,)(t) 2+@2 @x2(g()u(,tk+1))(x) 2NowweobservethatP(tk+1)u(xj,tk+1))]TJ /F4 11.955 Tf 11.95 0 Td[(Pk^ujk+1=u(xj,tk+1)k+^Pkk+1j,((xj,N(tk+1))u(xj,tk+1))]TJ /F5 11.955 Tf 11.95 0 Td[(kj^ujk+1)=(xj,Nk)[u(xj,tk+1))]TJ /F3 11.955 Tf 13.61 0 Td[(^ujk+1]+[(xj,Nk))]TJ /F5 11.955 Tf 11.95 0 Td[(kj]uk+1j=(xj,Nk)k+1j+0(xj,)kuk+1jCombiningtheaboveequationwecanhavethefollowingformoftheequation, k+1j)]TJ /F5 11.955 Tf 11.95 0 Td[(kj t+gjk+1j)]TJ /F4 11.955 Tf 11.95 0 Td[(gj)]TJ /F6 7.97 Tf 6.59 0 Td[(1k+1j)]TJ /F6 7.97 Tf 6.59 0 Td[(1 x (4) +(xj,Nk)k+1j+0(xj,)kuk+1j+ju(xj,tk+1)k+j^Pkk+1j (4) =)]TJ /F4 11.955 Tf 9.3 0 Td[(u(xj,tk+1)N0()@(xj,) @Nt)]TJ /F5 11.955 Tf 11.95 0 Td[((xj)u(xj,tk+1)P0()t)]TJ /F5 11.955 Tf 15.88 8.09 Td[(@2 @t2u(x,)(t) 2+ (4) @2 @x2(g()u(,tk+1))(x) 2 (4) 54

PAGE 55

Multiplyingbytonbothsideswearriveatthefollowingequation, k+1j)]TJ /F5 11.955 Tf 11.96 0 Td[(kj+t x(gjk+1j)]TJ /F4 11.955 Tf 11.95 0 Td[(gj)]TJ /F6 7.97 Tf 6.59 0 Td[(1k+1j)]TJ /F6 7.97 Tf 6.59 0 Td[(1)+ (4) t(xj,Nk)k+1j+t0(xj,)kuk+1j+tju(xj,tk+1)k+tj^Pkk+1j (4) =)]TJ /F4 11.955 Tf 9.3 0 Td[(u(xj,tk+1)N0()@(xj,) @N(t)2)]TJ /F5 11.955 Tf 11.95 0 Td[((xj)u(xj,tk+1)P0()(t)2 (4) c)]TJ /F5 11.955 Tf 15.88 8.09 Td[(@2 @t2u(x,)(t)2 2+@2 @x2(g()u(,tk+1))(x)(t) 2 (4) Fromthestructureoftheoriginalmodelwehavethatthederivativeofualongthecharacteristiclinesisnegative.Henceu(x,t)isboundedforallxandt.ThisimpliesN(t)isalsobounded,whichisobtainedbytheintegralofu(x,t).Sinceu(x,t)isapositivefunction,thederivativeofu(x,t)cannotbeunbounded,sincethatwillmakeunegativeforsomevaluesoftimet.ThisleadstodNalsobounded.Weletx=t,forsomeconstantk.Solvingfork+1jleadstoanequationoftheform, mjk+1j=kj+t xgj)]TJ /F6 7.97 Tf 6.59 0 Td[(1k+1j)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ ET BT /F1 11.955 Tf 433.45 -394.44 Td[((4) t0(xj,)kuk+1j)]TJ /F3 11.955 Tf 11.95 0 Td[(tju(xj,tk+1)k+((t)2) (4) wherewecallmj=1+gjt x+t(xj,Nk)+tj^Pk.Notemj0,hencetakingabsolutevalueonbothsideswehavethefollowingform, mjjk+1jjjkjj+t xgj)]TJ /F6 7.97 Tf 6.59 0 Td[(1jk+1j)]TJ /F6 7.97 Tf 6.59 0 Td[(1j+ (4) tj0(xj,)jjkjuk+1j+tju(xj,tk+1)jkj+((t)2) (4) 55

PAGE 56

Fromthebehavioroftheparametersweknowthatisboundedfrombelowby .LetTbethemaximumtimeandifnisthenumberofstepsinthenumericalsimulationswehave,mj1+gjt x+t +t ^PkThisreduces( 4 )tothefollowingform, (1+t( ))jk+1jjx+t(gjjk+1jj)]TJ /F4 11.955 Tf 17.93 0 Td[(gj)]TJ /F6 7.97 Tf 6.58 0 Td[(1jk+1j)]TJ /F6 7.97 Tf 6.58 0 Td[(1j) (4) jkjj+t xgj)]TJ /F6 7.97 Tf 6.59 0 Td[(1jk+1j)]TJ /F6 7.97 Tf 6.59 0 Td[(1j+tj0(xj,)jjkjuk+1j+ (4) tju(xj,tk+1)jkj+((t)2) (4) Wenotethatjisaboundedfunctionhencej .If0isunbounded,thatimpliesthepreypopulationdiesataninniteratewhichwillresultintheextinctionofthepreypopulationatanexponentialrate.Hence0 (constant).Summingoverallnwehave, (1+T n )jjk+1jj+t(gnjk+1nj)]TJ /F4 11.955 Tf 17.94 0 Td[(g0jk+10j) (4) jjkjj+C1tjkj+C2tjkj+((t)2). (4) Solvingforjjk+1jjresultsinthefollowingequation. jjk+1jjt (1+T n )(g0jk+10j)]TJ /F4 11.955 Tf 17.93 0 Td[(gnjk+1nj)+ (4) 1 (1+T n )(jjkjj+C1tjkj+C2tjkj)+((t)2) (4) Nowletusconsidertheboundarycondition,g0u(0,t)=Z10(x,N)u(x,t)dxWritingtheintegralindiscretetermswehave, 56

PAGE 57

g0u(0,tk+1)=nXj=1(xj,Nk+1)u(xj,tk+1)x(4)Thenumericalschemeisgivenby g0^uk+10=nXj=1k+1j^ukjx(4)Rewriting( 4 )andexpressingthefactorsintermsofkjwehavethefollowingstructure,(xj,Nk+1)u(xj,tk+1)=(xj,N(tk+t))u(xj,tk+t)=(xj,N(tk)+tN0())u(xj,tk+t)=[(xj,N(tk))+0(xj,N(tk))tN0()][u(xj,tk)+tu0(xj,)]=(xj,N(tk))u(xj,tk)+(t)Sinceand0areboundedfunctions. nXj=1(xj,Nk+1)u(xj,tk)x+(t)=g0u(0,tk+1)(4)Takingthedifferencebetween( 4 )and( 4 )wehave g0k+10=nXj=1k+1jkjx+(t)(4)Takingtheabsolutevaluesontheaboveequationwehave, g0jk+10j nXj=1jkjjx+(t)= jjkjj+(t) (4) Substitutingtheaboveformin( 4 )wehave 57

PAGE 58

jjk+1jj(t 1+T n +1 (1+T n ))jjkjj+C1tjkj+C2tjkj)+((t)2) (4) Letsnowfucusonthepredatorequation,P0=PZ10(x)u(x,t)dx)]TJ /F4 11.955 Tf 11.96 0 Td[(dPWediscretizethecontinuousequationtoobtainP(tk)=P(tk+t)=P(tk)+tP0(tk)+(t)2 2P00() P(tk+1))]TJ /F4 11.955 Tf 11.96 0 Td[(P(tk) t)]TJ /F3 11.955 Tf 13.15 8.08 Td[(t 2P00()=P0(tk)(4)Using( 4 )versionofthediscreteformintheoriginalequationwehave, P(tk+1))]TJ /F4 11.955 Tf 11.95 0 Td[(P(tk) t)]TJ /F3 11.955 Tf 13.15 8.09 Td[(t 2P00()= (4) P(tk)nXj=1(xj)u(xj,tk)x)]TJ /F4 11.955 Tf 11.96 0 Td[(d(P(tk+1))]TJ /F3 11.955 Tf 11.95 0 Td[(tP0()) (4) Hencewearriveatthefollowingequation, P(tk+1))]TJ /F4 11.955 Tf 11.96 0 Td[(P(tk) t)]TJ /F4 11.955 Tf 11.95 0 Td[(P(tk)nXj=1(xj)u(xj,tk)x+dP(tk+1 (4) =t 2P00()+dtP0() (4) Thenumericalschemecorrespondingtotheabovemethodisgivenby, 58

PAGE 59

^Pk+1)]TJ /F3 11.955 Tf 13.24 2.66 Td[(^Pk t)]TJ /F3 11.955 Tf 13.24 2.65 Td[(^PknXj=1jukj+d^Pk+1=0(4)Wetakethedifferencebetween( 4 )and( 4 )toobtain, k+1)]TJ /F5 11.955 Tf 11.96 0 Td[(k t)]TJ /F5 11.955 Tf 11.96 0 Td[((nXj=1jkjx)^Pk (4) )]TJ /F5 11.955 Tf 9.3 0 Td[(nXj=1ju(xj,tk)xk+dk+1 (4) =t 2P00()+dtP0()+(t) (4) Wesolvefork+1, (1+dt)k+1)]TJ /F5 11.955 Tf 11.95 0 Td[(k+((nXj=1jkjx)^Pk (4) +nXj=1ju(xj,tk)xk)t+(t) (4) (1+dt)k+1=k(1+nXj=1ju(xj,tk)xt)+ (4) (nXj=1jkjx)^Pkt+((t)2) (4) (4) Takingtheabsolutevalueonbothsideswehave, (1+dt)jjk+1jjjjkjj(1+nXj=1ju(xj,tk)xt)+ (4) (nXj=1jjkjjx)^Pkt+((t)2) (4) (4) Since^Pkisbounded,j and^Pk P, 59

PAGE 60

PNj=1ju(xj,tk)x ^Nk Nwhere^Nkisalsoboundedasprovedbefore. NXj=1jjkjjx jjkjj (4) Therefore,(1+dt)jjk+1jjjjkjj+t P jjkjj+(t) (4) Hence,jjk+1jj1+ Nt 1+dtjjkjj+ P t 1+dtjjkjj+(t2) (4) Notethat, N(tk)=NXj=1u(xj,tk)x (4) ^Nk=NXj=1^ukjx (4) )k=NXj=1kjx (4) )jkjjjkjj (4) Thisshowsthatweonlyneedtoworkwithkandk. jjk+1jj+jjk+1jj (4) (t 1+ t+1 1+ t+C1t+ P t 1+dt)jjkjj (4) +(1+ Nt 1+dt+C2t)jjkjj+(t2) (4) (Sincejjkjjjjkjj)NotewecanndC,dsuchthat, 60

PAGE 61

(t 1+ t+1 1+ t+C1t+ P t 1+dt)1+Ct 1+dt jjk+1jj+jjk+1jj (4) (1+Ct 1+dt)(jjkjj+jjkjj)+(t) (4) ((t))(1+p+p2+p3+....+pk)+ (4) pk+1(jj0jj+jj0jj) (4) (t)(pk)]TJ /F3 11.955 Tf 11.95 0 Td[(1 p)]TJ /F3 11.955 Tf 11.96 0 Td[(1)+pk+1(jj0jj+jj0jj) (4) wherep=1+Ct 1+dt,Notethat,usingtheinitialconditionsofthefunctionwehave,jj0jj=NXj=1j0jjx=NXj=1ju(xj,0))]TJ /F3 11.955 Tf 13.61 0 Td[(^uj0jx=NXj=1j(xj))]TJ /F5 11.955 Tf 11.95 0 Td[((xj)jx=0.Similarlyforthefunctionwehavejj0jj=jP0)]TJ /F3 11.955 Tf 13.24 2.65 Td[(^P0j=jP0)]TJ /F4 11.955 Tf 11.96 0 Td[(P0j=0.Thisshowsthat, 61

PAGE 62

jjk+1jj+jjk+1jj (4) (t)pk)]TJ /F3 11.955 Tf 11.96 0 Td[(1 p)]TJ /F3 11.955 Tf 11.96 0 Td[(1 (4) =T nM(1+CT=n 1+dT=N)k)]TJ /F3 11.955 Tf 11.96 0 Td[(1 1+CT=n 1+dT=N)]TJ /F3 11.955 Tf 11.96 0 Td[(1 (4) TM jc)]TJ /F4 11.955 Tf 11.96 0 Td[(djT(1+dT=N)(1+jC)]TJ /F4 11.955 Tf 11.96 0 Td[(djT N)Nt~Ct (4) where(t) tM,wheret=T=nHencetheconvergenceisofordert. 4.3SimulationsusingtheNumericalDiscretizationThenitedifferencemethod,denedabove,canbeusedtosimulatethePDEmodel.Tondparametersthatwouldproduceoscillations,welet=+wiandseparatetherealandimaginarypartin( 3 ).Welet=0.001.Wefurtherassignvaluestosomeoftheparameters.Thus,weassign=0.1,)]TJ /F4 11.955 Tf 9.3 0 Td[(F=1,andA=1.Wealsotake(x,N)=[A,1)e)]TJ /F10 7.97 Tf 6.59 0 Td[(cN.ForagivenwthesystemfortherealandtheimaginarypartbecomesalinearsysteminP,anddP.WesolvethatlinearsystemusingMathematica,andweobtainP=f1(w)anddP=f2(w)asfunctionsofw.Theparametricplotofthesetwofunctionsinthe(P,dP)planeisgiveninFigure 4-1 .Forw=3.96055weobtainthefollowingpositivevaluesforP=0.35anddP=16.3689.Thatgivesavalueford=46.7682857.Usingtheequationsfortheequilibria,wedeterminethat=9.62023,c=1,and=23.2591.Theparameterisdeterminedsothatpredator'sreproductionnumberislargerthanone.Inparticular,wetook,=1.TheresultingoscillationsofthepredatorandthepreyarepresentedinFigure 4-2 62

PAGE 63

Figure4-1. Parametricplotinthe(P,dP). Figure4-2. Theleftgureshowsthecycleinthe(N,P)planewithtimeasaparameter.Therightgureshowstheoscillationsinthetotalnumberofpreyandthepredatorasfunctionsoftimewhenthoseoscillationshavestabilized. 63

PAGE 64

CHAPTER5DISCUSSIONInthisarticleweintroduceanon-linearpredator-preymodelwherethepreyisstructuredbysize.Themainquestionthatweaddressiswhetherpredationonasize-structuredpreycanberesponsibleforthesustainedoscillationsobservedinthepredator-preydynamicsinnature.Weconsidertwomainaspects: 1. Weshowthatthepresenceofapredatorwhichpredatesonasize-structuredpreycandestabilizeanotherwisestableequilibriumoftheprey,evenifthepredationofthepredatorissizenon-specic. 2. Weshowthatsize-specicpredationiscapableofproducingoscillationsinthepredator-preydynamics.Wendthatcanbethecasebyexaminingascenariowhenthepredatorpredatesonthereproductivesizesofthepreyonly.ToaddressthemainquestionwerstexaminethedynamicsofthecorrespondingODEmodel,wherethepreyishomogeneouswithrespecttosize.WendthattheODEmodelhasanextinctionequilibrium,canpotentiallyhavemultipleprey-onlyequilibria,andauniquepredator-preycoexistenceequilibrium.Thecoexistenceequilibriumislocallyasymptoticallystableiftheprey'sgrowthratedecreaseswiththeincreaseofthepreypopulation.If,however,preygrowthrateexhibitsAlleeeffect,thenthepredator-preycoexistenceequilibriumcanbecomedestabilizedandoscillationsarepossible.Toruleoutthatthisscenarioisresponsiblefortheoscillationsinthesize-structuredmodel,weassumeinmostcasesthatthepreybirthratedeclineswiththepopulationsize.Thesize-structuredmodelalsohasanextinctionequilibriumandmultipleprey-onlyequilibria.Wecouldnotruleoutthepossibilitythatmultiplecoexistenceequilibriaexist.TointerpretconditionsforexistenceofequilibriawedenenetreproductionrateofthepreypopulationasafunctionofthepreypopulationsizeR(N),andintrinsicreproductionnumberofthepreypopulationR0,denedasthevalueofthenetreproductionratewhenthepreypopulationsizeiszero.WendthatifR0>1thereisalwaysatleastoneprey-onlyequilibrium.IfR0<1thentheremaybeno 64

PAGE 65

prey-onlyequilibria,ortheremaybeanevennumberofprey-onlyequilibria,iftheyareallsimple.Furthermore,wedenepredatorreproductionattheNjprey-onlyequilibrium.Conditionsonthepredatorreproductionnumberguaranteeexistenceofapredator-preycoexistenceequilibrium.WendthattheextinctionequilibriumislocallystableifR0<1andunstableotherwise.Theextinctionequilibriumcannotbegloballystablebecauseofthepresenceofsubthresholdequilibria.Furthermore,wendthatthejthprey-onlyequilibriumEjisunstableifthereproductionnumberofthepredatoratthejthonlyequilibriumRp,j>1.Oftheprey-onlyequilibriaforwhichRp,j<1holds,theonesforwhichthenetreproductionrateofthepreysatisesR0(Nj)>0arealsounstable.Thekeyresultonstabilityofprey-onlyequilibriasaysthatifthedeathrateissizeindependentandincreasingwithpreypopulationsize,whilethebirthrateisdecreasingwithpopulationbutmaydependonindividuals'sizeinanarbitraryfashion,thentheprey-onlyequilibriumisstable.Weinvestigatedthestabilityofthecoexistenceequilibriaintwocases.Intherstcaseallratesareconstantwithrespecttoindividuals'size,exceptthebirthratewhichmaybearbitrary.Forexponentialinsizebirthrateweshowthatthecoexistenceequilibriumislocallyasymptoticallystable.However,ifthepredatorpredatesuniformlyonallsizes,butonlymatureindividualsreproduce,theneveniftheprey'sbirthrateisdecreasingwiththeprey'stotalpopulationsize,thenthecoexistenceequilibriummaybecomeunstable,andHopfbifurcationoccurs.Wenotethatintheseconditions,ifallsizesofthepreyreproduceduniformly,thenoscillationswouldnothaveoccurred.Inthesecondcase,allratesareconstantwithexceptionofthepredationrateandpreybirthrate.Inthiscaseweallowthepredatortopredateonindividualsofreproductivesize.Weconcludethatthepredator-preyequilibriumcanbecomeunstableandoscillationsarepossible.Thus,theanswertoourmainquestionwhethersize-specicpredationcandestabilizethepredator-preydynamicsisyes.Theideaofsize-specicpredation 65

PAGE 66

destabilizingthedynamicsissomewhatparadoxalassizemaypermitsizerefugesforthepreyfrompredationwhichmayseemstabilizing.However,evolutionofthepreytomoreadvantageoussizes,notpreyedonbythepredator,happensonevolutionaryscalewhichmaybemuchslowerthanthetimethepredatorneedstoadapttopreyingondifferentsizes.InarecentarticleMougiandIwasa[ 9 ]ndthatifthepredator'straitevolvesfasterthantheprey's,oscillationsarepossibleandlikely.Inotherwords,sizeisonlyatemporaryescapemechanismforthepreyandassuchmayberesponsiblefortheoscillatorydynamics[ 10 ]. 66

PAGE 67

REFERENCES [1] L.J.S.Allen,AnintroductiontoMathematicalBiology,Pearson,NewJersey,2007 [2] A.Calsina,J.Saldana,Amodelofphysiologicallystructuredpopulationdynamicswithnonlinearindividualgrowthrate,JournalofMath.Biology,vol.33,pp.335-364,1995. [3] J.E.Childs,Size-dependentpredationonratsbyhousecatsinanurbansetting,JournalofMammology,vol.67,no.1,pp.196-199,1986. [4] J.Li,Dynamicsofage-structuredpredator-preypopulationmodels,JournalofMath.AnalysisandApplication,vol.152,pp.399-415,1990. [5] I.Lima,D.B.Olson,S.C.Doney,Intrinsicdynamicsandstabilitypropertiesofsize-structuredpelagicecosystemmodels,JournalofPlanktonRes,vol.24,no.6,pp.533-566,2002. [6] D.Lundvall,R.Svanback,L.Persson,P.Bystrom,Size-dependentpredationinpiscivores:interactionbetweenpredatorforagingandpreyavoidanceabilities,CanadianJournalofFisheriesandAquaticSys.,vol.56,no.7,pp.1285-1292,1999. [7] J.D.Logan,Phenologically-structuredpredator-preydynamicswithtemperaturedependence.BullitenofMath.Biology,vol.70,pp.1-20,2008. [8] J.D.Logan,G.Ledder,W.Wolesensky,TypeIIfunctionalresponseforcontinuous,physiologicallystructuredmodel,JournalofTechnologyandBusiness,vol.259,pp.373-381,2009. [9] A.Mougi,Y.Iwasa,Evolutiontowardsoscillationorstabilityinapredator-preysystem,Proc.R.Soc.B,(inpress). [10] A.M.deRoos,J.A.J.Metz,E.Evers,A.Leipoldt,Asizedependentpredator-preyinteraction:whopursueswhom?,JournalofMath.Biology,vol28,pp.609-643,1990. [11] M.L.Rosenzweig,Paradoxofenrichment:destabilizationofexploitationecosystemsinecologicaltime,Science,vol.171,pp.385-387,1971. [12] M.J.Tegner,L.A.Levin,Spinylobstersandseaurchins:Analysisofapredator-preyinteraction,JournalofExp.Math.BiologyandEcology,vol.73,no2,pp.125-150,1983. [13] J.Travis,W.H.Keen,J.Juilianna,Theroleofrelativebodysizeinapredator-preyrelationshipbetweendragonynaiadsandlarvalanurans,Oikos,vol.45,pp.59-65,1985. 67

PAGE 68

[14] EnricSala,MichaelH.GrahamCommunity-widedistributionofpredatorpreyinteractionstrengthinkelpforests,PNAS,vol93,no.6,pp.36783683,March2002. [15] M.Begon,C.Townsend,J.Harper,Ecology:Individuals,populationsandcommunities(Thirdedition),BlackwellScience,London,1996. [16] Simon.Jennings,Karema.J.Warr,Smallerpredator-preybodysizeratiosinlongerfoodchains,ProceedingsoftheRoyalSociety,vol270,pp.14131417,2003. [17] P.H.Warren,J.H.Lawton,Invertebratepredator-preybodysizerelationships:anexplanationforuppertriangularfoodwebsandpatternsinfoodwebstructure,Oceologia(Berlin),vol.74,pp.231-235,1987 [18] G.E.Hutchinson,HomagetoSantaRosaliaorwhywhyaretheresomanykindsofanimals?,TheAmericanNaturalist,vol.33,pp.145-159,1959 [19] R.M.May,Stabilityandcomplexityinmodelecosystems,PrincetonUniversitypress,Princeton,2001. [20] R.T.Paine,Foodwebcomplexityandspeciesdiversity,TheAmericanNatural-ist,vol.100,pp.65-75,1966 [21] M.S.Laska,J.T.Wootton,Theoreticalconceptsandempiricalapproachesmeasuringinteractionstrength,Ecology,vol.99,pp.461-476,1998 [22] R.D.Holt,Ontheintegrationofcommunityecologyandevolutionarybiology:historicalperspectivesandcurrentprospects,EcologicalparadigmsLost.RoutesofTheoryChange,K.Cuddington,B.E.Beisner,ElsevierAmsterdam,pp.235-271,2005. [23] D.N.Reznick,F.H.Shaw,F.H.Rodd,R.G.Shaw,Evaluationoftherateofevolutioninnaturalguppies(Poecillareticulata),Science,vol.275,pp.1934-1937,1997 [24] M.S.Zimmerman,Predatorcommunitiesassociatedwithbrookstickleback(Culaeainconstans)prey:pattersinbodysize,CanadianJournalofsheriesandaquaticscience,vol.63,pp.297-309,2006 68

PAGE 69

BIOGRAPHICALSKETCH SouvikBhattacharyareceivedhisBachelorofScienceinmathematicsfromSt.Xavier'sCollege,Kolkata(India)in2003andMasterofScienceinMathematicsfromIndianInstituteofTechnology,Kanpur(India)in2003.HejoinedtheDepartmentofMathematicsatUniversityofFloridainAugust,2006topursueaPhDinmathematics.HereceivedhisPh.DfromtheUniversityofFloridainthesummerof2011. 69