A Generalized Keller-Segel Model

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Title:
A Generalized Keller-Segel Model
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english
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Zuhr, Erica Leigh
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University of Florida
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Gainesville, Fla.
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Gopalakrishnan, Jay
Committee Co-Chair:
De Leenheer, Patrick
Committee Members:
Hager, William W
Keesling, James E
Peters, Jorg

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Subjects / Keywords:
chemotaxis -- formation -- keller -- pattern -- segel
Mathematics -- Dissertations, Academic -- UF
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Mathematics thesis, Ph.D.
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Abstract:
Motivated by the derivation of the standard Keller-Segel model, we introduce a generalized Keller-Segel model for chemotaxis. Our generalized system models an organism, the chemoattractant which the organism is chemotactically attracted to, and an arbitrary number of chemicals which interact with the chemoattractant. Our model accounts for the fact that some of these chemicals may be produced by the organism. We present results on the existence and linear stability of homogeneous steady states using a reduction to a finite dimensional system. These results significantly expand the class of chemical reaction networks which demonstrate chemotactically initiated pattern formation. In addition to theoretical results, we use the standard finite element method to demonstrate solutions to both the original and the generalized Keller-Segel models. In addition to demonstrating nonhomogeneous stationary solutions of the standard Keller-Segel model, we perform time simulations to demonstrate the stability of these stationary solutions. We also use a nonstandard finite element method to perform time simulations, and show that this nonstandard method reduces numerical instability. We prove that under certain conditions, this numerical method preserves positivity of solutions. Finally, we use a novel method of spectral bands to demonstrate directions of instability in the generalized model, and use these as a basis for further time simulations.
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by Erica Leigh Zuhr.
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Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Gopalakrishnan, Jay.
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Co-adviser: De Leenheer, Patrick.
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AGENERALIZEDKELLER-SEGELMODELByERICAZUHRADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012EricaZuhr 2

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ACKNOWLEDGMENTS Firstandforemost,IwouldliketothankmyadvisorsProfessorJayGopalakrishnanandProfessorPatrickDeLeenheer.Despitebeingacrossthecountryorevenoutofthecountryforalargeportionofmygraduatecareer,theywentaboveandbeyondtocontinuesupportingmyresearchandprofessionaldevelopment.ThanksalsogoouttoMargaretforbeingmythirdadvisorwhentheothertwowerenotinthestate.Iwouldliketothankmycommitteemembersfortheirtimeandvaluablefeedback.IwanttothankKevinforbeingsosupportive,andforalwaysbeingabletomakemesmilenomatterwhatIamdealingwithintheacademicworld.Ialsowanttothankmymom,dad,andstepdadforunconditionalloveandsupportineverythingIdo.Finally,thanksgoouttoallofmyfriendsinthemathdepartment,especiallymyofcemates.Imighthavebeenabletonishmydegreewithoutthem,butitwouldn'thavebeennearlyasmuchfun. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 10 2CHEMOTAXISALONGAFIXEDGRADIENT ................... 17 2.1AnExampleinOneDimension ........................ 17 2.1.1AnalyticalSteadyStateSolutions ................... 18 2.1.2NumericalVisualization ........................ 24 2.2TheMultidimensionalCase .......................... 24 2.2.1ExistenceofClassicalSolutions .................... 24 2.2.2PropertiesofNumericalSolutions ................... 25 2.2.3NumericalDifcultiesandMethodValidation ............. 27 3ASIMPLEKELLER-SEGELMODEL ........................ 32 3.1PositivityofExactSolutions .......................... 32 3.2PreliminariesforNumericalSteadyStateSolutions ............. 37 3.3NumericalSolutions .............................. 49 3.3.1SolutionsontheUnitSquare ..................... 49 3.3.2SolutionsonaDisk ........................... 50 4ANEXTENDEDKELLER-SEGELMODEL .................... 53 4.1TheGeneralizedModelForm ......................... 53 4.2ReactionsYieldingPositiveSolutions ..................... 55 4.3ExistenceofHomogeneousSteadyStateSolutions ............. 61 4.4StabilityofHomogeneousSteadyStateSolutions .............. 63 4.4.1ReductiontoFiniteDimension ..................... 64 4.4.2SufcientConditionsforInstability .................. 67 4.4.3SufcientConditionsforStability ................... 73 4.5Examples .................................... 79 5NUMERICALSOLUTIONS ............................. 87 5.1SpectralBands ................................. 87 5.2EigenfunctionsandPotentialPatterns .................... 91 5.3PositivityofNumericalSolutions ....................... 94 4

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5.3.1ImplementationoftheMethod ..................... 97 5.3.2DerivationoftheMethod ........................ 99 5.3.3ProofofPositivity ............................ 101 5.4Vectorization .................................. 104 5.5MethodValidation ............................... 105 6APPLICATIONSANDRESULTSOFSIMULATIONS ............... 108 6.1ApplicationtoTumorGrowth ......................... 108 6.2Keller-SegelTimeSimulations ........................ 112 6.2.1MinimalKeller-SegelModel ...................... 112 6.2.2FourEquationKeller-SegelModel ................... 116 6.3ParameterFittingintheFourEquationKeller-SegelModel ......... 122 REFERENCES ....................................... 129 BIOGRAPHICALSKETCH ................................ 132 5

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LISTOFTABLES Table page 3-1Errorforsteadystatesolutiononunitsquare ................... 49 3-2Errorforsteadystatesolutiononadisk ...................... 51 6-1ParametervaluesforfourequationKS ....................... 124 6

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LISTOFFIGURES Figure page 2-1Numericalsolutionto( 2 ) ............................. 24 2-2StandardFEMsolutionsto( 2 ) ......................... 30 2-3Exactsolutionto( 2 )onrenedmesh ..................... 31 3-1Solutionto( 3 )ontheunitsquare ........................ 50 3-2Solutionsto( 3 )onadisk ............................ 51 3-3Moresolutionsto( 3 )onadisk ......................... 52 5-1AnexampleofthespectralbandsofthematricesM(i)correspondingtothedimerizationreactionfor1i20. ........................ 89 5-2AnexampleofthespectralbandsofthematricesM(i)correspondingtothefullKeller-Segelmodelfor1i5. ........................ 91 5-3Eigenfunctioncorrespondingtotherstpositiveeigenvalueof( 5 )withu=170. .......................................... 94 5-4Eigenfunctionscorrespondingtothesecondpositiveeigenvalueof( 5 )withu=550. ....................................... 95 5-5Eigenfunctionsof( 5 ) ............................... 95 5-6MorespectralbandsofthefullKSmodel. ..................... 96 5-7Eigenfunctionsof( 5 )onadisk .......................... 96 5-8PositivitypreservingFEMsolutionsto( 2 ) ................... 107 6-1OnedimensionalFEMsolutionstoEquation( 6 ) ................ 110 6-2TwodimensionalFEMsolutionsforuofEquation( 6 ) ............. 111 6-3SpectralbandsoftheminimalKSmodel ...................... 114 6-4Blow-upoftheminimalKSmodel .......................... 115 6-5Convergent(u=11)versusblow-up(u=13)parameterrangesoftheminimalKSmodel .................................. 116 6-6Convergencetoanonhomogeneousstationarysolutiononadisk ........ 117 6-7ConvergencetoconstantsteadystateforfullKSmodel ............. 118 6-8DifferencebetweenstandardandpositivitypreservingFEMforfullKStimesimulations ...................................... 119 7

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6-9BlowupoffullKSmodelattimet=130 ...................... 120 6-10Behaviorofsolutionfromrandomlyperturbedinitialconditions ......... 121 6-11Alinearizedeigenfunctionofucorrespondingtoanegativeeigenvalue ..... 121 6-12SpectralbandsusingparametersfromTable 6-1 ................. 125 6-13Behaviornearu=0ofspectralbandsusingparametersfromTable 6-1 .... 125 6-14Spectralbandsforttedvalue .......................... 126 6-15Thelinearizedeigenfunctioncorrespondingtothelargestpositiveeigenvalue 127 6-16Othereigenfunctionscorrespondingtopositiveeigenvalues ........... 128 8

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AbstractofdissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyAGENERALIZEDKELLER-SEGELMODELByEricaZuhrMay2012Chair:JayGopalakrishnanCochair:PatrickDeLeenheerMajor:MathematicsMotivatedbythederivationofthestandardKeller-Segelmodel,weintroduceageneralizedKeller-Segelmodelforchemotaxis.Ourgeneralizedsystemmodelsanorganism,thechemoattractantwhichtheorganismischemotacticallyattractedto,andanarbitrarynumberofchemicalswhichinteractwiththechemoattractant.Ourmodelaccountsforthefactthatsomeofthesechemicalsmaybeproducedbytheorganism.Wepresentresultsontheexistenceandlinearstabilityofhomogeneoussteadystatesusingareductiontoanitedimensionalsystem.Theseresultssignicantlyexpandtheclassofchemicalreactionnetworkswhichdemonstratechemotacticallyinitiatedpatternformation.Inadditiontotheoreticalresults,weusethestandardniteelementmethodtodemonstratesolutionstoboththeoriginalandthegeneralizedKeller-Segelmodels.InadditiontodemonstratingnonhomogeneousstationarysolutionsofthestandardKeller-Segelmodel,weperformtimesimulationstodemonstratethestabilityofthesestationarysolutions.Wealsouseanonstandardniteelementmethodtoperformtimesimulations,andshowthatthisnonstandardmethodreducesnumericalinstability.Weprovethatundercertainconditions,thisnumericalmethodpreservespositivityofsolutions.Finally,weuseanovelmethodofspectralbandstodemonstratedirectionsofinstabilityinthegeneralizedmodel,andusetheseasabasisforfurthertimesimulations. 9

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CHAPTER1INTRODUCTIONChemotaxisisafascinatingbiologicalphenomenonwhichoccurswhenthemovementofacellororganismisaffectedbyachemicalintheenvironment[ 18 ].Positivechemotaxisoccurswhenachemicalcalledthechemoattractantattractstheorganism,andnegativechemotaxisoccurswhenaso-calledchemorepellentrepellstheorganism.Fromfertilizationofaneggintheearlieststatesofdevelopmenttoimmunesystemfunctiontocancergrowthandmetastasis,chemotaxisarisesinmanydifferentbiologicalprocesses[ 17 ].ThewellstudiedtwoequationKeller-Segel(KS)modelforchemotaxis,whichwewillreviewbelow,wasoriginallydevelopedbyKellerandSegelin[ 21 ].ThestronglycoupledtwoequationKSsystemmodelstwovariables,thespecieswhosemovementisaffectedandthechemoattractantorchemorepellent.Althoughgeneralizationsofthemodel(reviewedbelow)havebeenintroducedandstudiedpreviously,weintroduceourownformofageneralizedmodelwhichallowsformodelingofthespecieswhosemovementisbeingaffected,thechemoattractant,andanarbitrarynumberofchemicalswhichinteractwiththechemoattractantandmayormaynotbeproducedbythespecies.Inadditiontointroducingthemodelandgivingnewtheoreticalresultsonexistenceandstabilityofsteadystates,weapplytheniteelementmethod(FEM)tondnumericalsolutionsofboththegeneralizedmodelandthesimpletwoequationmodel.Asanextensionofresultsin[ 21 34 ],weshowthatadestabilizationmechanismwhichpotentiallyleadstopatternformationextendstomanychemicalreactionnetworks(CRNs)intheframeworkofourgeneralizedmodel.Beforecontinuing,werstintroducesomeimportantnotationthatwillbeusedthroughout.Weallowtheslightabuseofnotationthatvariablessuchasuandviwhenusedinanequationdenotetheconcentrationsofthespeciesbeingmodeled,butwhenthemeaningisclearincontextweoftenusethevariablesinsteadofreferringtothechemicalororganismbyname.Wedenotepartialderivativesbysubscripts,for 10

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example@tuisthepartialderivativeofuwithrespecttotimet.Whenconsideringadomain[0;T]Rn+1,thesymbolsranddenotethegradientandtheLaplacianwithrespecttothespatialvariablex2Rn.Whenusedwithderivatives,nor~nwilldenotetheoutwardpointingnormaltotheboundaryand@u @nrundenotesthedirectionalderivativeintheoutwardnormaldirection.AllintegralsaretakenwithrespecttothestandardLebesguemeasure.ByWk;p()wemeantheSobolevspaceconsistingofallfunctionsinLp()whosepartialderivativesuptoorderkexistintheweaksenseandareinLp().Asisstandardnotation,Hk()Wk;2().WedenotebyHk()NtheproductspaceHk()Hk()Hk(),wheretheproductistakenNtimes.BythefunctionspacesCn()foranyintegern0wedenotefunctionsonwhosen-thorderderivativesexistandarecontinuous.HenceC0()=C()isthespaceofcontinuousfunctionson.Finally,wegivethenotationwhichwillbeusedfortheFEManalysis.LetThdenoteatriangulationofthegivendomainwithglobalverticestypicallyindexedbyx`.ThelinearFEMspaceisdenedtobeSh=fw2C():wjTislinearoneverymeshtriangleT2Thg.Thefunction`2Shisdenedtobethepiecewiselinearfunctionwhichisoneatvertexx`andzeroatallotherverticesofthetriangulation.Tobetterunderstandthemotivationforourgeneralizedmodel,werstreviewthederivationoftheoriginalKeller-Segelmodel.In[ 21 ],KellerandSegelbasethederivationoftheirmodelontheslimemoldDictyosteliumdiscoideum.D.discoideumhasauniquelifecycle[ 16 18 ],andduetoitsinterestingdevelopmentisamodelorganismfortheNationalInstituteofHealth[ 18 ].Thelifecylebeginsastheamoebadivideandreproduceasexuallysolongasasufcientfoodsourceispresent.Oncethefoodsourceisdepleted,theamoebaenterastarvationmodeand,afteratime,onecellwillbegintoemitasignalofthechemicalcyclicAdenosineMonophosphate,orcAMP.Thissignalchemotacticallyattractstheotheramoeba,whichinturnbegintoemitacAMPsignal.Inadditiontheamoebaemitphosphodiesterase,anenzymethatdegradesthe 11

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cAMP.Thisfeedbackloopcausesthecellstoaggregateandformamulticellularslug,whichmovesasonebody.Oncetheslughasmovedtoabetterenvironment,thecellsdifferentiateintotwotypes,andformafruitingbodywhichemitsspores.Thesesporesgerminateintonewamoeba,andthelifecyclebeginsanew.TheKSmodelisbasedontheaggregationphaseofthelifecycle,whenthechemotacticfeedbackloopinitiatesaggregation.Inordertoderivethemodel,KellerandSegelmakethefollowingassumptionsin[ 21 ],asreviewedin[ 18 ].ConsiderthemodelonasufcientlysmoothdomainRnandfortimet>0.LetudenotetheconcentrationofD.discoideum,vdenotetheconcentrationofthechemoattractantcAMP,denotetheconcentrationoftheenzymephosphodiesterase,andctheconcentrationofacomplexformedbytheenzymeandthecAMP.AssumeadditionallythattheenzymedegradesthecampbymeansofaCRNrepresentedby v+ !c)166(!+degradedproduct:(1)Letthereaction( 1 )begovernedbythelawofmassactionwithforwardandbackwardrateconstantsfortherstreactiongivenbyk1andk2respectively,andthelastreactionrateconstantgivenbyk3.AssumealsothattheratesatwhichtheD.discoideumproducethecAMPandtheenzymearegivenbythenonnegativefunctionsf(v)andh(v;)respectively.Finally,assumethatallcomponentsdiffuseaccordingtoFick'slaw,andassumethattheuxofthecellhasanadditionaltermrepresentingthechemotacticcontribution.HencetheuxoftheamoebaisgivenbyJu(x;t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(D0ru(x;t)+(u;v)rv(x;t)whereDuisthediffusioncoefcientand(u;v)iscalledthechemotacticsensitivityfunction.Theseassumptionsalongwitheitherno-uxorNeumannboundaryconditions 12

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leadtothefullKeller-Segelmodelfrom[ 21 ], ut=r(D0ru)]TJ /F3 11.955 Tf 11.95 0 Td[((u;v)rv);x2;t>0 (1a)vt=D1v)]TJ /F3 11.955 Tf 11.95 0 Td[(k1v+k2c+uf(v);x2;t>0 (1b)t=D1)]TJ /F3 11.955 Tf 11.96 0 Td[(k1v+(k2+k3)c+uh(v;);x2;t>0 (1c)ct=D3c+k1v)]TJ /F4 11.955 Tf 11.95 0 Td[((k2+k3)c;x2;t>0 (1d)@u @n=@v @n=@ @n=@c @n=0;x2@;t>0: (1e)AssoonasKellerandSegelintroduce( 1 ),theyimmediatelyreduceittoatwoequationsystemwhichmodelsonlythechemoattractantandtheamoeba.Thisreductionusestheassumptionthatthetotalconcentrationoftheenzymeinbothfreeandboundformsisconstant,meaning+c=0:Thesecondassumptionisthatthecomplexcisinsteadystatewithregardstothereaction( 1 ),meaningthatk1v)]TJ /F4 11.955 Tf 11.96 0 Td[((k2+k3)c=0:Furtherassumptionsreviewedin[ 17 18 ]reducethetwoequationmodeltothecommonlystudiedminimalsystem, ut=r(D0ru)]TJ /F3 11.955 Tf 11.95 0 Td[(urv);x2;t>0 (1a)vt=D1v+u)]TJ /F3 11.955 Tf 11.95 0 Td[(v;x2;t>0 (1b)@u @n=@v @n=0;x2@;t>0 (1c)u(0;x)=u0(x);v(0;x)=v0(x);x2: (1d)In( 1 ),>0isaconstantdecayrateofthechemoattractant,>0istheconstantproductionrateperamoebaofthechemoattractant,andthechemotacticsensitivityfunction(u;v)from( 1a )isassumedtobeoftheformuforsomeconstant 13

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>0.Thesystem( 1 )hasbeenstudiedextensively,andresultsonexistenceanduniquenessofsolutions,positivity,blow-up,andstabilitycanbefoundin,forexample,[ 17 18 34 ].Blow-up,whichisconsideredtooccurwhentheL1normofeitheruorvbecomesunboundedinniteorinnitetime,hasattractedagreatdealofattentionbutnotwillnotbeaddressedinourworkexceptforthecitationofafewcomputationallyusefulresultsinSection 3.2 andsomenumericalresultsinChapter 6 .Theunreducedsystem( 1 )hasnotbeenthesubjectofagreatamountofstudy,andthispartiallymotivatesthegeneralizationoftheKSmodelwhichwillbeintroducedinChapter4.OneofthemostinterestingaspectsofourtheoreticalresultsonthegeneralizedKSmodelisthattheypointtochemotacticallyinducedpatternformationforverygeneralclassesofCRNs.Althoughthismechanismforpatternformationhasbeenshownin[ 34 ]andothersinthecaseoftheminimalmodel,wegeneralizeittoamuchlargerclassofsystems.ToemphasizehowchemotacticallyinducedinstabilitydiffersfromthetraditionalTuringdiffusionbasedinstability[ 41 ],wereviewanexampleofdiffusiveinstabilitiesasexplainedin[ 8 32 ].Forthisexample,consideraspatiallyhomogeneousstationarysolution(u;v)ofthepartialdifferentialequation(PDE) ut=D1u+f(u;v);x2;t>0 (1a)vt=D2v+g(u;v);x2;t>0 (1b)@u @n=@v @n=0;x2@;t>0 (1c)forsomesmoothfunctionsfandganddiffusioncoefcientsD1>0andD2>0.Thennoticethat(u;v)isalsoasolutionofthecorrespondingordinarydifferentialequation(ODE)system du dt=f(u;v);t>0 (1a) dv dt=g(u;v);t>0: (1b) 14

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Assumethat(u;v)isalinearlystablesolutionto( 1 ),thatisalleigenvaluesoftheJacobianJ=264@uf(u;v)@vf(u;v)@ug(u;v)@vg(u;v)375havestrictlynegativerealpart.Turingasksifitispossiblethatthesamesolution(u;v)islinearlyunstableinthesettingof( 1 ).Theanswerisyes,anditturnsoutthatnecessaryconditionsincluded16=d2andthattheJacobianmatrixhaveoneofthesignpatterns 264+)]TJ /F4 11.955 Tf -19.07 -23.91 Td[(+)]TJ /F10 11.955 Tf 9.3 41.36 Td[(375;264)]TJ /F4 11.955 Tf 19.26 0 Td[(+)]TJ /F4 11.955 Tf 19.26 0 Td[(+375;264++)-833()]TJ /F10 11.955 Tf 28.56 41.36 Td[(375;or264)-832()]TJ /F4 11.955 Tf .09 -23.91 Td[(++375:(1)OurchemotacticallyinducedstabilityresultsinSection 4.4 concerningthegeneralizedmodeldonotrequirerestrictionsonthediffusioncoefcients.AlthoughwedorequiresomerestrictionsonthesignsoftheentriesofaJacobianmatrix,thesesignpatternsaredifferentfromtheonesin( 1 ).Asmentionedpreviously,generalizationsoftheKSmodelforchemotaxishavebeenaddressedinotherrecentpublications[ 3 9 11 19 26 43 ].Mostoftheselookatmodelswithaspeciedsmallnumberofbothspeciesandchemotacticagents,forexample[ 3 9 11 26 ].In[ 3 ],globalexistenceofsolutionsforatwochemicalversionoftheKSsystemisdiscussed,alongwiththeexistenceofaLyapanovfunctional.In[ 9 ]ontheotherhand,aparabolic-ellipticsystemwithtwochemotacticspeciesbutonlyonechemoattractantisstudied.In[ 26 ]themulti-speciesconceptisfurthergeneralizedandasystemwithonechemoattractantbutanarbitrarynumberofspeciesisinvestigated.Oneadditionalchemicalisintroducedin[ 11 ]asequilibriumsolutionstoatwochemical,twospeciessystemareinvestigated.Howevernoneofthemodelsmentionedaboveaccountforanarbitrarynumberofchemicalsasourgeneralizedmodeldoes.In[ 19 43 ]twomodelswhichdoaccountforanarbitrarynumberofchemicalsareintroduced,andarebothsimilartothegeneralizedmodelweintroducehere.Inaddition 15

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toanarbitrarynumberofchemicalcomponents,bothmodelsaccountforanarbitrarynumberofspecieswhichchemotacticallyreact.Howeverthemodelin[ 43 ]doesnotallowfornontrivialchemicalreactionsbetweenthechemicalsbeingmodeled,whichisoneimportantcomponentofourinvestigation.Themodelin[ 19 ]isquitegeneral,andencompassesthesettingofourgeneralizedmodel.Howeverafterintroducingtheverygeneralsetup,themajorityoftheworkin[ 19 ]focusesonspecicexampleswhichlookatonlyoneortwochemotacticagentsandoneortwospecies.Exceptionsinclude[ 19 ,Section5],wheregeneralconditionsfortheexistenceofaLyapunovfunctionalforthegeneralmodelareshown,and[ 19 ,Section6]wheregeneralizationofsomeresultsfrom[ 34 ]onthesteadystateproblemaregiven.However[ 19 ]doesnotconsiderthegeneralizationoftheresultsondestabilizationofhomogeneousstationarysolutionsbasedonparameterrangesandthemeanoftheinitialcondition,whichisthefocusofoneofourmainresults. 16

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CHAPTER2CHEMOTAXISALONGAFIXEDGRADIENT 2.1AnExampleinOneDimensionInordertogainsomeinsightintothephenomenonofchemotaxisandthemathematicalmodelingofit,weinvestigateasimplepartialdifferentialequation(PDE)similartoonederivedin[ 25 ].Letu(x;t)representthedensityofabacterialpopulationatagivenpointx2[0;1]andtimet0.Itisassumedthatthebacteriaareattractedtoareasofhigherconcentrationofacertainchemicalsubstance.Thedensityofthissubstanceataanypointx2[0;1]isassumedtobeindependentoftimeandisdenotedbys(x).AsinthederivationoftheoriginalKeller-Segel(KS)model,itisassumedthattheuxofthebacteriahastwocomponents,therstarisingfromrandomdiffusivemovementandtheotherfromdirectedmovementtowardshigherconcentrationsofthechemoattractantsignals(x).Giventheseassumptions,theuxwilltaketheformJ(x;t)=D@ @xu(x;t))]TJ /F3 11.955 Tf 11.96 0 Td[(u(x;t)d dxs(x)where>0isaconstantrepresentingthechemotacticsensitivityandD>0isthediffusioncoefcient.CombiningthisuxtermwiththeconditionZS@u @t+rJdx=0foreverySleadstothePDE@tu=D@xxu)]TJ /F3 11.955 Tf 11.95 0 Td[(@x(us0(x)):ByrescalingthetimevariabletoallowD=1andapplyingno-uxboundaryconditions,wearriveattheone-dimensionalPDE @tu=@xxu)]TJ /F3 11.955 Tf 11.96 0 Td[(@x(u(x;t)s0(x))x2(0;1);t>0 (2a)@xu)]TJ /F3 11.955 Tf 11.96 0 Td[(u(x;t)s0(x)=0x2f0;1g;t>0: (2b) 17

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2.1.1AnalyticalSteadyStateSolutionsItispossibletoanalyticallysolveforsteadystatesof( 2 ).Thesteadystateproblemwillbe@xxu)]TJ /F3 11.955 Tf 11.96 0 Td[(@x(us0(x))=0;andintegratinggives@xu)]TJ /F3 11.955 Tf 12.6 0 Td[(us0(x)=kforsomeconstantk.Applyingtheboundaryconditions( 2b )atx=0orx=1showsthatforanycontinuoussteadysolutionwemusthavek=0,meaningthatanysteadystatesolutionmustsatisfy @xu)]TJ /F3 11.955 Tf 11.95 0 Td[(us0(x)=0:(2)Since( 2 )isalinearequationinu,itcanalwaysbesolvedanalyticallywiththegeneralsolutionbeinggivenbyu(x)=Ces(x);aonedimensionalsetofsolutionsparametrizedbytheconstantC.Inordertobeabletocompletelyclassifysteadystatesolutionsandtheirstability,weturntothespecicexamplewheres(x)=ax.Wewillcallastationarysolutionlinearlystableifalleigenvaluesofthelinearizedproblemaboutthesteadystatehavestrictlynegativerealpart.Inthissectionwewillshowthatallstationarysolutionsof( 2 )inthecasewheres(x)=axarelinearlystablewithrespecttoperturbationsUsatisfyingR10U(x)dx=0.Toseethereasonfortherestrictionontheperturbations,noticethatsolongasasolutionuto( 2 )issufcientlysmoothwecanuseintegrationbypartsandtheboundaryconditiongivenbyEquation( 2a )tosee @ @tZ10u(x;t)dx=Z10@tudx=Z10@x(@xu)]TJ /F3 11.955 Tf 11.95 0 Td[(us0)dx=[@xu)]TJ /F3 11.955 Tf 11.96 0 Td[(us0]10=0: 18

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Sincethetotalmassinthedomainofanysolutionremainsconstantovertime,toshowstabilityweonlyconsiderperturbationsUawayfromthesteadystatewhichhavezeromeanandthereforedonotchangethemassofthesolution.Wewilldenotethedifferentialoperatorin( 2a )withs(x)=axasL:=@2 @x2)]TJ /F3 11.955 Tf 11.95 0 Td[(@ @xwhere:=a.HencewehaveL[u]=@xxu)]TJ /F3 11.955 Tf 11.95 0 Td[(@xu:Thentodeterminestabilityofanysteadystatewhere@tu=L[u]=0,weneedtosolvetheeigenvalueproblem L[u]=u:(2)ThisbecomestheODE d2u dx)]TJ /F3 11.955 Tf 11.95 0 Td[(du dx)]TJ /F3 11.955 Tf 11.95 0 Td[(u=0:(2)Thecharacteristicequationfor( 2 )isr2)]TJ /F3 11.955 Tf 11.96 0 Td[(r)]TJ /F3 11.955 Tf 11.96 0 Td[(=0,whichwillhaverootsr1;2=p 2+4 2:Inordertodeterminethesignofforpossiblesolutionsto( 2 ),werstwillshowthatalleigenvaluesmustbereal,andthenwillconsiderthreecasesbasedonthevalue2+4,thediscriminantofthecharacteristicequation.Weconsiderwhathappenswhenhasnonzeroimaginarypart.Since26=0isrealandIm()6=0,itfollowsthat2+46=0.Hencer1andr2aredistinctandsolutionsof( 2 )areoftheformU(x)=c1er1x+c2er2xwherer1andr2maybecomplex.SubstitutingthisexpressionforUands0(x)=aintotheboundarycondition( 2b )andevaluatingatx=0andx=1respectivelygivesthe 19

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system c1(r1)]TJ /F3 11.955 Tf 11.96 0 Td[()+c2(r2)]TJ /F3 11.955 Tf 11.95 0 Td[()=0 (2a)c1er1(r1)]TJ /F3 11.955 Tf 11.95 0 Td[()+c2er2(r2)]TJ /F3 11.955 Tf 11.95 0 Td[()=0: (2b)Writingthisinmatrixform,wesee( 2 )becomesA~c:=264r1)]TJ /F3 11.955 Tf 11.96 0 Td[(r2)]TJ /F3 11.955 Tf 11.95 0 Td[(er1(r1)]TJ /F3 11.955 Tf 11.96 0 Td[()er2(r2)]TJ /F3 11.955 Tf 11.95 0 Td[()375264c1c2375=26400375:HenceanonzeroeigenfunctionexistsifandonlyifthematrixAissingular,soifandonlydetA=0,or (r1)]TJ /F3 11.955 Tf 11.96 0 Td[()(r2)]TJ /F3 11.955 Tf 11.95 0 Td[()(er2)]TJ /F3 11.955 Tf 11.96 0 Td[(er1)=0:(2)Weconsiderthethreecaseswherer1=,r2=,ander2)]TJ /F3 11.955 Tf 12.2 0 Td[(er1=0.Considerrstthecasewherer1=.Thensincer1+r2=,r2=0anditfollowsfromthedenitionofr1;2that=0,whichisacontradictiontothefactthathasnonzeroimaginarypart.Asimilarargumentshowswecannothaver2=.Itremainstoconsiderthecasewhereer1=er2.Ifwewritethepossiblycomplexrootsasr1=a1+ib1andr2=a2+ib2,thener1=er2holdsifandonlyifa1)]TJ /F3 11.955 Tf 11.96 0 Td[(a2=0andb1)]TJ /F3 11.955 Tf 11.95 0 Td[(b2=2kforsomeintegerk.Writethecomplexnumberp 2+4asei=p 2+4forsomereal0andreal.Since2isrealandhasnonzeroimaginarypart,p 2+46=0andsoinfact>0.Recallingtheoriginaldenitionsofr1;2,wecanwriter1=1 2(+ei)andr2=1 2()]TJ /F3 11.955 Tf 11.96 0 Td[(ei): 20

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Thenrecallingthatisrealwecancalculate0=a1)]TJ /F3 11.955 Tf 11.95 0 Td[(a2=Re(r1))]TJ /F1 11.955 Tf 11.96 0 Td[(Re(r2)=1 2+1 2Re(ei))]TJ /F10 11.955 Tf 11.96 16.86 Td[(1 2)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2Re(ei)=Re(ei)=cos():Since>0,itmustbethatcos()=0and=(2`+1) 2forsomeinteger`.Inotherwords,weknowthatei=p 2+4ispurelyimaginary.Howeverthisimpliesei=i=p 2+4sothat2i2=)]TJ /F3 11.955 Tf 9.3 0 Td[(2=2+4:RecallingthatandarerealnumbersthisimpliesIm()=0whichisacontradictiontotheassumptionthathadnonzeroimaginarypart.Hence( 2 )hasonlyrealeigenvalues.Nowthatwehaveshownanyeigenvaluemustbereal,therearethreepossiblecaseswhichcanbeinvestigatedseparately.Firstweconsiderthecasewhere2+4>0.Inthiscasetherearetwodistinctrealrootsr1andr2,andtheformofthesolution,asinthecasewithimaginary,isU(x)=c1er1x+c2er2x:Againapplyingtheno-uxboundarycondition,c1andc2mustsatisfythesystem( 2 )inorderforU(x)torepresentaneigenfunction.Again,thisimpliesthateitherr1=,r2=,orer1=er2.Sincer1andr2arerealanddistinct,wecannothaveer1=er2.Ifr1=thenasinthepreviouscaseweconcludethatr2=0and=0.Similarlyifr2= 21

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thenr1=0and=0.Weclaimthat=0violatesthezeroaveragecondition.Ifthisisthecase,thenwehaveshownnoapplicableeigenfunctionsexistinthecasewhere2+4>0.Indeed,notethatif=0theneitherr1=0orr2=0.Ifforexampler1=0,thenEquation 2a andr2=implythatc1=0.HoweverthenU(x)=c2er2x;soU(x)doesnothavezeroaverageunlessc2=0aswell.Thesameargumentholdsforr2=0,andsonoapplicableeigenfunctionsexistinthiscase.Thenextcaseis2+4=0,whichimpliesthatthecharacteristicequationhasonedoublerootrandthatUisoftheformU(x)=c1erx+c2xerx:Applicationoftheno-uxboundaryconditionatx=0andx=1forthisformofUleadstothesystem c1r+c2)]TJ /F3 11.955 Tf 11.96 0 Td[(c1=0 (2a)c1rer+c2rer+c2er)]TJ /F3 11.955 Tf 11.96 0 Td[(c1er)]TJ /F3 11.955 Tf 11.95 0 Td[(c2er=0: (2b)Cancelinger6=0fromEquation( 2b )andthensubtractingEquation( 2a )givesc2(r)]TJ /F3 11.955 Tf 11.96 0 Td[()=0:Sincer==2itcannotbethatr)]TJ /F3 11.955 Tf 12.16 0 Td[(=0,andsoweconcludethatc2=0.Then( 2a )becomesc1(r)]TJ /F3 11.955 Tf 11.95 0 Td[()=0;andwesimilarlyconcludethatc2=0andthattherearenoeigenfunctionsinthiscase. 22

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Wenallyturntothecasewhere2+4<0,whichimpliesthesolutionstothecharacteristicequationaretwocomplexconjugateimaginaryroots,r1;2= 2ip )]TJ /F3 11.955 Tf 9.3 0 Td[(2)]TJ /F4 11.955 Tf 11.96 0 Td[(4 2:Letting:=p )]TJ /F5 7.97 Tf 6.59 0 Td[(2)]TJ /F8 7.97 Tf 6.59 0 Td[(4 2,UisoftheformU(x)=e 2x(c1cos(x)+c2sin(x)):Applyingtheno-uxboundaryconditionatx=0andx=1respectivelygives c2+ 2c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c1=0 (2a)cos()c2+ 2c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c1+sin())]TJ /F3 11.955 Tf 9.3 0 Td[(c1+ 2c2)]TJ /F3 11.955 Tf 11.96 0 Td[(c2=0: (2b)SubstitutingEquation( 2a )into( 2b )andsimplifyinggivessin())]TJ /F3 11.955 Tf 9.3 0 Td[(c1)]TJ /F10 11.955 Tf 11.95 13.28 Td[( 2c2=0:If)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F3 11.955 Tf 9.3 0 Td[(c1)]TJ /F10 11.955 Tf 11.95 9.69 Td[()]TJ /F5 7.97 Tf 6.68 -4.98 Td[( 2c2=0thentogetherwithEquation( 2a )thisleadstothelinearsystem264)]TJ /F3 11.955 Tf 9.29 0 Td[()]TJ /F5 7.97 Tf 10.5 4.71 Td[( 2)]TJ /F5 7.97 Tf 10.5 4.71 Td[( 2375=264c1c2375=26400375whichhasdeterminant2 4+2.However2 4+2=)]TJ /F3 11.955 Tf 9.3 0 Td[(>0inthiscase,sothesystemhasnononzerosolutionsforc1andc2.Ontheotherhandifsin()=0thentheboundaryconditionsarealsosatised.So,valuesof=kforsomeintegerkproduceeigenvaluesoftheformk=)]TJ /F3 11.955 Tf 9.3 0 Td[(2 4)]TJ /F3 11.955 Tf 11.96 0 Td[(k22:WehavenowshownalleigenvaluesofLarenegativeexceptfor=0,andsoanystationarysolutionof( 2 )islinearlystablewithrespecttoperturbationsU(x)satisfyingR10U(x)dx=0. 23

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Figure2-1. Numericalsolutionto( 2 ) 2.1.2NumericalVisualizationUsingMATLAB'sPDEsolverpdepe,wecandemonstratethatsolutionsquicklyconvergetoasteadystatewhichmimicstheshapeofthesignals(x)foravarietyofdifferentsignals.Figure 2-1 showsthecasewheres(x)=sin(2x)andtheinitialconditionistheconstantu(0)1. 2.2TheMultidimensionalCase 2.2.1ExistenceofClassicalSolutionsNextweconsiderthemultidimensionalversionof( 2 )onsomedomainRN.ThePDEisgivenby @tu=u)-222(r(urs(x));x2;t2[0;T] (2a)0=@u @n)]TJ /F3 11.955 Tf 11.96 0 Td[(u@s @n;x2@;t2[0;T] (2b)whereu:RN!Rands:RN!R.Ifsandaresufcientlysmooth,thenstandardtheoryofparabolicPDEsaysthatauniquesolutionwillexistforeverysufcientlysmoothinitialconditionsolongasthatinitialconditionvanishesinsome-neighborhoodof@[ 12 ,Theorem5.3.2].AsimilarresultforellipticPDEholds,whichimpliestheexistenceofsteadystatesolutionsto( 2 ). 24

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2.2.2PropertiesofNumericalSolutionsIndimensionshigherthanoneitisnotastrivialtondanalyticsolutionsto( 2 )ineitherthetimedependentorindependentcase.Forthisreasonwenowturntoniteelementanalysistondsolutions.Inthissection,wedevelopthegeneralframeworkthatwewillusethroughouttocalculatesolutionsnumerically.ConsideraboundeddomainwithLipschitzcontinuousboundary@.Multiplyingthroughbyatestfunction'2H1(),integratingbyparts,andusingtheboundaryconditions( 2b )gives Z(@tu)'=)]TJ /F10 11.955 Tf 11.29 16.27 Td[(Zrur'+Zursr';(2)theweakformulationof( 2 ).Usingthenotationexplainedintheintroduction,webeginbyformulatingtheproblemusingthestandardniteelementmethodinthepiecewiselinearniteelementspaceShwithbasisfunctionsf`gL0`=1.Werstdiscretizeonlyinspace,andforanyxedtimetwritetheapproximatingfunctionuh2Shofuasu(x;t)uh(x;t)=L0X`=1U`(t)`(x):Inordertogiveamatrixformulationoftheproblem,denethebilinearformsa(u;v)=)]TJ /F10 11.955 Tf 11.29 16.28 Td[(Zrurv+Zursrvandb(u;v)=Zuv:foranyu,v2H1().Forbasisfunctions`,m2ShdenethematricesAandBbyA`m=a(m;`)andB`m=b(m;`): 25

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Thensubstituteuuh=P`U`(t)`(x)and'=mforeachbasisfunctionmofShinto( 2 )toobtainthespatiallydiscretizedweakformulationof( 2 ).Inmatrixformthisbecomesthesystemofordinarydifferentialequations Bd~U(t) dt=A~U(t):(2)GiveninitialconditionsU0(t),system( 2 )canbesolvedexactlysincethematricesAandBareindependentoft[ 6 ].Hencepiecewiselinearapproximatingsolutionsaregivenintermsofthematrixexponentialby ~U(x;t)=U0(x)etB)]TJ /F12 5.978 Tf 5.76 0 Td[(1L:(2)NotethatBisinvertible,asareallGrammatricesofinnerproductsofabasis.Indeed,letybeinthenullspaceofBandlet(x):=P`y``bethefunctioninShrepresentedbyxunderthecurrentchoiceofbasis.Thenkk2=<;>==X`y`Xmym<`;m>=0wherethelastlinefollowsfromthe`-throwoftheexpansionofBy=0.Butkk2=0ifandonlyif=0,whichimpliesthatitsvectorrepresentationy=0,andsothenullspaceofBconsistsofonlythezerovectorandBisinvertible.OnceweturntothefullKeller-Segelmodel,thematrixAfromEquation( 2 )willnolongerbeindependentoftandsoanexplicitformulasuchasEquation( 2 )isnotingeneralpossibletond[ 6 ].Forthisreason,wecontinuethenumericalanalysisanddiscretize( 2 )intimeaswellasinspace.Addingindicestoournotationtoaccountfortime,wewritetheexpansionoftheapproximatingfunctionofunh2Shafternstepsat 26

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timetn=ntasu(x;tn)unh=L0X`=1Un``:Thenapproximatingthetimederivativebyduh dtun+1h)]TJ /F3 11.955 Tf 11.95 0 Td[(unn tandusinganimplicitscheme,weobtaintheformula~Un+1=(B)]TJ /F4 11.955 Tf 11.96 0 Td[(tA))]TJ /F8 7.97 Tf 6.59 0 Td[(1B~Un=~btondasolutionaftern+1timestepswhenthesolutionafterntimestepsisknownandwheneverB)]TJ /F4 11.955 Tf 13.1 0 Td[(tAisinvertible.IngeneralitisnotthecasethatB)]TJ /F4 11.955 Tf 13.1 0 Td[(tAisinvertible,howeveritcanbeshowninsomecasesunderadditionalrestrictionsontheboundaryconditionsand,forexample,s.SomeresultsforthesimpleKeller-SegelmodelusingastandardniteelementmethodsimilartotheonejustderivedwillbeshowninSection 3.3 2.2.3NumericalDifcultiesandMethodValidationItiswellknownthatsolutionstoconvection-diffusionproblemssimilarto( 2 ),especiallythoseforwhichtheconvectivetermisrelativelylarge,canshownumericalinstabilitiesusingthestandardniteelementmethod(FEM)[ 20 44 ].InthissectionwedemonstratethesepotentialnumericalproblemsandatthesametimevalidatetheaccuracyofthematricesusedinourstandardFEMschemes.ConsidertheconvectiondiffusionPDE Du+ru=f(x;y);(x;y)2 (2a)@u @n=0;(x;y)2@ (2b) 27

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whichismotivatedbyanexamplein[ 20 ].Wewillsolve 2 for2R2usingthestandardniteelementmethod.Theweakformulationof( 2 )is )]TJ /F3 11.955 Tf 11.95 0 Td[(DZrur'+Z(ru)'=Zf'(2)forall'2H1().Substitutingabasisfunction`ofShfor'in( 2 )aswellastheFEMexpansionofu,weobtain NXj=1)]TJ /F3 11.955 Tf 9.29 0 Td[(DZrjri+Z(rj)iUj=Zf`;1iN:(2)ThenletthestiffnessmatrixLbedenedbyLij=Zrjri;denethematrixFbyFij=Z(rj)i;andnallydenethevectorbbybi=Zf(x;y)i:UsingthisnotationtheniteelementmatrixapproximationofEquation( 2 )is )]TJ /F3 11.955 Tf 11.95 0 Td[(DL~U+F~U=~b:(2)Uponinspectionitiseasytoseethatforanyconstantfunctionu,thelefthandsideofEquation 2 iszero.ThisimpliesthatthelefthandsideofEquation( 2 )isalsozerosoif~Uisequaltothevectorofallonesoranyconstantmultipleofit,then~Uisinthenullspaceof)]TJ /F3 11.955 Tf 9.3 0 Td[(DL+F.Hence,asexplainedintheprevioussection,weaddtherestrictionthatanyconstantsolutionmustbezero,whichisequivalenttosettingthevalueatonenodeequationequaltozero,oreliminatingthecorrespondingrowandcolumnfromDL+F.Eliminatingtherstnodefromthematrix,wendnumericallythat 28

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thisnewreducedmatrixisinvertibleandmaysolveforasolution~Uusingthismethodwhichisaccurateuptoadditionofaconstantvalue.WetesttheniteelementmatricesfromEquation 2 bysolvingthesystemforafunctionf(x;y)forwhich( 2 )hasaknownsolution.Considerthefunctionfdenedbyf(x;y)=)]TJ /F4 11.955 Tf 10.49 8.08 Td[(2aea(2x)]TJ /F8 7.97 Tf 6.59 0 Td[(1)(3ea(2x)]TJ /F8 7.97 Tf 6.58 0 Td[(1))]TJ /F4 11.955 Tf 11.96 0 Td[(1) (1+ea(2x)]TJ /F8 7.97 Tf 6.58 0 Td[(1))3andthefunctionudenedbyu(x;y)=1 1+ea(2x)]TJ /F8 7.97 Tf 6.59 0 Td[(1)onthedomain=[0;1][0;1].Directcomputationveriesthatforthegivenf,usolvesthePDE( 2a )forD=)]TJ /F8 7.97 Tf 6.58 0 Td[(1 aand,forexample,=(1;0).Clearly@u=@y=0.Infactsince@u @x=)]TJ /F4 11.955 Tf 9.3 0 Td[(2aea(2x)]TJ /F8 7.97 Tf 6.58 0 Td[(1) (1+2aea(2x)]TJ /F8 7.97 Tf 6.58 0 Td[(1))2weseethat@u @xx=0=)]TJ /F4 11.955 Tf 9.3 0 Td[(2a ea(e)]TJ /F8 7.97 Tf 6.58 0 Td[(2a+2e)]TJ /F5 7.97 Tf 6.59 0 Td[(a+1)and@u @xx=1=)]TJ /F4 11.955 Tf 9.3 0 Td[(2aea e2a+2ea+1sothatclearlylima!1@u @xx=0=lima!1@u @xx=1=0:Henceforlargeenoughvaluesofa,uhasapproximatelyNeumannboundaryconditions.Usingthestandardniteelementmatrixformulationtosolveforu,weseegoodconvergencetothesolutionasthemeshsizegetsveryne,butforaroughmeshthesolutionshowsoscillationsnearthesteepgradientseenintheexactsolutioninFigure 2-3 aroundx=0:5.TheFEMsolutionsfortwodifferentmeshsizescanbeseeninFigure 2-2 .Forcomparison,theexactsolutionuto( 2 )isshowninFigure 2-3 .TheoscillatorybehaviorseenwhenusingthestandardFEMtosolvethistestproblemis 29

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Figure2-2. StandardFEMsolutionsto( 2 ) partofthemotivationforanonstandardniteelementmethodwhichwewillintroduceinSection 5.3 30

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Figure2-3. Exactsolutionto( 2 )onrenedmesh 31

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CHAPTER3ASIMPLEKELLER-SEGELMODEL 3.1PositivityofExactSolutionsInthissectionwewillinvestigatesomepropertiesofthetwoequationKeller-Segel(KS)modelderivedinChapter1.Foreaseofreference,werestatethesystemhere. @tu=Dr(ru)]TJ /F3 11.955 Tf 11.96 0 Td[(urv);x2;t>0 (3a)@tv=~Dv)]TJ /F3 11.955 Tf 11.96 0 Td[(v+u;x2;t>0 (3b)@u @n=@v @n=0;x2@;t>0 (3c)u(x;0)=u0;v(x;0)=v0x2 (3d)In( 3d ),u0andv0arenon-negativesmoothfunctionsonwhichareassumedtonotbeidenticallyzero.Fortheremainderofthechapter,letbeabounded,connectedopensubsetofRnwithsmoothboundary@.ForanyxedT>0,weletT:=(0;T]betheparaboliccylinder.Strictpositivityofexactsolutionsisconsideredstandard,butasitisdifculttondacareful,completeproof,weincludethedetailsforcompleteness.ThemainresultonpositivityisTheorem 3.3 ,butinordertoproveitwerstneedsomepreliminarytheoremstatementsandarguments.WebeginwithstatementsofHopf'sLemmaandastrongmaximumprincipleforparabolicpartialdifferentialequations(PDE).ThroughoutthischapterweassumethatLisauniformlyellipticdifferentialoperatoroftheform Lu=)]TJ /F10 11.955 Tf 11.29 11.35 Td[(Xi;jaij(x;t)@xixju(x;t)+Xibi(x;t)@xiu(x;t)+c(x;t)u(x;t)(3)withaij,biandcallcontinuousandbounded.Part(i)ofthefollowingversionofHopf'sLemmaforparabolicPDEisstatedandprovedin[ 27 ],andpart(ii)followsbyapplyingpart(i)to)]TJ /F3 11.955 Tf 9.3 0 Td[(z.Theorem 3.1 isaversionofthestrongmaximumprinciplefromSection7.1in[ 10 ]. 32

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Lemma3.1. Let>0andR>0besomexedconstants,andx(y;s)2RN+1.Supposethecoefcientsaij,bjandcareallboundedandthattheoperator@t+LisuniformlyparabolicinthelowerparabolicfrustumPF=f(x;t)2Rn+1:jx)]TJ /F3 11.955 Tf 11.96 0 Td[(yj2+2(s)]TJ /F3 11.955 Tf 11.95 0 Td[(t)z(x;t);(x;t)2PFwithjx)]TJ /F3 11.955 Tf 11.96 0 Td[(yjR=2 (3b)c(x;t)z(x1;s)0;(x;t)2PF: (3c)Ifn=(x1)]TJ /F3 11.955 Tf 11.95 0 Td[(y)=jx1)]TJ /F3 11.955 Tf 11.95 0 Td[(yjistheoutwardunitnormalofPFatx1,then @z @n>0(3)andzz(x1;s)inPF. Theorem3.1. (StrongMaximumPrincipleforc0)Assumethefunctionu(x;t)iscontinuousonTandthatthederivatives@xiu;@xixjuand@tuexistandarecontinuousonT.Supposealsothat@t+Lisuniformlyparabolicwithc0inTandthatisconnected. 33

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(i) If@tu+Lu0inTanduattainsanonnegativemaximumoverTatpoint(x0;t0)2Tthenuconstantont0. (ii) Similarlyif@tu+Lu0inTanduattainsanonpositiveminimumoverTatpoint(x0;t0)2Tthenuconstantont0.ThefollowinglemmaisinChapter7,corollary2.3in[ 36 ],butherewesignicantlysimplifytheprooffrom[ 36 ]byusingLemma 3.1 Lemma3.2. Let@ @t+Lbeauniformlyparabolicoperatorwithc0.Ifviscontinuouson[0;T]andsatises @tv+Lv0in(0;T] (3) (x)v+@v @n=0on@(0;T] (3) v=gonft=0g (3) whereg0butgnotidenticallyzeroand(x)0iscontinuouson,thenv(x;t)>0forall(x;t)2(0;T]. Proof. Werstshowthatv0on(0;T].Bywayofcontradiction,assumenot.Thenthereexistssomepoint(x0;t0)2(0;T]suchthatv(x0;t0)<0.Thenmin[0;T]v<0andmustbeachievedatsomepoint(x;t)2[0;T].Wemusthavet>0becauseoftheinitialconditionsg0.Now,ifx2with0
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ofLemma 3.1 toconclude@v(x;t) @n<0.Butthenbecause0andv(x;t)<0wehave(x)v(x;t)+@v(x;t) @n<0whichisacontradictiontothefactthatvsatises( 3 ).Wehavenowestablishedacontradictioninallcases,andsowehaveshownv(x;t)0forall(x;t)2[0;T].Itonlyremainstoshowv>0on(0;T].Bywayofcontradiction,againsupposenot.Sincewehavealreadyshownv0,thentheremustexistsomepoint(x0;t0)2(0;T]suchthatv(x0;t0)=0Thentheminimumover[0;T]isachievedat(x0;t0),soifx02thestrongmaximumprinciple(Theorem 3.1 )impliesv0on[0;T]whichisagainacontradictiontov(x;0)=g(x)andtheassumptionthatgisnotidenticallyzero.Ontheotherhandifx02@thenasinthepreviousargumenttoshowv0,wecanassumethattheminimumisattainedonlyontheboundaryofandapplyLemma 3.1 toobtainacontradictiontotheboundaryconditions( 3 ). Nextwewillprovealemmawhichissimilartoproblem7.7in[ 10 ]. Lemma3.3. Supposeuisasmoothsolutionof @tu+Lu=0in(0;T] (3a)@u @n=0on@(0;T] (3b)u=gonft=0g: (3c)where@t+Lisauniformlyparabolicoperator.Ifg0,gisnotidenticallyzero,andthefunctioncisboundedbelow,thenu>0on(0;T]. Remark3.2. ForourapplicationweonlyneedtoseethatthisLemma 3.3 holdsforsolutionson[0;T]forTniteandbounded,inwhichcasetheboundednessofcisclearfromthecontinuityassumptiononallcoefcientsoftheoperator.Howevertheproofwillholdexactlyaswrittenforasolutionufor( 3 )on[0;1)ifcisbounded. 35

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Proof. Letm=inf(0;T]c(x;t)>sincecisassumedtobeboundedbelow.Denev(x;t):=emtu(x;t).Thennoticethatforanypartialderivatives,wehave @xiv=emt@xiu(3)and @xixjv=emt@xixju:(3)Then( 3 )and( 3 )implythatforouroperatorLwehaveLv=emtLu,orrewritingthat Lu=e)]TJ /F5 7.97 Tf 6.59 0 Td[(mtLv:(3)Thenusing( 3a ),( 3 )andthedenitionofvwecalculate@tv(x;t)=emtut(x;t)+memtu(x;t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(emtLu(x;t)+mv(x;t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(emte)]TJ /F5 7.97 Tf 6.58 0 Td[(mtLv(x;t)+mv(x;t)toseethatvsatisesthePDE@tv+Lv)]TJ /F3 11.955 Tf 12.61 0 Td[(mv=0.Noticealsothatbytheboundaryandinitialconditionsforuandthedenitionofv,wehavethatvsatises( 3b )on@[0;T]andv=gonft=0g.DeninganewoperatorKv:=Lv)]TJ /F3 11.955 Tf 12.6 0 Td[(mv,weseethat@t+Kisalsoauniformlyparabolicoperatorandthe\c"termofthisoperatorisc(x;t))]TJ /F3 11.955 Tf 12.1 0 Td[(m0bythedenitionofm.ThenwecanapplyLemma 3.2 (with0)tosaythatv>0on(0;T].Thenwehavethatu(x;t)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(mtv(x;t)>0aswell. WenowhavethetoolstoprovestrictpositivityofexactsolutionstotheminimalKSsystem. Theorem3.3. Let(u;v)beasmoothsolutionto( 3 )ontheinterval[0;T]withu0;v0notidenticallyzero,0,and0.Thenu(x;t)>0andv(x;t)>0on(0;T]. 36

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Proof. Werstlookattheequationforu.From( 3a )andtheproductrulewehave@tu)]TJ /F4 11.955 Tf 11.95 0 Td[(u+rurv+uv=0:ThenforLu:=)]TJ /F4 11.955 Tf 9.3 0 Td[(u+rurv+uv,@ @t+Lisauniformlyparabolicoperatorwithsmoothaij,biandc,andsincec=visacontinuousfunctionoverthecompactset[0;T],itisboundedbelow.Thensince@tu+Lu=0on(0;T]wecanapplyLemma 3.3 tosaythatu>0on(0;T].Nextlookatv.Equation( 3b )implies@tv)]TJ /F4 11.955 Tf 11.95 0 Td[(v+v=u:Againdeningadifferentialoperatorbasedonthis,ifLv:=)]TJ /F4 11.955 Tf 9.3 0 Td[(v+vthen@ @t+Lisauniformlyparabolicoperator.Wejustshowedu0,sosince0and0,wehavethatvt+Lv=u0.Wealsohavec=0sothatLemma 3.2 with=0maybeappliedtoconcludev>0on(0;T]aswell. 3.2PreliminariesforNumericalSteadyStateSolutionsOurrstnumericalexperimentsarecarriedouton( 3 )withallconstantsandthelinearsensitivityfunctionassumedtobeone.Thisreducesthetimedependentproblemtoasystemfrom[ 18 33 ]whichis @tu=r(ru)]TJ /F3 11.955 Tf 11.95 0 Td[(urv);x2;t>0 (3a)@tv=v)]TJ /F3 11.955 Tf 11.96 0 Td[(v+u;x2;t>0 (3b)0=@u @n=@v @n;x2@;t>0: (3c)Thesteadystateproblemthenbecomes 0=r(ru)]TJ /F3 11.955 Tf 11.95 0 Td[(urv);x2;t>0 (3a)0=v)]TJ /F3 11.955 Tf 11.96 0 Td[(v+u;x2;t>0 (3b)0=@u @n=@v @n;x2@;t>0: (3c) 37

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Inordertocomputesolutionsto( 3 ),wewillrstreducethesystemtoonepartialdifferentialequationwhichisequivalenttotheoriginalsystem.Thisapproachissummarizedin[ 18 ],andwellinthedetailshereforcompleteness.WerstneedstatementsofHopf'sLemmaandthestrongmaximumprincipleforellipticPDE.ThefollowingtwotheoremstatementsaretheellipticanaloguesofLemma 3.1 andTheorem 3.1 ,andwiththeexceptionofpart(ii)ofLemma 3.4 ,arefrom[ 10 ].Part(ii)ofLemma 3.4 followsfromapplyingpart(i)tothefunction)]TJ /F3 11.955 Tf 9.3 0 Td[(u. Lemma3.4. Assumeu2C2()\C()andthecoefcientcfromEquation 3 satisesc=0in.Supposealsothatisopen,connected,andbounded. (i) SupposeLu0inandthereexistsapointx02@suchthatu(x0)>u(x)forallx2U:Finally,assumesatisestheinteriorballconditionatx0,meaningthereexistsanopenballBwithx02@B.Then@u @n(x0)>0;wherendenotestheouterunitnormaltoBatx0. (ii) SimilarlysupposeLu0inandthereexistsapointx02@suchthatu(x0)
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Theorem3.5. Letbeboundedand@beC2.Assumeu;v2C2()\C1()andareniteeverywhere.Then(u;v)satises( 3 )ifandonlyifvsatises 0=v+kev)]TJ /F3 11.955 Tf 11.95 0 Td[(v;x2 (3a)@v @n=0;x2@;t>0 (3b)andu=kevforsomeconstantk. Proof. Fortheforwardimplication,assumethatu;v2C2()\C1()satisfy( 3 ).Consideranewfunction (x)denedby (x)=u(x)e)]TJ /F5 7.97 Tf 6.59 0 Td[(v(x).Noticethat alsohasNeumannboundaryconditions.Thensinceuandvarebothniteeverywhere,weseeu= ev.Substitutingthisinto( 3a ),applyingtheproductruletwiceandcancelingoutthenon-zerofunctionevthengives +rvr =0:ConsidertheellipticoperatorLdenedbyL := +rvru,sothatcoefcientsbjoftherstderivativepartoftheoperatoraregivenbytherstorderpartialsoftheknownfunctionv.Thensincev2C2()\C1(),thecoefcientsofLareboundedbyboundedandarealsocontinuous.SinceL 0in,wemayapplythestrongmaximumprincipleofTheorem 3.4 andHopf'sLemma 3.4 .So,either attainsitsmaximumoverataninteriorpoint,whichbythestrongmaximumprincipleimplies kforsomeconstantk,or achievesitmaximumonlyontheboundaryofthedomainwhichimpliesbyHopf'sLemmathat@ @n>0.HoweverthislastinequalityisacontradictiontotheNeumannBoundaryconditionsof .Thusitmustbethat kin,andsou=kev.Wenowsubstituteu=kevinto( 3b )toobtain( 3a ).Forthereversedirection,wenoticethatifvisasolutionof( 3 )andwedeneu=kev,thenNeumannboundaryconditionsforufollowfromtheboundaryconditionsonv.Substitutinguandvinto( 3a )-( 3b ),theequationshold. 39

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Noticethatforanyxedconstantu>0,thesystem( 3 )hasexactlyonespatiallyconstantsolution(u;v)withv=u.Asexpected,thesameresultfollowsifoneconsidersahomogeneoussolutionvto( 3 ).Then( 3a )reducestov=kev,andsinceu=kevisanassumed,weseethatindeedu=vistheonlystationarysolution,andadditionallywecansolvetoseethatk=v ev.ThestabilityofsuchhomogeneousstationarysolutionstoEquation 3 isfullycharacterizedasaspecicexampleofamoregeneralcaseanalyzedin[ 34 ].ThestabilityresultusesatechniquewhichwewillexplainandgeneralizeinSection 4.4 .Thetechniqueinvolves1<0,thenonzeroeigenvalueoftheLaplacianonwithNeumannboundaryconditionswhichisclosesttozero.Thenitisshownin[ 34 ]thatifu<1)]TJ /F3 11.955 Tf 12.46 0 Td[(1thehomogeneousstationarysolution(u;v)islinearlystableandifu>1)]TJ /F3 11.955 Tf 12.47 0 Td[(1itislinearlyunstable.Asthatanalysisfullycharacterizesconstantsolutionsto( 3 ),weturntonumericalmethodstosearchfornonconstantsolutions.Inordertondsuchsolutionsto( 3 ),werstprovethefollowinglemmawhichisexplainedin[ 18 ].Fromhereonwedenotethemeanofafunctionoverthedomainbym(z):=1 jjRz(x)dx. Lemma3.5. Letv2C2().Thenvisasolutionto( 3 )withm(v)=ifandonlyifz=v)]TJ /F3 11.955 Tf 11.96 0 Td[(isasolutionof z)]TJ /F3 11.955 Tf 11.95 0 Td[(z+ez m(ez))]TJ /F4 11.955 Tf 11.95 0 Td[(1=0;x2 (3a)@z @n=0;x2@ (3b)withm(z)=0. Proof. Werstassumethatv2C2()isasolutionto( 3 ).Thendenez=v)]TJ /F3 11.955 Tf 12.03 0 Td[(m(v).Clearlym(z)=0,andweclaimthatzsolves( 3 ).Itiseasytoseethat( 3b )holdsbytheNeumannboundaryconditionsonv.Toseethat( 3a )holds,substitutev=z+m(v)into( 3a )toobtain z)]TJ /F4 11.955 Tf 11.95 0 Td[((z+m(v))+kez+m(v)=0:(3) 40

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Integratingbothsidesof( 3 )overandusingintegrationbypartsandNeumannboundaryconditions,alongwiththefactthatRz=0gives m(v)=kem(v)m(ez):(3)Rewriting( 3 )using( 3 )thengives0=z)]TJ /F3 11.955 Tf 11.96 0 Td[(z)]TJ /F3 11.955 Tf 11.96 0 Td[(kem(v)m(ez)+kem(v)ez=z)]TJ /F3 11.955 Tf 11.96 0 Td[(z+kem(v)m(ez)ez m(ez))]TJ /F4 11.955 Tf 11.96 0 Td[(1=z)]TJ /F3 11.955 Tf 11.96 0 Td[(z+m(v)ez m(ez))]TJ /F4 11.955 Tf 11.96 0 Td[(1whichis( 3a )with=m(v).Forthereversedirection,assumethatzsolves( 3 )withm(z)=0.Letv:=z+andnoticethatvstillhasNeumannboundaryconditions.Thensubstitutingz=v)]TJ /F3 11.955 Tf 12.43 0 Td[(into( 3a )andsimplifyingyieldsv)]TJ /F3 11.955 Tf 11.95 0 Td[(v+ m(ez)eev=0;whichis( 3a )withk= m(ez)e. Hencetondasolution(u;v)to( 3 )itissufcienttondasolutionzto( 3 )withm(z)=0.Wemaychoosearbitrarily,andoncewehavesolvedforzwithaxedthenv=z+;u=ez m(ez)solve( 3 ).Noticethat=m(v)=m(u).WewanttolookforasolutioninthesubspaceofH1()consistingoffunctionswithzeroaverage.MotivatedbythiswedenefH1():=fw2H1():Zw(x)dx=0g: 41

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Asin[ 18 ],wetrytoshowtheexistenceofanontrivialsolutionusingthefunctional J(z)=1 2Zjrzj2+jzj2dx)]TJ /F3 11.955 Tf 11.96 0 Td[(jjlog1 jjZezdx:(3)LetL(H1();R)denotethefunctionspaceofboundedlinearoperatorsmappingH1()toR.ThenwewishtoshowthatJisawelldenedfunctional,andcalculatetheFrechetderivativeofJ,whichisafunctionJ0:H1()!L(H1();R).WerstnotethatfortheremainderofthissectionweabusenotationandallowvaluesoftheconstantCtovarydependingonthecontextbeingused.Inordertodothis,werstneedapreliminaryresult.Itrequiresthatisboundedandsatisestheconecondition,whichaccordingto[ 14 ,Theorem1.2.2.2]isequivalentto@locallyLipschitzandcompact.Ifsatisestheseconditions,thenwehavefrom[ 40 ,Theorem2]thattheembedding H1(),!L()(3)iscontinuouswhere(t)=et2)]TJ /F4 11.955 Tf 12.07 0 Td[(1andL()isthecorrespondingOrliczspacenormedbykukL()=inffk>0j(u=k)1g:Thecontinuousembedding( 3 )impliesthatthereexistssomeconstantC>0withkukL()CkukH1()forallu2H1().Weusetheexistenceofsuchaconstanttoprovethefollowingresult. Lemma3.6. ThereexistsconstantC>0suchthatforallu2H1()and2R,ZeueCkuk2H1()2=4(1+jj): Proof. Consideraminimizingsequenceki!kukL().Thenforevery>0thereexistskkisuchthatkukL()kkukL()+ 42

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andbythedenitionofkkL()Z(u=k)=Ze(u=k)2)]TJ /F4 11.955 Tf 11.95 0 Td[(1dx1whichimpliesZe(u=k)21+jj:Now,foranyrealthearithmetic-geometricmeaninequalityimpliesthatr u k(k)=u 2k+k 2oru 1 4u k2+k22 4u k2+k22 4:Henceeue(u=k)2+(k=2)2andsoZeue(k 2)2Ze(u=k)2e(k 2)2(1+jj)e2 4(kukL()+)2(1+jj)e2 4(CkukH1()+)2(1+jj)forevery>0.Thenletting!0,theresultfollows. Notice 3.6 immediatelyshowsthatJiswelldened.WenowcontinuetocalculateJ0(z)byndingaboundedlinearoperatorAz=J0(z)suchthatlimh!0jJ(z+h))]TJ /F3 11.955 Tf 11.96 0 Td[(J(z))]TJ /F3 11.955 Tf 11.95 0 Td[(Az(h)j khkH1existsandisequaltozero.Usingthislimitdenition,foranyz2H1()wecancalculatethatthederivativeevaluatedatagiven2fH1()is J0(z)()=(rz;r)+(z;))]TJ /F3 11.955 Tf 24.28 8.09 Td[( m(ez)(ez;):(3) 43

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ToverifythederivativeofthersttwotermsofJ,weletJ1:=1 2)]TJ 5.48 -.05 Td[(Rjrzj2+jzj2dxandcalculateJ1(z+h))]TJ /F3 11.955 Tf 11.95 0 Td[(J1(z)=1 2Zjrz+rhj2)-221(jrzj2+jz)]TJ /F3 11.955 Tf 11.96 0 Td[(hj2)-222(jzj2dx=1 2Zjrzj2+2rzrh+jrhj2)-222(jrzj2+jzj2)]TJ /F4 11.955 Tf 11.96 0 Td[(2zh+jhj2)-221(jzj2dx=Z(rzrh))]TJ /F3 11.955 Tf 11.95 0 Td[(zhdx+khk2H1():forafunctionh2H1()withkhkH1()small.Motivatedbythiscalculation,deneforanyxedz2H1()thelinearfunctionalA1z:H1()!RbyA1z()=(rz;r)+(z;)forany2H1().ThenjJ1(z+h))]TJ /F3 11.955 Tf 11.95 0 Td[(J1(z))]TJ /F3 11.955 Tf 11.95 0 Td[(A1z(h)j=khk2H1()sothatclearlylimh!0jJ1(z+h))]TJ /F3 11.955 Tf 11.95 0 Td[(J1(z))]TJ /F3 11.955 Tf 11.95 0 Td[(A1z(h)j khkH1=0:ThisveriesthederivativeofthersttwotermsofJ.Tocomputethederivativeofthethirdandnalterm,wewillusethechainruleforFrechetderivatives[ 15 ]onthefunctionsg()=log():R!Randf()=1 jjRe:H1()!R.ItiseasytoseethattheFrechetderivativeoffunctionsmappingR!Risjustthelinearmaptakingt2Rtottimesthederivativeofthefunction,meaningthatifprimedenotestheFrechetderivative,theng0(x)(t)=1 xt:ThentocompletelyverifythederivativeofthefunctionalJitonlyremainstocomputethederivativeoffunctionalsoftheformf(z)=Rez.Thisleadstothefollowinglemma. 44

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Lemma3.7. Foranyconstantiff(z)=RezthentheFrechetderivativeoffatanyfunctionz2H1()isBz():=Zezdx: Proof. Wecalculatejf(z+h))]TJ /F3 11.955 Tf 11.96 0 Td[(f(h))]TJ /F3 11.955 Tf 11.96 0 Td[(Bz()j=Zez)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(eh)]TJ /F4 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(hkezkL2()keh)]TJ /F4 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(hkL2()=kezkL2()Xm2(h)m m!L2()kehkL2()Xm2jjm m!khmkL2()=kezkL2()Xm2jjm m!khkmL2m(): (3)Now,byLemma 3.6 ,kezkL1()C1(kzkH1())isboundedbyaconstantdependentonkzkH1()butindependentofkhkH1().Wenextturntoestimatingthesumin 3 .ByaspecicversionoftheSobolevembeddingtheoremseenin[ 37 ],thereexistsaconstantC>0suchthatforallv2Lp()kvkLp()Cp1=2kvkH1()forp2onLipschitzdomains.Henceforeachm0wehave jjm m!khkmL2m()jjmCm(2m)m=2 m!khkmH1():(3)ThenaskhkH1()!0,xEkhkH1()sothatwemaywrite Xm2jjm m!khkmL2m()khk2H1()Xm2jjmCm(2m)m=2Em)]TJ /F8 7.97 Tf 6.59 0 Td[(2 m!:(3)WeneedtoshowjjmCm(2m)m=2Em)]TJ /F8 7.97 Tf 6.59 0 Td[(2 m! 45

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isnite,andthenbythelimitdenitionofthederivativewearedone.Toseethis,let:=jjCp 2eE p mandnotethat<1forlargeenoughm.ThenrecallStirling'sapproximationform!,thatis 1 m!
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where(u;v)=Ruvand(ru;rv)=Rrurv.In[ 18 ]Horstmannexplainstheideaofaproofofexistenceofasolutionto( 3 )inthecasewherek>4basedontheMountainPassTheorem.AlthoughwewillnotendupusingtheMountainPassTheorembecausetheparameterrangesweareinterestedarenotintherequiredrange,westateaversionofthetheoremfrom[ 10 ]belowforreference. Theorem3.6. ConsideranabstractfunctionalI[]2C1(H;R)forsomerealHilbertspaceHsuchthatI0[u]existsforeachu2Handiscontinuous.AdditionallyassumethatIsatisesthePalais-Smalecompactnesscondition.Supposealsothat 1. I[0]=0, 2. thereexistconstantsr;a>0suchthatI[u]aifkuk=r,and 3. thereexistsanelementv2Hwithkvk>randI[v]0.Dene)-277(:=fg2C([0;1];H)jg(0)=0;g(1)=vg:Thenc:=infg2)]TJ /F4 11.955 Tf 8.73 7.43 Td[(max0t1I[g(t)]isacriticalvalueofI.In[ 18 ],thesequenceoffunctions u0:=log2 (2+jx)]TJ /F3 11.955 Tf 11.96 0 Td[(x0j2)2)]TJ /F3 11.955 Tf 11.96 0 Td[(mlog2 (2+jx)]TJ /F3 11.955 Tf 11.95 0 Td[(x0j2)22fH1()(3)wherex0isanyxedpointin@isusedtondanappropriatefunctionvforcondition3ofTheorem 3.6 .Althoughwedidnotuseparameterswithintherangeforapplicationofthisproof,westillusedthesequenceu0tondinitialguessesforthenumericalsimulationsasdiscussedinSection 3.3 .Inordertodetermineforwhichvaluesofwearelikelytondnonhomogeneoussolutionsto( 3 ),werecallthefollowingpreviouslymentionedresultfrom[ 34 ]that 47

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spatiallyconstantstationarysolutions(u;v)to( 3 )willbelinearlystablewithrespecttoperturbationsUsatisfyingRUdx=0wheneveru<1)]TJ /F3 11.955 Tf 12.62 0 Td[(1andlinearlyunstableifu>1)]TJ /F3 11.955 Tf 12.98 0 Td[(1where1<0isthenonzeroeigenvalueofsmallestnormsatisfying!1=1!1onwithzeroNeumannboundaryconditions.ToseethereasonfortherestrictionRUdx=0,noticethatforsufcientlysmoothuwemaycalculateusingEquation( 3a )@tZu(x;t)dx=Z@tu(x;t)dx=Zr(ru)]TJ /F3 11.955 Tf 11.95 0 Td[(urv)dx=Z@(ru)]TJ /F3 11.955 Tf 11.95 0 Td[(urv)n=0byNeumannboundaryconditions.Hencethemeanofu(x;t)isaconstantu0whichisindependentoftime.Itthereforemakessensetochooseonlyperturbationswithzeromeanand,additionally,tochooseastartingpointfortheiterationforwhichm(u0)=fallsintotherangewheretheconstantsteadystateisunstable;thisallowsforahigherpossibilityofconvergencetoamoreinterestingnonconstantsteadystate.Inadditiontochoosinginarangewheretheconstantsteadystateisunstable,wewouldalsoliketoensurethatsolutionsdonotblowup.Horstmannsummarizesresultsonblow-upandtimeasymptoticbehaviortotheminimalsysteminTable4.4of[ 18 ],andamongtheresultsisthatsolutionstothesystem( 3 )existgloballyintime,haveboundedL1norm,andconvergetoastationarysolutionast!1ifZu0<4:Henceifwechose=1 Ru0=m(u)sothat1)]TJ /F3 11.955 Tf 11.95 0 Td[(1<<4 jj; 48

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wehaveagoodchanceofndinganonconstantsteadystatesolutionof( 3 )orequivalentlyaconstantsolutionto( 3 ). 3.3NumericalSolutions 3.3.1SolutionsontheUnitSquareBeforemovingontointroduceourgeneralizedKeller-Segelmodel,weshowsomenumericalsolutionsto( 3 )foundbycalculatingsolutionsto( 3 )usingthestandardFEMintroducedinSection 2.2.2 .AllnumericalsolutionsinthecurrentsectionwerefoundusingaNewtoniterationtondzerosofthediscretizedformof( 3 ).Webeginwithsolutionsontheunitsquare=[0;1][0;1].SincetheeigenvaluesoftheLaplacianontheunitsquarearen;m=)]TJ /F4 11.955 Tf 9.3 0 Td[((n+m)22forn;m=0;1;2;:::wearelikelytondnonconstantsolutionsto( 3 )forvalueswhichsatisfy1+2<<4:Indeed,onlywhen>1+2waschosenwereweabletodemonstrateconvergencetoanontrivialfunctionz,andinmanycaseswealsoneeded<4.Weexperimentedwithinitialguessfunctionsoftheformcu0fromequation( 3 )fordifferentvaluesofx02@,>0small,andconstants0
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Figure3-1. Solutionto( 3 )ontheunitsquare 3.3.2SolutionsonaDiskWenextlookatsolutionsonadisk.Ouroriginalcomputationswereperformedontheunitdisk,butwehaddifcultyndinginitialconditionsoftheform( 3 )thatdidnoteitherblowuporconvergetotheconstantsolution.Indeed,noticethatonaadiskofradiusone,thesmallestnonzeroeigenvalueoftheLaplacianis1)]TJ /F4 11.955 Tf 22.83 0 Td[(3:38996.Hencetherearenovaluesofwhichsatisfy1)]TJ /F3 11.955 Tf 11.95 0 Td[(14:38996<<4=4 jjandthereforenovaluesforwhichweareassuredofhavinginstabilityofconstantsteadystatesaswellasnoblowup.ThereforeweturntocomputationonadomainequaltoadiskofsomeradiusR,forwhichthenonzeroeigenvalueclosesttozerosatises1)]TJ /F4 11.955 Tf 23.16 0 Td[(3:38996=R2.Inordertoensureanonzerorangeofinitialconditionssatisfyingtherequiredinequality,itisnecessarytohave1)]TJ /F3 11.955 Tf 11.95 0 Td[(11+3:38996 R2<4 R2;thatisR<0:7811:Thusweproceedtocalculatesolutionsto( 3 )onadiskofradius0:5,againusinginitialguessesoftheform 3 .Asolutionform(u0)==15andx0=(0;0)is 50

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Figure3-2. Solutionsto( 3 )onadisk showninFigure 3-2 ,andconvergenceofthesolutionforvariousmeshsizesisseeninTable 3-2 .Varyingtheinitialcondition( 3 )bychangingthevalueofx02@wecaninfactdemonstrateaninnitenumberofsolutionswithpeaksanywhereontheboundaryof@,forexampleasinFigure 3-3 Table3-2. Errorforsteadystatesolutiononadisk Numberofnodeshvaluekzh)]TJ /F3 11.955 Tf 11.96 0 Td[(zh=2k2H1() 1450.14.175450.051.3421130.0250.30583210.01250.047533,0250.00625 51

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Figure3-3. Moresolutionsto( 3 )onadisk 52

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CHAPTER4ANEXTENDEDKELLER-SEGELMODEL 4.1TheGeneralizedModelFormWewouldliketoextendtheresultsofthepreviouschapterstomoregeneralsystemssimilartotheKeller-Segel(KS)model.Todothis,againconsiderapopulationofsomespeciesofdensityuclimbingthegradientofachemicalofconcentrationvN.Biologically,itislikelythatthischemicalwillbeinvolvedinreactionswithothercompoundsintheenvironment,someofwhichmayalsobeproducedbythespecies.Sointhismoregeneralsetting,wealsomodeltheconcentrationsofN)]TJ /F4 11.955 Tf 13.15 0 Td[(1otherchemicalsv1;v2;:::;vN)]TJ /F8 7.97 Tf 6.59 0 Td[(1whichinteractwitheachother.Thenwriting~v=[v1;:::;vN]t,itcanbeassumedthatthechemicalreactionnetwork'scontributiontothechangeof~vwithrespecttotimeismodeledby~g(~v)forsomefunction~g:RN+!RN.Inourapplicationsthisfunction~gwilllikelybedeterminedbymassactionkinetics.Notethatgmayincludedecayofthechemicals.AssumealsothatforsomesubsetA=fi1;i2;:::;ikgf1;2;:::;Ngthepopulationproducesthechemicalvijatarateij0.Let~2RNbethevectorwhichhasiinthei-thcomponentifi2Aand0otherwise.AssumethatthediffusioncoefcientforuisgivenbytheconstantD>0,andthediffusioncoefcientsforeachviaregivenbyconstantseDi>0.LeteDbethediagonalmatrixwithelementseDialongthediagonalanddenotethechemotacticcoefcientby.ThenapplyingNeumannboundaryconditionsandnonnegativeinitialconditions,weobtainthefollowingsystemofequations. ut=Du)-222(r(urvN)x2;t>0 (4a)~vt=eD~v+u+~g(~v)x2;t>0 (4b)u(x;0)=u0(x);~v(x;0)=~v0(x)x2 (4c)@u(x;t) @n=@vi(x;t) @n=0;x2@;t0;1iN (4d) 53

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Nowthattheformofthegeneralizedmodelhasbeenexplained,wegivesomeexamplesofsituationswhichcanbemodeledusing( 4 ). Example4.1. Webeginwithasimpletheoreticalcasewithtwochemicalspeciesinordertobetterunderstandtheframeworkofthemodel.Dimerizationisthechemicalreaction2v1 !v2;wherethereactionmayormaynotbereversible.Iftheforwardandbackwardsrateconsantsforthedimerizationreactionaregivenbyk1>0andk20respectivelyandthedecayratesofeachchemicalaredenotebyi,thenusingthelawofmassactionthefunction ~g(~v)=0B@)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v21+k2v2)]TJ /F3 11.955 Tf 11.96 0 Td[(1v1k1v21)]TJ /F3 11.955 Tf 11.95 0 Td[(k2v2)]TJ /F3 11.955 Tf 11.96 0 Td[(2v21CA(4)describesthekineticsoftheCRN.Hencethetheoreticalscenariowhereanorganismemitsachemicalv1whichundergoesdimerizationtoformthechemicalv2,whichinturnisthechemoattractantofucanbemodeledby( 4 )for~gasin( 4 )and~=0B@101CAforsomeconstant1>0. Example4.2. ConsidertheunsimpliedfourequationKSsystem( 1 ).Notethatassumingmassactionkineticsonthechemicalsinquestion,constantratesofproduction,andachemotacticsensitivityfunctionoftheform(u;v)=uforsomeconstant,( 1 )withpossibledecayofallchemicalscanbeexpressedintheframeworkof( 4 )bysetting~=0BBBB@1031CCCCA 54

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and~g(~v)=0BBBB@)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v1v3+(k2+k3)v2)]TJ /F3 11.955 Tf 11.95 0 Td[(1v1k1v1v3)]TJ /F4 11.955 Tf 11.95 0 Td[((k2+k3)v2)]TJ /F3 11.955 Tf 11.96 0 Td[(2v2)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v1v3+k2v2)]TJ /F3 11.955 Tf 11.96 0 Td[(3v31CCCCA:InthisnewframeworkudenotesthedensityoftheamoebaDictyosteliumdiscoideum,v1denotesthedensityofthephosphodiesteraseenzymeemittedbytheamoeba,v3denotethedensityofthechemoattractantcAMP,andv2thedensityofthecomplexformedfromtheenzymeandthecAMP.Allki,1and3areassumedtobepositiveconstants,whiletheiareassumedtobenonnegative. 4.2ReactionsYieldingPositiveSolutionsInthissectionwewillshowconditionsunderwhichsufcientlysmoothsolutionsto( 4 )remainpositive,oratleastnonnegative,iftheinitialcondtionsu0andv0arenonnegativeandnotidenticallyzero.Thefactthatu>0willfollowwithoutanyfurtherassumptionsexactlyasinTheorem 3.3 forthecaseofthesimpleKeller-Segelmodel.Inordertoprovenonnegativityfor~vacertaininwardpointingconditionisrequiredofthefunction~gontheboundaryofthepositiveorthant.Aswewillsee,thisassumptionarisesnaturallyif~gisderivedusingthelawofmassactionkinetics.ThemainresultofthissectionisTheorem 4.4 ,butweneedapreliminarytechnicalresultbeforeitcanproven.Lemma 4.1 wasinspiredbyanargumentin[ 42 ]. Lemma4.1. Supposethereisan"0>0suchthatforall0<"<"0,thefunctionf:R[0;T]7!Rsatises f()]TJ /F3 11.955 Tf 9.3 0 Td[(";x;t)>0;(x;t)2[0;T](4) 55

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Letw:[0;T]!Rbecontinuouslydifferentiablewithrespecttot,twicecontinuouslydifferentiablewithrespecttox,andsatisfy @tw=~Dw+f(w;x;t);x2;t2(0;T] (4a)w(x;t)0x2;t=0; (4b)togetherwiththefollowingboundarycondition:Eitherw(x;t)0;x2@;t2(0;T]; (4c)or@w @n0;x2@;t2(0;T]: (4d)Thenw(x;t)0forallx2andt2[0;T]. Proof. Ifnot,thereisan(~x0;~t0)2[0;T]suchthatw(~x0;~t0)<0.Thenw(~x0;)isthennegativeat=~t0butnonnegativeat=0dueto( 4b ).Hence,thereare(smallenough)valuesof"2(0;"0)suchthatwattainsthevalueof)]TJ /F3 11.955 Tf 9.3 0 Td[("atoneormore2(0;~t0).Fixsuchan",andlett0bethersttimewhenwattainsthevalueof)]TJ /F3 11.955 Tf 9.3 0 Td[(".Letx0beanypointinsuchthatw(x0;t0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(".Then,asincreasestot0andissufcientlyclosetot0,thefunctionw(x0;)cannotincrease,so @tw(x0;t0)0:(4)Notealsothat min(x;t)2[0;t0]w(x;t)=w(x0;t0)=)]TJ /F3 11.955 Tf 9.3 0 Td[("(4)(becauseifwtookavaluelesserthan)]TJ /F3 11.955 Tf 9.3 0 Td[("in[0;t0],thent0wouldnotbethersttimewhenwattains)]TJ /F3 11.955 Tf 9.3 0 Td[(").Theremainderoftheproofissplitintwoparts.First,suppose( 4c )holds.Then,x062@.Inviewof( 4 ),wethereforehave w(x0;t0)0:(4) 56

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Combining( 4 )and( 4 ),wendthat0[@tw)]TJ /F4 11.955 Tf 15.38 3.03 Td[(~Dw](x0;t0)=f()]TJ /F3 11.955 Tf 9.3 0 Td[(;x0;t0).Thiscontradicts( 4 )andnishestheproofinthecaseoftheboundarycondition( 4c ).Tocompletetheproofforthecaseoftheboundarycondition( 4d ),rstnotethatifx0isaninteriorpointof,thenweobtainacontradictionusing( 4 )and( 4 )asabove,soweneedonlyconsiderx02@.By( 4a )and( 4 ),theinequality@tw)]TJ /F4 11.955 Tf -434.13 -20.88 Td[(~Dw=f>0holdsat(x0;t0),andsobycontinuity,itholdsinaclosedneighborhood0[t1;t0](0;t0],where0isaballwhoseboundarycontainsx0.By( 4 ),weknowthattheminimumofwin0[t1;t0]isattainedat(x0;t0).Ifthisminimumisalsoattainedatanotherpoint(x00;t00)inthesameneighborhood,thenx00isaninteriorpointof,so( 4 )and( 4 )nishtheproofasbefore.Henceitonlyremainstoconsiderthesituationwhenx02@0isthesolepointin0[t1;t0]wherethemaximumisattained.Butinthissituation,allconditionsofpart(ii)ofLemma 3.1 aresatisedinasufcientlysmallparabolicfrustumcontainedin0(t1;t0].Hence,( 3 )impliesthat@()]TJ /F3 11.955 Tf 9.3 0 Td[(w)=@n>0atx0whichcontradicts( 4d ). Inordertostrengthenthisresulttoamoreusefulform,weneedaresultonexistenceanduniquenessofaPDEoftheform( 4 ).Thefollowingisacombinationofversionsof[ 12 ,Theorems6and10inSection7.4],modiedtotourapplication.Beforestatingthetheorem,werecallthatafunctionf(w;x;t)issaidtobelocallyHoldercontinuousin(x;t)ifthereisaCand0<<1suchthatjf(w;x1;t1))]TJ /F3 11.955 Tf 11.95 0 Td[(f(w;x2;t2)jCj(x1;t1))]TJ /F4 11.955 Tf 11.95 0 Td[((x2;t2)jforall(x1;t1)and(x2;t2)ineveryclosedboundedsubsetBofT. Theorem4.3. Consider( 4a )togetherwiththeinitialandboundarycondition w(x;t)=w0(x;t);(x;t)2(ft=0g)[(@[0;T])(4)forsomesmoothw0.Assumethatf(w;x;t)isLipschitzcontinuousinw,uniformlywithrespecttoboundedsubsetsofR[0;T],andalsolocallyHoldercontinuous 57

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withrespectto(x;t).Suppose@tw0=~Dw0+f(w0;x;0)on@.Thenthereissome00suchthat maxijfi(~y;x;t))]TJ /F3 11.955 Tf 11.95 0 Td[(fi(~z;x;t)jMfmaxijyi)]TJ /F3 11.955 Tf 11.95 0 Td[(zij(4)forall~y;~z2Dandall(x;t)2[0;T].Usingthisterminologyweproveonemorelemmabeforeprovingthedesirednalresult.Lemma 4.2 wasinspiredbyastatementinlecturenotesfromChrisCosner(AnaloguesofMaximumPrinciplesforSystems,privatecommunication)aboutstrengtheningtheresultsof[ 42 ],buttheresultwasnotexplicitlyproveninthenotes. Lemma4.2. Supposethat~f:RN[0;T]islocallyLipschitzcontinuousin~y,uniformlyin(x;t),anduniformlyonboundedsubsetsofRN.Suppose~fisalsolocallyHoldercontinuousin(x;t).Finally,assumethatforall(x;t)2[0;T], fi(y1;y2;:::;yi)]TJ /F8 7.97 Tf 6.58 0 Td[(1;0;yi+1;:::;yN;x;t)0;i=1;2;:::;N(4)wheneveryj0forallj6=i.Let~w:RN[0;T]!RNbeasmoothsolutionof @twi=~Diwi+fi(~w;x;t);x2;t2(0;T];i=1;2;:::;N; (4a)~w(x;t)0x2;t=0; (4b)togetherwiththefollowingboundarycondition:Either~w(x;t)0;x2@;t2(0;T]; (4c)or@~w @n0;x2@;t2(0;T]: (4d) 58

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Then~w(x;t)0forallx2andt2[0;T]. Proof. TheideaistoconstructanewfunctionFifromfisuchthatLemma 4.1 canbeapplied.Tothisend,rstleta+=max(a;0).Itcanbeeasilyveried,casebycase,thatforanytwonumbersaandb, ja+)]TJ /F3 11.955 Tf 11.96 0 Td[(b+jja)]TJ /F3 11.955 Tf 11.96 0 Td[(bj:(4)Next,deneFi:R[0;T]!RbyFi(v;x;t)=fi(w1(x;t);:::;wi)]TJ /F8 7.97 Tf 6.59 0 Td[(1(x;t);v+;wi+1(x;t);:::;wN(x;t);x;t)+v+)]TJ /F3 11.955 Tf 11.96 0 Td[(v:By( 4 )and( 4 ),FiislocallyLipschitzcontinuousinv,uniformlyin(x;t).Wenowprovethat,foranarbitrary">0, Fi()]TJ /F3 11.955 Tf 9.3 0 Td[(";x;t)>0;8(x;t)2K)]TJ /F5 7.97 Tf 6.58 0 Td[("1i(4)forany"1<"=Mf.HereKi=f(x;t)2[0;T]:wj(x;t)forallj6=ig:Beforeprovingthis,notethatalthough( 4 )andthedenitionofFimmediatelyimplytheinequalityFi()]TJ /F3 11.955 Tf 9.3 0 Td[(";x;t)",thisinequalityholdsingeneralonlyfor(x;t)2K0i.Toobtainasimilarinequalityinalargerset,weuse( 4 ).So,forany"1>0andany(x;t)2K)]TJ /F5 7.97 Tf 6.59 0 Td[("1i,fi(~0;x;t))]TJ /F3 11.955 Tf 11.96 0 Td[(fi(w1;:::;wi)]TJ /F8 7.97 Tf 6.59 0 Td[(1;0;wi+1;:::;wN;x;t)Mf"1:Sincethersttermaboveisnonnegativedueto( 4 ),thisimpliesthatFi()]TJ /F3 11.955 Tf 9.3 0 Td[(";x;t)=fi(w1;:::;wi)]TJ /F8 7.97 Tf 6.59 0 Td[(1;0;wi+1;:::;wN;x;t)+")]TJ /F3 11.955 Tf 21.92 0 Td[(Mf"1+":and( 4 )follows.Accordingly,wex"1<"=Mfandproceed.Toprovethelemmabywayofcontradiction,suppose~w60.Thenthereexistssome"2,sufcientlysmall,andchosensothat0<"2"1,suchthatatleastoneofthecomponentsof~wattainthevalue)]TJ /F3 11.955 Tf 9.3 0 Td[("2.Lett1>0bethersttimethatanyofthecomponentsof~wattainthevalue)]TJ /F3 11.955 Tf 9.3 0 Td[("2,andletiandx12besuchthat 59

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wi(x1;t1)=)]TJ /F3 11.955 Tf 9.29 0 Td[("2.Then(cf.( 4 )) minimin(x;t)2[0;t1]wi(x;t)=wi(x1;t1)=)]TJ /F3 11.955 Tf 9.3 0 Td[("2:(4)Clearly,thisimpliesthat [0;t1]K)]TJ /F5 7.97 Tf 6.59 0 Td[("2iK)]TJ /F5 7.97 Tf 6.58 0 Td[("1i:(4)Now,letvibethesolutionto @tvi=~Divi+Fi(vi;x;t);x2;t2(0;t1]; (4a)vi=wi;(x;t)2(ft=0g)[(@[0;t1)): (4b)ByTheorem 4.3 andtheaforementionedcontinuitypropertiesofeachFi,viexistsinaninterval[0;t2],wheret2>0isthemaximaltimeofexistenceofthesolution.Theremainderoftheproofissplitintotwocases:t2t1andt2t2byagaininvokingTheorem 4.3 .Thisisacontradictiontothemaximalityoft2andnishestheproof. 60

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Theorem4.4. Supposeasmoothsolution(u;~v)to( 4 )existsontheinterval[0;T]forsomeT>0forsmoothinitialconditionsu0and~v0whicharenonnegativeandnotidenticallyzeroon.Assume@issmooth,~g:RN!RNisuniformlyLipschitzoncompactsubsetsofRN,andthat~gsatises gi(v1;v2;:::;vi)]TJ /F8 7.97 Tf 6.59 0 Td[(1;0;vi+1;:::;vN)0;i=1;2;:::;N(4)whenevervj0forallj6=i.Thenon[0;T],u>0andeachcomponentof~visnonnegative. Proof. Theargumentthatu>0on[0;T]isexactlyasinTheorem 3.3 .Toshowthat~visnonnegative,write( 4 )intheformof( 4a )toseethatfi(~v;x;t)=iu(x;t)+gi(~v(x;t)):Sinceuiissmoothonthecompactset[0;T]andgiwasassumedtobeuniformlyLipschitzin~voncompactsets,itfollowsthatthefunctionfiislocallyLipschitzcontinuousin~v,uniformlyin(x;t).By( 4 ),i0,andu(x;t)>0weseethat( 4 )holdson[0;T]whenevervj0.ThereforewemayapplyLemma 4.2 toconclude~visnonnegativeon[0;T]. Remark4.5. Noticethatif~gisderivedaccordingtothelawofmassactionkinetics,then~gwillbeasmoothfunctionwhichsatises( 4 ). 4.3ExistenceofHomogeneousSteadyStateSolutionsAgainconsiderthegeneralizedKSmodel( 4 ).Asteadystate(u;~v)ofthesystemwillsatisfy Du)-222(r(urvN;)=0 (4a)eD~v+~u+~g(~v)=0: (4b) 61

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Ifsuchasteadystateisconstantthen( 4 )willreduceto ~u+~g(~v)=0:(4)Givenanon-zeroconstantu,thereisaquestionastowhetherornotanon-zeroconstantsteadystate(u;~v)existsforsomeconstant~v2RN,andifitdoesexistifthehomogeneoussteadystateisstable.Beforecontinuingwiththeanalysisonthissubject,someterminologyonmatricesneedstobeintroducedandoneresultwillbestated.Thefollowingdenitionsandsubsequentresultareallfrom[ 2 ].Wenotethatinthefollowinganalysiswhenwewrite~v0andA0foravectorormatrixwemeancomponentwisenonnegativity. Denition4.1. AMetzlermatrixA2RNRNisamatrixwherealloffdiagonalentriesarenonnegative,ieforall1i;jNAij0;i6=j: Denition4.2. AmatrixA2RNRNisanM-matrixifforall1i;jNitsatises 1. Aij0;i6=j 2. Aii0 3. WhenAisexpressedintheformA=sI)]TJ /F3 11.955 Tf 12.25 0 Td[(Bfors>0andBnonnegative,then(B)sThefollowingcharacterizationofanonsingularM-matrixisfrom[ 2 ,Theorem6:2:3,conditionN38]. Lemma4.3. A2RNRNisanonsingularM-matrixifandonlyifAisinversepositive,thatis,A)]TJ /F8 7.97 Tf 6.59 0 Td[(1existsandA)]TJ /F8 7.97 Tf 6.59 0 Td[(10.Usingthepreviousresults,wecannowgiveasufcientconditionforexistenceofnonnegativehomogeneoussteadystatesinthecasewhere~gisalinearfunction. 62

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Lemma4.4. Assumethat~g(~v)=)]TJ /F3 11.955 Tf 9.3 0 Td[(A~vforsomematrixA2RNRN.ThenAisanonsingularM-matrixifandonlyif( 4 )hasanonnegativehomogeneoussteadystatesolutionforeachconstantu>0andvector~0. Proof. FirstassumethatAisanonsingularM-matrix,andxanyu>0and~0.Thendening ~v=uA)]TJ /F8 7.97 Tf 6.58 0 Td[(1~;(4)~vsatises( 4 ).ByLemma 4.3 ,~v0andtherefore(u;~v)isanonnegativehomogeneoussteadystatesolutionto( 4 ).Furthermore,noticethat~visunique.Indeed,if~wsatises( 4 )thenA~w=u~whichimpliesthat~w=uA)]TJ /F8 7.97 Tf 6.59 0 Td[(1~=~vby( 4 ).Toprovetheconverse,let(u;~v(i))beanonnegativehomogeneoussteadystateobtainedbythehypothesisusingu=1and~v=~ei,theithcoordinatevector.Thenby( 4 ),wehaveA~v(i)=~ei.DeningthematrixC=[~v(1)j~v(2)jj~v(N)];wehavethatAC=I.ItfollowsthatAisinvertible,andsinceA)]TJ /F8 7.97 Tf 6.58 0 Td[(1C0,byLemma 4.3 ,AisinversenonnegativeandthereforeanonsingularM-matrix. 4.4StabilityofHomogeneousSteadyStateSolutionsHavingproventheexistenceofhomogeneousstationarysolutionsundercertainconditions,wenowassumethattheyexistandturntodeterminingtheirstability.Inordertoanalyzethestabilityofagivenhomogeneoussolutionto( 4 ),weusethemethodsofSchaafin[ 34 ]toobtainsomegeneralresults.Thegeneralmethodinvolveslinearizingaboutaconstantsteadystateandusingtheeigenvaluesofthelinearizedproblemtodeterminestability.Thenforexampleasin[ 22 ]wegivethefollowingdenitionsforasteadystateuofthegenericPDE ut=F(u)on(0;T)(4) 63

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whereu(t)takesvaluesinsomeBanachspace(X;kk). Denition4.3. WesaythatuislinearlystableifalleigenvaluesofthelinearizedeigenvalueproblemDFju(U)=Uhavenegativerealpartanddonothavezeroasalimitpoint.Itissaidtobelinearlyunstableifthereexistsaneigenvaluewithpositiverealpart.Assumethataconstantsteadystate(u;~v)of( 4 ),orequivalentlyasolutionof( 4 ),exists.Wewishtondtheeigenvaluesofthelinearizedsystemaboutthatsteadystateinordertodeterminelinearstability.Supposethatisaneigenvalueofthelinearizedsystem,meaningthatsatises DU)]TJ /F3 11.955 Tf 11.96 0 Td[(uVN=U;x2;t>0 (4a)~D~V+~U+J~V=~V;x2;t>0 (4b)@U @n=@Vi @n=0;1iN;x2@;t>0 (4c)forsomenontrivialfunctions(U(x);~V(x))on.In( 4b ),JistheNNJacobianmatrixof~gwithrespectto~vevaluatedatthesteadystate~v,meaningJij=@gi @vj~v:Wewishtostudytheeigenvalueproblem( 4 )inordertodetermineconditionsunderwhichspatiallyhomogeneousstationarysolutionsareunstable. 4.4.1ReductiontoFiniteDimensionTosimplifythestabilityanalysis,wewillreducetheinnitedimensionaleigenproblem( 4 )toacountablenumberofnitedimensionalmatrixeigenproblems.Inordertoallowourresultstobeasgeneralaspossible,welookattheweakformulationofthe 64

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eigenproblemandsearchfor(u;~v)2H1()N+1whichsatisfy )]TJ /F10 11.955 Tf 11.29 16.27 Td[(ZDrU r'+ZurVN r'=ZU '; (4a))]TJ /F10 11.955 Tf 11.29 16.27 Td[(Z~DkrVk r'+kZU '+NXl=1JklZVl '=ZVk ';k=1;2;:::;N; (4b)forall'2H1().Notethatinthecaseofsufcientlysmoothdomainboundary@andeigenfunctions(u;~v),( 4 )canberecoveredfrom( 4 )bymultiplyingthroughbyatestfunction,integratingbyparts,andusingboundaryconditions.Werstreduce( 4 )toaseriesofnitedimensionaleigenproblemsbygeneralizingamethoddueto[ 34 ].Toproceed,weletf!igi1beasetoffunctionsinH1()normalizedtohaveL2()normequaltooneandwhichareweakeigenfunctionsoftheLaplacianoperatorwithNeumannboundaryconditions.Writetheeigenvaluesofeach!iasiandordersothat0=0>123;.Hencethe!isatisfy )]TJ /F10 11.955 Tf 11.95 16.27 Td[(Zr!i r'=iZ!i ';8'2H1();i=0;1;2;::::(4)Thensubstitutingany!iinfor'in( 4 )andusing( 4 )tosimplify,weobtainthesystem DiZU!i)]TJ /F3 11.955 Tf 11.96 0 Td[(uiZVN!i=Zu!i (4a)~DkiZVk!i+kZU!i+NXl=1JklZVl!i=ZVk!i (4b)Toexpressthelefthandsideof( 4 )moreconciselyinmatrixform,xi0andletx0=RU!iandxk=RVk!ifor1kN.Thenif~x=[x0;x1;:::;xN]t,( 4 )reduces 65

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tothe(N+1)(N+1)matrixsystem 266666666664Di000)]TJ /F3 11.955 Tf 9.3 0 Td[(iu1J11+i~D1J12J1;N)]TJ /F8 7.97 Tf 6.59 0 Td[(1J1N2J21J22+i~D2J2;N)]TJ /F8 7.97 Tf 6.59 0 Td[(1J2N..................NJN1JN2JN;N)]TJ /F8 7.97 Tf 6.59 0 Td[(1JNN+i~DN377777777775~x=~x(4)Denotethematrixonthelefthandsideof( 4 )byA(i).ThenthestabilityofthesteadystatedependsontheeigenvaluesofA(i)accordingtothefollowingtheorem. Theorem4.6. Thenumbersolvestheweakeigenproblem( 4 )ifandonlyifitsolves( 4 )forsomei,i0. Proof. Firstassumesolves( 4 )fornontrivialeigenfunctions0B@U~V1CA2L2().Thensincethef!igi0formacompleteorthonormalsetinL2(),itispossibletoxisothateitherRU!i6=0orRVk!i6=0forsome1kN.Thenforthisi,thepreviouscalculationsshowthatsolves( 4 )withx6=0.Toshowtheconverse,assumethatisaneigenvalueof( 4 )witheigenvectorx6=0forsomexedeigenvalueioftheLaplacian.Let!ibetheeigenfunctionoftheLaplacianwhichcorrespondstoi.Denethenontrivialfunction0B@U~V1CA2L2()N+1byU=x0!iand~V=0BBBBBBB@x1!ix2!i...xN!i1CCCCCCCA: 66

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Substitutingthesedenitionsintothelefthandsideof( 4 )andusing( 4 )and( 4 )showsthatuandvasdenedaboveindeedsatisfy( 4 ).Forexamplefor( 4a )weuse( 4 )tocalculate)]TJ /F10 11.955 Tf 11.29 16.27 Td[(ZDrU r'+ZurVN r'=)]TJ /F3 11.955 Tf 9.3 0 Td[(DZr(x0!i) r'+Zur(xN!i) r'=Dx0iZ!i ')]TJ /F3 11.955 Tf 11.96 0 Td[(uxNiZ!i '=(Dx0i)]TJ /F3 11.955 Tf 11.96 0 Td[(uxNi)Z!i ':Thennoticingthattherstrowof( 4 )givesDx0i)]TJ /F3 11.955 Tf 11.95 0 Td[(uxNi=x0;itfollowsthat( 4a )holds.Equation( 4b )followssimilarlysubstitutingineachVk=xk!iintothelefthandsideofequation( 4 )andthenusingthekthrowof( 4 )tosimplify. 4.4.2SufcientConditionsforInstabilityInordertoanalyzetheeigenproblem( 4 )anddeterminesufcientconditionsforinstabilityofhomogeneoussteadystates,weneedsomestandardresultsfrommatrixtheoryandlinearalgebra.Thefollowingdenitionsandresultsarestatedandprovedin[ 2 ].Wealsonotethatthroughoutthissection,apositivevectorwillrefertooneinwhicheachcomponentispositive,whileanonnegativevectormeansthateachcomponentisnonnegative. Denition4.4. TheassociateddirectedgraphG(A)ofanNNmatrixA,consistsofNverticesP1;P2;:::;PNwhereandedgeleadsfromPitoPjifandonlyofaij6=0. Denition4.5. AdirectedgraphGisstronglyconnectedifforanyorderedpair(Pi;Pj)ofverticesofG,thereexistsasequenceofedges(apath)whichleadsfromPitoPj. Denition4.6. AmatrixAiscalledirreducibleifG(A)isstronglyconnected. Theorem4.7(Perron-Frobenius). LetAbeasquare,nonnegativematrix. 67

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(i) IfAispositive,meaningeachelementofAisstrictlygreaterthanzero,thenthespectralradius(A)isasimpleeigenvalueofA,greaterthanthemagnitudeofanyothereigenvalue. (ii) IfAisnonnegativeandirreducible,then(A)isasimpleeigenvalue,anyeigen-valueofAofthesamemodulusisalsosimple,Ahasapositiveeigenvectorxcorrespondingto(A),andanynonnegativeeigenvectorofAisamultipleofx. Theorem4.8(Spectralradiusbounds). LetAbeanon-negativeirreducibleNNmatrix.LetsidenotethesumoftheelementsoftheithrowofA.LetS=max1iNsiands=min1iNsi.Thenthespectralradius(A)satisess(A)S:Usingthesestandardresults,wenowcanstateandproveatheoremgivingsufcientconditionsforinstabilityofahomogeneoussteadystate. Theorem4.9. Assumethesystem( 4 )hasapositivehomegeneoussteadystatesolution(u;~v).LetJdenotetheJacobianof~gevaluatedat(u;v),andassumealsothatthematrixJandthevector~satisfythefollowingconditions. 1. Thereexistssome1iNsuchthati>0. 2. Jisirreducible. 3. JisMetzlerTheniftheproductuoriissufcientlylarge,(u;~v)willbeunstable. Remark4.10. Biologicallytherstconditionmeansthatthespeciesinquestionmustproduceoneofthechemicalsbeingmodeled,whichiswhatcreatestheinterestingfeedbackmechanismaswellasthecouplingofthesystem.Thesecondconditionisatechnicalassumptionwhichwillberelaxedinthenexttheorem.Thethirdconditionisthemostrestrictiveofthethree,howevertherearelargeclassesofchemicalreactionnetworksforwhichtheassumptionholds.ThesewillbeexploredinSection 4.5 ,alongwithanexampleinwhichconditionthreedoesnothold,buttheconclusionofthetheoremremainstrue. 68

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Proof. UsingthetechniqueofShaafin[ 34 ]thatwasoutlinedinthepreviousSection 4.4.1 ,recallthatthelinearizedeigenvalueproblemaboutasteadystatesolutionof( 4 )canbereducedtothecountablenumberof(N+1)(N+1)matrixeigenproblemsA(i)~x=~xfrom( 4 ).Inthefollowingcalculationswexi>0sothati<0anddenoteisimplyas.Denotethematrixin( 4 )asAA()andletK:=)]TJ /F3 11.955 Tf 9.3 0 Td[(u>0betheentryintheupperrighthandcornerofA.Wecanchooser>0,dependentonbutindependentofKand~,largeenoughsothatB:=A+rIhaspositivediagonalentries.ThensinceJwasassumedtobeMetzler,K>0and~isanonnegativevector,Bisanon-negativematrix.Infact,Bisanonnegativeirreduciblematrix.Toseethis,considertheassociateddirectedgraphG(B)onN+1verticesv0;v1;:::;vN.ThegraphcanbeconstructedbyaugmentingG(J),theassociateddirectedgraphofJ,withthevertexv0correspondingtotherstrowandcolumnofBandassumingthateachvertexhasalooptoaccountforthepositivediagonalentries.SinceJisirreducible,G(J)isstronglyconnected.Thensince~6=0thereexistsapathfromvjtov0forany1jN,andsinceK>0thereexistsapathfromv0tovjforany1jN.HenceG(B)isstronglyconnected.Wenowclaimthat limK!1(B)=limi!1(B)=1(4)If( 4 )holdsthenthetheoremholdsbythefollowingargument.SinceBisnonnegative,byTheorem 4.7 (B)isaneigenvalueofB.Thenif( 4 )holds,wemaychooseK(ori)largeenoughsothat(B)>r.Then(B))]TJ /F3 11.955 Tf 11.38 0 Td[(r>0isaneigenvalueofA=B)]TJ /F3 11.955 Tf 11.38 0 Td[(rIandso(u;~v)islinearlyunstable.So,weconcludetheproofbyverifyingequation( 4 ).Firstconsiderthecasewhereuisarbitrarilylarge.Beforecontinuing,wenotethatforanyintegerm1 ((Bt)m)=((Bt))m;(4) 69

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soitisenoughtoshowthat limK!1((Bt)m)=1(4)forsomem1,takethemthroot,anduse(B)=(Bt).DenotetheentriesofthetransposeofthematrixBbyBt=(bt)ij:Toshow( 4 ),werstobservetheformulaformatrixmultiplication (Bt)nij=XIbtii1bti1i2bti2i3:::btin)]TJ /F12 5.978 Tf 5.75 0 Td[(1j(4)whereIisthesetofallmulti-indicesi1;i2;:::;in)]TJ /F8 7.97 Tf 6.59 0 Td[(1witheachindexiksuchthat0ikN.NotethatherewecontinueindexingtheN+1entriesofBtby0;1;:::;Ninsteadofthestandard1;2;:::;N+1sotheindicesoftheverticesmatchwiththeircorrespondingvariable.Next,recallthatG(B)andhenceG(Bt)isstronglyconnected.Thereforethereisapathbetweeneverypairofverticesand,morespecically,foreach1iNthereisapathfromvitov0.Byconstructionthereisanedgefromvertexitovertexjifandonlyifbtij>0,soalsodenotethisedge,ifitexists,bybtij.FinallyobservethatsinceeveryvertexofG(Bt)hasaloop,ifthereisapathoflengthmfromonevertextoanother,thereisapathofanylengthlongerthanmaswell.UsingirreducibilityofBtalongwiththeobservationsabove,wecanndaxedintegerm>0suchthatthereisapathoflengthm)]TJ /F4 11.955 Tf 12.04 0 Td[(1fromvitovNforevery0iN.ThenaddingtheedgebtN0fromvNtov0toeachofthesepaths,wehaveapathoflengthmfromvitovNwhichisoftheformbti0i1bti1i2bti2i3:::btim)]TJ /F12 5.978 Tf 5.76 0 Td[(1im>0 70

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wherei0=iandbtim)]TJ /F12 5.978 Tf 5.75 0 Td[(1im=btN0=b0N=K.Theproductispositivebythedenitionofthegraphandhypothesis4.LetCi(K)=bti0i1bti1i2bti2i3:::btim)]TJ /F12 5.978 Tf 5.75 0 Td[(2im)]TJ /F12 5.978 Tf 5.76 0 Td[(1;0iNbeaconstantdependentonlyonK,andnoticethatCiwillonlyincreaseasKincreases,sinceitsdependenceonKmustbelinearorhigher.Thenbyformula( 4 )formatrixmultiplication,thereexistsatermCi(K)inthe(i;0)entryof(Bt)mfor0iN.AsinTheorem 4.8 ,letsibethesumoftheentiresinthei-throwof(Bt)m.Denes=min0iNsiandletC(K)=min0iNCi(K).C(K)alsoincreaseswithKsinceeachCi(K)does.BecauseBtisanonnegativematrix,allofthetermsineachentryof(Bt)mwillnonnegative,sossatisess=min0iNsimin0iN((Bt)m)i0min0iNCi(K)K=C(K)K:ThenbyTheorem 4.8 ,thespectralradiusof(Bt)msatises((Bt)m)C(K)K:LettingK!1,( 4 )and( 4 )followforthecaseofu.Inthecasewherethereissomexediwith1iNforwhichiisarbitrarilylarge,theprooffollowsintheexactsamemannerbyconsideringthegraphG(Bt)andndingforeach0iNapathoflengthmfromvitoviwhichendsintheedgei. InordertomakeoneofthehypothesesofTheorem 4.9 lessrestrictive,weneedtointroducesomemoreterminology.Recallthatastronglyconnectedcomponentofagraphisamaximalstronglyconnectedsubgraph.Thenconsiderthefollowingequivalencerelationonverticesofagraph.Lettwoverticesviandvjbeequivalent 71

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ifthereisadirectedpathfromvitovjandapathfromvjtovi.Thennoticetheequivalenceclassescreatedbythisequivalencerelationconsistexactlyoftheverticesofthestronglyconnectedcomponentsofthegraph.Usingthislanguage,westatethefollwingdenitionfrom[ 2 ] Denition4.7. TheclassesofanonnegativeNNmatrixAarethedisjointsubsetsfi1;i2;:::;ikgf1;2;:::;NgcorrespondingtoverticesoftheequivalenceclassesofG(A).Wealsoidentifyaclasswithitsstronglyconnectedcomponent. Denition4.8. TheclassesofanonnegativematrixAaretheequivalenceclassesofitsassociateddirectedgraphG(A)asdescribedabove.AsdescribedinSection2.3of[ 2 ],notethatbyareorderingofverticesapermutationmatrixPcanbefoundsothatT=PAPtisalowerblocktriangularmatrixwiththeblockscorrespondingtotheclassesofA.SincetheclassesofG(A)arestronglyconnected,itfollowsthatthediagonalblocksofTareirreducible.UsingthesefactswecandroptheirreducibilityassumptioninTheorem 4.9 andreplaceitwithaweakerassumptiononpathsinthechemicalreactionnetworkasfollows. Theorem4.11. Assumethesystem( 4 )hasapositivehomogeneoussteadystateso-lution(u;~v),andthatthethatthematrixJthevector~satisythefollowingconditions. 1. Thereexistssome1iNsuchthati>0andthereisadirectedpathfromvitovNinG(Jt). 2. JisMetzlerThenifeithertheproductuoriissufcientlylarge,(u;~v)islinearlyunstable. Proof. AsintheproofofTheorem 4.9 ,theproblemisreducedtotheseriesofeigenvalueproblemsin( 4 ).Againxingi<0,denotingthematrixin( 4 )as 72

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AA(i)andchoosingr>0largeenoughsothatB:=A+rIhaspositivediagonalentries,wecanconsiderthegraphG(Bt)asdescribedintheproofofTheorem 4.9 .Fixiforwhichi>0andthereisapathfromvitovN.TheninG(Bt)itfollowsthatthereisanedgefromv0tovisinceBt0i=i>0.AlsosinceBtN0=K>0thereisanedgefromvNtov0.ThepathfromvitovNalongwiththetwoedgesmentionedpreviouslycreatesacycleandshowsthatv0,vi,andvNarepartofthesamestronglyconnectedcomponent.ThereforethereexistsapermutatationmatrixPsuchthatT=PBtPtwhereTisanupperblocktriangularmatrixwithsomeblockTjcorrespondingtothestronglyconnectedcomponentcontainingvNandvi.Sincethiscomponentisstronglyconnected,itfollowsthatTjisanirreduciblematrix.ThenviewingtheverticesinG(Tj)astheirownchemicalreactionnetwork,thematrixTjfallsintothecaseoftheproofofTheorem9andhencehasanarbitrarilylargepositiveeigenvalueasK!1andasi!1,specicallyonelargerthanr.Howeversincetheeigenvaluesofablocktriangularmatrixaresimplytheeigenvaluesofthediagonalblocks,thisimpliesthatThasaneigenvaluelargerthanrwhichimpliesBtandBdoaswell.HenceA=B)]TJ /F3 11.955 Tf 12.27 0 Td[(rIhasapositiveeigenvalue. 4.4.3SufcientConditionsforStabilityAsmentionedintheintroduction,theideaofinitiationofpatternformationbydestabilizationofahomogeneoussteadystatedatesbacktoTuring'sclassicpaper[ 41 ].Wehaveshownnecessaryconditionsforaconstantsteadystatetobelinearlyunstable,sowhatwouldmakethisresultofevenmoreinterestwithregardstopatternformationisifthesamehomogeneoussteadystateisunstablewithoutthechemotacticfeedfackterm.Inthissectionweshownecessaryconditionsforasteadystatetobelinearlystablewithrespecttocertainreasonableperturbations.Asimilaranalysiswasexplainedin[ 34 ]forthecaseofthesimpleKSmodelofChapter3. 73

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Therestrictionofperturbationsfollowsfromthefactthatifusatises( 4a )witheitherNeumannboundaryconditionsonuandvNornouxboundaryconditionsonu,thenitmustbethatRu(x;t)dx=Ru0(x)dxisconstantintime.Thiscanbeseenusing( 4a ),integrationbypartsandboundaryconditionsasfollows.d dtZu(x;t)dx=Zut(x;t)dx=ZDu)-221(r(urvN)dx=DZ@@u @n)]TJ /F3 11.955 Tf 11.96 0 Td[(u@vN @ndx=0SosinceRudxisconstantforanysolution,whendetermininglinearstabilityitmakessensetolookonlyatperturbationsUwhichsatisfyRU(x;t)dx=0.Weconsiderstabilityinthecasewhere~g(~v)in( 4b )isalinearfunction,andletA2RNRNbethematrixsuchthat~g(~v)=A~v.Thislimitsthechemicalreactionnetworkstoreactionsoftheformvi!vjorvi!;,butthesecomponentreactionsmaybecombinedarbitrarilyandstillgeneratealinearfunctionunderthelawofmassaction.Considertheprevioustworeactiontypesinthesettingofmassactionkinetics,andlettherateconstantforthereactiongoingtochemicalifromchemicaljbekij0.Similarlylettherateconstantforthedecayreactionvi!;bedenotedbyi0.Thenforevery1iN,thefunction~g(~v)willbegivenbygi(~v)=NXj=1;j6=ikijvj)]TJ /F5 7.97 Tf 23.37 14.95 Td[(NXj=1;j6=ikjivi)]TJ /F3 11.955 Tf 11.96 0 Td[(ivi: 74

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Thenthematrixformulationof~gwillbe~g(~v)=0BBBBBBBBBB@)]TJ /F3 11.955 Tf 9.29 0 Td[(1)]TJ /F10 11.955 Tf 11.95 8.97 Td[(Pj6=1kj1k12k13k1Nk21)]TJ /F3 11.955 Tf 9.3 0 Td[(2)]TJ /F10 11.955 Tf 11.96 8.96 Td[(Pj6=2kj2k23k2Nk31k32)]TJ /F3 11.955 Tf 9.3 0 Td[(3)]TJ /F10 11.955 Tf 11.95 8.96 Td[(Pj6=3kj3k3N...............kN1kN2kN3:::kNN1CCCCCCCCCCA0BBBBBBBBBB@v1v2v3...vN1CCCCCCCCCCAwhichimpliesthattheJacobianof~gwithrespecttovis J=0BBBBBBBBBB@)]TJ /F3 11.955 Tf 9.3 0 Td[(1)]TJ /F10 11.955 Tf 11.95 8.96 Td[(Pj6=1kj1k12k13k1Nk21)]TJ /F3 11.955 Tf 9.3 0 Td[(2)]TJ /F10 11.955 Tf 11.96 8.96 Td[(Pj6=2kj2k23k2Nk31k32)]TJ /F3 11.955 Tf 9.3 0 Td[(3)]TJ /F10 11.955 Tf 11.95 8.97 Td[(Pj6=3kj3k3N...............kN1kN2kN3:::kNN1CCCCCCCCCCA(4)Beforeprovingatheoremonstabilityinthelinearcase,weneedtwopreliminaryresults. Lemma4.5. Assumethatthekineticsofasystemofchemicalsofdensitiesv1;v2;vNarerepresentedbyalinearfunction~g:RN!RNwhichisderivedusingthelawofmassactionkinetics.AssumealsothatthematrixJ,theJacobianof~gwithrespecttov,isirreducible,andthatatleastonechemicaldecaysataratei>0.ThentheeigenvaluesofJwillallhavestrictlynegativerealparts. Proof. Ifthechemicalreactionnetworkisasdescribed,thentheJacobianJwillbeoftheform( 4 ).ThisisaMetzlermatrixsinceeachkij0,andsothereexistssomerealnumbers>0suchthatJ+sIisanonnegativeirreduciblematrix.ThenwecanapplythePerronFrobeniusTheoremtoconcludetheexistenceofaneigenvaluep>0ofJ+sIcorrespondingtoapositiveeigenvectorxpsuchthat(J+sI)=p.Sincep>0isthedominanteigenvalue,noticethisimpliesalleigenvaluesofJ+sIareina 75

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discofradiuspcenteredat0inthecomplexplaneandsowillsatisfy Re())]TJ /F3 11.955 Tf 9.3 0 Td[(1)]TJ /F3 11.955 Tf 9.3 0 Td[(2)]TJ /F3 11.955 Tf 35.21 0 Td[(Nxp=(p)]TJ /F3 11.955 Tf 11.95 0 Td[(s)NXi=1xiwhichimpliesthatp)]TJ /F3 11.955 Tf 11.95 0 Td[(s<0sincePxi>0.Combiningthisfactwith( 4 ),weseeRe()0forsome1iN,<0isxed,and~Disanonnegativediagonalmatrix,thenJ+~DisalsoaMetzlermatrixwithallnegativeeigenvalues.Furthermore,everyeigenvalueofJ+~D 76

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satisesRe()
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aneigenvalueofJ+~DRe()0forsome1iN.If=0,then(u;v)islinearlystablewithrespecttoperturbationssatisfyingRu(x;t)dx=0. Proof. UsingTheorem 4.6 andthetechniqueofSchaafin[ 34 ]outlinedatthebeginningofthissection,recallthatwecanreducetheproblemofndingeigenvaluesofthelinearizedgeneralKSsystem( 4 )aboutanonnegativehomogeneoussteadystate(u;v)tothefamilyofeigenvalueproblems( 4 ).Inthisspeciccase,weseefrom( 4 )thatwearesearchingforeigenvaluesofthefamilyofmatricesA(i)=266666666664Di00001J11+i~D1J12J1;N)]TJ /F8 7.97 Tf 6.59 0 Td[(1J1N2J21J22+i~D2J2;N)]TJ /F8 7.97 Tf 6.59 0 Td[(1J2N..................NJN1JN2JN;N)]TJ /F8 7.97 Tf 6.59 0 Td[(1JNN+i~DN377777777775;1iN:SinceeachA(i)isablocklowertriangularmatrix,theeigenvaluesofAiwillbeDi0andtheeigenvaluesofJ+i~D.Weseefrom( 4 )thatJisMetzler.HencebyLemma 4.6 fori=0;1;2;:::theeigenvaluesofJ+i~Dallhavestrictlynegativerealpart.AlsobyLemma 4.6 anyeigenvalueofJ+i~Disboundedawayfromzerouniformlyin 78

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ibyRe()<)]TJ /F4 11.955 Tf 14.83 0 Td[(max1jNj+imin1jN~Dj)]TJ /F4 11.955 Tf 27.45 0 Td[(max1jNjRecallthat0=0<12sothatDi1<0foralli1.SowiththeexceptiontheeigenvalueD0=0,alleigenvaluesofeveryA(i)havestrictlynegativerealpartanddonothavezeroasalimitpoint.So,itonlyremainstoshowthatanyeigenfunction0B@uv1CAcorrespondingtotheeigenvalue=0violatestheconditionRu(x;t)dx=0.Toshowthis,let~x=[x0x1xN]t6=0betheeigenvectorofA(0)correspondingtotheeigenvalue=0.Noticethatifx0=0,thenitmustbethat[x1x2xN]t6=0andsoJ266666664x1x2...xN377777775=0:Howeverthisimpliesthat0isaneigenvalueofJ,whichisacontradictionbasedonLemma 4.5 ,soweconcludex06=0.ByTheorem 4.6 ,theeigenfunctioncorrespondingto=0is0B@uv1CAwhereu=x0!1isanonzerofunction.HoweverZu(x;t)dx=Zx0!1(x;t)dx=Zx0=jjx06=0: 4.5ExamplesInthissectionwereviewsomeexampleswhichweexploredin[ 8 ].TherstexampleisalargeclassofCRNstowhichwemayapplyTheorem 4.11 ,whilethesecondisaspeciccaseinwhichthesametheoremmaybeapplied.Thenal 79

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exampleshowsacasewhereoneofthehypothesesofTheorem 4.11 isviolated,buttheconclusionstillremainstrue. Example4.13. Againconsiderthecasewhere~g(~v)in( 4b )isalinearfunctionandletA2RNRNbethematrixsuchthat~g(~v)=A~v.WithterminologyasinSection 4.4.3 ,recallthattheJacobianof~gwillbegivenby( 4 ).Wewanttolookatthestabilityofpositivesteadystatesundertheseconditions.Now,sinceAisMetzler,itfollowsthat)]TJ /F3 11.955 Tf 9.3 0 Td[(AisoftheformofanM-matrix,soifitisinfactanonsingularM-matrixthenLemma 4.4 showstheexistenceofnonnegativehomogeneoussteadystates.Inthissettingwehavethefollowingcorrolaryonthestabilityofpositivehomogeneoussteadystates. Corollary4.1. Assumethat( 4 )hasapositivehomogeneoussteadystatesolution(u;~v),andthatthefunction~gislinearandobtainedbythelawofmassactionkinetics.Ifforsomechemicalvithereexistsapathinthechemicalreactionnetworkfromvi!vNandi>0,then(u;~v)islinearlyunstablewheneveruoriissufcientlylarge. Proof. WemustverifytheconditionsofTheorem 4.11 .ThepathintheCRNfromvitovNimpliesthatthereexistspositivereactionrateconstantskii1;ki2i1;ki3i2;;kNin)]TJ /F12 5.978 Tf 5.75 0 Td[(1forsomepositiveintegern.By( 4 ),thiscorrespondsexactlytoapathinG(Jt)fromvitovN.Furthermore,byhypothesisi>0,sotherstconditionissatised.Sinceeachi0andkij0,by( 4 )JisMetzlerandthesecondconditionissatised. Furthermore,noticethatinthecasewhereJisirreducibleTheorem 4.12 willapply.Hencepositivehomogeneoussteadystatesintheirreduciblecasearealwaysstablewhen=0,andcanalwaysbedestabilizedbystrongenoughchemotacticfeedbackaccordingtoCorollary 4.1 Example4.14(Dimerizationwithdecay). WereturnagaintoExample 4.1 toexploretheexistenceandstabilityofpositivehomogeneoussteadystates.Recallthatthe 80

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dimerizationreactionisgivenby2v1 !v2:Additionally,assumedecayofchemicalsv1andv2atrates10and2>0respectively.Denotetherateconstantfortheforwardreactionbyk1andthereversereactionbyk2.Thenaccordingtothelawofmassactionkinetics,thefunction~g(~v)describingtheCRNkineticswillbe~g(~v)=0B@)]TJ /F3 11.955 Tf 9.29 0 Td[(k1v21+k2v2)]TJ /F3 11.955 Tf 11.96 0 Td[(1v1k1v21)]TJ /F3 11.955 Tf 11.95 0 Td[(k2v2)]TJ /F3 11.955 Tf 11.96 0 Td[(2v21CA:Wealsoassumethatthespeciesofdensityuproducesv1atarate1>0andthatthechemicalv2isnotproducedsothat1=0.Considering( 4 )inthissetting,weknowby( 4 )thathomogeneoussteadystatesaresolutionstothesystem 1u)]TJ /F3 11.955 Tf 11.95 0 Td[(k1v21+k2v2)]TJ /F3 11.955 Tf 11.95 0 Td[(1v1=0; (4a)k1v21)]TJ /F4 11.955 Tf 11.95 0 Td[((k2+2)v2=0: (4b)Wesolve( 4 )toverifythatpositivesteadystatesexist.Adding( 4a )to( 4b )toobtainanewrstequationgivestheequivalentsystem 1u=1v1+2v2; (4a)v2=k1 k2+2v21: (4b)Thensubstituting( 4b )into( 4a )showsthatv1mustbeasolutiontothequadraticequation k12 k2+2v21+1v1)]TJ /F3 11.955 Tf 11.95 0 Td[(1u=0:(4)Itcanbedirectlyveriedusingthequadraticequationthat( 4 )hasauniquepositiverootwheneveru>0.Denotingthisrootbyr(u),weseethatthereisafamilyofpositivehomogeneoussteadystatesolutionsthatcanbeindexedbyu.Sogivenanyconstant 81

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u>0,apositivesteadystatesolutionto( 4 )isgivenbyu>0;v1=r(u)>0;v2=k1 k2+2r2(u)>0:Henceweseethatpositivehomogeneousstationarysolutionsexist,andwewillnowshowthatTheorem 4.11 appliestothisexample.TheJacobianof~gwillbegivenbyJ=264)]TJ /F4 11.955 Tf 9.3 0 Td[(2k1v;1)]TJ /F3 11.955 Tf 11.96 0 Td[(1k22k1v;1)]TJ /F3 11.955 Tf 9.3 0 Td[(k2)]TJ /F3 11.955 Tf 11.95 0 Td[(2;375whichisaMetzlermatrix.Alsosincek2>0and2k1v;1>0,itfollowsthatthereisapathinG(Jt)fromv1tov2andfromv2tov1.HencethehypothesesofTheorem 4.11 areveried,andweconcludethatthepositivehomogeneoussteadystatesinthisexamplewillbelinearlyunstableforlargeenoughvaluesuor1. Example4.15. Asourlastexampleinthissection,wedemonstrateacasewherethehypothesesofTheorem 4.11 arenotsatisedbuttheconclusionstillremainsvalid.Thisshowsthatwhiletheconditionsofthetheoremaresufcient,theyarenotnecessary.Considerthereactionv1+v2 !v3:Weagainassumethatthespeciesuproducestherstchemicalv1atarate1>0,butdoesnotproducetheothertwochemicalssothat1=2=0.Alsoassumethedecayrateofv1is1>0whilethedecayratesofv2andv3satisfy2=3=0.Inthiscasethefunction~g(~v)willbegivenby~g(~v)=0BBBB@)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v1v2+k2v3)]TJ /F3 11.955 Tf 11.96 0 Td[(1v1)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v1v2+k2v3k1v1v2)]TJ /F3 11.955 Tf 11.96 0 Td[(k2v31CCCCA: 82

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Againwerstverifythatpositivehomogeneoussteadystatesexist.By( 4 ),thismeanswemustndpositivesolutionstothesystem )]TJ /F3 11.955 Tf 9.3 0 Td[(k1v1v2+k2v3)]TJ /F3 11.955 Tf 11.96 0 Td[(1v1+u=0; (4a))]TJ /F3 11.955 Tf 9.3 0 Td[(k1v1v2+k2v3=0 (4b)k1v1v2)]TJ /F3 11.955 Tf 11.95 0 Td[(k2v3=0: (4c)Noticethat( 4b )and( 4c )areequivalent,andsoweonlyneedtosearchforsolutionsto( 4a )( 4b ).Thensubtracting( 4b )from( 4a )weobtaintheequivalentsystem 1v1=u; (4a)k1v1v2=k2v3: (4b)Henceweseethatthereareaninnitenumberofpositivehomogeneoussteadysolutions,indexedbyarbitrarypositiveconstantsu>0andv2>0.Thesestationarysolutionsaregivenbytheformulasu>0;v1= 1u;v2>0;v3=k1 1k2uv2:Wenowturntodeterminingthestabilityofthesepositivehomogeneousstationarysolutions.SincetheJacobianevaluatedatsomepositivehomogeneoussteadystate(u;~v)isJ=266664)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v;2)]TJ /F3 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v;1k2)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v;2)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v;1k2k1v;2k1v;1)]TJ /F3 11.955 Tf 9.3 0 Td[(k2377775:whichisnotingeneralaMetzlermatrix,wecannotdirectlyapplyanyofourstabilityresults.However,wemaystillapplyTheorem 4.6 toreducethelinearizedeigenvalue 83

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problemtondingtheeigenvaluesofthematrices A()=266666664D00)]TJ /F3 11.955 Tf 9.3 0 Td[(u1)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v;2+~D1)]TJ /F3 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v;1k20)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v;2)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v;1+~D2k20k1v;2k1v;1)]TJ /F3 11.955 Tf 9.3 0 Td[(k2+~D3377777775(4)foreigenvaluesoftheLaplacianwithNeumannboundaryconditions.Asinpreviouscalculations,wedeneK=u<0.WewillrstconsiderthecasewhenK=0,ndconditionsontheeigenvalues,andthenshowthattheseconditionswillchangeasKbecomesarbitrarilylarge.Fixsomeeigenvalue<0oftheLaplacianasin( 4 ).FortheK=0case,thematrixA()isblockdiagonalwitheigenvaluesD<0andtheeigenvaluesofthelowerright33submatrix.Thissubmatrixcanbewrittenas 266664)]TJ /F3 11.955 Tf 9.29 0 Td[(a)]TJ /F3 11.955 Tf 11.95 0 Td[(d1)]TJ /F3 11.955 Tf 9.3 0 Td[(bc)]TJ /F3 11.955 Tf 9.3 0 Td[(a)]TJ /F3 11.955 Tf 9.3 0 Td[(b)]TJ /F3 11.955 Tf 11.96 0 Td[(d2cab)]TJ /F3 11.955 Tf 9.3 0 Td[(c)]TJ /F3 11.955 Tf 11.95 0 Td[(d3377775(4)usingthepositivenumbersa=k1v;2;b=k1v;1;c=k2;d1=1)]TJ /F3 11.955 Tf 12.19 0 Td[(~D1;d2=)]TJ /F3 11.955 Tf 9.3 0 Td[(~D2;andd3=)]TJ /F3 11.955 Tf 9.3 0 Td[(~D3.Thencharacteristicequationof( 4 )is3+b22+b1+b0:=3+(a+b+c+d1+d2+d3)2+(a(d2+d3)+b(d1+d3)+c(d1+d2)+d1d2+d1d3+d2d3)+(ad2d3+bd1d3+cd1d2+d1d2d3)=0: 84

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WeapplytheRouth-Hurwitzcriteria(seeforexample[ 13 ]),whichsaysthatalleigenvaluesofamatrixhavenegativerealpartifandonlyiftheinequalities b2>0 (4a)b0>0 (4b)b1b2)]TJ /F3 11.955 Tf 11.95 0 Td[(b0>0 (4c)hold.Theinequalities( 4a )and( 4b )areimmediate,and( 4c )followsbydirectcalculation.Thuswhenever<0andK=0alleigenvaluesofA()havenegativerealpart.ThiswillnotbetrueforsufcentlylargevaluesofK>0,aswenowshow.ConsideringthecasewhereK>0and<0isstillxed,weusecofactorexpansionontherstrowof( 4 )tocomputedet(A())=C)]TJ /F3 11.955 Tf 11.96 0 Td[(Kdet2666641)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v;2+~D1)]TJ /F3 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v;10)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v;2)]TJ /F3 11.955 Tf 9.29 0 Td[(k1v;1+~D20k1v;2k1v;1377775whereC>0isthedeterminantofthematrixAA()inthecaseK=0,andisthusindependentofK.ThenexpandingdeterminantofthesubmatrixtoinvestigatetheKdependence,weseethedeterminantissomefunctionofKgivenbyf(K):=det(A(;K))=C+K1k1v;2~D2whereK1k1v;2~D2<0.HenceforsomesufcientlylargevalueofK0ofK,wehavef(K0)<0whileforK=0,F(0)>0.WeclaimthisimpliesthatforsomeK1with0
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parts,thenbecausetheeigenvalueswillappearinconjugatepairswhichhavepositiveproduct,itfollowsthatexactlytwooftheeigenvaluesmustberealandhaveoppositesignsinorderforthedeterminanttoeverbenegative.WehavenowshownthatforsufcientlylargevaluesofK,theinitiallystablepositivehomogeneoussteadystatewillbecomelinearlyunstable.Alongwiththecasewhere~gisalinearfunctionandtheJacobianof~gisirreducibleatthesteadystate,wenowhavetwoexamplesofhowchemotacticfeedbackinageneralCRNcandestabilizehomogeneoussteadystates.So,bothoftheseexamplesarecaseswherechemotacticallyinducedinstabilitycouldpotentiallyleadtopatternformation. 86

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CHAPTER5NUMERICALSOLUTIONSAsinthetwoequationcase,weemploytheniteelementmethod(FEM)tosearchfornonhomogeneousstationarysolutionsofEquation 4 andtoimplementtimesimulations.InthischapterwegiveageneralizationofthemethodsusedinSection 3.3 tondnonhomogeneousstationarysolutions.WealsopresenttimesimulationsusingbothastandardFEMandamethodusinganon-standarddiscretizationinordertopreservepositivityandreducenumericalinstability.WewillprovenumericalnonnegativityinthecaseofthesimpletwoequationKeller-Segel(KS)model,andalsointhegeneralizedmodelundercertainconditionsonthefunction~grepresentingthechemicalreactionnetwork(CRN).Throughoutthischapterweassumethatisapolygonaldomainsothatitcanbemeshedexactly. 5.1SpectralBandsRecallthatinSection 3.2 weusedresultsfrom[ 34 ]and[ 18 ]todeterminethestabilityofspatiallyhomogeneoussolutions(u;~v)to( 3 ).Thereexistsarangeofvaluesforwhichuislargeenoughsothatthehomogeneoussteadystateof( 3 )isunstable,butsmallenoughtoavoidblow-upsolutions.Onlywithinthisrangeofparameterswereweabletondconvergencetononhomogeneoussteadystates.InspiredbyTheorem 4.11 ,wehopetogeneralizethismethodtosolutionsofthesystem( 4 )todetermineforwhichvaluesoftheconstantsteadystateueigenvaluesofthelinearizedproblembecomepositive.Wethensearchforsolutionsintheneighborhoodwherethesignchangetakesplace.Inordertovisualizethespectrumoflinearizedoperatorsoftheform 4 ,weonceagainturntoTheorem 4.6 toreducetheinnitedimensionaleigenproblemtoanindexedseriesof(N+1)(N+1)matriceswhereNisthenumberofchemicalsbeingmodeled.Wemaycalculatealleigenvaluesof 4 aboutagivenhomogeneoussteadystate(u;~v)bynumericallyndingfori=1;2;3;:::eigenvaluesof 87

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thematricesA(i)in( 4 ).Recallthatinmanyofourexamples,wefoundthatpositiveornonnegativesteadystatescouldbeindexedbythevalueoftheconstantu.Foranyoftheseexampleswecanproducegraphsvisualizingthespectrumusingthefollowingprocedure. 1. Fixsomeu0anddeterminethepositivehomogeneoussteadystatecorrespondingtothisu. 2. ForsomenitenumberI,calculatealleigenvaluesfjgN+1j=1ofthematrixA(i;u)fori=0;1;:::I. 3. Letm(u;i)=max1jN+1Re(j). 4. Foreachi=0;1;:::;I,plotthepoint(u;m(u;i)). 5. Repeatforsomenitenumberofindicesiforuinarange[ustart;ustop].Byplottingtherst10,20ormorebandsovervariousranges[ustart;ustop],wecanseewherebandscrossthexaxisanddeterminepossiblevaluesofuforwhichthelinearizedeigenproblemmayproduceeigenfunctionswhichleadtospatiallynonhomogeneousstationarysolutions.Wenowreturntosomepreviousexamplesinordertoseetheresultsofthespectralbandcomputations.ConsideragainExample 4.1 where~garisesfromadimerizationreaction.InExample 4.14 wefurtherexploredthisexampletoshowtheexistenceofaninnitefamilyofsteadystateswhichcanbeindexedbyu>0andaregivenbyu>0;v1=r(u)>0;v2=k1 k2+2r2(u)>0wherer(u)istheuniquepositiverootofaspecicquadraticequation.Followingthealgorithmaboveforfori=1;:::;20,performingallcalculationsontheunitsquareandsettingallnonzeroconstantsequaltounity,Figure 5-1 isobtained.Uponcloserinspection,itcanbeseenthatthelinecorrespondingtothemaximumrealpartsofeigenvaluesofm(u;1)crossesthex-axisatarounducrit=22:14. 88

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Figure5-1. AnexampleofthespectralbandsofthematricesM(i)correspondingtothedimerizationreactionfor1i20. TurningnexttothefullfourequationKSmodeldiscussedinExample 4.2 ,wecanobtainasimilarspectralbandplotgivenafewassumptionsontheparameters.Thevalidityoftheseassumptions,aswellascomputationsdonewithparametervaluesapproximatedfromtheliterature,willbefurtherexploredinSection 6.3 .Forthetimebeingweassume2=3=0andsoconsider( 4 )with~ggivenby~g(~v)=0BBBB@)]TJ /F3 11.955 Tf 9.29 0 Td[(k1v1v3+(k2+k3)v2)]TJ /F3 11.955 Tf 11.95 0 Td[(1v1k1v1v3)]TJ /F4 11.955 Tf 11.96 0 Td[((k2+k3)v2)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v1v3+k2v21CCCCAand~givenby~=0BBBB@1031CCCCA:Werstwanttoshowthatthefull,fourequationKSmodelhaspositivehomogeneoussteadystatesolutions.By( 4 )thisproblemreducestoshowingthatpositivesolutions 89

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u;~vof 1u)]TJ /F3 11.955 Tf 11.95 0 Td[(k1v1v3+(k2+k3)v2)]TJ /F3 11.955 Tf 11.95 0 Td[(1v1=0 (5a)k1v1v3)]TJ /F4 11.955 Tf 11.95 0 Td[((k2+k3)v2=0 (5b)3u)]TJ /F3 11.955 Tf 11.95 0 Td[(k1v1v3+k2v2=0 (5c)exist.Adding( 5a )and( 5b ),weimmediatelyseethatgivenaxedconstantu>0,wehavev1=1 u>0:Similarly( 5b )and( 5c )implythatv2=3 k3u>0:Thensubstitutingthesetwovaluesforv1andv2into( 5b )andassumingthat>0andk3>0,weseethatsolongasourxedu>0,v3=(k2+k3)3 1k1k3>0;aconstantwhichisindependentofthevaluechosenforu.Henceweseethatforanygivenconstantvalueu>0,thereexistsauniquepositivevector~v2R3suchthat(u;~v)isahomogeneoussteadystatesolutionofthefourequationKSmodel.Inthisparticularcasethe44matrixA(i)from( 4 )willbegivenby 266666664Di00)]TJ /F3 11.955 Tf 9.3 0 Td[(ui1)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v3)]TJ /F3 11.955 Tf 11.95 0 Td[(+i~D1k2+k3)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v10k1v3)]TJ /F4 11.955 Tf 9.3 0 Td[((k2+k3)+i~D2k1v13)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v3k2)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v1+i~D3377777775:(5)Settingallnonzeroconstantsequaltounityandperformingtheproceduredescribedaboveonthematrix( 5 ),weobtainFigure 5-2 90

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Figure5-2. AnexampleofthespectralbandsofthematricesM(i)correspondingtothefullKeller-Segelmodelfor1i5. 5.2EigenfunctionsandPotentialPatternsUsingthemethoddescribedinSection 5.1 ,wecanndexactvaluesforpositiveeigenvaluesofthelinearizedsystem( 4 )aboutahomogeneoussteadystate(u;~v).Wenextusethestandardniteelementdiscretizationof( 4 ),asexplainedinSection 3.3 ,tosolveforapproximateeigenvectorscorrespondingtothesepositiveeigenvalues.Sincethepositiveeigenvaluesrepresentdirectionsofinstability,theseeigenfunctionswillrepresentpotentialpatternsformedbythechemotacticinstability.Wealsohopethattimesimulationswillshowsomeofthesemayindicatepatternsofnonconstantsteadystates.WeexplorethislastideafurtherinSection 6.2.2 ,butfornowwesimplydemonstrateeigenfunctionsarisingfromthepositiveeigenvaluesseeninFigures 5-1 and 5-2 .FornowwewillassumethatallconstantsareequaltounityinordertoqualitativelyunderstandpatternsinasettingsimilartothatoftheminimalsysteminEquation( 3 ).InSection 6.3 ,wefurtherinvestigatebiologicallyrealisticparametervaluesandexploretheeffectsofchangingparametervalues.Toshowhowtocomputetheseeigenfunctions,weagainreturntothefullKSmodelintroducedinExample 4.2 with2=3=0.Consideringthisspeciccase,theweak 91

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eigenvalueproblemintheformof 4 isgivenby )]TJ /F10 11.955 Tf 11.29 16.27 Td[(ZDrUr+uZrV3r=ZU (5a))]TJ /F4 11.955 Tf 12.04 3.02 Td[(~D1ZrV1r+1ZU)]TJ /F4 11.955 Tf 11.96 0 Td[((k1v3+)ZV1+(k2+k3)ZV2)]TJ /F3 11.955 Tf 11.96 0 Td[(k1v1ZV3i=ZV1 (5b))]TJ /F4 11.955 Tf 12.05 3.03 Td[(~D2ZrV2r+k1v3ZV1)]TJ /F4 11.955 Tf 11.95 0 Td[((k2+k3)ZV2+k1v1ZV2=ZV2 (5c))]TJ /F4 11.955 Tf 12.04 3.03 Td[(~D3ZrV3r+3ZU)]TJ /F3 11.955 Tf 11.96 0 Td[(k1v3ZV1+k2ZV2)]TJ /F3 11.955 Tf 11.95 0 Td[(k1v1ZV3=ZV3: (5d)UsingnotationasintheintroductionandSection 3.3 ,considerthelinearniteelementsubspaceShwithbasisf`gL0`=1,andletLbetheL0L0stiffnessmatrixdenedbyLij=Rrjri.SimilarlyletMbetheL0L0massmatrixdenedbyMij=Rji.DenetheapproximationsuhandvhinShbyuuh=X`Un``;vivnh;i=X`Vni;``:Thenthematrixformulationofthesystem( 5 )forasolution(uh;~vh)is C266666664~U~V1~V2~V3377777775=E266666664~U~V1~V2~V3377777775(5) 92

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wherethe4L04L0matricesCandEaredenedby C=266666664)]TJ /F3 11.955 Tf 9.29 0 Td[(DL00uL1M)]TJ /F4 11.955 Tf 12.04 3.02 Td[(~D1L)]TJ /F4 11.955 Tf 11.95 0 Td[((k1v3+)M(k2+k3)M)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v1M0k1v3M)]TJ /F4 11.955 Tf 12.05 3.03 Td[(~D2L)]TJ /F4 11.955 Tf 11.95 0 Td[((k2+k3)Mk1v1M3M)]TJ /F3 11.955 Tf 9.3 0 Td[(k1v3Mk2M)]TJ /F4 11.955 Tf 12.04 3.02 Td[(~D3L)]TJ /F3 11.955 Tf 11.96 0 Td[(k1v1M377777775(5)and E=266666664M0000M0000M0000M377777775:(5)WithknowledgefromthespectralbandcomputationsofthevalueofpositiveeigenvalueswecanusetheMATLABcommandeigstondtheeigenvectorsofthegeneralizedeigenproblem( 5 )correspondingtopositiveeigenvalues.Weseethat,aswehoped,valuesof(u;~v)withuslightlylargerthanucritdoindeedreturnpositiveeigenvaluesinthespatiallydiscretizedsettingapproximatingtheexacteigenvaluescalculatedpreviously.Ifweusetheeigscommandoptiontoreturneigenvectorsaswell,thenwehavetheniteelementrepresentationoftheeigenfunction,andhencethedirectionofinstability,correspondingtoaneigenvaluewithpositiverealpart.InthecaseofthefourequationKeller-Segelmodeleigenproblem( 5 )withallparameterssettounity,Figure 5-2 showsthatthersteigenvaluebecomespositivearounducrit=165:6.Thenxingu=170slightlylargerthanucritandsolvingforeigenvaluesneartheoneobtainedintheexactcalculation,weobtainedapositiveeigenvalue10:016withthecorrespondingeigenfunctionseeninFigure 5-3 .Thiseigenvaluehasmultiplicitytwo,withasecondeigenfunctionsimilartotheoneseeninFigure 5-3 butwiththepeaksandvalleysinoppositecorners.Anotherspectralbandbecomespositivearoundu=525,solinearizingaroundthesteadystatecorrespondingtou=550,wehavetwopositiveeigenvaluesof( 5 ),1 93

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Figure5-3. Eigenfunctioncorrespondingtotherstpositiveeigenvalueof( 5 )withu=170. 0:0463and20:474.Thenew,smallesteigenvaluecorrespondstotheeigenfunctionpicturedinFigure 5-4 .Noticethatqualitatively,thisnewpatternisthesameasthesteadystatesolutiontotheminimalmodelshowninFigure 3-1 .Furtherincreasingusothattwomorebandscrossthex-axis,eigenfunctionsoftheucomponentonlycorrespondingtothenewlypositiveeigenvaluesareseeninFigure 5-5 .Alsoofnoteisthatthebandsofthersttwoeigenvaluescross,sothattheeigenvaluecorrespondingtotheeigenfunctionofFigure 5-4 becomestheeigenvaluewithlargestpositiverealpart.ThebandsforthishigherrangeofuvaluescanbeseeninFigure 5-6 .ResultsofsimilarcomputationscandoneonadiskcanbeseeninFigure 5-7 ,whichshowstheeigenfunctionsofuonlycorrespondingtotherstfourspectralbandstocrossthex-axis. 5.3PositivityofNumericalSolutionsHavingfoundpotentialinitialconditionsforwhichpatternformationmayoccur,itisofinteresttonextperformtimesimulationstoseeifpatternsdodevelopandif 94

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Figure5-4. Eigenfunctionscorrespondingtothesecondpositiveeigenvalueof( 5 )withu=550. Figure5-5. Eigenfunctionsof( 5 ) 95

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Figure5-6. MorespectralbandsofthefullKSmodel. Figure5-7. Eigenfunctionsof( 5 )onadisk nonhomogeneoussteadystatesarefound.InthecaseofthesimpleKSmodel,wehavealreadydemonstratednonhomogeneousstationarysolutionssotimesimulationswilldemonstratewhethersuchstationarysolutionsarestable,andthereforepotentiallyobservedinnature.Aswehaveshownthatclassicalsolutionsof( 3 )and( 4 )remainatleastnonnegativeconsideringnonnegative,nonzeroinitialconditions,wewishtoknowifournumericalschemesmaintainnonnegativityaswell.Forthestandardniteelementmethodwehavenosuchguarantee,butinthissectionweintroduce 96

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anonstandarddiscretizationwhichinmanycasesensuresthatsolutionsremainnonnegativeandadditionallyminimizesthenumericalinstablityforconvectiondiffusionproblemsshowninSection 2.2.3 5.3.1ImplementationoftheMethodAsbeforeweletThdenoteatriangulationoftheboundeddomainR2.WeagainworkintheLagrangeniteelementspaceSh.Letf~x`gL0`=1denotetheglobalverticesofThand`2Shthecorrespondingbasisfunctions.Wewritetheexpansionsoftheapproximatingfunctionsafternstepsattimetn=ntasunh,vnh;i2Shofuandviasu(x;tn)unh=L0X`=1Un``;vi(x;tn)vnh;i=L0X`=1Vni;``:Thenifthesolutions~Unand~Vniareknown,wemaysolveforsolutionsattimetn+1by 1 kM(~Un+1)]TJ /F3 11.955 Tf 13.03 3.02 Td[(~Un)=)]TJ /F3 11.955 Tf 9.3 0 Td[(DA(vnh;N)~Un+1 (5a)1 kM(~Vn+1i)]TJ /F3 11.955 Tf 12.83 3.03 Td[(~Vni)=)]TJ /F4 11.955 Tf 12.04 3.03 Td[(~DiL~Vn+1i+iM~Un+~Gi(~vnh) (5b)whereMisadiagonalmatrixobtainedafterlumpingthemasses[ 38 ],meaningM``=Xm~M`m;~M`m=Z`m;andthestiffnessmatricesareLlm=Zrmrl;Alm(zh)=XKAK;zhj(l);j(m)wherej(l)2f0;1;2gdenotesthelocalvertexnumberofthelthglobalvertex.AK;zhand~Gi(~vnh)willbedenedshortly. 97

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WenowdescribethelocalstiffnessmatrixAK.Forthelocalindicesjuseadditionmodulothreesothatif,forexamplej=2thenAKj;j+1=AK2;0.Foranyzh2Sh,wesetcj=1 jEjjZEje)]TJ /F8 7.97 Tf 6.59 0 Td[((=D)zhwhereEjistheedgeoppositetothejthvertex,anddenetheentriesofthematrixAK;zhjkAKjkby AKjj=wj+1pj cj+1+wj+2pj cj+2 (5a)AKj;j+1=)]TJ /F3 11.955 Tf 11.96 0 Td[(wj+2pj+1 cj+2 (5b)AKj;j+2=)]TJ /F3 11.955 Tf 11.96 0 Td[(wj+1pj+2 cj+1 (5c)wherepj=e)]TJ /F8 7.97 Tf 6.59 0 Td[((=D)zj,zjequalsthevalueofzhatthejthvertex,andwj=(1=2)cotjwherejistheinteriorangleofKatthejthvertex.Asmentionedpreviously,thediscretizationleadingtotheabovedenedAKjkwasproposedin[ 44 ]andwillbederivedforourparticularcaseinSection 5.3.2 .The`thcomponentofvector~Gi(vnh)shouldbeanapproximationofRgi(~vnh)`,socalculateGni;`[~Gi(vnh)]`=Zgi(~vnh)`Z"L0Xm=1gi(~v(~xm;tn))m#`=L0Xm=1gi(Vn1;m;Vn2;m;:::;VnN;m)Zm`=[~M~hi]`where~hiistheconstantvectordenedforeach1iNby [~hi]m=gi(Vn1;m;Vn2;m;:::;VnN;m):(5) 98

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Inordertomaintainsomepositivitypreservingpropertiesweagainusethemasslumpingtechniqueasin[ 38 ]andingeneraldenethevector~Gni,1iN,by ~Gni=M~hi:(5) 5.3.2DerivationoftheMethodAsinSection 5.3.1 ,letThbeatriangulationofaboundeddomainR2andletShH1()bethesubspaceoffunctionswhicharecontinuousandpiecewiselinearwithrespecttoTh.Letqj,j2f0;1;2gdenotethelocallyindexedverticesofatriangleT2Th,andconsideradditionoverthelocalindicesjtobemod3fortheremainderofthissection.LetaKijdenotethelocalstiffnessmatrixonagivenelementK,ieaKij=ZTrjriwheretheiarethelocallinearbasisfunctionsonK.LetE=Eijdenotetheedgebetweentheithandjthverticesandwithendpointsqiandqj.DeneEf=f(qi))]TJ /F3 11.955 Tf 11.96 0 Td[(f(qj)foranyfunctionfdenedonE.Finally,letKEdenotetheinteriorangleofthevertexoppositeofedgeE=Eij.Aderivationofthefollowingimportantfactisgivenintheappendixof[ 44 ].Givenalocalbilinearform a(uh;vh)=ZKruhrvh(5)denedforuh;vh2Sh,wecanusethesymmetryofaijandthefactthataKii=)]TJ /F10 11.955 Tf 11.29 8.97 Td[(P3j6=iaKijalongwithageometricderivationtorewrite 5 as a(uh;vh)=XEK!KEEuhEvh(5)where!KE=1 2cot(KE): 99

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Notethatfori6=j,!KEij=)]TJ /F3 11.955 Tf 9.3 0 Td[(aKij.WewillreturntoEquation( 5 )later.Wenowturntothespeciccaseof( 4a ).Wewishtoapproximatethevariationalformulationof( 4a )insuchawaythatwemayuse( 5 )toconstructthelocalstiffnessmatricesandthenuserestrictionsonanglesizeinthetriangulationtoensurepositivity.In[ 29 ]andlaterin[ 44 ],thisisdonebyapproximatingtheuxof( 4a )byaconstantovereachtriangleK.Recallthattheuxof( 4a )is~J(u)=Dru)]TJ /F3 11.955 Tf 11.96 0 Td[(urvN;andsowemaywritethevariationalformulationof( 3a ),assumingNeumannboundaryconditionsonbothuandv,as Zut'=)]TJ /F10 11.955 Tf 11.29 16.27 Td[(Z~J(u)r'(5)foralltestfunctions'2H1().Inordertodiscretize( 5 ),weperformthefollowingcalculationsfrom[ 44 ],modiedtotourboundaryconditionsandspecicequation.Denethefunction EonanedgeEby E=()]TJ /F3 11.955 Tf 9.29 0 Td[(=D)vNforvN2Shandperformthefollowingcalculations.HereweabusenotationanddenotethederivativeoveranedgeEinthetangentialdirectionEby@ E @E.Wecalculatee)]TJ /F5 7.97 Tf 6.59 0 Td[( E@(e Eu) @E=e)]TJ /F5 7.97 Tf 6.59 0 Td[( Ee E@u @E+ue E@ E @E=1 jEjhruE)]TJ /F3 11.955 Tf 14.51 8.09 Td[( DurvNEi=1 DjEj~J(u)E: 100

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Thenmultiplyingthroughbye E,integratingovertheedgeE,andsubstitutinginthedenitionof Egives E(e)]TJ /F8 7.97 Tf 6.58 0 Td[((=D)vu)=1 DjEjZEe)]TJ /F8 7.97 Tf 6.59 0 Td[((=D)vN(~J(u)E)ds:(5)Now,approximatingtheuxonanygiventriangleKwithsomeconstantvector~Jc(u),wecanobtainfrom( 5 )theequality ~Jc(u)E=DE(e)]TJ /F8 7.97 Tf 6.59 0 Td[((=D)vNu)1 jEjZEe)]TJ /F8 7.97 Tf 6.59 0 Td[((=D)vNds)]TJ /F8 7.97 Tf 6.59 0 Td[(1:(5)Using( 5 )alongwith~Jc(u)=r(~Jc(u)~x)andE(~Jc(u)~x)=~Jc(u)E;wecanobtaintheapproximation )]TJ /F10 11.955 Tf 11.95 16.27 Td[(ZK~J(u)r')]TJ /F10 11.955 Tf 23.91 16.27 Td[(ZK~Jc(u)r'=)]TJ /F10 11.955 Tf 13.13 11.36 Td[(XEKZE!TE(Jc(u)E)E':(5)Finally,bycombining( 5 )with( 5 )wesee )]TJ /F10 11.955 Tf 11.95 16.27 Td[(ZK~J(u)r')]TJ /F3 11.955 Tf 21.92 0 Td[(DXEKZE!TEE(e)]TJ /F8 7.97 Tf 6.59 0 Td[((=D)vu)1 jEjZEe)]TJ /F8 7.97 Tf 6.59 0 Td[((=D)vds)]TJ /F8 7.97 Tf 6.59 0 Td[(1E':(5)Noticethat( 5 )reducesto( 5 )withu=j,'=i,andjEj=jEj. 5.3.3ProofofPositivityWerstshowpositivityofuhfornumericalsolutionsof( 4 )calculatedusing( 5 ). Theorem5.1. SupposethatThisatriangulationofsuchthatallmeshtriangleshavenon-obtuseinteriorangles.Ifu00thenforanystepsizet>0wehaveun0h0onforalltimet=n0(t)suchthatsolutionsvnhexistforalln
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SincebydenitionMisadiagonalmatrixwithpostivediagonalentries,itfollowsthatif~Un0thenM~Un0.Hencetoprovethat~Un+10oneonlyneedtoshowthatBM+(t)DAisinversepositive,oneoftheequivalentconditionstobeinganinvertibleM-matrix.By[ 2 ,theequivalenceM35()N38],itisenoughtoshowthatBhaspositivediagonalentriesandthatBisstrictlydiagonallydominant,meaningB``>Xm6=`jBm`j:Toshowthenecessaryconditions,werstobservethatthediagonalelementsofBaregivenbyB``=M``+(t)DAll>0:ClearlyM``>0,D>0andt>0byassupmtion.FurthermoreA``>0followsfrom( 5a )andthefactthat!j0since0
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by( 5b )( 5c ).Usingtheaboveobservations,calculateB``=M``+kDXm6=`()]TJ /F3 11.955 Tf 9.3 0 Td[(Am`)=M``+kDXm6=`jAm`j>kDXm6=`jAm`j(asM``>0)=Xm6=`jBm`j(asMm`=0):HenceBisanonsingularM-matrix,soB)]TJ /F8 7.97 Tf 6.59 0 Td[(10.Sinceu0implies~U0,theresultholdsbyinduction. Wenextprovepositivityofthemethoddenedby( 5 )intheparticularcasewhere~gisalinearfunction. Theorem5.2. LettheassumptionsonTh,t,andtn=ntbeasinTheorem 5.1 .Additionallyassumethat~g=R~visalinearfunction,withthematrixRsuchthatI+(t)Risanonnegativematrix.Ifu0andeverycomponentof~v0arenonnegativefunctionson,thenvn0h;i0forall1iNsolongasvnh;iexistsforalln
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SinceUn0byTheorem 5.1 andMisanonnegativediagonalmatrix,thersttermof( 5 )isclearlyanonnegativevector.Usingthestandardfactthatforanylocalbasiselements'iand'jonatriangleKwehaveRKrirj=)]TJ /F4 11.955 Tf 9.29 0 Td[((1=2)cot(Eij)andtherestriction=2itisclearthatoff-diagonalentriesofLarenonpositive.AlsoobviouslydiagonalentriesofLarepositive.ThereforewemayarguejustasintheproofofTheorem 5.1 that(M+t~DiL)isanM-matrixandhenceinversepositive.Itfollowsthat~Vn+1isnonnegativeinallcomponentswhenever~Vnisnonnegative,andhencetheresultfollowsbyinduction. Remark5.3. NoticethatwecanconsiderthesimpleKSmodel( 3 )asaspeciccaseof( 4 )withN=1andg(v)=)]TJ /F3 11.955 Tf 9.3 0 Td[(valinearfunctionsothatTheorems 5.1 5.2 showpositivityofuhandvhforapproximatesolutionsof( 3 )usingthemethodin( 5 ). 5.4VectorizationWemakesomenalcommentsaboutimplementationofniteelementmethodtimesimulationstondapproximatesolutionsof( 4 ).Thedifcultyarisesfromtheterm )-221(r(urvN)(5)in( 4a )whichishighlynonlinearandalsocontributestothestrongcouplingoftheequationsforuand~v.ConsiderthelinearniteelementspaceShwithabasisofL0functions.Ifthestandardniteelementformulationisused,theterm( 5 )leadstoaL0L0submatrixofthefull(N+1)L0(N+1)L0matrixusedtosolvethediscretizedformof( 4 ).Bythenonlinearityoftheterm,thismatrixmustbedependentoneitherthevalueofuhorvh;Nattheprevioustimestep,andsomustbecalculatedanewateachtimeiteration.Forexample,onepossiblediscretizationoftheterm( 5 )istodenethematrixD(u)byD(u)ij=Zur'jr'i:ThenifuhP`U``andvh;NP`VN;`aretheapproximationsofuandvNintheniteelementspaceatanyxedtime,( 5 )isapproximatedinmatrixformbyD(uh)~VN. 104

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Evenwhenthepositivitypreservingmethodisused,noticethat( 5 )stilldependsonthevaluesofvNattheprevioustimestepandsomustalsobecalculatedwithinthetimesteploopateachiteration.Forthisreasonitisveryimportanttobeasefcientaspossiblenotonlywhenperformingthematrixinversiontosolveforthenextstepbutalsowhenconstructingthematrices.AsourprogrammingwasprimarilydoneusingMATLAB,weusedavectorizationtechniquefrom[ 5 ]toensureefciencyinmatrixconstruction.TheideabehindthevectorizationtechniqueistouseMATLAB'sbuiltinstructuresforsparsematricesalongwithefcientcalculationsonvectors.Forexample,considertwovectors~aand~bandtheoperation~c=~a:~b,wherethe:isabuiltinMATLABfunctionthatperformselementwisevectorormatrixmultiplication.Itismuchfastertorunthelineofcode~c=~a:~bthantoruntheloop>>fori=1:length(~a)>>c(i)=a(i)b(i)>>end,evenwhenstoragefor~cispreallocated[ 5 ].Inourcode,weapplythisideatotheconstructionofstiffnessmatricesasin[ 5 ]sothatonlytwonestedloopsofthreeiterationseachareruntobuildthelocalstiffnessmatrices.Withineachnestedloopavectorizedcalculationbuildsthelocalstiffnessmatrixinoneline.TheselocalmatricesarethenplacedintotheglobalmatrixusingMATLAB'ssparsematrixnotation. 5.5MethodValidationInordertovalidatetheaccuracyofthematrixAconstructedasdescribedinSection 5.3.1 ,weagainreturntothetestproblem( 2 )fromSection 2.2.3 .Inadditiontovalidatingtheaccuracyofthepositivitypreservingmatrix,thetestproblemwillalsoexemplifyhowsolutionsobtainedusingthisnewmethodarebetterbehavedwhenexactsolutionshavesharpgradients.RecallthatA(z)wasbederivedsothattheFEMdiscretizationofthetermDu)-222(rurv; 105

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withNeumannboundaryconditionsandweakformulation )]TJ /F3 11.955 Tf 11.95 0 Td[(DZruri+Zurvri(5)isgivenby)]TJ /F3 11.955 Tf 9.3 0 Td[(DA(rv)~U:WewillusethematrixAA(v)forv(x;y)=x.Thenwehaverv=(1;0)=.SubstitutingthisvandtheFEMapproximationu=PiuiiintotheweakformulationgivenbyEquation( 5 )becomesXj)]TJ /F3 11.955 Tf 9.3 0 Td[(DZrjri+Zjriuj1iN:ThuswemusthavethatAijZrjri)]TJ /F3 11.955 Tf 14.51 8.09 Td[( DZj(ri)forv(x;t)=x,orinotherwords)]TJ /F3 11.955 Tf 9.3 0 Td[(DAij)]TJ /F3 11.955 Tf 21.92 0 Td[(DZrjri+Zj(ri);whichisexactlythetransposeofthematrixformulationgivenby( 2 )for=1.Whenusingthismethodtosolvefor~U,thenon-standardniteelementsolutiongivesagoodapproximationoftheactualsolution,evenforroughmesheswherethestandardmethodshowedoscillation.SolutionsusingthemethoddescribedaboveareshowninFigure 5-8 .BycomparingthesesolutionstothesolutionsfromthestandardFEMinFigure 2-2 fromSection 2.2.3 ,weseethatthepositivitypreservingmethoddoesindeedreducetheoscillatorybehavior,evenforthecoarsemesh. 106

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Figure5-8. PositivitypreservingFEMsolutionsto( 2 ) 107

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CHAPTER6APPLICATIONSANDRESULTSOFSIMULATIONS 6.1ApplicationtoTumorGrowthAlthoughitseffectsarenotyetfullyunderstood,chemotaxisisbelievedtobeafactorintumorgrowthandmetastasis.Inthissectionwelookatatumorgrowthmodelproposedin[ 4 ]whichisbasedonhaptotaxis,orchemotaxisinwhichthechemoattractantisboundandthereforedoesnotdiffuse.Themodelaccountsfortumorcellproductionoractivationofmatrixdegradingenzymes(MDEs),whichcausedecayofcomponentsoftheextracellularmatrix(ECM).Thedegradationofthismatrixgivestumorcellsroomtogrow,andatthesametimethetumorcellsclimbgradientsoftheECMinordertosupporttheirgrowth.Fortheremainderofthissectionweleturepresentthecelldensity,v1representtheconcentrationoftheMDEandv2representtheECMconcentration.It[ 4 ]itisassumedthattheMDEisproducedataconstantrate1>0percancercell,andthatitalsodegradeswithrateconstant0.ThenusingthesamechemotacticuxtermastheKeller-Segelmodel,thefollowingsystemofequationsisgivenin[ 4 ]. @tu=D0u)]TJ /F3 11.955 Tf 11.96 0 Td[(r(urv2);x2;t>0 (6a)@tv1=~D1v1+1u)]TJ /F3 11.955 Tf 11.96 0 Td[(v1;x2;t>0 (6b)@tv2=)]TJ /F3 11.955 Tf 11.96 0 Td[(v1v2;x2;t>0: (6c)Theauthorsof[ 4 ]alsoassumethatthecancercellsandtheMDEremaininsidethedomainandthereforeimposeno-uxboundaryconditionsonuandv1,whichresultsintheboundaryconditions )]TJ /F3 11.955 Tf 9.29 0 Td[(D0@u @n+u@v2 @n=0;x2@;t>0 (6a)@v1 @n=0;x2@;t>0: (6b) 108

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NoticethatuptoboundaryconditionsthismodeltstheframeworkofthegeneralizedKSmodel( 4 )withthefunction~ggivenby~g0B@v1v21CA=0B@)]TJ /F3 11.955 Tf 9.3 0 Td[(v1)]TJ /F3 11.955 Tf 9.3 0 Td[(v1v21CA;thevector=[1;0]andthediffusionmatrixD=264~D1000375:IfweassumetheMDEdegradestheECMbymeansofareactionoftheformv1+v2)166(!v1+degradedproductwithreactionrateconstantandthatv1)166(!;withrateconstant,then~gcanberecoveredusingthelawofmassactionkinetics.Inadditiontoprovidingarealbiologicalexamplewhichtstheframeworkofourgeneralizedmodel,weareinterestedin( 6 )becausewemayuseittovalidateourtimesimulationsbyreproducingsimulatedresultsfrom[ 4 ].Theinitialconditionsusedin[ 4 ]arebasedontheideaofanalreadyformedtumorwithcelldensitygivenbyu(x;0)=8><>:exp)]TJ /F5 7.97 Tf 6.58 0 Td[(x2 ;x2[0;0:25]0;x2(0:25;1]:TheythenassumethattheinitialMDEconcentration,asitisemittedbythetumorcells,willbeproportionaltothetumorcelldensity,anddenev1(x;0)=0:5u(x;0).AssumingtheMDEfromthepreviouslyformedtumorhasalreadydegradedtheECM,theyconsiderv2(x;0)=1)]TJ /F4 11.955 Tf 12.26 0 Td[(0:5u(x).Usingtheseinitialconditionsandastandardniteelementmethod(FEM)withpiecewiselinearbasisfunctionsandNeumannboundary 109

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Figure6-1. OnedimensionalFEMsolutionstoEquation( 6 ) conditions,wesolvedthesystem( 6 )withboundaryconditions( 6 )inonedimensionuntiltimet=20.TheresultsareshowninFigure 6-1 ,andappeartocloselymatchthesolutionsfrom[ 4 ].Fortwodimensionstheinitialconditionsagainconsiderapreviouslyformedtumor,thistimecenteredinthemiddleoftheunitsquaredomain.Theinitialconditionforthetumorcellsisgivenbyu(x;0)=8><>:exp)]TJ /F5 7.97 Tf 6.58 0 Td[(r2 ;r2[0;0:1]0;x2(0:1;1]wherer2=(x)]TJ /F4 11.955 Tf 12.05 0 Td[(0:5)2+(y)]TJ /F4 11.955 Tf 12.04 0 Td[(0:5)2.TheinitialconditionsfortheMDEandECMaregivenasintheonedimensionalcase,thatisv1(x;0)=0:5u(x;0)andv2(x;0)=1)]TJ /F4 11.955 Tf 12.38 0 Td[(0:5u(x). 110

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Figure6-2. TwodimensionalFEMsolutionsforuofEquation( 6 ) OurstandardFEMsolutionintwodimensionsalsoappearstomatchthesolutionsin[ 4 ],includingthehigherdensityringofcellswhichdevelopsneartheedgeofthetumorandcontinuestoinvadetheECM.ThetumordensitiesforvarioustimesareshowninFigure 6-2 .Asanadditionalverication,itwascheckedthatthetotalcelldensityoverthedomainisconserved,whichshouldbeimpliedbytheno-uxboundaryconditionsandlackofbirthordeathtermsinEquation( 6a ).Thishelduptomachineprecision,or10)]TJ /F8 7.97 Tf 6.58 0 Td[(17.Thefollowingparameterswereusedforboththeone-dimensionalandtwo-dimensionalsimulations,asstipulatedin[ 4 ]:D0=0:001,~D1=0:001,=0:005,=10,1=0:1,and=0.Fortheonedimensionalsimulation,=0:01whileforthetwodimensionalsimulation=0:0025. 111

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6.2Keller-SegelTimeSimulations 6.2.1MinimalKeller-SegelModelInthissectionwenumericallyreproducesomewellknownresultsontheminimalKeller-Segel(KS)modelinordertofurthervalidateournumericalmethodsandhaveabasisforcomparisonfortheresultsoftimesimulationsofthefullKSmodel.WerstreviewsomeresultsmentionedinSection 3.2 onthestabilityandglobalbehaviorofsolutionstoEquation( 3 ).Recallthatasin[ 18 ],wedeneblowupofasolution(u;v)to( 3 )ashavingoccuredwheneitherku(x;t)kL1()orkv(x;t)kL1()becomesunboundedineitherniteorinnitetime.In[ 18 ],itisstatedthatsolutionsto( 3 )existgloballyintimeandhaveboundedL1()normiftheinitialconditionforusatisesZu0(x)dx<4:Ifontheotherhand48thentherestillexistinitialconditionsforwhichblow-upoccurs,itisjustnotnecessarilyontheboundaryof[ 18 ].Thenextresult,from[ 34 ],involvestheeigenvalue1<0oftheLaplacianoperatoronthedomainwhichisclosesttozero.Ifjj(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1)>Zu0(x)dxthenthespatiallyhomogeneousstationarysolutionto( 3 )islinearlystable. 112

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Weperformedtimesimulationsofsolutionsto( 3 )usingboththestandardFEMdescribedinSection 2.2.2 andthepositivitypreservingmethoddescribedinSection 5.3 .Thefollowingdiscussionisforresultsusingthedomainequaltotheunitsquare.Unlessotherwisenoted,simulationsinthissectionwerecomputedusingameshsizeofaroundh0:03andtimestepst=0:01.Thechoiceoftwasbasedonwhattimestepsizewasneededforconvergencetotheknownspatiallyconstantsteadystatesfromrandomlyperturbedinitialconditions.Foranyconstantu>0,noticethatifv=u,then(u;v)isaspatiallyhomogeneousstationarysolutiontothesystem( 3 ).Thenforu<(1)]TJ /F3 11.955 Tf 11.99 0 Td[(1)jj=1+210:87[ 34 ]saysthat(u;v)islinearlystable.Thisstabilityresultwasconrmedinnumericaltimesimulationexperimentsusinginitialconditionsoftheconstantsolution(u;v)perturbedbothrandomlyandbyperturbationsinthedirectionofeigenfunctionscorrespondingtopositiveeigenvalueswhenu>1+2.TheseeigenfunctionsweredeterminedusingthespectralbandsmethoddescribedinSection 5.1 .ThespectralbandsfortheminimalKeller-Segel(KS)systemcanbeseeninFigure 6-3 .Noticethisplotalsonumericallyandvisuallyconrmsthetheoreticalresultfrom[ 34 ]thatthestationarysolutionto( 3 )ontheunitsquarebecomesunstablearoundu=11.Wenextinvestigatetimesimulationofsolutionsto( 3 )whenu>412:57,inparticularforu=13.Recallthatforthisparameterrangethetheorypredictsthatthespatiallyconstantsteadystateisunstableandthatsolutionswhichexhibitblow-upbehaviorontheboundaryexist.Weagainusedasmallperturbationawayfromthespatiallyhomogeneousstationarysolution(u;v)astheinitialconditionforthetimesimulation.Forthestandardniteelementmethod,thesolutionpropogatesawayfromthestationarysolutionwhenperturbedslightly,justaspredictedbythetheory.Howeverafteraperiodofgrowthawayfromthestationarystate,thesolutionbeginstoblow-upinanumericallyunstableway.ItshowslargeoscillationsandattainsnonphysicalnegativevaluesasseeninFigure 6-4 .Thisnumericalinstabilitymotivatetheneed 113

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Figure6-3. SpectralbandsoftheminimalKSmodel forthepositivitypreservingmethod.Figure 6-4 alsoshowsthepositivitypreservingmethoddoesindeedpreservepositivity,andthesolutionobtainedusingthepositivitypreservingmethodcontinuestoconvergetowardswhatappearstobea-functionsingularityinacornerofthedomainevenafterthestandardmethodbeginstoshownumericalinstability.Asmentionedin[ 18 ],a-functioninuistheexpectedbehaviorofsolutionswhenblowupoccurs.ThesolutionsshowninFigure 6-4 areallforthefunctionu,butsolutionsofvarequalitativelysimilar.Bothsolutionswerefortimestepsofsizet=0:01andmeshsizeh0:03.Wenallyturntovaluesofuontheunitsquareforwhich1+20.Infactresultscitedin[ 18 ]statethatsolutionsshouldconvergetoastationarysolutionast!1.Sincetheuniquespatiallyconstantstationarysolutionisunstableforu>1+2,thisimpliesaspatiallynonhomogeneousstationarysolutionexists.Althoughwehavefoundsomesuchsolutions,picturedinSection 3.3 ,usingaNewtoniteration,wehavenotbeenableto 114

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Figure6-4. Blow-upoftheminimalKSmodel showconvergencetothestationarysolutionthroughtimesimulations.HoweverasseeninFigure 6-5 ,thetimebeforeblow-upandnumericalinstabilityoccursislongerwhen1+24.Thisgiveshopeforfutureworkthatifmeshsizeortimestepsaresufcientlysmall,theconvergencebehaviorcanbedemonstratednumerically.Wenotethatforcomputationsontheunitcircle,similarresultswereobtained.Convergencetothestationarysolutionwasshownnumericallyintheappropriateparameterrange,whileblow-upandnumericalinstabilitywasobservedintheparameterrangewheresuchsolutionsaretheoreticallyguaranteedtoexist.Howeverweinthiscasewewereabletoshowconvergencetoanonhomogeneousstationarysolutionwithintherangewherethespatiallyconstantstationarysolutionisunstablebutsolutionsaretheoreticallyguaranteedtoexistforalltime.TheerrorforconvergenceinthisrangeisshowninFigure 6-6 115

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Figure6-5. Convergent(u=11)versusblow-up(u=13)parameterrangesoftheminimalKSmodel 6.2.2FourEquationKeller-SegelModelHavingveriedsomestandardtheoreticalresultsinthecaseoftheminimalKSmodel,wenowturntocomparingnumericalsimulationstoournewtheoreticalresultsfromChapter4.WewillrefertothetwoequationKSmodelstudiedintheprevioussectionastheminimalmodel,andtothefourequationKSmodelwhichwenumericallyinvestigateinthissectionasthefullorfourequationKSmodel.ThisspecicfullKSmodelshouldnotbeconfusedwiththegeneralKSmodelintroducedinChapter4asitisinfactaspecialcaseofthatmodel.Aswiththesimulationsoftheminimalmodel,westudythecasewhereallconstantparametersaresettounity.Whiletheseranges 116

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Figure6-6. Convergencetoanonhomogeneousstationarysolutiononadisk areclearlynotbiologicallyrealistic,theyshouldstillallowforacomparisonbetweenthequalitativebehavioroftheminimalandfullKSmodels.Alsoasinthelastsection,unlessotherwisenotedallsimulationswereperformedusingaFEMonameshsizeofh0:03witht=0:01.RecallthatinSection 5.1 wefoundthatthersteigenvalueofthelinearizedeigenproblemaboutthespatiallyhomogeneousstationarysolutiontothefullKSmodelbecamepositivearounducrit=165:6.HencewersttestedtimesimulationsinthecaseofthefullKSmodelforperturbationaroundthespatiallyconstantstationarysolutioncorrespondingtou=150
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Figure6-7. ConvergencetoconstantsteadystateforfullKSmodel toaneigenvaluethatwilllaterbecomepositive.Thisisthesameperturbationthatwillbeusedwhenweshowdivergencefromthestationarysolutionforvaluesofu>ucrit,andwasdeterminedusingthemethoddescribedinSection 5.2 .Asourtheoreticalresultsandspectralbandsplotsdonotgiveanyinformationaboutblow-upornonblow-up,wecanonlyusenumericaltimesimulationstotestbehaviorforvaluesofu>ucrit.Weperformedtimesimulationswithinitialconditionsequaltosmallperturbationsfromthespatiallyconstantstationarysolution.Webeginwithresultsobtainedfromsimulationsofinitialconditionscorrespondingtoperturbationsinthedirectionoftherstunstableeigenfunction,obtainedasdescribedinSection 5.2 .Unlikeinthecaseofu
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Figure6-8. DifferencebetweenstandardandpositivitypreservingFEMforfullKStimesimulations method.Astheseresultsaccuratelyshowedthebehaviorofblowupinthetwoequationcase,theyshouldgiveanaccurateportrayalofblow-upinthiscaseaswell.Similarlytothecaseoftheminimalmodel,thepositivitypreservingmethodforthefullmodelshowsblow-upofu(x;t)intheformofa-functionontheboundary.Thesolutionsforallfourfunctionswhenblow-upoccursinucanbeseeninFigure 6-9 .So,asinthecaseoftheminimalKSmodel,itappearsthatwhenblow-upoccursinthefullKSmodelitis,atleastinthesesimulations,ofa-functionsingularitytype.AlthoughwehavenotbeenabletondconvergenceintimetoaspatiallynonhomogeneousstationarysolutioninthecaseofeithertheminimalorfullKSmodel,wedohavecalculationsthatshowthepatternsarisingfromtheeigenfunctionscorrespondingtopositiveeigenvaluesarestableinsomesense.Toseethisstability,weusedtimesimulationsusinganinitialconditionofthespatiallyconstantstationarysolutionperturbedbyasmallrandomfunctioninsteadofperturbinginthedirectionoftheunstableeigenfunctionofthestationarysolution.Underthesetimesimulationsusingtheconstantstationarysolutioncorrespondingtou=180,boththestandardand 119

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Figure6-9. BlowupoffullKSmodelattimet=130 positivitypreservingmethodsquicklyconvergetoapatternsimilartotheoneofthedominanteigenvaluebeforeslowlystartingtoincreaseinthedirectionofthiseigenvalueandeventuallydemonstratingeithernumericalinstability(inthecaseofthestandardmethod)or-functionblow-up(inthecaseofthepositivitypreservingmethod).SolutionssimulatedusingthestandardFEMfromtheserandomlyperturbedinitialconditionsatvarioustimescanbeseeninFigure 6-10 .Thearbitraryinitialconditionattimet=0istherstimage,andveryquicklyafterthatsmallaggregationcentersbegintoappearasattimet=0:02.Ofinterestisthataroundtimet=0:15,foratimethepatternresemblesoneoftheeigenfunctionscorrespondingtoaneigenvaluewhichisnegativeforlinearizationabouttheconstantsteadystatecorrespondingtou=180butbecomespositive,andthereforeadirectionofinstability,forlargervaluesofu.TheformofuforthiseigenfunctioncanbeseeninFigure 6-11 ,andtheeigenfunctionsofv1,v2andv3arequalitativelysimilar. 120

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Figure6-10. Behaviorofsolutionfromrandomlyperturbedinitialconditions Figure6-11. Alinearizedeigenfunctionofucorrespondingtoanegativeeigenvalue Insummary,thenumericalexperimentsinthissectionhaveshownthatimmediatelybeforeandafterdestabilizationofthespatiallyconstantstationarysolution,theminimalKSmodelandthefullKSmodelhaveverysimilarbehaviorinthecasewhereallconstantsaresettounity.Bothsystemsshowconvergencetothespatiallyconstantsteadystatewhenuisbelowacriticalvalue,andpropagateinthedirectionofaqualitativelysimilareigenfunctionwhenuisslightlyaboveacriticalvalue.Anopen 121

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questionforfurtherexperimentationisifthebehaviorofthetwomodelsremainsthesameforlargervaluesofuwhendifferenteigenvalues,andhencedifferenteigenfunctionscorrespondingtothem,becomedominant.ThenumericaltechniqueofndingadirectionofinstabilityfromahomogeneousstationarysolutionusingaspectralbandsplotcanalsobegeneralizedtoaddressanynumberofotherapplicationswhichmaytthegeneralizedKSmodel,whichisanotheropendirectionforfutureresearch. 6.3ParameterFittingintheFourEquationKeller-SegelModelInalloftheprevioussectionswehavetakentheapproachof[ 33 ]bysettingallnonzeroconstantsequaltounityandobservingthequalitativebehaviorofthesystem.InthissectionwewillapplythespectralbandandtimesimulationmethodstothefullKSmodelwithmorerealisticparametervalues.AsummaryoftheparametersusedforthecalculationsinthissectionisgiveninTable 6-1 .Wedonotclaimthatthesevaluesareexactlyaccurate,andinfactmanyareveryroughestimates,butwehopethattheyprovidesomeideaofthecorrectbiologicalscaleinordertoseerelevantresults.Somebriefcommentsonthevaluesarenecessary.Diffusioncoefcientsforthreeofthefourcomponentswerefoundintheliterature,butavaluecouldnotbefoundforadiffusioncoefcientofthecomplexconsistingoftheenzymeboundwiththecAMP.Forthisreason,theparameterD2wasestimatedusingtheStokes-Einsteinequation(seeforexample[ 31 ]), D=kBT 6r:(6)In( 6 )kBistheBoltzmannconstant,Tistheabsolutetemperature,istheviscosity,andristheradiusoftheparticle,whichinthiscaseisamoleculeofthechemicalcompound.Ifallotherconstantsareassumedtobethesameforthechemicalsinquestionandtheeffectiveradiusofthecomplexisassumedtobesumoftheradiiofthe 122

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enzymeandthecAMP,thenusing( 6 )D2maybeestimatedbyD2=1 D1+1 D3)]TJ /F8 7.97 Tf 6.59 0 Td[(1:WealsonotethatthemorecommonlystudiedparameterintheliteratureforchemicalreactionsseemstobetheratioKd=k2 k1,knownasthedissociationconstant.Itwasassumedthatk2shouldbeonthescaleofk3=9:310)]TJ /F8 7.97 Tf 6.59 0 Td[(4,theotherrateconstantwithunitsofs)]TJ /F8 7.97 Tf 6.59 0 Td[(1.Henceweproducedspectralbandsplotsforvaluesofk2varyingfrom7:510)]TJ /F8 7.97 Tf 6.59 0 Td[(6to7:5102andthecorrespondingk1values10)]TJ /F8 7.97 Tf 6.59 0 Td[(14to10)]TJ /F8 7.97 Tf 6.59 0 Td[(6ascalculatedfromKd.Withinthisrangethespectralbandsdidnotchangevisibly,andsoitseemsreasonabletoassume,aswasstatedintheliterature,thatthevalueofKdistheimportantratioandthatsolongasitisconstanttheresultswillnotbesignicantlyaffected.Inordertokeepallvalueswithinacomputablerange,wechosethevalueofk2=0:75.ThenusingKdandthechosenvalueofk2,k1wascalculatedtobe10)]TJ /F8 7.97 Tf 6.59 0 Td[(9.Finally,inordertoguaranteetheexistenceofahomogeneoussteadystatesolutionwhichmaybecalculatedexactly,wefoundthatweneed2=3=0and1>0.In[ 24 ],itismentionedthatstarvedcellsexcreteaproteinwhichinhibitstheactivityoftheirextracellularphosphodiesterase[enzyme].Ifthiseffectisestimatedbydecayoftheenzymeatapositiverateconstant,then1>0wouldbeaplausiblebiologicalassumption.Additionally,ifnodecayoftheenzymeisassumedthennospatiallyhomogeneousstationarysolutionsexistforanyvalueofu>0.ThiswouldbeadenitequalitativedifferencebetweenthebehaviorofsolutionstothefullKSsystemandtheminimalKSsystem,andsowouldplacefurtherimportanceonthestudyofthegeneralmodelwedevelopinthiswork,ifnotnecesarilythecasestudiedinthissectionwhere1>0.Sincenovaluecouldbefoundintheliterature,weexprimentedwithvaluesof1between1=10)]TJ /F8 7.97 Tf 6.59 0 Td[(6and1=10andfoundveryfewvisibledifferencesbetweenthespectralbandsplots.Henceweestimate1tobeontheorderofthedecayrateconstantk3=9:310)]TJ /F8 7.97 Tf 6.59 0 Td[(4andchoose1=10)]TJ /F8 7.97 Tf 6.59 0 Td[(4. 123

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Table6-1. ParametervaluesforfourequationKS VariableValueReferences D0=DDicty410)]TJ /F8 7.97 Tf 6.59 0 Td[(8cm2=sec[ 28 ]D1=Denzyme6:110)]TJ /F8 7.97 Tf 6.58 0 Td[(7cm2=sec[ 39 ]D2=Dcomplex5:810)]TJ /F8 7.97 Tf 6.58 0 Td[(7cm2=secEstimatedwithStokes-EinsteinD3=DcAMP9:710)]TJ /F8 7.97 Tf 6.58 0 Td[(6cm2=sec[ 28 ]k110)]TJ /F8 7.97 Tf 6.58 0 Td[(9cm2/(molecule-sec)CalculatedfromKdandk1k20:75sec)]TJ /F8 7.97 Tf 6.58 0 Td[(1Chosenforscalek39:310)]TJ /F8 7.97 Tf 6.58 0 Td[(4s)]TJ /F8 7.97 Tf 6.58 0 Td[(1[ 23 ]1:210)]TJ /F8 7.97 Tf 6.58 0 Td[(14cm4/(molecule-sec)[ 28 30 ]110)]TJ /F8 7.97 Tf 6.58 0 Td[(4s)]TJ /F8 7.97 Tf 6.58 0 Td[(1Ontheorderofk3[ 24 ]13000molecules/secpercell[ 1 ]34105molecules/secpercell[ 28 ]Kd=k2=k17:5108molecules/cm2[ 16 ] UsingthevaluesinTable 6-1 andthemethodofspectralbandsinSection 5.1 ,weobtainFigure 6-12 .Itatrstappearedthat,unlikeinthecasewhereallparametersweresettounity,assoonasthesteadystatevalueubecomespositive,positiveeigenvaluesoccur.Howeverexperimentationwithparametervaluesshowedthatinfactthesamebehaviorisexhibited,wesimplyneededtozoominmuchclosertozero.AspectralbandplotinwhichthedestabilizationbehaviorisvisibleisshowninFigure 6-13 .Figure 6-13 showsthattheconstantstationarysolutionbecomesunstablearoundu=0:002.Astheunitsofuareincells/cm2,thisimpliesthatanyrealistichomogeneoussteadystatewouldbeunstable.Forthisreason,wetryadifferentapproachtotheparametervalues.Otherthan1,whichaccordingtonumericalexperimentsdoesnotappeartogreatlyaffectthecriticalvalueatwhichdestabilizationoccurs,theparametervalueforwhichtheestimateisthemostroughis.Thevalueforwasapproximatedfromliteraturethatviewsthechemotacticsensitivityfunctionuaseitherastepfunctioninuoraconstantfunction[ 28 30 ],notalinearfunction,andnoliteraturecouldbefoundonwhatanappropriatelinearchemotacticsensitivityfunctionwouldbe.AsthelinearformofthechemotacticsensitivityfunctionisoneofthemostcommonlystudiedformsoftheKSmodel[ 18 ],itisofinteresttoobtainamoreaccurateapproximatevaluefor. 124

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Figure6-12. SpectralbandsusingparametersfromTable 6-1 Figure6-13. Behaviornearu=0ofspectralbandsusingparametersfromTable 6-1 125

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Figure6-14. Spectralbandsforttedvalue Theinformationwedohaveisthataggregationfailstooccurwhentheconcentrationoftheamoebaislessthan4000-5000cells/cm2[ 7 ].Ifchemotacticfeedbackisindeedwhatdestabilizestheamoebaandinitiatesaggregation,wemayadjustthevalueofsothatthecriticalvalueofuisbetween4000and5000cells/cm2.Usingtheparameterttingconcept,weseethatavalueof=5:410)]TJ /F8 7.97 Tf 6.59 0 Td[(19withallotherparametervaluesasinTable 6-1 givesthevalueucrit4270.UsingthemethodsofSection 5.1 with=5:410)]TJ /F8 7.97 Tf 6.58 0 Td[(19weobtainthespectralbandsplotfromFigure 6-14 .Astheorderofmagnitude10)]TJ /F8 7.97 Tf 6.59 0 Td[(19isbelowmachineprecision,itshouldalsobenotedthatinordertohavecomputablevalues,weconvertedalloftheparametersfortheviequationstounitsof106molecules/cm2insteadofmolecules/cm2.Althoughthisscalingdoesnotaffect,forexample,anyofthediffusioncoefcients,itwillaffectparametervalueswhichinludemoleculesintheunits.Thevaluesthatareaffectedbecomek1=10)]TJ /F8 7.97 Tf 6.59 0 Td[(3cm2=(106molecules-sec),=5:410)]TJ /F8 7.97 Tf 6.59 0 Td[(13cm4=(106molecules-sec),1=0:003cm4=(106molecules-sec),and3=0:4cm4=(106molecules-sec)afterthescaling.Aswouldbeexpected,thisparameterrescalingdidnotaffectthecomputationsinanywayotherthanscalingofthesolutions. 126

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Figure6-15. Thelinearizedeigenfunctioncorrespondingtothelargestpositiveeigenvalue WecannowusethemethodsofSection 5.2 tondeigenfunctionscorrespondingtothepositiveeigenvaluesandthereforevisualizepotentialpatternsformedjustafterdestabilizationbychemotaxis.Wefoundtheeigenfunctionsoftheeigenproblemlinearizedabouttheconstantstationarysolutioncorrespondingtou=4500molecules/cm2.Thelargestpositiveeigenvaluewas1:1310)]TJ /F8 7.97 Tf 6.59 0 Td[(7,andtheeigenfunctionsforuandallvicorrespondingtothiseigenvaluecanbeseeninFigure 6-15 .Inthisparameterrange,unlikeinthecasewhereallparametersweresettounity,theshapeoftheeigenfunctionforv1isexactlytheoppositeoftheshapesoftheotherthreeeigenfunctions.Inotherwords,theeigenfunctionforv1haspeakswhentheotherthreeeigenfunctionshavevalleys,andvalleyswhentheothershavepeaks.Biologicallythechemicalv1reducestheamountofchemoattractantintheenvironment,sothismirroringbehaviormakessense.OthereigenfunctionscorrespondingtopositiveeigenvaluescanbeseeninFigure 6-16 .Theseeigenfunctionsalldemonstratethemirroringbehavioroftheeigenfunctioncorrespondingtothedominanteigenvalue. 127

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Figure6-16. Othereigenfunctionscorrespondingtopositiveeigenvalues WeconcludebyemphasizingthatinthecaseofthefullKSmodelwehavejustshownthatparametervaluescanqualitativelychangethenatureofthestationarysolutions.Anopenquestionforfutureresearchisiftimesimulationsalsodifferinthiscase.AssolutionstothefourequationKSmodeldidtendtoconvergetopositiveeigenfunctions,itislikelythatparametervaluesaffectthenatureoftimedependentsolutionsaswell.So,althoughbehavioroftheminimalKSsystemandthefullKSsystemisqualitativelysimilarforthecasewhereallconstantsaresettounity,theresultsofthisnalsectionshowthatfurtherinvestigationisneededtoshowthatinallcasesthebehaviorofsolutionstothetwosystemsissimilar. 128

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BIOGRAPHICALSKETCH EricaZuhrwasbornandraisedinOakRidge,Tennessee.ShegraduatedfromOakRidgeHighSchoolin2003,andwentontoattendtheUniversityofNorthCarolinaatChapelHillandmajorinmathematics.ShegraduatedfromtheUniversityofNorthCarolinawithhonorsin2007,andatthattimewasawardedtheArchibaldHendersonPrizeinMathematics.ShewasthenofferedfullsupportintheformofanAlumniFellowshiptopursueadoctoraldegreeinmathematicsattheUniversityofFlorida. 132