Examples of Reaction-Diffusion Equations in Biological Systems

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Examples of Reaction-Diffusion Equations in Biological Systems Marine Protected Areas and Quorum Sensing
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Inman, Jessica
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Doctorate ( Ph.D.)
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University of Florida
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Mathematics
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De Leenheer, Patrick
Committee Co-Chair:
Hagen, Stephen James
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Boyland, Philip Lewis
Pilyugin, Sergei S
Zhang, Lei
Osenberg, Craig Warren

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diffusion -- model -- mpa -- quorum -- reaction -- sensing
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Reaction-diffusion models are widely used to describe physical phenomena, with applications as varied as epidemic spread and self-regulated pattern formation in animal embryos. In this document, we present three reaction-diffusion models. The first is a model of fish movement into and out of a marine protected area (MPA), an area of coastline wherein fishing is restricted or prohibited. MPAs are promoted as a tool to protect over-fished stocks and increase fishery yields. Previous models suggested that adult mobility modified effects of MPAs by reducing densities of fish inside reserves, but increasing yields (i.e. increasing densities outside of MPAs). Empirical studies contradicted this prediction: as mobility increased, the relative density of fishes inside MPAs (relative to outside) increased or stayed constant. To attempt to explain these empirical results through modeling, we examined the effects of differential movement inside versus outside the reserve as well as the effects of a movement bias at the boundary of the reserve. We found that differential movement could not explain empirical findings, but a movement bias model could. The second and third models describe quorum sensing systems. In a quorum sensing system, bacteria synthesize small diffusible chemicals called autoinducers. Once a critical concentration of autoinducer is reached, the bacterial colony undergoes a shift in gene expression. The second model describes a colony of genetically modified bacteria that respond to but cannot produce autoinducer. Our model contains a minimal set of components necessary to describe experimentally observed patterns of cell response to a diffusing autoinducer signal in a spatially extended system. Our model incorporates diffusion of the signal, logistic growth of the bacteria and a cooperative (Hill function) response to the signal. We observe and predict cell response to the diffusing signal over distances of ~1 cm on time scales of ~10 h. Our model and experiments display patterns that are qualitatively dissimilar from simple diffusion: the observed response is surprisingly insensitive to the distance the signal has traveled. The third model describes an intact quorum sensing system in the bacterium Aliivibrio fischeri. Our model describes only the autoinducer signal concentration and the autoinducer synthase concentration, and incorporates diffusion of the autoinducer signal and auto-feedback in the production of this signal. This model is able to describe a quorum sensing shift in gene expression on the colony level, which appears as a traveling wave. We give a proof of the existence of a traveling wave solution to a class of models that includes our quorum sensing model. We also use the conditions of this theorem to determine parameter ranges over which our quorum sensing model admits a traveling wave solution.
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by Jessica Inman.
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Thesis (Ph.D.)--University of Florida, 2013.
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EXAMPLESOFREACTION-DIFFUSIONEQUATIONSINBIOLOGICALSYSTEMS:MARINEPROTECTEDAREASANDQUORUMSENSINGByJESSICALANGEBRAKEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013JessicaLangebrake 2

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ForMatt 3

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ACKNOWLEDGMENTS Iwouldliketothankmyadvisor,Dr.DeLeenheer,andmyco-advisor,Dr.Hagen,fortheirguidanceandendlesspatience.IwouldliketothankDr.Osenbergforgivingmeperspectiveontheapplicationofmathematicstobiology.IwouldliketothankGabeDilanjiforbeingagreatcollaborator;formaintaininghisexcitementthroughallthosehourshespentwithmeinthelabandforhelpingmeseethebeautyinherentinquorumsensing.Iwouldalsoliketothankmyfriendsandfamilyfortheirsupport,especiallymybrother,Chris,sister-in-law,Heather,andparents,BethandLarry.Lastly,Iwouldliketothankmyhusband,Matt,forhisconstantencouragementandunderstanding. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 10 CHAPTER 1BACKGROUND ................................... 12 1.1Reaction-DiffusionModelsinBiologicalSystems .............. 12 1.2MathematicalPreliminaries .......................... 12 1.2.1LinearAlgebra ............................. 12 1.2.2ComplexAnalysis ............................ 13 1.2.3Analysis ................................. 14 1.2.4OrdinaryDifferentialEquations .................... 16 2DIFFERENTIALMOVEMENTANDMOVEMENTBIASMODELSFORMARINEPROTECTEDAREAS ................................ 19 2.1Introduction ................................... 19 2.2Model ...................................... 22 2.3StabilityoftheSteadyStateSolution ..................... 28 2.4QualitativeAnalysisoftheSteadyStateSolution .............. 29 2.5AMovementBiasModel ............................ 31 2.6ConstantDiffusionandSmoothMortalityRate ............... 34 2.7Discussion ................................... 37 3QUORUMSENSINGBACKGROUND ....................... 41 3.1QuorumSensing ................................ 41 3.2LuxR-LuxISystem ............................... 42 3.3PreviousModels ................................ 43 4ASPATIALLYEXPLICITQUORUMSENSINGMODEL .............. 46 4.1ExperimentalConguration .......................... 49 4.2MathematicalModel .............................. 50 4.3ParameterEstimation ............................. 54 4.4ResultsofLaneExperiments ......................... 58 4.4.1DiffusionofaDye ............................ 58 4.4.2LuxR-LuxISystemResponsetoAHLDiffusion ........... 59 4.5DiscussionandModelSimulations ...................... 62 4.6Methods ..................................... 65 5

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4.6.1Bacterialcultures ............................ 65 4.6.2Well-platemeasurements ....................... 66 4.6.3Laneapparatusandimaging ..................... 67 5SIGNALPROPAGATIONINAQUORUMSENSINGSYSTEM .......... 69 5.1MathematicalModel .............................. 70 5.2TravelingWaveSolutionof( 5 ),( 5 ) ................... 72 5.3TheExistenceofaTravelingWaveSolutiontoaClassofReaction-DiffusionSystems ..................................... 77 5.3.1Preliminaries .............................. 77 5.3.2TheWaveSpeedc ........................... 80 5.3.3Ra20F(U,VG(U))dU=0 ........................ 83 5.3.4Ra20F(U,VG(U))dU>0 ........................ 86 5.3.4.1TheSetsP1andP2 ..................... 91 5.3.4.2P16=; ............................. 92 5.3.4.3P26=; ............................. 98 5.3.4.4P1andP2areOpenandDisjoint .............. 106 5.3.4.5TheExistenceofaHeteroclinicConnection ........ 108 5.3.4.6TheExistenceofaTravelingWave ............. 111 5.3.5Ra20F(U,VG(U))dU<0 ........................ 112 5.3.6StatementofExistenceTheorem ................... 113 6FUTUREWORK ................................... 114 APPENDIX APROOFOFTHEOREM2.2 ............................. 116 BSKETCHOFTHEPROOFOFTHEOREM2.4 .................. 121 CCONTINUITYOFTHESTABLEMANIFOLDWITHRESPECTTOPARAMETERS 123 C.1Assumptions .................................. 123 C.2Outline ...................................... 123 C.3FixtheDimensionofStableManifold ..................... 125 C.4P(c)VariesContinuouslywithRespecttoc ................. 126 C.5ConstructionoftheStableManifold ...................... 129 C.5.1Preliminaries .............................. 129 C.5.2ExistenceoftheStableManifold ................... 130 C.5.2.1:Cb!C .......................... 131 C.5.2.2:Cb!Cb .......................... 135 C.5.2.3isaContraction ...................... 136 C.5.2.4ExistenceoftheStableManifold .............. 136 C.6(t,a,c)isContinuouswithRespecttoc .................. 139 C.7TheStableManifoldisContinuouswithRespecttoc ............ 147 REFERENCES ....................................... 148 6

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BIOGRAPHICALSKETCH ................................ 156 7

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LISTOFTABLES Table page 4-1Asummaryofvariablesandparametersusedinthemodel( 4 )-( 4 ). ... 55 5-1Asummaryofvariablesandparametersusedinthemodel( 5 ),( 5 ). .... 72 8

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LISTOFFIGURES Figure page 2-1MPAsdistributedevenlyalonganinnitecoastline. ................ 22 2-2Simulationofmodel( 2 ) .............................. 24 2-3Plotsofbiologicallysignicantquantities ...................... 31 2-4Plotsofbias ...................................... 33 2-5Plotsofbiologicallysignicantquantitiesforstrongbias ............. 35 3-1AliivibrioscheriMJ11LuxR-LuxIdiagram .................... 43 4-1Chapter 4 methodology ............................... 47 4-2Experimentalcongurationforlaneexperiments ................. 48 4-3Experimentaldatafromwell-plateandtstomodel( 4 )-( 4 ) ........ 56 4-4Diffusionofuoresceindyeinagarlane ...................... 59 4-5Responseofthesensorstrain(E.coli+pJBA132)todiffusingAHL ....... 60 4-6BioluminescenceresponseofluxI-decientA.scheriVCW267todiffusingAHL .......................................... 61 4-7PatternsofexpressionpredictedfortheE.coli+pJBA132sensorstraininresponsetodiffusingAHL .............................. 63 5-1Nullclinesofsystem( 5 ),( 5 ) .......................... 73 5-2PlotsofVF(u)andVG(u) .............................. 81 5-3Theregion. ..................................... 91 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyEXAMPLESOFREACTION-DIFFUSIONEQUATIONSINBIOLOGICALSYSTEMS:MARINEPROTECTEDAREASANDQUORUMSENSINGByJessicaLangebrakeAugust2013Chair:PatrickDeLeenheerCochair:StephenJ.HagenMajor:MathematicsReaction-diffusionmodelsarewidelyusedtodescribephysicalphenomena,withapplicationsasvariedasepidemicspreadandself-regulatedpatternformationinanimalembryos.Inthisdocument,wepresentthreereaction-diffusionmodels.Therstisamodelofshmovementintoandoutofamarineprotectedarea(MPA),anareaofcoastlinewhereinshingisrestrictedorprohibited.MPAsarepromotedasatooltoprotectover-shedstocksandincreasesheryyields.PreviousmodelssuggestedthatadultmobilitymodiedeffectsofMPAsbyreducingdensitiesofshinsidereserves,butincreasingyields(i.e.increasingdensitiesoutsideofMPAs).Empiricalstudiescontradictedthisprediction:asmobilityincreased,therelativedensityofshesinsideMPAs(relativetooutside)increasedorstayedconstant.Toattempttoexplaintheseempiricalresultsthroughmodeling,weexaminedtheeffectsofdifferentialmovementinsideversusoutsidethereserveaswellastheeffectsofamovementbiasattheboundaryofthereserve.Wefoundthatdifferentialmovementcouldnotexplainempiricalndings,butamovementbiasmodelcould.Thesecondandthirdmodelsdescribequorumsensingsystems.Inaquorumsensingsystem,bacteriasynthesizesmalldiffusiblechemicalscalledautoinducers.Onceacriticalconcentrationofautoinducerisreached,thebacterialcolonyundergoesashiftingeneexpression. 10

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Thesecondmodeldescribesacolonyofgeneticallymodiedbacteriathatrespondtobutcannotproduceautoinducer.Ourmodelcontainsaminimalsetofcomponentsnecessarytodescribeexperimentallyobservedpatternsofcellresponsetoadiffusingautoinducersignalinaspatiallyextendedsystem.Ourmodelincorporatesdiffusionofthesignal,logisticgrowthofthebacteriaandacooperative(Hillfunction)responsetothesignal.Weobserveandpredictcellresponsetothediffusingsignaloverdistancesof1cmontimescalesof10h.Ourmodelandexperimentsdisplaypatternsthatarequalitativelydissimilarfromsimplediffusion:theobservedresponseissurprisinglyinsensitivetothedistancethesignalhastraveled.ThethirdmodeldescribesanintactquorumsensingsysteminthebacteriumAliivibrioscheri.Ourmodeldescribesonlytheautoinducersignalconcentrationandtheautoinducersynthaseconcentration,andincorporatesdiffusionoftheautoinducersignalandauto-feedbackintheproductionofthissignal.Thismodelisabletodescribeaquorumsensingshiftingeneexpressiononthecolonylevel,whichappearsasatravelingwave.Wegiveaproofoftheexistenceofatravelingwavesolutiontoaclassofmodelsthatincludesourquorumsensingmodel.Wealsousetheconditionsofthistheoremtodetermineparameterrangesoverwhichourquorumsensingmodeladmitsatravelingwavesolution. 11

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CHAPTER1BACKGROUND 1.1Reaction-DiffusionModelsinBiologicalSystemsAreaction-diffusionequationisanequationoftheformdu dt=Dd2u dx2+f(u)whereDisthediffusionconstant,x2Rrepresentspositionandt2Rrepresentstime.Dd2u dx2isthediffusiontermandf(u)isthereactionterm.u(x,t)isacontinuousfunctionthatrepresentsaquantityofinterest,suchaschemicalconcentrationorpopulationdensity.Scientistsusesystemsofreaction-diffusionequationstodescribeaplethoraofbiologicalsystems.Modelsexisttodescribetumorgrowth[ 32 ],epidemicspread[ 65 ],animaldispersal[ 64 ],andevenself-regulatedpatternformationinananimalembryo.[ 52 ]Inthistext,wepresentthreereaction-diffusionmodels.Therst,appearinginChapter 2 ,isamodeldescribingthemovementofshintoandoutofamarineprotectedarea,anareaofcoastlinewhereshingisrestrictedorprohibited.Chapters 4 and 5 containreaction-diffusionmodelsthatdescribequorumsensingsystems.Inaquorumsensingsystem,bacteriasynthesizesmalldiffusiblechemicalscalledautoinducers.Onceacriticalconcentrationofautoinducerisreached,thebacterialcolonyundergoesashiftingeneexpression.ThemodelpresentedinChapter 5 describesanintactquorumsensingsystem,whilethemodelinChapter 4 describesacolonyofgeneticallymodiedbacteriathatrespondtobutcannotproduceautoinducer. 1.2MathematicalPreliminariesHere,wegiveseveraldenitions,lemmasandtheoremsthatwillbeusedlaterinthetext. 1.2.1LinearAlgebraThefollowingtheoremcanbefoundinFriedbergetal.(2003).[ 34 ] 12

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Theorem1.1(PrimaryDecompositionTheorem). LetTbealinearoperatoronann)]TJ /F10 11.955 Tf 9.3 0 Td[(dimensionalvectorspaceVwithcharacteristicpolynomialf(t)=()]TJ /F8 11.955 Tf 9.3 0 Td[(1)n(1(t))n1(2(t))n2(k(t))nk,wherethei(t)'s(1ik)aredistinctirreduciblemonicpolynomialsandtheni'sarepositiveintegers.ThenV=K1K2KkwhereKi=fx2Vj(i(T))p(x)=0forsomepositiveintegerpg(1ik). 1.2.2ComplexAnalysisThefollowingdenitionsandtheoremcanbefoundinBrownandChurchill(2009).[ 7 ] Denition1. Acontour,orpiecewisesmootharc,isanarcconsistingofanitenumberofsmootharcsjoinedendtoend.Ifz=z(t),atb,representsacontour,z(t)iscontinuousandz0(t)ispiecewisecontinuous.Whenonlytheinitialandnalvaluesofz(t)arethesame,acontourCiscalledasimpleclosedcontour.Suchacurveispositivelyorientedwhenitisinthecounterclockwisedirection. Denition2. Afunctionfofthecomplexvariablezisanalyticatapointz0ifithasaderivativeateachpointinsomeneighborhoodofz0. Denition3. Apointz0iscalledasingularpointofafunctionfifffailstobeanalyticatz0butisanalyticatsomepointineveryneighborhoodofz0.Asingularpointz0issaidtobeisolatedif,inaddition,thereisadeletedneighborhood0
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Denition4. Whenz0isanisolatedsingularpointofafunctionf,thereisapositivenumberR2suchthatfisanalyticateachpointzforwhich0
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Theorem1.3(ImplicitFunctionTheorem). Ifx=(x1,x2,...,xn)2Rnandy=(y1,y2,...,ym)2Rm,let(x,y)denotethepoint(orvector)(x1,x2,...,xn,y1,y2,...,ym)2Rn+m.Letf(x,y)beaC1mappingofanopensetERn+mintoRn,suchthatf(x0,y0)=0forsomepoint(x0,y0)2E.AssumethatDfxisinvertible.ThenthereexistopensetsURn+mandWRmwith(x0,y0)2Uandy02Whavingthefollowingproperty:Toeveryy2Wcorrespondsauniquexsuchthat(x,y)2Uandf(x,y)=0.Ifthisfisdenedtobeg(y),thengisaC1mappingofWintoRn,g(y0)=x0,f(g(y),y)=0(y2W),andDg(y0)=)]TJ /F8 11.955 Tf 9.3 0 Td[((Dfx))]TJ /F7 7.97 Tf 6.59 0 Td[(1Dfy. Theorem1.4(DominatedConvergenceTheorem). Letbeameasure,Ebeameasur-ablesetandffngbeasequenceofmeasurablefunctionssuchthatfn(x)!f(x)foreachx2Easn!1.IfthereexistsafunctiongthatisintegrableonEsuchthatjfn(x)jg(x)forallnandforeachx2E,thenfisintegrableandlimn!1ZEfnd=ZEfd.ThefollowinglemmacanbefoundinLogemannandRyan(2004).[ 56 ] 15

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Lemma1(Barbalat'sLemma). Supposef(t)2C1(a,1)andlimt!1f(t)=where2R.Iff0isuniformlycontinuous,thenlimt!1f0(t)=0. 1.2.4OrdinaryDifferentialEquationsThefollowingdenitions,lemmasandtheoremscanbefoundinChicone(2006).[ 13 ] Lemma2. IfAisannnmatrix,thenetAisamatrixwhosecomponentsare(nite)sumsoftermsoftheformp(t)etsintandp(t)etcostwhereandarerealnumberssuchthat+iisaneigenvalueofA,andp(t)isapolynomialofdegreeatmostn)]TJ /F8 11.955 Tf 11.96 0 Td[(1. Theorem1.5. SupposethatAisannn(real)matrix.Thefollowingstatementsareequivalent. (1) ThereisanormkkaonRnandarealnumber>0suchthatforallv2Rnandallt0,etAvae)]TJ /F14 7.97 Tf 6.59 0 Td[(tkvka. (2) IfkkgisanarbitrarynormonRn,thenthereisaconstantC>0andarealnumber>0suchthatforallv2Rnandallt0,etAvgCe)]TJ /F14 7.97 Tf 6.59 0 Td[(tkvkg. (3) EveryeigenvalueofAhasnegativerealpart.Moreover,if)]TJ /F9 11.955 Tf 9.3 0 Td[(exceedsthelargestofalltherealpartsoftheeigenvaluesofA,thencanbetakentobethedecayconstantin( 1 )or( 2 ).Inthefollowingtwodenitionsandtheorem,let(X,kk)beanormedvectorspaceanddenetheinducedmetricdonXbyd(x,y)=kx)]TJ /F3 11.955 Tf 11.95 0 Td[(yk. Denition5. Apointx02Xisaxedpointofafunction:X!Xif(x0)=x0. 16

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Denition6. Supposethat:X!X,andisarealnumbersuchthat0<1.Thefunctioniscalledacontractionwithcontractionconstantifd((x),(y))d(x,y)forallx,y2X. Theorem1.6(ContractionMappingTheorem). Ifthefunctionisacontractiononthecompletemetricspace(X,d),thenhasauniquexedpointx2X. Theorem1.7(Gronwall'sInequality). Supposethata
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Denition8. SupposethattisaowonRnandp2Rn.ApointxinRniscalledanomegalimitpoint(!-limitpoint)oftheorbitthroughpifthereisasequenceofnumberst1t2t3suchthatlimi!1ti=1andlimi!1ti(p)=x.Thecollectionofallsuchomegalimitpointsisdenoted!(p)andiscalledtheomegalimitset(!-limitset)ofp. Lemma4. Consider_x=f(x),x2Rn.Letx(t)beasolutionsuchthatlimt!1x(t)=x.Thenf(x)=0. Proof. Sincelimt!1x(t)=x,fxgisthe!-limitsetofx.Since!-limitsetsareinvariantundertheowandthis!-limitsetconsistsofasinglepoint,wehavethatf(x)=0. Lemma5. Ifthemapf:J!Rninthedifferentialequation_x=f(t,x,)iscontinuouslydifferentiable,t02JR,x02Rn,and02Rm,thenthereareopensetsJ0J,0and0suchthat(t0,x0,0)2J000,andauniqueC1function:J000!Rngivenby(t,x,)!(t,x,)suchthatt7!(t,x,)isasolutionofthedifferentialequation_x=f(t,x,)and(0,x,)=x.Inparticular,t7!(t,x0,0)isasolutionoftheinitialvalueproblem_x=f(t,x,0),x(t0)=x0. 18

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CHAPTER2DIFFERENTIALMOVEMENTANDMOVEMENTBIASMODELSFORMARINEPROTECTEDAREAS 2.1Introduction1Overshinghasreducedmarineshstocksanddegradedhabitats[ 79 80 ].Asaconsequence,sheriesmanagementhasbecomeamajoreconomicandenvironmentalchallenge.Marinereserves(ormarineprotectedareas,MPAs)arefrequentlyadvocatedasanefcientmanagementtooltorestorehabitatsandprotectover-harvestedstocks[ 15 41 44 79 80 ].MPAsoffertwopotentialbenets.First,theycanlocallyincreasethedensitiesofharvestedspecies[ 16 41 ],butsee[ 71 ].Secondly,theycanincreaseshingyieldsoutsideofthemarinereserveviaspilloverand/orlarvalexport[ 39 76 79 ](spilloverisdenedasthenetmovementofadultshfromthereservetotheshinggrounds,whichresultsinabiomassexport).DespitetheevidencesupportinglocalbenetsofMPAs,uncertaintiesremain[ 44 71 79 ].Forexample,theoreticalstudieshavesuggestedthatthelocaleffectivenessofanMPAdecreasesasadultmobilityincreases[ 38 59 63 73 95 ].Empiricaldatadonotsupportthistheoreticalexpectation.Forexample,inarecentmeta-analysisofMediterraneanMPAs,Claudetetal.(2010)[ 16 ]calculatedtherelativedensitiesofshinsidevs.outsideMPAs,andcomparedtheresultsforspecieswithlow,mediumorhighadultmobility.Contrarytothetheoreticalexpectation,theyfoundthatmoremobilespeciesshowedgreaterincreasesindensityinsideofMPAs(relativetooutside).Toexplaintheirsurprisingresults,Claudetetal.suggestedthatmobilespeciescouldbenetmorefromMPAsthanexpectediftheybiasedtheirmovementinfavorofthe 1ReproducedwithpermissionfromJ.Langebrake,L.Riotte-Lambert,C.W.Osenberg,andP.DeLeenheer.Differentialmovementandmovementbiasmodelsformarineprotectedareas.J.Math.Biol.,64(4):667696,2012. 19

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reserve.SuchabiascouldresultiftheMPAalteredhabitatavailabilityorquality[ 80 ]andthetargetspeciespreferredthismodication.Incontrasttotheexpectednegativerelationshipbetweenincreasedlocaleffects(onegoalofMPAs)andmobility,modelsgenerallyindicatethatshingyields(thesecondgoal)shouldincreasewithshmobilityasaresultofincreasedspillover.[ 38 54 63 ]Therearenoavailableempiricaldatatoevaluatewhetherthisexpectationalsoiscontradicted.TheconictbetweenempiricalandtheoreticalpredictionsabouttherelationshipbetweenmobilityandlocaleffectsofMPAs,aswellastheimportanceofspilloverforproducingincreasedsheriesyields,suggeststhatweneedtoexaminetheeffectsofmobilityinnewways.Themainpurposeofthischapteristoproposeseveralmodelsthatcouldreconcilemodelpredictionsandempiricalresults.WewillstartbyintroducingamodelthatexamineshowdifferentialmovementinsideversusoutsidetheMPAcanaffecttheefcacyofMPAs.Todatetherehavebeenonlylimitedstudiesofthisphenomenon.Forexample,Rodwelletal.(2003)[ 77 ]developedatwopatchmodelwhereadultmovementwasdescribedbyanannualtransferfromthemostpopulatedpatchtotheother,i.e.fromthereservetotheshinggrounds.Wemodelshmovementasadiffusionprocessandassumethatthediffusionparameterissmallerinsidethereservethanoutside.Mathematically,themodelisaboundaryvalueproblemwithpiecewiseconstantparametersindifferentspatialregions.Oneachregionthesteadystateequationislinearsothatitcanbesolvedexplicitly.Thesolutionsneedtobematchedattheinterfaceoftheregions,atechniquethatiswell-known,seeforinstance[ 8 81 ].ThenweinvestigatehowdifferentialdiffusionaffectstheexpectedbenetsofMPAs.Wefocusonfourmeasures:abundanceofshintheshinggrounds(i.e.theamountofshintheshedarea),totalabundance(i.e.theamountofshcontainedintheMPAandshedareacombined),thelocaleffect(i.e.,logoftheratioofthedensityinsidevs.outsideoftheMPA,whichisacommonmeasureoftheeffectof 20

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anMPA),andsheriesyield(i.e.,theamountofshcaughtbyshersperunitoftime).Wewillshowthatifthetwodiffusionparametersarescaledasmobilityincreases,yettheirratioremainsconstant,themeasuresvaryinawaythatisinaccordancewiththetheoreticalpredictionsfromtraditionalmodels.NextweintroduceamodelthatincorporatesamovementbiastowardstheMPAthatislocalizedtotheMPAboundary.Itarisesasthelimitofarandomwalkmodelwheretherandomwalkisonlytrulyrandomiftherandomwalkerisnotlocatedontheboundary,butbiasedifheis.Weshowthatthereisacriticalvalueforthebiasparameterthatcontrolsthedependenceofthefourmeasuresonincreasedmobility.Forsmallbias,theresultsareinlinewithwhattraditionalmodelspredict,butforlargebiasvalues,onceagain,weareabletoreconciletheoryanddata.Theresultsforthismodelonlydependonthebiasvalue,andtheyremainvalidwhetherornotweassumedifferentialdiffusioninsideandoutsidetheMPA.Finally,weproposeasimpliedmodelwithhomogeneousdiffusioneverywhere,butwithsmooth-asopposedtopiecewiseconstant-mortalityrates.Weshowthatoncemore,itispossibletounitedataandtheory,atleastonthelevelofoneofourmeasures,namelytheabundanceofshintheshinggrounds.Ourresultssuggestthatexplanationsofdatadependontheunderlyingmodelassumptionsinaverysubtleway.Theyseemtoindicatethatvariousexplanationsarepossibleandthatfurtherresearchisrequiredtoelucidatethisproblem.Therestofthischapterisorganizedasfollows.InSection 2.2 wepresentourmodelandshowthatithasauniquesteadystate.WeexaminethestabilityofthesteadystateinSection 2.3 .Section 2.4 introducesvariousmeasuresthatquantifytheeffectoftheMPA,andweinvestigatehowincreasedmobilityaffectsthesemeasures.InSection 2.5 weinvestigateamovementbiasmodel,andinSection 2.6 weconsideramodelwithhomogeneousdiffusionandsmoothmortality.WeconcludeourchapterinSection 2.7 withadiscussion.ProofsoftwoofourresultsareinAppendices A and B 21

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2.2ModelMarineProtectedAreas(MPAs)areportionsofcoastlineinwhichshingisrestrictedordisallowed.IthasbeentheorizedthatestablishingMPAsperiodicallyalongacoastlinewillincreasetheoverallpopulationofshaswellasincreasetotalshingyield.Inthischapter,weexaminethecasewhereMPAsaredistributedevenlyandperiodicallyalongastraightcoastline.Thecoastlinecanthereforebesplitupintoseveral(or,infact,innitelymany)identicalsections,eachcontaininganMPAsurroundedbyunprotectedwaters,calledshinggrounds.Toexaminethissituation,weallowonesectionofcoastlinetoberepresentedbytheinterval[)]TJ /F8 11.955 Tf 9.3 0 Td[(1+2k,1+2k]andtheMPAtobe[)]TJ /F3 11.955 Tf 9.3 0 Td[(l+2k,l+2k],where0
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additionalsharebeingremovedbyshermenoutsidetheMPA.Thus,weletthemortalityrateinsidetheMPAbeiandthemortalityrateoutsidetheMPAbeowherei><>>:Dox2()]TJ /F8 11.955 Tf 9.29 0 Td[(1+2k,)]TJ /F3 11.955 Tf 9.29 0 Td[(l+2k)[(l+2k,1+2k)Dix2()]TJ /F3 11.955 Tf 9.3 0 Td[(l+2k,l+2k)and(x)=8>><>>:ox2()]TJ /F8 11.955 Tf 9.3 0 Td[(1+2k,)]TJ /F3 11.955 Tf 9.3 0 Td[(l+2k)[(l+2k,1+2k)ix2()]TJ /F3 11.955 Tf 9.3 0 Td[(l+2k,l+2k)UsingMATLAB,wecanplotthesolutiontothissystemastimeprogresses.InFigure 2-2 ,wecanseetheprogressionofthesystem(graphedontheinterval[0,1],forreasonsthatwillbecomeclearbelow)throughtimeandseetheplotsbecomemoreandmoresimilartothesteadystatesolutioncalculatedlaterinthissection.Weareinterestedinndingsteadystatesolutionsto( 2 ).Steadystatesolutionsarefunctionsn(x)thatareindependentoft,non-negative,continuousandthatsatisfy0=(D(x)n0)0+R)]TJ /F9 11.955 Tf 11.95 0 Td[((x)n,wheretheprime0standsford=dx.Weadditionallyrequirethatn(x)havecontinuousuxandbeperiodic.Thatis,theux)]TJ /F3 11.955 Tf 9.3 0 Td[(D(x)n0(x)mustbecontinuousand n(x+2)=n(x)forallx2R.(2) 23

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At=0.1 Bt=0.5 Ct=2 Dt=5 Et=10 FSteadyStateSolutionFigure2-2. Simulationofmodel( 2 ).TheabovegraphswereproducedinMATLABusingtheparametersl=3=16km,Do=2km2 year,Di=0.04km2 year,R=0.5thousandsofsh (year)(km),i=0.251 year,o=0.51 yearandInitialConditionn(x,0)=1thousandsofsh kmforallx. NotethatsinceD(x)isdiscontinuousattheMPAboundaries,requiringcontinuousuximpliesthatn0(x)mustalsobediscontinuousthere.Werestrictoursearchforsteadystatesolutionstoonlythosethataresymmetricwithrespecttox=0,thatis,functionsn(x)suchthat n(x)=n()]TJ /F3 11.955 Tf 9.3 0 Td[(x)forallxnotontheMPAboundaries.(2)Theserequirementscreateadditionalconditionsthatn(x)mustsatisfy.Inordertohavecontinuousdensityn(x)andcontinuousux)]TJ /F3 11.955 Tf 9.3 0 Td[(D(x)n0(x),wemustforcetheleft-andright-handlimitsofthesefunctionstomatchattheboundariesbetweentheMPAandunprotectedwaters.Thus,wehavethefollowingmatchingconditions: n)]TJ /F8 11.955 Tf 7.09 1.8 Td[((l+2k)=n+(l+2k),n)]TJ /F8 11.955 Tf 7.09 1.8 Td[(()]TJ /F3 11.955 Tf 9.3 0 Td[(l+2k)=n+()]TJ /F3 11.955 Tf 9.3 0 Td[(l+2k) (2) 24

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Din0)]TJ /F8 11.955 Tf 7.09 2.95 Td[((l+2k)=Don0+(l+2k),Don0)]TJ /F8 11.955 Tf 7.08 2.95 Td[(()]TJ /F3 11.955 Tf 9.29 0 Td[(l+2k)=Din0+()]TJ /F3 11.955 Tf 9.3 0 Td[(l+2k)foreveryk2Z.Here,thesubscripts)]TJ /F1 11.955 Tf 12.63 0 Td[(and+indicatetheleftandrightlimitrespectively.Wewillshowthattheproblemcanbesubstantiallysimplied.InsteadofsolvingthesteadystateequationonR,itwillsufcetosolvetheproblemon[0,1]withNeumannboundaryconditions.Toseethis,noterstthattakingthederivativewithrespecttoxin( 2 )yieldsthat: n0(x)=)]TJ /F3 11.955 Tf 9.3 0 Td[(n0()]TJ /F3 11.955 Tf 9.3 0 Td[(x)forallxnotontheMPAboundaries.(2)Inparticular,settingx=0impliesthat: n0(0)=0(2)Similarly,takingderivativesin( 2 )andsettingx=)]TJ /F8 11.955 Tf 9.29 0 Td[(1showsthatn0(1)=n0()]TJ /F8 11.955 Tf 9.3 0 Td[(1)Buttogetherwith( 2 ),evaluatedatx=1,thisimpliesthat n0(1)=0(2)Thus,foreverysolutiontooursteadystateproblem,thereisnouxinthepointsx=0andx=1.Let'sassumefornow(wewillactuallyprovethisbelow)thatwecanndanon-negativefunctionn(x)satisfying: 0=(D(x)n0)0+R)]TJ /F9 11.955 Tf 11.96 0 Td[((x)n,x2[0,1](2)thatiscontinuousin[0,1],differentiablein[0,1],exceptperhapsinx=l,wherethefollowingmatchingconditionshold: n)]TJ /F8 11.955 Tf 7.09 1.8 Td[((l)=n+(l)andDin0)]TJ /F8 11.955 Tf 7.09 2.96 Td[((l)=Don0+(l)(2) 25

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andwithNeumannboundaryconditions n0(0)=n0(1)=0(2)Thenitisnothardtoseethatthefunctionn(x)canbeextendedtoRandthattheresultingextensionisasolutiontoouroriginalsteadystateproblemthatsatisesalltheconstraintsweimposed.Indeed,rstweextendthefunctionn(x)denedon[0,1]to[)]TJ /F8 11.955 Tf 9.3 0 Td[(1,+1]bydeningn()]TJ /F3 11.955 Tf 9.3 0 Td[(x)=n(x).Itiseasilyveriedthatthisextensionsatisesthesteadystateequationon[)]TJ /F8 11.955 Tf 9.3 0 Td[(1,0].Also,bytheverydenitionofthisextension,itautomaticallysatisesthesymmetryconstraint( 2 )ontheinterval[)]TJ /F8 11.955 Tf 9.3 0 Td[(1,1],andthematchingconditions( 2 )atx=)]TJ /F3 11.955 Tf 9.3 0 Td[(l.Secondly,weextendthisextendedfunctionn(x),whichisnowdenedon[)]TJ /F8 11.955 Tf 9.3 0 Td[(1,1],periodicallytoR,bydening:n(x+2k)=n(x),forallk2Z.Itiseasilyveriedthattheresultingextensionisasolutiontoouroriginalproblem.Whatremainstobeprovedisthefollowing: Theorem2.1. Theboundary-valueproblem( 2 )with( 2 )and( 2 )hasauniquenon-negativesolutionn(x)whichiscontinuousin[0,1]andcontinuouslydifferentiablein[0,1]nflg. Proof. Solvingtheequationon[0,l)and(l,1]andusingtheNeumannboundaryconditions( 2 ),wendthat: n(x)=8>><>>:ccosh(ix)+R i,x2[0,l)dcosh(o(x)]TJ /F8 11.955 Tf 11.96 0 Td[(1))+R o,x2(l,1],(2) 26

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wherecanddareconstantsdeterminedbelow,andwherewehaveintroducedthefollowingpositiveparameters: i=r i Di,o=r o Do.(2)Tondcanddweusethematchingcondition( 2 ):ccosh(il)+R i=dcosh(o(1)]TJ /F3 11.955 Tf 11.95 0 Td[(l))+R ocDiisinh(il)=)]TJ /F3 11.955 Tf 9.3 0 Td[(dDoosinh(o(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l)),or,usingmatrixnotation:0B@cosh(il))]TJ /F8 11.955 Tf 11.29 0 Td[(cosh(o(1)]TJ /F3 11.955 Tf 11.95 0 Td[(l))sinh(il)Doo Diisinh(o(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l))1CA0B@cd1CA=0B@R1 o)]TJ /F7 7.97 Tf 14.73 4.71 Td[(1 i01CAThissetofequationshasauniquesolutionifandonlyifthedeterminantofthematrixontheleft,isnonzero.Wecalculatethisdeterminant:=det0B@cosh(il))]TJ /F8 11.955 Tf 11.29 0 Td[(cosh(o(1)]TJ /F3 11.955 Tf 11.95 0 Td[(l))sinh(il)Doo Diisinh(o(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l))1CA=Doo Diicosh(il)sinh(o(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l))+sinh(il)cosh(o(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l)), (2)andseethatitisalwayspositive,sincebothtermsofthesumalwaysare.Thesetoflinearequationsthereforehasauniquesolution: 0B@cd1CA=1 R1 i)]TJ /F8 11.955 Tf 16.28 8.09 Td[(1 o0B@)]TJ /F5 7.97 Tf 10.49 4.71 Td[(Doo Diisinh(o(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l))sinh(il)1CA(2)Inparticular,weseethatc<0andd>0.Also,noticethatpluggingthesevaluesofcanddbackinto( 2 ),wendthattheuniquesteadystatesolutionisadecreasingfunctionofx(becauseitsderivativeisnegativeeverywhereexceptinx=lwhereitisnotdened,butwhereniscontinuous). 27

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Finally,weneedtoverifythatn(x)0forallx2[0,1].Butsincen(x)isdecreasing,itsminimalvalueisachievedatx=1,wheren(x)equalsd+R owhichispositive.Thus,n(x)0forallx2[0,1]asrequired. 2.3StabilityoftheSteadyStateSolutionThesimulationinFigure 2-2 suggeststhatthesteadystaten(x)determinedanalyticallyin( 2 )with( 2 )isasymptoticallystable.Stabilitypropertiesofsteadystatesareoftenestablishedusingalinearizationargument.InthisSectionwewillstudytheeigenvalueproblemthatariseswhenthesystemislinearizedatthesteadystate.Wewillshowthatalltheeigenvaluesarenegative,providingfurtherevidenceofthestabilityofthesteadystate.Linearizingmodel( 2 )atthesteadystateyieldsthefollowingeigenvalueproblem: w=(D(x)w0)0)]TJ /F9 11.955 Tf 11.95 0 Td[((x)w,w0(0)=w0(1)=0(2)Solutionsofthisproblemareeigenvalue-eigenfunctionpairs(,w(x))withw(x)6=0.Wedenotetheoperatorontheright-handsideoftheequation( 2 )byL[w].Itsdomainconsistsoffunctionswthatarecontinuouson[0,1],withcontinuouslydifferentiableux)]TJ /F3 11.955 Tf 9.3 0 Td[(D(x)w0,andsatifyingNeumannboundaryconditioninx=0andx=1.Integrationbypartsshowsthatthisoperatorisself-adjoint,i.e.(L[u],v)=(u,L[v])foralluandvinthedomainofL,where(u,v)denotestheinnerproductR10uvdx.Consequently,theeigenvaluesofLarereal.Wewillprovethatinfacteveryeigenvaluemustbenegative.Toseethis,assumethatthereisaneigenvalue0andcorrespondingeigenfunctionw(x)6=0,satisfying( 2 ).UsingtheNeumannboundarycondition,thesolutionw(x)takesthefollowingform:w(x)=8>><>>:Acosh(ix),x2[0,l)Bcosh(o(x)]TJ /F8 11.955 Tf 11.95 0 Td[(1)),x2(l,1], 28

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whereAandBareconstantsdeterminedbelow.Theparametersiandoare:i=r i+ Diando=r o+ Do,andtheyarepositivebecause0.TodetermineAandBwematchthevaluesofw(x)andoftheuxesD(x)w0(x)atx=l:Acosh(il)=Bcosh(o(l)]TJ /F8 11.955 Tf 11.96 0 Td[(1))ADiisinh(il)=BDoosinh(o(l)]TJ /F8 11.955 Tf 11.95 0 Td[(1))SinceAandBcannotbezero(otherwisew(x)wouldbezero),wecandividebothequations,whichyields:Diitanh(il)=Dootanh(o(l)]TJ /F8 11.955 Tf 11.95 0 Td[(1)).Butthisequationcannotholdssinceiandoarepositive,sothattheleft-handsideisalwayspositive,whereastheright-handsideisalwaysnegativesincel<1.Thus,therecannotbeaneigenvalue0.Theabsenceofanonnegativeeigenvaluesuggestsstability. 2.4QualitativeAnalysisoftheSteadyStateSolutionInordertoanalyzethesteadystatesolutionqualitatively,weintroduceandexaminefourquantities: 1. FishingGroundsAbundance(FGA): Io=Z1ln(x)dx(2)wheren(x),thesteadystatesolutionto( 2 )givenin( 2 )and( 2 )representsthedensityofshintheshinggrounds.TheFGAisthetotalamountofshintheshinggrounds,atthestatestate. 2. Yield Y=Z1l(o)]TJ /F9 11.955 Tf 11.96 0 Td[(i)n(x)dx,(2) 29

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whereo)]TJ /F9 11.955 Tf 12.11 0 Td[(irepresentstheshingrate.Hence,theYieldrepresentsthenumberofshcaughtbyshermenintheshinggroundsperunitoftime,atthesteadystate. 3. TheTotalAbundance A=Z10n(x)dx,(2)whichrepresentsthetotalnumberofshinboththeMPAandintheshinggroundscombinedatthesteadystate. 4. TheLogRatio L=ln 1 lRl0n(x)dx 1 1)]TJ /F5 7.97 Tf 6.59 0 Td[(lR1ln(x)dx!,(2)thenaturallogoftheratiooftheaverageabundanceofshintheMPAandintheshinggroundsevaluatedatthesteadystate.WeareinterestedinwhathappensasbothDiandDoincrease,yettheirratioDi Doremainsconstant.Forconvenience,wedene Di=D,Do=1 D(2)andweletDvary,whileallotherparameters,including,remainconstant.WesummarizethebehaviorofIo,Y,AandLasfunctionsofDasfollows: Theorem2.2. Assumethat( 2 )holdsforsomeconstant>0.AsthediffusioncoefcientDincreases,theFGAIoandYieldYarenon-decreasing,whereasboththeTotalAbundanceAandLogRatioLarenon-increasing.TheproofcanbefoundinAppendix A .Figure 2-3 includesgraphsofthesequantitiesforchosenparameters.Wealsoinvestigatedwhathappensifinsteadoftheratio,thedifferenceofDoandDiremainsconstant,yetbothincreaselinearlywithD.TheconclusionsofTheorem 2.2 remainthesame,butsincetheproofisverysimilar,ithasbeenomitted. 30

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AFishingGroundsAbundanceIo BYieldY CTotalAbundanceA DLogRatioLFigure2-3. TheabovegraphswereproducedinMATLABusingtheparametersl=3=16km,R=0.5thousandsofsh (year)(km),i=0.251 year,o=0.51 year.Thevaluesfordifferinthethreedifferentcolorplots,where=0.5,1,2fortheblue,redandgreenplots,respectively. 2.5AMovementBiasModel2InOvaskainenandCornell(2003)[ 72 ]amodelisproposedthatincorporatesamovementbiastowardstheMPA.Thismodelisobtainedasthelimitofabiasedrandomwalkmodel.ThebiasoccurswhentherandomwalkerissituatedontheboundaryoftheMPA,becausetakingthenextsteptowardstheMPAispreferred.Itisassumedthattheprobabilitytomovetotherightis(1+z)=2,andtheprobabilitytomovetotheleftis(1)]TJ /F3 11.955 Tf 12.52 0 Td[(z)=2,whereztakesavaluein()]TJ /F8 11.955 Tf 9.3 0 Td[(1,1)asameasureofthedegreeofbias.Notethatz=0correspondstoacasewithoutbias.Whentherandomwalkerisnotontheboundary,theprobabilityofmovingleftorrightis1=2.Thesteadystateproblem 2Intheoriginalpublication,thissectioncontainedanerrorinthematchingconditions( 2 ).Ithasbeencorrectedhere. 31

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correspondingtothemovementbiasmodelfromOvaskainenandCornell(2003),appliedinthesetupofanMPA,isasfollows: 0=(D(x)n0)0+R)]TJ /F9 11.955 Tf 11.95 0 Td[((x)n,0
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2. Ifz=z(criticalbias),thenn(x)ispiecewiseconstantwithajumpdiscontinuityatx=l,givenbytherstmatchingconditionin( 2 ). 3. If)]TJ /F8 11.955 Tf 9.3 0 Td[(1><>>:c1cosh(ix)+R i,x2[0,l)d1cosh(o(x)]TJ /F8 11.955 Tf 11.96 0 Td[(1))+R o,x2(l,1](2)whereiandoweredenedin( 2 ),andwherec1andd1areobtainedusingthematchingcondition( 2 ): 0B@c1d11CA=1 R(z)0B@)]TJ /F5 7.97 Tf 10.5 4.71 Td[(Doo Diisinh(o(1)]TJ /F3 11.955 Tf 11.95 0 Td[(l))sinh(il)1CA(2)where R(z)= 1 i)]TJ /F4 11.955 Tf 11.96 18.18 Td[(r Do Di1)]TJ /F3 11.955 Tf 11.95 0 Td[(z 1+z1 o!R(2)and =Doo Diisinh(o(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l))cosh(il)+r Do Di1)]TJ /F3 11.955 Tf 11.96 0 Td[(z 1+zcosh(o(1)]TJ /F3 11.955 Tf 11.95 0 Td[(l))sinh(il)(2) 33

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whichispositivebecauseof( 2 ).Therefore,thesignofc1andd1isdeterminedbythesignofR(z),andthelatterchangessignwhenzcrossesz.Thisinturnimpliesthethreedistinctivecasesfortheshapesofthegraphsofn(x).Finally,weneedtocheckthatn(x)isnon-negativeinallthreecases.Forz=zthisisobviousbecauseinthiscase,n(x)iseitherequaltoR=iortoR=o,andbothvaluesarepositive.Ifz0.Asthediffusionco-efcientDincreases,theFGAIoandYieldYarenon-decreasing(non-increasing),whereasboththeTotalAbundanceAandLogRatioLarenon-increasing(non-decreasing),providedthatz0intheentiredomain[0,1],andwithnonconstant,positive,smoothandincreasingmortalityrate(x),sayinC1[0,1]withd=dx>0.ThisreectsthatthevaluesofthemortalityratearehigheroutsidethaninsidetheMPAduetoshing. 34

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AFishingGroundsAbundanceIo BYieldY CTotalAbundanceA DLogRatioLFigure2-5. TheabovegraphsforthecaseofstrongbiaswereproducedinMATLABusingtheparametersl=3=16km,=0.02,R=0.5thousandsofsh (year)(km),i=0.02361 year,o=0.51 year,z=)]TJ /F8 11.955 Tf 9.3 0 Td[(0.99999. Weconsiderthesteadystateproblem: Dn00+R)]TJ /F9 11.955 Tf 11.96 0 Td[((x)n=0,0
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Io(D)isnotnecessarilynondecreasingwithDasinTheorem 2.2 ,atleastforsufcientlysmallMPAsizes: Theorem2.5. Thereexistsl2(0,1)suchthatifllimD!1Io(D)(2) Proof. Inordertoestablish( 2 ),itsufcestoshowthat Z1l1 (x)dx>1)]TJ /F3 11.955 Tf 11.95 0 Td[(l R10(x)dx(2)holds,by( 2 ).Tothatend,wedenethefollowingsmoothauxiliaryfunction:F(l):=Z10(x)dxZ1l1 (x)dx)]TJ /F8 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(l)Wehavethat: 1. F(0)>0.Indeed,thisconditionfollowsfromanapplicationoftheCauchy-SchwarzinequalityinL2[0,1]tothefunctionsp (x)and1=p (x)(theinequalityisstrictbecausep (x)and1=p (x)arelinearlyindependentsincebyassumption(x)isnotaconstantfunction). 2. F(1)=0,whichisimmediatefromthedenitionofF. 3. dF dl(1)>0.Indeed,wehavethat:dF dl(l)=Z10(x)dx)]TJ /F8 11.955 Tf 17.68 8.08 Td[(1 (l)+1,andsince(x)isincreasingon[0,1],thereholdsthat(x)<(1)forx<1,sothatdF dl(1)=Z10(x)dx)]TJ /F8 11.955 Tf 18.9 8.09 Td[(1 (1)+1<(1))]TJ /F8 11.955 Tf 18.9 8.09 Td[(1 (1)+1=0.Thesethreefactsimplytheexistenceofsomelin(0,1)suchthatF(l)=0,andF(l)>0foralll2[0,l).Butthisimpliesthatforthesevaluesofl,( 2 ),andhence( 2 )holds. 36

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2.7DiscussionPreviousmodelssuggestedthatincreasingmobility(e.g.,asreectedinanincreasingdiffusionparameter)would:(1)reducethelocaleffectofanMPA(i.e.,reducetherelativedisparityindensityinsidevs.outsideoftheMPA);and(2)increasetheyield(i.e.,increasethecatchbyshersintheunprotectedregion).Spillover,themovementofadultsfromtheMPAintotheshedregion,contributestobothphenomena.Empiricaldatacontradicttherstexpectation:moremobileshesshowagreaterrelativedensityinsideofMPAscomparedtomoresedentaryspecies.Wehypothesizethatthismightbetheresultofspillin,drivenbydifferentialmovementofshintotheMPA:i.e.,ifshdiffuseatdifferentratesinsideversusoutsidetheMPA(asreectedbytheparameterinourmodel),thenwehypothesizethatincreasedoverallmovement(reectedintheparameterD)wouldleadtoagreaterbuildupofshinsidetheMPA.Wedescribedmovementandmovementbiasviaadiffusionprocessandadiscontinuityindiffusionparameters.Wewereabletoshowthatourmodelhadaunique,nonnegative,continuoussteadystatesolution.Atthissteadystate,asshmobility(D)increased,abundancesintheshinggroundsandyieldsincreased,whereastotalabundancesandlog-responseratiosdecreased(Figure 2-3 ).ThesequalitativeresultswereindependentoftheratioofthediffusionconstantsinandoutsidetheMPA.Thisresultisconsistentwithpastmodels(usingonediffusionparameter,evenifthesemodelsdidnotcalculateexplicitlythelogratio,ameasurecommonlyusedinempiricalstudies),butitisinconsistentwiththeempiricalresultsofClaudetetal.(2010).[ 16 ]InastudyofMPAsintheFloridakeys,EgglestonandParsons(2008)[ 26 ]observedspill-inoflobstertoMPAs,presumablyresultingfromgreatermovementoflobstersintheshedregionsandlessmovementinsidetheMPAs.Thus,differentialdiffusion(asdenedinourmodel),intheabsenceofmovementbias,cannotexplaintheseinterestingempiricalresults,norcanotherexistingmodelswithevensimplerdiffusiondynamics.However,theincorporationofastrongmovementbias 37

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(Figure 2-5 ,z
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researchisneededtodetermineifourconstantrecruitmentassumptionwillaltertheconclusionsaboutthequalitativeeffectsofadultmovement.Similarly,moreanalysisisrequiredtoestablishwhethertheformofdensity-dependenceaffectspredictionsaboutotheraspectsofMPAs,includingtherelationshipbetweenyieldandMPAsize.WehavealsopresentedamovementbiasmodelinwhichthebiasonlyoccursontheMPAboundaryandnowhereelse.Inmanyothermovementbiasmodels,thebiasoccurseverywhere.Forinstance,thereisarecentbodyofworkonadvectiondiffusionmodelsinthetheoreticalecologyliterature[ 10 11 ].Advectionmayoccurthroughdifferentmechanisms.Anobviousoneiswhenoceancurrentsmoveshpopulations,butmoresophisticatedwaysarepossiblesuchasmovementofshinthedirectionofaresourcegradient.Populationswillcrowdinregionswheretherearelotsofresourceswhenmovementduetothisadvectivesourcedominatesdiffusion.Thisiscomparabletotheblow-upphenomenoninchemotaxissystems[ 66 ]liketheKellerSegelmodel[ 49 ],whichincorporatesmovementofcellsinthedirectionofachemicalsubstancethattheysecretethemselves.Finally,followingthesuggestionofananonymousreviewer,weinvestigatedasimplemodelwithsmoothparameters.Weassumedthatdiffusion,andrecruitmentarespatiallyuniform,butthatmortalityisnonuniformandmonotonicallyincreasingsothatthemortalityrateishigheroutsidethaninsidetheMPA.Itturnsoutthattheshinggroundsabundanceisnotnecessarilyincreasingwithincreasedmobility,providedthattheMPAsizeissmallenough.Thismodelthereforeprovidesyetanotherpossibleexplanationfortheempiricaldata.Moretargetedeldresearchwillbeneededtoelucidatewhichofthesemodelsisthemoreaccurateone,orwhatmodicationsthemodelsshouldbesubjectto.AcknowledgmentsWethankBenBolkerforhisinvaluablediscussionsandtheOceanBridgesProgram(fundedbytheFrench-AmericanCulturalExchange)andtheQSE3IGERTProgram 39

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(NSFawardDGE-0801544)forfacilitatingthiscollaboration,whichwasinitiatedduringLouiseRiotte-Lambert'sinternshipattheUniversityofFlorida.Wearealsoverygratefultotwoanonymousreviewerswhosesuggestionsallowedustomakesignicantimprovementstoanearlierversionofthepaper.TherstreviewersuggestedthatwetrytoestablishtheresultsinSection 2.5 .TheresultinSection 2.6 isentirelycreditedtothesecondreviewer. 40

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CHAPTER3QUORUMSENSINGBACKGROUND 3.1QuorumSensingQuorumSensing(QS)isameansbywhichbacteriacancontrolgeneexpression.InaQSsystem,diffusiblechemicalscalledautoinducersaresynthesizedandaccumulateinthelocalenvironment.Whenacriticalautoinducerconcentrationisreached,ittriggersapopulation-wideshiftingeneexpression.OneoftherstdescriptionsofQSwasasamethodofbioluminescenceregulationinthebacteriaAliivibrioscheri(formerlyVibrioscheri[ 89 ])andVibrioharveyiinthe1970's.[ 67 68 74 ]Quorumsensingwasthenthoughttobeamechanismbywhichbacteriacoulddetecttheirpopulationdensity,andthatapopulation-widereactionwouldtakeplaceonlywhenacertaindensity,orquorum,wasreached.[ 36 ]QSisnowrealizedtobemuchmoreversatile,allowingbacteriatoregulatesymbioticinteractionsandpotentiallydetectchangesintheirenvironment.[ 25 42 75 ]TherearemanyexamplesofQSsystems.OnesuchisfoundinPseudomonasaeruginosa,abacteriumthatchronicallyinfectsthelungsofmostcysticbrosispatients.P.aeruginosahasatleasttwoQSsystems,oneofwhichregulatesamultitudeofvirulencefactors,includingthecreationofabiolm,anextracellularmatrixofpolysaccharidesthatencasesbacteriaandprotectsthemfromantimicrobialtreatments.[ 18 ]AnotherexampleisSinorhizobiummeliloti,anitrogen-xingbacteriathatformsasymbioticrelationshipwithsomelegumes.S.meliloticontrolstheestablishmentofsymbiosiswithMedicagosativathroughaQSsystem.[ 40 ]TheQSsysteminA.scheriwasoneofthersttoberecognizedandisstillanareaofferventresearch.RegulationofbioluminescenceinA.scheriiscontrolledbyatleastthreeQScircuits,theAinS-AinR,LuxS-LuxP/QandLuxR-LuxIsystems.[ 58 62 ]TheLuxR-LuxIsystemisusedasaparadigmforQSsystemsinmanyGram-negativebacteria,includingP.aeruginosa,describedabove.LuxR-LuxIhomologuesexist 41

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toregulategenesinvolvedinbioluminescence(asinthecaseofVibrioharveyi),symbiosis,pathogenesis,biolmformation,geneticcompetence,motilityandantibioticproduction.[ 24 35 62 74 ]AsLuxR-LuxIhomologuesarerelativelycommoninnature,wechosethissystemasthebasisforthemodelswepresentinthefollowingchapters.AmorecompletedescriptionoftheLuxR-LuxIsystemfoundinA.scheriappearsinSection 3.2 3.2LuxR-LuxISystemAliivibrioscheriisaGram-negativebacteriafoundbothfree-livinginmarineenvironmentsandasasymbiontwiththeHawaiianbobtailsquid(Euprymnascolopes).Asasymbiont,A.scheriinhabitsthelightorganofE.scolopes,anutrient-richenvironment.Inreturn,A.scheriequipsE.scolopeswithcounterilluminationviaQS-regulatedbioluminescence.LightemittedbysymbioticA.scherihidestheshadowofthehostsquid,whichprovidesE.scolopesanadditionaldefenseagainstpredation.[ 93 ]BioluminescenceinA.scheriisregulatedbyatleastthreeQScircuits,theAinS-AinR,LuxS-LuxP/QandLuxR-LuxIsystems.TheLuxR-LuxIsystemisthecoreofA.scheriQS-regulatedbioluminescence,aswedescribebelow.TheAinS-AinRandLuxS-LuxP/QsystemsinuencebioluminescencebyregulatingproductionofasRNAthattranscriptionallyrepresseslitRtranscript.ThetranscriptionalregulatorLitRenhancesluxRexpressionwithoutalteringexpressionoftheotherluxgenes.[ 62 ]IntheLuxR-LuxIsystem(Figure 3-1 ),theluxRgeneencodesthetranscriptionfactorLuxR.TheluxICDABEGoperonencodestheLuxIenzymeaswellascomponentsnecessaryforsynthesisoftheluciferase,thelight-producingenzyme,andproductionofitssubstrates.LuxIcatalyzesthesynthesisoftheacyl-homoserinelactone(AHL)3-oxo-C6,anautoinducerthattranscriptionallyactivatesLuxR.TheLuxR/3-oxo-C6complexactivatestheexpressionoftheluxICDABEGoperon,creatingapositivefeedbackloop.[ 24 35 62 ] 42

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Figure3-1. MJ11isawildtypeA.scheriwithanintactluxoperonforsynthesis(viaLuxI)anddetection(viaLuxR)ofAHLsignalandproductionofbioluminescence.ReproducedwithpermissionfromDilanjietal.[ 22 ].Copyright2012AmericanChemicalSociety. Thispositivefeedbackloopactsasaswitch,ippinggeneticexpressionfromanunactivatedstatetoanactivatedstate.InthecaseoftheLuxR-LuxIsysteminA.scheri,theactivatedstateischaracterizedbybioluminescence,whichtheunactivatedstatelacks.SinceAHLsarefreelydiffusiblethroughthecellmembrane,theirconcentrationsarelocallyapproximatelyequalextracellularlyandintracellularly.[ 35 ]Hence,weexpectthatifonebacteriumisexperiencinganAHLlevelhighenoughtobeactivated,thensoareitsnearestneighbors.Asmorebacteriabecomeactivated,wewillseeaswitchonthecolonylevelfromtheunactivatedstatetotheactivatedstate. 3.3PreviousModelsQSsystemshavebeenmodeledextensivelyusingdifferentialequationsandcomputationalmodels,mostlyforspatiallyhomogeneoussystems.[ 12 37 46 70 91 ]Forexample,Gardeetal.(2010)presentakineticdifferentialequationsmodeloftheQSsystemfoundinAeromonashydrophila.ThemodelsystemisanEscherichiacolistrainthathasbeengeneticallymodiedtocontainthegenesnecessaryforautoinducerdetection,butnotthosenecessaryforproduction.WhenthisE.colistrainisactivated(duetohighautoinducerconcentrations),thesynthesisofgreenuorescentprotein(GFP)drasticallyincreases.TheconcentrationofGFP(measuredviauorescence)thengivesameasureofbacterialresponsetoAHL.AschemelikethisisacommonmethodofexperimentallystudyingQSsystems.Suchgeneticallymodiedstrainsof 43

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bacteria(commonlyE.coli)arecalledsensorstrains.Gardeetal.explicitlymodeltheconcentrationsoftranscriptionfactorproteins,activatedtranscriptionfactors,activatedreceptorsites,nonmatureGFPandmatureGFP.TheyuseMichaelis-MentenkineticstodescribethedecayofGFP,aswedoinChapter 4 .Unlikeourmodel,however,Gardeetal.assumethatthesignal(autoinducer)moleculesarewell-mixedandtheyneglectauto-feedbackbymakingaquasi-steadystateapproximationintheequationdescribingtheconcentrationofactivatedtranscriptionfactors.[ 37 ]Thesesimplicationseliminatethecomplexspatialpatternsourmodelelicits,asdiscussedinChapter 4 .WhenauthorsdescribeaspatiallyexplicitQSsystem,themodelsaretypicallyverycomplexandarenotanalyzedanalytically.[ 43 ]AcommonthemeamongmanyofthesemodelsisthedescriptionofaQSsystemthatregulatesbiolmproduction.[ 4 14 23 33 51 90 ]TwoexamplesofotherapplicationsofspatiallyextendedmodelsareNetoteaetal.(2009)andMelkeetal.(2010).Netoteaetal.developanagent-basedcomputationalmodelofQSregulatedswarminginP.aeruginosa.[ 69 ]Melkeetal.formulateamodelofQSthatallowsforthediffusionofautoinducer.TheysimulatebacterialQSresponseunderseveraldifferentenvironmentalgeometriesandareabletoelicitQSactivationinasparselypopulatedyetconninggeometry.[ 60 ]Inaspatiallyextendedsystem,theQSmodulatedcolony-widechangeingeneexpressionmayappearasapropagatingwave,asweexploreinChapter 5 .Someauthorshavedelvedintothisphenomenon,bothexperimentallyandthroughmodeling.AswedoinChapter 5 ,bothDaninoetal.(2010)andWardetal.(2003)explicitlyincorporatethediffusionofautoinducermoleculesintheirmodels.Daninoetal.constructedmicrouidicdevicesinwhichamodiedE.colistrainexhibitedanoscillatinguorescenceresponseunderanautoinducerow.Underlowowrates,theyobservedaspatiallypropagatingwaveofuorescence.Daninoetal.formulateacomplimentarysystemofdelaydifferentialequationstodescribetheirexperimentsandgiveseveralcomputationalsimulations.[ 20 ]Wardetal.modelaQSsystemthatincorporatesbiolm 44

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production.Theirsimulationsdemonstrateapropagatingwaveofup-regulationthroughthecolony.Theyfurtherinvestigatethiswavebyexaminingthescenariowhereanup-regulatedbiolmisarticiallyintroducedtoasignicantlylargerdown-regulatedbiolm.Thescenarioisfurtherrestrictedbytheassumptionthatbacteriadonotproduceanyautoinduceruntiltheyareup-regulated.Thoughtheydonotmathematicallyprovetheexistenceofatravelingwavesolutiontothissimpliedmodel,certainnecessaryconditionsareexplored.[ 90 ]InChapter 5 ,weintroduceasimpleQSmodelintendedtorepresenttheLuxR-LuxIQSsystemseeninA.scheri.OurmodelisabletodescribeaQSshiftingeneexpression,whichappearsasatravelingwave.Theadvantagetoourmodelisthatweareabletomathematicallyprovetheexistenceofthistravelingwave.Morerecently,stochasticQSmodels,whichcomparelarge-scaleandsingle-celldynamics,haveappeared.[ 94 ]ThesehaveaccompaniedtheemergenceofexperimentalstudiesthatexaminetheQSdynamicsofasinglecellusingmicrouidicdevices.[ 61 82 ] 45

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CHAPTER4ASPATIALLYEXPLICITQUORUMSENSINGMODEL1AsdiscussedinSection 3.3 ,muchofthepreviousQSmodelingefforthascenteredaroundeitherspatiallyhomogeneoussystemsorbiolms.However,allbacteria,notjustthosethatproduceabiolm,liveinaspatiallyextendedworld.Inthischapter,wepresentasimple,spatiallyexplicitQSmodelbasedontheLuxR-LuxIsystem(seeSection 3.2 )withanaimtoexaminespatialpatternsinQS.WeconstructedamathematicalmodelfortheactivationofthequorumsensingcircuitinthesensorstrainE.coli+pJBA132inresponsetodiffusingAHL.ThissensorstrainisEscherichiacoliMT102harboringplasmidpJBA132,constructedbyAndersenetal.[ 2 ]andcontainingthesequenceluxR-PluxI-gfp(ASV).(Figure 4-2 (A))ThestrainwasprovidedbyDr.FatmaKaplan.E.coli+pJBA132containsthesequence(luxR)necessarytosynthesizetheproteinLuxR.TheLuxR/AHLcomplexbindstothepromoterregionPluxIandup-regulatestranscriptionofthesequence(gfp(ASV))necessarytosynthesizeGFP.SinceE.coli+pJBA132synthesizesGFPinplaceofLuxI,itwillrespondtoAHLbutcannotsynthesizeAHL.TheconcentrationofGFP(measuredviauorescence)givesameasureoftheresponseofE.coli+pJBA132toAHL.Thesequencegfp(ASV)encodesavariantofGFPwithashorthalflife(1h),whichpreventsGFPfromaccumulatingindenitelyduringthemeasurements.[ 3 ]Ourmodelingeffortsarecloselyentwinedwithcomplementaryexperiments.InSection 4.1 ,wedetailtheexperimentthatourmodelattemptstocapture.Inthisexperiment,diffusingAHLinducesGFPproductioninE.coli+pJBA132.TheGFPthendecaysovertime.InSection 4.2 ,wedevelopourmodel.Next,inSection 4.3 ,we 1ReproducedinpartwithpermissionfromG.E.Dilanji,J.B.Langebrake,P.DeLeenheer,andS.J.Hagen.Quorumactivationatadistance:Spatiotemporalpatternsofgeneregulationfromdiffusionofanautoinducersignal.J.Am.Chem.Soc.,134(12):56185626,2012. 46

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estimateparametersforourmodelusingaspatiallyhomogeneousE.coli+pJBA132experiment.Wearethenreadytotestourparameterizedmodelpredictionsagainstspatiallyextendeddata,whichwedescribeinSection 4.4 .Therstoffoursetsofexperiments(Section 4.4.1 )exploresthespatialpatterncreatedbyadiffusingdye.Theotherthreespatiallyextendedexperiments(Section 4.4.2 )investigatethespatialpatternselicitedbythreestrainsofbacteria(E.coli+pJBA132,A.scheristrainVCW267(-luxI)andA.scheriwild-typestrainMJ11)whenexposedtoadiffusingAHLsignal.A.scheristrainMJ11isawild-typestrainwithanintactluxoperonforAHLsynthesisandresponse.VCW267isamutantA.scherithatlackstheAHLsynthase(LuxI).(Figure 4-2 (A))Lastly,inSection 4.5 ,wecompareourparameterizedmodelpredictionsagainstallfoursetsofspatiallyextendedexperimentsandprovidesomediscussion.(Figure 4-1 ) Figure4-1. Chapter 4 methodology.Beginningintheupperleft(Modeling)box,themodelinginthischapterprogressesviathesolidarrows.Theleft-hand(Experiment)boxesgiveinputtomodelingboxesviadashedanddottedarrows.Thedashedarrowssignifydatainputwhilethedottedarrowsigniesexperimentalsetupinput. 47

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Figure4-2. Experimentalconguration:(A)Strainsusedinthisstudy.MJ11isawildtypeA.scheriwithanintactluxoperonforsynthesis(viaLuxI)anddetection(viaLuxR)ofAHLsignalandproductionofbioluminescence;VCW267isamutantA.scherilackingtheAHLsynthase(LuxI);ThepJBA132sensorstrainofE.colihasagfpreporterundercontroloftheluxIpromoter,butlacksluxI;(B)Thelightdomeprovideshighlyuniform,diffuseexcitationlightforimagingGFPuorescenceofbacteriaembeddedinagar.Thesameopticalcongurationallowsustomeasurebioluminescenceandopticaldensityofthesamplesinsitu;(C)Bacteria/agarmixtureisloadedintoaframecontainingfourparallel,independentlanes.Adropletofautoinducerdepositedattheterminusofeachlanediffusesdownthelane,generatingapatternofQSactivation;(D)RepresentativeuorescenceimagesshowingimagescollectedfromatypicalE.coli+pJBA132experiment. 48

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4.1ExperimentalCongurationWemodelanexperimentperformedbyGabrielDilanjiwhereinAHL(concentrationC(x,t)nM)isloadedintotheterminusofalong(32mm)agarlanepopulatedbyE.coli+pJBA132.TheAHListhenallowedtodiffusedownthelanewithdiffusionconstantD(mm2=h).Thebacteriaareheldstationarybytheagarandthereforedonotdiffuse.[ 19 ](Figure 4-2 (B))Wewillmodelthislaneasone-dimensional(x,pointingdownthelaneandwherex=0isthesource)undertheassumptionthatthelaneishomogeneousinthetransversaldirection.Forthedurationoftheexperiment,wemeasurebothcellpopulationdensity(n(t),cellspercm3)andtheconcentrationofuorescentGFPperunitvolumeofagar(G(x,t)).Thecelldensityisassumedtobeproportionaltotheexperimentally-measuredopticaldensityoftheagarandthusismeasuredinODunits.AswedetectGFPthroughitsuorescence(percamerapixel),G(x,t)ismeasuredinunitsofcountsperpixel.SinceE.coli+pJBA132respondstoexogenousAHLbysynthesizingGFP,thespatio-temporalpatternofuorescenceinthelaneisanalogoustothepatternofQSup-regulation.WeperformsimilarlaneexperimentswithauorescentdyeaswellastwostrainsofAliivibrioscheri,MJ11andVCW267(-luxI).A.scheristrainMJ11isawild-typestrainwithanintactluxoperonforAHLsynthesisandresponse.VCW267isamutantA.scherithatlackstheAHLsynthase(LuxI).(Figure 4-2 (A))AsbothofthesestrainsbioluminesceinthepresenceofAHL,wemeasureluminescence(countsperpixel)inlieuofuorescence.Inadditiontotheselaneexperiments,weperformedawell-plateexperimentwhereinE.coli+pJBA132wasgrowninagarinthepresenceofvariedconcentrationsofexogenousAHL(0-500nM).EachofsixteenwellscontainedhomogeneouslydistributedAHLataxedconcentrationC.Wemeasuredeachofcelldensity(n)andGFPuorescence(G)ineachwelloveraperiodof25h.Thisexperimentwasusedto 49

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parameterizethemodeldescribedinSection 4.2 .WedetailtheparameterizationofthismodelinSection 4.3 .Foradditionaldetailsregardingtheseexperiments,seeMethods,Section 4.6 4.2MathematicalModelTomodeltheinteractionbetweenspatialdiffusionoftheAHL(acylhomoserinelactoneautoinducer)andtheexpressionofGFP,weconsidereachofcellpopulationdensity(cellspercm3),AHLconcentrationandGFPconcentration.Thenumberofbacterialcellsperunitvolumeintheagarlaneisdenotedbyn(t),whichispresumedtobeproportionaltotheexperimentally-measuredopticaldensityoftheagarandisthereforemeasuredinODunits.Inourexperiments,theinitialcelldensityisthesameeverywhereandthecellsareimmobilizedinagar[ 19 ].Hence,nisindependentofone-dimensionalspacexandisafunctiononlyoftimet.Wedescribethechangeincelldensitywithrespecttotimeasalogisticfunction: dn dt=n1)]TJ /F3 11.955 Tf 14.83 8.09 Td[(n K(4)whereKisthecarryingcapacityand=ln(2)istheintrinsicgrowthrate(doublingsperhour)[ 21 ].Inawell-plateexperiment(seeMethods,Section 4.6 ),exogenousAHLisprovidedinthegrowthmediumandiswell-mixed.AsthesensorstrainE.coli+pJBA132cannotproduceAHL,theAHLconcentrationCisthenconstant.Howeverinalaneexperiment,exogenousAHLissuppliedatthelaneterminus(x=0)anddiffusesoutwardwithtime.ThenCisafunctionofspaceandtime,C(x,t).AsAHLischemicallystableforourexperimentalconditionsandtimescales[ 27 ],C(x,t)evolvesaccordingtothediffusionequation, @C @t=D@2C @x2(4)whereDisthediffusionconstant. 50

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WealsoconsidertheconcentrationoftheuorescentproteinGFPperunitvolumeofagar.Followingexpressionofthegfpgene,theGFPpolypeptideisnotuorescentuntilithasundergoneamaturationprocessthatinvolvesfolding,cyclization,dehydrationandoxidation[ 88 ].WemodelthisprocesswiththreeformsofGFP:U1representsthenewlysynthesized,non-uorescentpolypeptide,U2representsafoldedbutnon-uorescentprotein,andGisthematureuorescentprotein.AsGisdetectedthroughitsuorescence(percamerapixel)theunitsofmeasurementforallthreeformsarecounts=pixel.TheGFPisanunstablevariant(GFP(ASV)[ 3 ])andwemodelthedegradationofeachofitsforms(U1,U2,G)asacompetitive,Michaelis-Mentenprocess[ 37 55 ]: g(V)=k1V k2+U1+U2+G(4)whereVcanbeanyofU1,U2orG.Hence,allformsofGFParedegradedatequivalentratesandthetotalrateofproteindegradationis k1U1 k2+U1+U2+G+k1U2 k2+U1+U2+G+k1G k2+U1+U2+G=k1T k2+T(4)whereT=U1+U2+G.Itisnoweasytoseethatk1isthemaximumdegradationrate(h)]TJ /F7 7.97 Tf 6.58 0 Td[(1)ofGFPwhilek2(counts=pixel)istheMichaelisconstantof( 4 ).SinceGFPissynthesizedasU1,thetimederivative@U1=@tdependsexplicitlyontherateofgfpexpressioninresponsetoAHL.ThisratedependsonboththeconcentrationofAHL(C)andonthegrowthstage,aswell-plateexperimentsshowsynthesisslowingasn!K.WeuseaHillfunction( 4 )tomodeltheAHL-dependenceofU1synthesis.TheHillfunctionprovidescooperativeswitchingfromthesynthesis-offtothesynthesis-onstates,viatwoparametersaandm:f(C)=Cm(x,t) am+Cm(x,t) (4)Hereaisthehalf-activationconstant(unitsofnM)andmistheHillcoefcient(dimensionless). 51

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Tomodelthegrowth-ratedependenceofGFPproductionourmodelalsoincludesthelogisticgrowthfunction( 4 )intherateofGFPproduction:n1)]TJ /F3 11.955 Tf 14.83 8.09 Td[(n K.Finallythemodelalsorequiresaproportionalityfactor(counts=pixel)thatdeterminestheoverallrateofGFPproduction.ThentheproductionofU1proceedsataratef(C)n1)]TJ /F3 11.955 Tf 14.83 8.09 Td[(n K.WedescribethetransformationofU1intoU2withtheconstantpercapitaratem1(h)]TJ /F7 7.97 Tf 6.58 0 Td[(1),andthetransformationofU2intoGwiththeconstantpercapitaratem2(h)]TJ /F7 7.97 Tf 6.59 0 Td[(1).Combiningalloftheseprocesses,wehave@U1 @t=f(C)n1)]TJ /F3 11.955 Tf 14.83 8.08 Td[(n K)]TJ /F3 11.955 Tf 11.95 0 Td[(m1U1)]TJ /F3 11.955 Tf 11.96 0 Td[(g(U1) (4)@U2 @t=m1U1)]TJ /F3 11.955 Tf 11.96 0 Td[(m2U2)]TJ /F3 11.955 Tf 11.95 0 Td[(g(U2) (4)@G @t=m2U2)]TJ /F3 11.955 Tf 11.96 0 Td[(g(G) (4)wheref(C)isdenedin( 4 )andg(V)isdenedin( 4 ).Nowthatwehavethesystemof( 4 ),( 4 ),( 4 ),( 4 ),and( 4 ),wespecifyinitialconditions.Theinitialcelldensityisconstanteverywhereandwedenoteitbyn0.n(0)=n0Inthelane(diffusingsignal)experiments,thecellsgrowinanarrowchanneloflengthL(mm).AHLisinitiallydepositedattimet=0ontoaregionoflength(mm)atoneterminus(x=0)ofthelane.TheamountofAHLinitiallyloadedischaracterizedbytheconcentrationitproduceswhenfullydiffused(t!1)throughoutthelane,C1(nM). 52

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HencetheinitialconditionforC(x,t)isC(x,0)=8>><>>:C1L 0x00Insummary,ourmodelis:dn dt=n1)]TJ /F3 11.955 Tf 14.83 8.08 Td[(n K (4)@C @t=D@2C @x2 (4)@U1 @t=f(C)n1)]TJ /F3 11.955 Tf 14.83 8.09 Td[(n K)]TJ /F3 11.955 Tf 11.95 0 Td[(m1U1)]TJ /F3 11.955 Tf 11.95 0 Td[(g(U1) (4)@U2 @t=m1U1)]TJ /F3 11.955 Tf 11.96 0 Td[(m2U2)]TJ /F3 11.955 Tf 11.96 0 Td[(g(U2) (4)@G @t=m2U2)]TJ /F3 11.955 Tf 11.96 0 Td[(g(G) (4)n(0)=n0 (4)C(x,0)=8>><>>:C1L 0x0
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U2(x,0)=0 (4)G(x,0)=0 (4)@C @x(0,t)=@C @x(L,t)=0forallt>0 (4)wheref(C)=Cm(x,t) am+Cm(x,t) (4)and g(V)=k1V k2+U1+U2+G.(4)Table 4-1 listsalloftheparametersandvariablesandtheirestimatedvalues.Themethodforestimationisdescribedinthefollowingsection. 4.3ParameterEstimationAllparameterestimationwasperformedwithMATLABR[ 86 ].Weobtainedparametersforourmodelbyanalyzingdatafromexperimentsconductedinastandard48-well-plate.E.coli+pJBA132wasgrowninagarinthepresenceofdifferentconcentrationsofexogenousAHL(3-oxo-C6-HSL,seeMethods,Section 4.6 ),whileopticaldensityanduorescencewererecordedoveraperiodof25h.Wetdatafromsixteenwells,withAHLconcentrationsrangingfrom0nMto500nM.SincetheAHLwasmixedintotheagaratthestartofthismeasurement,( 4 )-( 4 )reducetoaspace-independentsystem(G(x,t)!G(t),U1(x,t)!U1(t),etc.)whereCisconstantwithineachwell.TheexperimentdoesnotprovidetheAHLdiffusionconstantD.However,forthe3-oxo-C6-HSLautoinducerusedinthepresentstudyandanaqueousmedium,theliteraturesuggestsD'5.510)]TJ /F7 7.97 Tf 6.59 0 Td[(6cm2=s=2mm2=h[ 45 84 ].Wedonottthedataatlatertimesonthegrowthcurve,t13h,wherethedegradationofGFPslowsandtheslopeofOD(t)indicatesweakornegativegrowththatisinconsistentwiththelogistic( 4 ). 54

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Table4-1. Asummaryofvariablesandparametersusedinthemodel( 4 )-( 4 ). Variable/DenitionValues/UnitsParameter ncellconcentrationopticaldensity(dimensionless)intrinsiccellgrowthrate0.8h)]TJ /F7 7.97 Tf 6.58 0 Td[(1Kcellcarryingcapacity0.9(well-plate),0.4(lane)opticaldensity(dimensionless)CAHLconcentrationnMDAHLdiffusionconstant2mm2=hU1unfoldedGFPcounts=pixelproportionalityfactor6.9107counts=pixelfcooperativeswitchfunctionunitlessahalf-activationcoefcient1.5nMmHillcoefcient1.66unitlessm1foldingrateofGFP0.67h)]TJ /F7 7.97 Tf 6.58 0 Td[(1g(V)degradationofGFPinformVcounts=(pixelh)k1maximumdegradationrate2.4105counts=(pixelh)k2Michaelisconstant6.2104counts=pixelU2foldedbutnon-uorescentGFPcounts=pixelm2maturationrateofGFP29h)]TJ /F7 7.97 Tf 6.58 0 Td[(1GuorescentGFPcounts=pixeln0initialcellconcentrationopticaldensity(dimensionless)C1fully-diffusedconcentrationofAHLnMLlengthofagarlane32mmlengthofAHLloadingregion2mm Weperformedsomeadjustmentstothewell-platedatapriortotting.AswellscontainingdifferentAHLconcentrationsdidnotreachpeakgrowthrateatexactlythesametime,weappliedsmallhorizontalshiftstotheOD(t)(andcorrespondingG(t))curvesuntilalldn=dtdatareachedtheirpeakatthesametimet.Thisoffsetwasontheorderof0.04-0.65h,whichissmallwithrespecttothedurationoftheGFPproductionphaseofinterest.FurthermoretheOD(t)valuesaremeasuredrelativetoOD(0),notabsolute.ThiscreatesanambiguityintheinitialODforeachwell,orequivalentlyinn0.Thereforeweestimatedn0foreachwellbyrequiringthateachn(t)(anditsdn=dt)talogisticgrowthmodelatearlytimes:( 4 )requiresthatdn=dt!0asn!0atearlytimes.Therefore 55

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Figure4-3. Experimentaldatafromwell-plateandtstomodel.Curvesareoffsetverticallyforclarity.(A)GFPuorescence,(B)@G=@t,(C)OD(t),and(D)dOD(t)=dtareshownforE.coli+pJBA132growinginLBagarinawell-plateinthepresenceoftheindicatedexogenousAHLconcentrations.Data(solid)werettothemodelof( 4 )-( 4 )(dashed). weaddedtoeachOD(t)curveasmalloffsetsufcienttogiveOD!0asdOD=dt!0.Thatis,weusethelogisticgrowthmodeltosetthe(unknown)offsetOD(0)inthedata.WethenusetheOD(t)astheexperimentalvaluesforn(t).Wethenobtainedthegrowthratebytting( 4 )toall16opticaldensitycurves,usingaglobalnonlinearleastsquaresmethod,whichconsistsofaNelder-Meadsimplex(directsearch).Theresultinggrowthrate'0.8h)]TJ /F7 7.97 Tf 6.59 0 Td[(1istypicalunderourgrowthconditions.[ 37 87 ]WendahighercarryingcapacityK'0.9(ODunits)inthewell-platethaninthelaneexperiments,wherewendK'0.4,asthewell-platesamplesaredeeperthantheagarlanes.(Figure 4-3 ) 56

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Wethent( 4 )-( 4 )toobtaintheremainingparameters.WerstdenedasetofscaledvariablesfU1=U1 ,fU2=U2 ,eG=G ,ek1=k1 ,ek2=k2 inordertoeliminatetheconstantfactorinthemodelequations.Thisreducesthenumberofparameterstobet:@fU1 @t=f(C)n1)]TJ /F3 11.955 Tf 14.83 8.09 Td[(n K)]TJ /F3 11.955 Tf 11.96 0 Td[(m1fU1)]TJ /F4 11.955 Tf 48.62 11.24 Td[(ek1fU1 ek2+fU1+fU2+eG (4)@fU2 @t=m1fU1)]TJ /F3 11.955 Tf 11.95 0 Td[(m2fU2)]TJ /F4 11.955 Tf 48.62 11.24 Td[(ek1fU2 ek2+fU1+fU2+eG (4)@eG @t=m2fU2)]TJ /F4 11.955 Tf 50.33 11.24 Td[(ek1eG ek2+fU1+fU2+eG (4)wheref(C)isdenedin( 4 ).RecallthatC,andhencef(C),isconstantforeachwell.WeusedliteraturedatatomakeaninitialguessfortheHillparametersaandm.(Theguesswaslaterrenedasdescribedbelow.)Wethenusedaglobalnonlinearleastsquaresroutinetot( 4 )-( 4 )andfoundvaluesform1,m2,ek1andek2.TheroutineemployedaNelder-Meadsimplex(directsearch)method,minimizingthesumofsquareerrorsbetweentheGFPdataandtherescaledmodelpredictioneG,whereisthebestconstantsolutiontotheequationeG=Gdata.Leastsquareerrorswerefoundforeachof16well-platedatasetswithnonzeroAHLconcentrationandsummedtogivetheglobalerror.Werenedourestimateforaandmbyrepeatingtheabovet(form1,m2,ek1andek2)formanyvaluesofaandmandlookingforaglobalminimumtothesumofsquares.Wefoundthematurationratestobem1'0.67h)]TJ /F7 7.97 Tf 6.58 0 Td[(1andm2'29h)]TJ /F7 7.97 Tf 6.59 0 Td[(1.ThesevaluesimplyaGFPmaturationtimem)]TJ /F7 7.97 Tf 6.58 0 Td[(11+m)]TJ /F7 7.97 Tf 6.59 0 Td[(12'1.5h.ThusforgrowthinagarweobtainslightlyslowermaturationfortheunstablegreenuorescentproteinGFP(ASV)maturationtimethanthe'0.67hthatwasobservedinliquidmedium.[ 37 ] 57

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Tocompareourdegradationparametersk1(=ek1'2.4105counts=(pixelh))andk2(=ek2'6.2104counts=pixel)toliteraturevaluesmeasuredinunitsofh)]TJ /F7 7.97 Tf 6.59 0 Td[(1,wecanexaminethetypicaldegradationratespresentinourmodelsystem.FromthetotalreporterproteinT=U1+U2+Gpresentinthemodel,theeffectivedecayrateatagiventimetisk1T(t) k2+T(t)1 T(t)=k1 k2+T(t)Thisyieldsa(t-dependent)decayrate0.09-3.7h)]TJ /F7 7.97 Tf 6.59 0 Td[(1,whichspansliteraturevaluesof0.3)]TJ /F8 11.955 Tf 11.95 0 Td[(0.4h)]TJ /F7 7.97 Tf 6.58 0 Td[(1.[ 3 37 55 ] 4.4ResultsofLaneExperimentsTheexperimentsdescribedhereinareattributedtoGabrielDilanji.Toexperimentallyexplorespatio-temporalpatternsinQSregulation,weperformedfoursetsoflaneexperiments.Intheseexperiments,bacteriawereembeddedinalongagarlane,theneitheruorescentdyeorexogenousAHLwasloadedintothelaneterminus(x=0)andallowedtodiffusedownthelane.Asthedyediffused,wemeasuredtheuorescenceofthelane.AstheexogenousAHLdiffused,wemeasuredbothcelldensity(opticaldensity)andeitheruorescenceorluminescence,dependingonthenatureoftheQSresponseoftheembeddedbacteria. 4.4.1DiffusionofaDyeThesimplestpatternwemayexpecttoseeinthelaneexperimentsisthatofone-dimensionaldiffusion,asgovernedby( 4 ).Toexaminethispatternandtocreateabenchmarkforanalyzingoursubsequentresults,weloaded1Lofauoresceinsolution(0.2Minwater)intotheterminus(x=0)ofalongagarlanepopulatedbyE.coli+pJBA132.Thisuorescentdyeisthenallowedtodiffusedownthelane.TheresultingpatterninC(x,t)(measuredviauorescence)isshowninFigure 4-4 (A-C).Weseestronginterdependenceofspaceandtime:theconcentrationofdyechangesrapidlyclosetothedyedropletandatearlytimes,whileitchangesmoreslowlyfarther 58

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fromthedropletandatlatertimes.Thisischaracteristicbehaviorofdiffusivespreading,describedbyx22Dt.Thesamepatternsareobservedregardlessofwhetherbacteriaarepresentintheagarlane. Figure4-4. Diffusionofuoresceindyeinagarlane.Fluorescenceisplottedasafunctionof(A)timetand(B)distancexfromdyedroplet.Thecontourlinesinamapofuorescencevs.xandtshowthex22Dtbehaviorthatischaracteristicofsimplediffusivespreading;(D)Simulationof( 4 )bythenitedifferencemethodinMATLAB,basedonD=1.510)]TJ /F7 7.97 Tf 6.58 0 Td[(6cm2/sandusingthesamedyeconcentrationandboundaryconditionsasin(C). 4.4.2LuxR-LuxISystemResponsetoAHLDiffusionFigure 4-5 showstheresponseG(x,t)ofthesensorstrainE.coli+pJBA132toadiffusingAHLsignal.Weobservedaspatiallypropagatingresponsethatextendedontheorderof1cmoverthecourseof10h.ThisdistanceissignicantwhencomparedtothesizeofasingleE.colibacterium,whichis2)]TJ /F8 11.955 Tf 11.95 0 Td[(4minlength.Thepatternofactivationweobserved(Figure 4-5 )isqualitativelydifferentfromstandarddiffusion(Figure 4-4 ).Theobserveduorescenceresponseisremarkablyself-similarwhenexaminedatxedpointsinspaceoratxedtimes.EachoftheseslicesofthesurfaceG(x,t)arequalitativelysimilar,butvaryinmagnitude.Whenweconsideraxedpointinspacex,weseeastrongerresponseclosertothelaneterminus(x=0).Ifweconsideraxedpointintime,themagnitudeoftheresponse 59

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increasesuntilt'10h,thenthesignalbeginstofade.Thispatterniscausedbythecell'snonlinearactivationresponse,whichtapersoffwhencellsenterthestationaryphase.ThatG(x,t)displaysqualitativepatternsthatareindependentofpositionandtimeischaracteristicallydissimilarfromdiffusion. Figure4-5. Responseofthesensorstrain(E.coli+pJBA132)todiffusingAHL.(A)and(D)showG(x,t),thespatiotemporalpatternofreporteruorescence,followingdepositionofanAHL(3-oxo-C6-HSL)dropletattheterminus(x=0)ofanagarlaneatt=0.TheamountofAHLintroducedwassufcienttoproduceanal(fullydiffused)concentrationofC1=0.4nM(A-C)or4nM(D-F)throughoutthelane.(B)and(E)showslicesthroughG(x,t)atxeddistancesx,while(C)and(F)showslicesthoughG(x,t)atxedtimest. Figure 4-6 showstheresponseofluxI-decientA.scheristrainVCW267todiffusingAHL.VCW267containsthegenesnecessaryfordetectionofAHLandforbioluminescence,butnotthosenecessaryfortheproductionoftheAHLsynthaseLuxI.(SeeMethods,Section 4.6 ;Figure 4-2 )WeseeaqualitativelysimilarresponseasintheexperimentswithE.coli+pJBA132(Figure 4-5 ),thoughthesensitivityofthe 60

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responseisreduced.Weagainobservesignalingdistancesontheorderof1cmandindependenceoftheresponseG(x,t)withrespecttopositionandtime. Figure4-6. BioluminescenceresponseofluxI-decientA.scheriVCW267todiffusingAHL.(A)and(D)showthebioluminescencevs.xandt,followingdepositionofa3-oxo-C6-HSLdropletattheterminus(x=0)ofanagarlaneatt=0.TheamountofAHLintroducedwassufcienttoproduceanal(fullydiffused)concentrationofC1=400nM(A,B,C)or2M(D,E,F).(B)and(E)showslicesthroughthedataatxeddistancesx,while(C)and(F)showslicesatxedt. Lastly,weexaminedtheresponseofwild-typeA.scheristrainMJ11todiffusingAHL.MJ11containsanintactLuxR-LuxIsystem,andthusisabletodetectandsynthesizeAHL,andtobioluminesce.(SeeMethods,Section 4.6 ;Figure 4-2 )TheabilityofMJ11tosynthesizeAHLallowsforauto-feedbackinthisprocess.Infact,inourexperiments,thebacterialproductionofAHLoverwhelmedtheexogenouslyintroducedAHL(C1=0,1,6,or60nM)byt'5h,atwhichtimetheentirelaneluminescedbrightly.However,forsmallxandearlyt,thepatternsofresponsewerequalitativelysimilarto 61

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thosedescribedaboveforE.coli+pJBA132andforA.scheriVCW267.(Datanotshown) 4.5DiscussionandModelSimulationsQSisaninherentlyspatialprocessasitdependsontheproductionandtransportofsmallautoinducermolecules.NowthatthereisawealthofQSmodelinginspatiallyhomogeneoussystems,whichneglectautoinducertransportbyassumingthatthemediumiswell-mixed(asdiscussedinSection 3.3 ),wearewell-equippedtoexploretherealmofspatialinhomogeneities.ThisisofpracticalimportanceasnaturalQSsystemsoftenoccurinhighlyheterogeneousenvironments,suchastherhizosphere,theareaofsoilveryclosetotherootofaplant.[ 42 ]Wehavealreadymentioned(inSection 3.1 )anexampleofthisverycircumstance:S.melilotiisanitrogen-xingbacteriathatoccursbothfree-livinginthesoilandasasymbiontwithsomelegumes.TheestablishmentofthissymbiosisoccurspartiallyintherhizosphereandisfacilitatedbyacomplexQSsystem.[ 40 ]Thoughourmodeldoesnotattempttoincorporateenvironmentalheterogeneity,itdoesallowforspatialheterogeneityinautoinducerconcentration.Ourmathematicalmodel( 4 )-( 4 )givesasimplespatially-explicitdescriptionofaQScircuit.Wemodeledthespatio-temporalpatternsofgeneregulationduetoaQScircuitviathediffusionofanautoinducer.Wemodeledthetranscriptionalresponsetothisautoinducerwithanonlinear(Hill)functionmodulatedbylogisticbacterialpopulationgrowth)]TJ /F5 7.97 Tf 6.68 -4.97 Td[(dn dt.OurmeasureofresponseisthemagnitudeofuorescenceofGFP,whichexistsintwoimmature(non-uorescent)andonemature(uorescent)state,eachdegradedbyaMichaelis-Mentenprocess.Asabaselineagainstwhichtocompareourmodelsimulationsandexperiments,werstexaminedthesimplediffusionofadye.(Figure 4-4 (A-C)givestheexperimentalresults.)Figure 4-4 (D)showsanumericalsimulationofthediffusionequation( 4 ).TheinitialconcentrationofdyeC(x,t=0)ismodeledasastepfunctiontoemulateourexperimentalsetup.ThecalculatedC(x,t)patternisverysimilartothatobtained 62

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experimentallyanddisplaysthesamex22Dtbehavior,acharacteristicofdiffusivespread.ComparingthisnumericalsolutiontothedataallowsustoestimateD'1.510)]TJ /F7 7.97 Tf 6.59 0 Td[(6cm2/sfordiffusionofuoresceinin0.75%agar.Thegoodagreementbetweenthesimulationandexperimentsuggeststhattheagarlanemayindeedbeapproximatedbyonespatialdimension. Figure4-7. PatternsofexpressionpredictedfortheE.coli+pJBA132sensorstraininresponsetodiffusingAHL.Thepatternsweregeneratedbysimulationusingthemodel( 4 )-( 4 ),assuminganalAHLconcentrationofC1=0.4nM(A)or4nM(B,C).(D)showstheconcentrationofdiffusingAHL,C(x,t)(inM),assuminganalAHLconcentrationofC1=4nMandadiffusionconstantD=2mm2=h. WeusedtheparametersfoundinSection 4.3 alongwithourliteratureestimateforthediffusionconstantoftheAHL3-oxo-C6-HSL(summarizedinTable 4-1 )tosimulatethespatio-temporalresponse(G(x,t))ofthesensorstrainE.coli+pJBA132todiffusingexogenousAHL(C(x,t)).WesimulatedtwodifferentAHLloadingscenarios,oneinwhichthenalAHLconcentrationC1=0.4nM(Figure 4-7 (A))andanotherinwhichC1=4nM(Figure 4-7 (B-D)).ThesesimulationscorrespondtotwolaneexperimentsdescribedinSection 4.4.2 andshowninFigure 4-5 .Bothofoursimulationsarequalitativelyandquantitativelysimilartotheexperimentaldata.Inboth 63

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experimentsandbothsimulations,thepeakinresponseoccursat10)]TJ /F8 11.955 Tf 12.24 0 Td[(12h.Inboththeexperimentandthesimulation,theC1=0.4nMloadingresultedinaresponseextending4mmandtheC1=4nMloadingresultedinaresponseextending1cm.Asinthecorrespondingexperiments,ourmodelpredictionofG(x,t)isqualitativelyindependentofspaceortime,thoughthemagnitudeoftheresponseisnot.ThisexcellentagreementbetweenmodelandexperimentssuggeststhatweareabletocapturetheessentialelementsoftheLuxR-LuxIsystemresponsebyincorporatingadiffusivesignal,non-linearcellresponsetoAHL,andlogisticcellgrowthinourmodel.WhencomparedtolaneexperimentsconductedwithA.scheriVCW267(-luxI),ourmodelsimulationspredictaqualitativelysimilarresponsepattern,thoughthemagnitudeoftheresponsediffers.(Figures 4-6 4-7 )Similarly,beforetheexogenouslyintroducedAHLisoverwhelmedbyendogenousAHL,ourmodelsimulationsqualitativelydescribetheQSresponseofwild-typeA.scheriMJ11.ItisunsurprisingthatourmodeldivergesfromtheseMJ11laneexperimentsasourmodeldoesnotincorporatebacterialproductionofAHL,animportantfeatureofthewild-typestrainMJ11.ThegoodqualitativeagreementbetweenourmodelpredictionsandtheA.scherilaneexperiments(atleastatearlytimesandsmallxinthecaseofthewild-typestrainMJ11)suggeststhateventhoughA.schericontrolsbioluminescencethroughthreeentwinedQScircuits(AinS-AinR,LuxS-LuxP/QandLuxR-LuxI,asdescribedinSection 3.2 ),theminimalsetofcomponentsinourmodelcharacterizingtheLuxR-LuxIsystemissufcienttodescribethebulkoftheQSbioluminescenceresponseofA.scheri.Ourexperimentalresultsandmodelpredictionsshowedtwointerestingfeaturesthatwarrantfurthermention.First,weobservedandpredictedanextended,coordinatedcellresponseonthemillimeterorcentimeterscale.Theseresponsedistancesarelargerthanwewouldexpectfromamoleculediffusingatarateof2mm2=h.Second,weobservedandpredictedatemporalsynchronizationofresponsewithinthebacterialcolony,animportantfeatureofQSsystems.Sinceautoinducerisdiffusingawayfrom 64

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apointsource,theamountoftimeittakesfortheautoinducerconcentrationtoreachagivenvalueatagivendistancexfromthesourcescalesasx2.Thus,wemayexpectthatbacteriaslightlyfartherfromthesourcewouldrespondmuchlater.However,weseethatthequalitativetemporalpatternofresponseremainsconstantasthedistancefromthesourcexincreases,thoughtheintensitydecreases(Figures 4-5 4-6 4-7 ).Ourexperimentsraiseaninterestingquestion:Inasinglebacterialculture,whatistheQSresponsetomultipleautoinducersthathavedifferentdiffusioncoefcients?S.melilotimakesacasefortherelevanceofthisquestion.S.melilotihasacomplexQSsysteminvolvingmanydifferentautoinducers.Theseautoinducershavecarbonchainsrangingfrom8carbonsto18carbonsinlength.[ 40 ]Theseshort-andlong-chainautoinducershavearangeofdiffusioncoefcients,andthuswewouldexpectS.melilotitorespondtothemindifferentways.However,thedifferenceindiffusioncoefcientalonemaynotbesufcienttocharacterizethedifferenceinresponse.Thestrengthoftheauto-inductiveQSresponsetoaparticularautoinducermayvaryindependentlyofthemagnitudeofdiffusioncoefcient.Thus,aslower-diffusingautoinducermayinduceafarther-reachingresponseiftheQScircuitismoresensitivetoit.Thisisanareaofcurrentresearch.Inthefollowingchapter,weformulateasimpliedversionofthemodel( 4 )-( 4 )tofurtherexaminepatternsinQSgeneactivation.Specically,westudytheexistenceofatravelingwavesolutiontooursimpliedmodel.Thislineofresearchcouldpotentiallygiveinsightastothespeedofsignalpropagationasacomplimenttothischapter'scommentsonthedistanceofsignalpropagation. 4.6MethodsThissectionandtheexperimentsdescribedhereinareattributedtoGabrielDilanji. 4.6.1BacterialculturesFigure 4-2 showstheQSbacterialstrainsusedinthiswork.ThequorumsensorstrainisEscherichiacoliMT102harboringplasmidpJBA132,constructedbyAndersen 65

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etal.[ 2 ]andcontainingthesequenceluxR-PluxI-gfp(ASV)(Figure 4-2 ).ThestrainwasprovidedbyDr.FatmaKaplan.CulturesinexponentialphasewerepreparedbygrowingtheE.colitoOD600=0.3inLuria-Bertani(LB)medium,approximatelypH7,at37C.Theculturewasprewarmedfor15sat50CCelsiusinordertopromotesurvivalinwarmagar[ 5 ],andthendiluted100intomolten0.75%LBagarat50C.250Loftheagarmixwasthenquicklypipettedintoeachofthefourparallellanesintheobservationdevice(describedbelow).Thelanedevicewassandwichedbetweentwoglasscoverslipsastheagarcooled.Theupperglasscoverslipwasthencarefullyremoved,leavingaveryatanduniformslabofagarwithineachlane.Thedevicewasincubatedatroomtemperaturefor1.5hbeforemeasurementsbegan.AliivibrioscheristrainVCW267isasynthase-decient(-luxI)mutantproducedfromanES114wildtypebackgroundandwasprovidedtousbyDr.EricStabb.A.scheristrainMJ11isawildtypestrainthatwasderivedfromitssymbiotichostshMonocentrisjaponicusandprovidedtousbyDr.MarkMandel.BothstrainsweregrowntoOD600=0.3incommercialphotobacteriummedium(No.786230,CarolinaBiological),approximatelypH6.9,atroomtemperatureandthenpreparedasabovefortheagarlanes.Thephotobacteriummediumisarichmediumcomposedofyeastextract,tryptone,phosphatebuffer,andglycerolinarticialseawater. 4.6.2Well-platemeasurementsInordertoobtainparametersforourmathematicalmodelforE.coli+pJBA132growthanditsresponsetotheAHL,wemeasuredtheopticaldensityanduorescenceofthisstraininthepresenceofvariousautoinducerconcentrationsinamultiwellplate,usinganautomatedplatereader(BiotekSynergy2).Adiluteculturewasloadedintoindividualwellscontaining0.1%agarinLBmediumandAHL(3-oxo-C6-HSL,N-(3-oxohexanoyl)-L-homoserinelactone,CAS143537-62-6,SigmaChemicalCo.)atconcentrationsrangingfrom0to500nM.GFPuorescenceandopticaldensityofeachwellweremeasuredoveraperiodof25hatroomtemperature.Asdescribedinthe 66

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SupportingInformationwethenttheresultingmultidimensionaldataset(opticaldensityandGFPuorescence[AHL]time)tothemodelthatisformulatedinSection 4.2 4.6.3LaneapparatusandimagingWestudiedspatiotemporalpatternsofQSregulationinbacteria/agarmixturesthatwereloadedintorectangularlanesoflength32mmandcrosssection3.5mm2mm(widthdepth).Theseagarlaneswerepreparedbycastingtheagarmixtureintoablack-anodizedaluminumcomborframethatdenedfourparallelchannels(Figure 4-2 ).Theframerestedonaglasscoverslipthatwascoatedwithathin,transparentsiliconeelastomersealant(Sylgard184,DowCorningInc.).Thehumidityoftheagarwasmaintainedbycoveringthelaneswithaclearpolycarbonatelidduringmeasurements.GFPuorescenceexcitationwasprovidedbyblueLEDlightpassingthroughanexcitationlter(ThorlabsMF469-35)anddiffusivelyscatteredtowardthesamplebyalightdome(Figure 4-2 ).Thelightdomewasaplastichemisphere(15cmdiameter)whoseinteriorwascoatedwithahigh-reectance,non-uorescingBaSO4paint[ 85 ].Multiplescatteringoftheexcitationlightwithinthedomeyieldedhighlyuniformilluminationoftheagarlanes:Thevariationinilluminationacrosstheimageeldwaslessthan3%.Noexcitationlightwasrequiredforthebioluminescencemeasurements.LuminescenceandODprobelight(seebelow)fromthefourparallellaneswerecollectedthroughthesameopticalpath(Figure 4-2 )andimagedonaCCDcamera.Thelane/coverslipassemblywasseatedonablackanodizedaluminumbaseplatethatcontainedanarrayofpinholes(0.7mmdiameter),allowinginsitumeasurementsoftheagaropticaldensity(OD):greenLEDlightwasdirectedupwardthroughthepinholes(frombeneaththebaseplate)andthroughtheagartoproduceatransmittedlightimageonthecamera.Usingatimercircuittoswitchbetweentwolightsources(blueGFPuorescenceexcitationversusgreenpinholelightforOD)inalternateexposures,we 67

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collectedasequenceofODanduorescence/bioluminescenceimagesofeachlaneoverthemeasurementperiod.WeintroducedexogenousAHLintoanagarlanebydepositing1LofaconcentratedAHL(3-oxo-C6-HSL)solutionontothesurfaceoftheagaratoneterminusofthelane.(C1inthegurelabelsreferstothefullydiffused,t!1,AHLconcentrationthatresultedfromthisinitialloading.)TogeneratethesimplediffusionpatternshowninFigure 4-4 ,weuseduoresceindye(CASNo.2321-07-5,SigmaChemicalCo.)insteadofautoinducersolution.ImageswererecordedonaCCDcamera(13001030arrayof6.7mpixelswith12-bitreadout,cooledto)]TJ /F8 11.955 Tf 9.29 0 Td[(10C,MicroMax,PrincetonInstruments)througha2achromaticdoubletlens(MAP1075150-A,ThorLabs)andaGFPemissionlter(MF525-39,Thorlabs).CCDimageswerecollectedwithexposuretimes1-10sandarepetitionrateof0.004Hzoverperiodsof20-24h.TheCCDimageswerehardware-binnedby5pixelsinthey-direction(transversetodiffusion)andby2pixelsinthex-direction(alongthedirectionofdiffusion).Theimageframecaptureda13.8mmlengthalongeachofthefourlanes,ornearlyhalfofeach32mmlane. 68

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CHAPTER5SIGNALPROPAGATIONINAQUORUMSENSINGSYSTEMInthischapter,weintroduceasimple,spatiallyextendedQSmodelintendedtorepresenttheLuxR-LuxIsystempresentinAliivibrioscheri.Inthissystem,theluxRgeneencodesthetranscriptionfactorLuxR.TheluxICDABEGoperonencodestheLuxIenzymeaswellasthecomponentsnecessaryforbioluminescence.LuxIcatalyzesthesynthesisofanAHLthattranscriptionallyactivatesLuxR.TheLuxR/AHLcomplexactivatestheexpressionoftheluxICDABEGoperon,thuscreatingapositivefeedbackloop.(SeeSection 3.2 foramorecompletedescriptionoftheLuxR-LuxIsystem.)Asinthepreviouschapter,wemodelacolonyofbacteria(inthiscase,A.scheri)embeddedinagarinalongrectangularlaneenvironment.Weagainassumethatthelaneishomogeneousacrossitswidthandthereforedescribethelaneinonespatialdimensionx.However,inthischapter,weassumethatthelanehasinnitelength,thatis,thatx2R.OurmodelisabletodescribeaQSshiftingeneexpressiononthecolonylevel,whichappearsasatravelingwave.AswementionedinSection 3.3 ,Daninoetal.andWardetal.bothstudiedwavesofgeneexpressioninQSsystemsviaexperimentsandmodeling.[ 20 90 ]However,neitheroftheseteamsofauthorsprovestheexistenceofatravelingwavesolutiontotheirmodels,aswedo.Ourmathematicalproofoftheexistenceofasolutionthatexhibitsthecharacteristicsofacolony-levelshiftingeneexpressiongivescredencetothemodelweusetodescribetheQSsysteminA.scheri.Furthermore,ourmathematicalrigorguaranteesthatthesolutionwedescribeisunaffectedbyanynumericalerrororartifactsofcomputationaltechnique.Inthefollowingsection,wepresentourmodeloftheLuxR-LuxIsysteminA.scheri.InSection 5.2 ,wegiveparameterconditionsunderwhichthereexistsatravelingwavesolutiontoourmodel.InSection 5.3 ,wegiveageneraltheoremand 69

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proofoftheexistenceofatravelingwavesolutiontoaclassofreaction-diffusionsystemsthatincludesourmodel. 5.1MathematicalModelTomodeltheLuxR-LuxIsystemfoundinA.scheri,weconsideronlyAHLconcentrationandLuxIconcentrationinaninniteone-dimensionaldomainx.ThoughtheconcentrationofLuxRplaysaroleintheLuxR-LuxIsystem,weassumethatitisnotalimitingfactorinoursystemanddonotmodelitexplicitly.WealsodonotmodelthepopulationdensityofA.scheri,thoughthegrowthphaseofbacteriaplaysaroleintheregulationofproteinsynthesis.[ 22 90 ]Thus,weexpectthatourmodelwillonlybevalidforshorttimescalesrelativetocellgrowth.ThefollowingmodelislargelyinspiredbythatinSection 4.2 ,buthasbeensimpliedtofacilitatethemathematicalexplorationofatravelingwave.AsAHLisfreelydiffusiblethroughtheA.schericellmembrane,wedescribethespatialspreadofAHL(concentrationA(x,t),nM)bythediffusionequation:@A @t=D@2A @x2whereD(mm2/h)isthediffusionconstant.[ 35 48 ]SinceLuxIcatalyzesA.s-cherisynthesisofAHL,theproductionrateofAHLdependsexplicitlyontheLuxIconcentrationL(unitsofnM).WedenotetheperunitLuxIproductionrateofAHLby(h)]TJ /F7 7.97 Tf 6.59 0 Td[(1).Finally,weassumethatAHLdegradesataconstantper-capitarate(h)]TJ /F7 7.97 Tf 6.58 0 Td[(1).ThoughAHLisstableonthescaleofhoursatneutralpH,AHLdegradesatanon-negligiblerateunderalkalineconditionsandinthepresenceofquorum-quenchingenzymes.[ 1 27 45 ]Hence,wedescribetheAHLconcentrationA(x,t)by@A @t=D@2A @x2+L)]TJ /F9 11.955 Tf 11.95 0 Td[(A.TherateofchangeoftheconcentrationofLuxIperunitvolumeofagarisexplicitlydependentontheconcentrationofAHLthroughtheactivationoftheexpressionof 70

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theluxICDABEGoperonbytheLuxR/AHLcomplex,modeledherewithaHillfunction( 5 ).TheHillfunctionenablesacooperativeswitchfromthesynthesis-offstatetothesynthesis-onstateviatheparametersaandm:f(A)=hAm am+Am (5)wherea(nM)isthehalf-saturationconstant,m(unitless)istheHillcoefcientandh(nMperh)isthemaximumLuxIproductionrate.Wedenotetheenzymaticper-capitaLuxIdecayrateby(h)]TJ /F7 7.97 Tf 6.58 0 Td[(1).Thus,theconcentrationofLuxIisgivenby@L @t=f(A))]TJ /F9 11.955 Tf 11.95 0 Td[(Lwheref(A)isgivenin( 5 ).Ourcompletemodelis@A @t=D@2A @x2+L)]TJ /F9 11.955 Tf 11.96 0 Td[(A (5)@L @t=f(A))]TJ /F9 11.955 Tf 11.95 0 Td[(L (5)wheref(A)=hAm am+Am.Table 5-1 givesasummaryofparameterandvariabledenitions.Wenotethatinourfutureexperiments,wewillnotexplicitlydetecttheconcentrationofLuxI.Instead,wewillassumethatthemeasuredluminescenceofthesystemisproportionaltotheconcentrationofLuxIasfollows:Sincethetranscriptionoflux-ICDABEGisnecessarytosynthesizebothLuxIandthecomponentsnecessaryforbioluminescence,weassumethattheproductionrateofLuxI(duetotheAHLconcentrationA)isproportionaltotherateofincreaseofluminescenceofthesystem.AssumingthatthesystembeginsdevoidofbothLuxIandluminescentcompounds,theconcentrationofLuxIisthenproportionaltothemeasuredluminescenceofthesystem. 71

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Table5-1. Asummaryofvariablesandparametersusedinthemodel( 5 ),( 5 ). Variable/DenitionUnitsParameter AAHLconcentrationnMDAHLdiffusionconstantmm2=hAHLproductionrateperunitofLuxIh)]TJ /F7 7.97 Tf 6.59 0 Td[(1AHLper-capitadegradationrateh)]TJ /F7 7.97 Tf 6.59 0 Td[(1LLuxIconcentrationnMfcooperativeswitchfunctionhmaximumLuxIproductionratenM=hahalf-activationcoefcientnMmHillcoefcient(unitless)per-capitadecayrateofLuxIh)]TJ /F7 7.97 Tf 6.59 0 Td[(1 5.2TravelingWaveSolutionof( 5 ),( 5 )Wewillnowndparameterrangesunderwhich( 5 ),( 5 )admitsatravelingwavesolution.Letthedomainsofxandtbeinnite.Mathematically,atravelingwavesolutionof( 5 ),( 5 )isasolutionoftheform(A(),L())where=x+ctforsomerealnumberc,calledthewavespeed,andthereexistsomeniterealnumbersA0
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(iii) E0andE2arestableandE1isasaddle.Thenthereexistssomec2Rsuchthat(A(x+ct),L(x+ct))=(A(),L())isabistablemonotoneincreasingtravelingwavesolutionto( 5 ),( 5 )withlim!(A(),L())=E0andlim!1(A(),L())=E2.Furthermore,thewavespeedchasthesamesignastheintegralRA201 f(A))]TJ /F14 7.97 Tf 13.21 5.26 Td[( AdA. Remark1. ItisinterestingtonotethatthesignofthewavespeedcisdeterminedbyanintegralwithintegrandequaltothedifferenceoftheL)]TJ /F10 11.955 Tf 12.62 0 Td[(andA)]TJ /F10 11.955 Tf 9.3 0 Td[(nullclinesofsystem( 5 ),( 5 ).WedefertheproofofTheorem 5.1 untilthefollowingsection(x 5.3 ),wherewegiveageneraltheoremandproofoftheexistenceofatravelingsolutiontoaclassofreaction-diffusionequationsthatincludessystem( 5 ),( 5 ).WenowexamineparameterrangesunderwhichConditions( i )-( iii )aresatised.Recallthat>0,h>0,a>0,m>0,>0and>0.Condition( i )requiresthatf(A)becontinuouslydifferentiableforA2[0,1).For0
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Weassumefortheremainderofthissectionthatm1.WenowexamineCondition( ii ).Clearly,(0,0)isazeroofsystem( 5 ),( 5 ).Thenullclinesof( 5 )and( 5 )aregivenbyL= AandL=1 f(A),respectively.(Figure 5-1 )Inordertoshowthatthereexisttwostrictlypositivesteadystatesof( 5 ),( 5 ),itsufcestoshowthatthereexist(atleast)twopositivesolutionsA1andA2to A=1 f(A) A=1 hAm am+Am h=Am)]TJ /F7 7.97 Tf 6.59 0 Td[(1 am+Am:=g(A). (5)Firstsupposethatm=1.Theng(A)=(a+A))]TJ /F7 7.97 Tf 6.59 0 Td[(1,g(0)=1 a,limA!1g(A)=0,g(A)>0forallA>0andd dAg(A)<0forallA>0.Thus,( 5 )hasexactlyonepositivesolutionandCondition( ii )doesnothold.Supposenowthatm>1.Theng(0)=0,limA!1g(A)=0,g(A)>0forallA>0nite,d dAg(A)>0forA20,(m)]TJ /F8 11.955 Tf 11.96 0 Td[(1)1 ma,andd dAg(A)<0forA>(m)]TJ /F8 11.955 Tf 11.96 0 Td[(1)1 ma.LetB:=g(m)]TJ /F8 11.955 Tf 11.95 0 Td[(1)1 ma>0denotethevalueofgevaluatedatitsuniquepositivecriticalpoint.Then( 5 )hasexactlytwopositivesolutionswhen0< hB.Thus,Condition( ii )maybesatisedonlyif0< h1onlyif
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Wehaveshownthatiftheaboveconditionholds,thenthereactionsystem( 5 ),( 5 )hasexactlythreexedpoints,E0=(0,0),E1=(A1,L1)andE2=(A2,L2)whereE1andE2arestrictlypositiveand00andthattherightmostfactoroftheaboveequationisthedifferenceoftheslopesoftheA)]TJ /F1 11.955 Tf 9.3 0 Td[(nullclineandtheL)]TJ /F1 11.955 Tf 9.3 0 Td[(nullclineofsystem( 5 ),( 5 ).WenowclaimthatifCondition( ii )holds,thendet(J(0,0))>0,det(J(A1,L1))<0,anddet(J(A2,L2))>0.Indeed,sinceCondition( ii )holds,m>1andd dA1 f(A)jA=0=0< .Then A>1 f(A)forsmallpositiveA.Notethatf(A)isasigmoidfunctiononthedomainA2[0,1).Specically,f(A)ispositive,strictlyincreasing,hasexactlyoneinectionpoint,andisbounded.ThensincetheA)]TJ /F1 11.955 Tf 12.62 0 Td[(andL)]TJ /F1 11.955 Tf 9.3 0 Td[(nullclinesintersectexactlytwiceforA>0(atA1 75

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andA2),itmustbetruethat8>>>>>><>>>>>>: A>1 f(A)forA2(0,A1) A<1 f(A)forA2(A1,A2) A>1 f(A)forA2(A2,1)andso <1 d dAf(A)A=A1and >1 d dAf(A)A=A2.(See,forexample,Figure 5-1 .)Thenby( 5 ),det(J(0,0))>0,det(J(A1,L1))<0,anddet(J(A2,L2))>0.FirstconsiderE0=(0,0).SincethedeterminantofJ(0,0)istheproductofitseigenvalues,andsincethisquantityispositive,theeigenvaluesofJ(0,0)musteitherberealandhavethesamesignorbeacomplexconjugatepair.SincethetraceofJ(0,0)isthesumofitseigenvaluesandtr(J(0,0))=)]TJ /F9 11.955 Tf 9.29 0 Td[()]TJ /F9 11.955 Tf 10.98 0 Td[(<0,therealpartsoftheeigenvaluesofJ(0,0)mustbenegative.Similarly,theeigenvaluesofJ(A2,L2)havenegativerealpart.ThenE0=(0,0)andE2=(A2,L2)arestable.NowconsiderJ(A1,L1).Sincedet(J(A1,L1))<0,J(A1,L1)musthaveonepositiverealeigenvalueandonenegativerealeigenvalue.ThenE1=(A1,L1)isasaddle.ThuswehaveshownthatifCondition( ii )holds,thenCondition( iii )holds.SinceCondition( ii )holdsifandonlyif( 5 )holds,Theorem 5.1 impliesthatif>0,h>0,a>0,m>1,>0and0<
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5.3TheExistenceofaTravelingWaveSolutiontoaClassofReaction-DiffusionSystemsHere,weshowtheexistenceofatravelingwavesolutiontothesystem8>><>>:@u @t=D@2u @x2+F(u,v)@v @t=G(u,v) (5)where 1. F,G2C1(R2+,R)andthereexistthreepointsE0=(0,0)T,E1=(a1,b1)TandE2=(a2,b2)Twith00,Gv(u,v)<0andFu(u,v)0for(u,v)2[0,a2][0,b2],Gu(u,v)>0for(u,v)2R2+nf(0,0)gandGu(u,v)j(0,0)0. 4. IfGu(u,v)j(0,0)=0andRa20F(u,VG(u))du>0whereVG(u)satisesG(u,VG(u))=0forallu2[0,a2],thenG(u,v)satisesG(u,v)=(u,v))]TJ /F3 11.955 Tf 12.79 0 Td[(mvwherem>0,(u,v)>0forallu>0,v2Randbounded.IfGu(u,v)j(0,0)=0andRa20F(u,VG(u))du<0,thenG(u,v)satisesG(u,v)=)]TJ /F9 11.955 Tf 9.3 0 Td[((a2)]TJ /F3 11.955 Tf 9.86 0 Td[(u,b2)]TJ /F3 11.955 Tf 9.87 0 Td[(v)+m(b2)]TJ /F3 11.955 Tf 9.86 0 Td[(v)wherem>0,(u,v)>0forallu
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functionVF(u)suchthatF(u,VF(u))=0forallu2[0,a2],VF(0)=0,VF(a1)=b1andVF(a2)=b2. Proof. BytheImplicitFunctionTheorem(Theorem 1.3 ),sinceG(0,0)=0andGv<0byConditions 1 and 3 ,thereexistssome0<1andauniquecontinuouslydifferentiablefunctionVG(u)suchthatG(u,VG(u))=0forallu2[0,1)andVG(0)=0.If1>a2,thenourconstructionofVG(u)iscomplete.Otherwise,1a2.WeclaimthatwecanextendVG(u)totheclosedintervalsuchthatG(u,VG(u))=0forallu2[0,1].Inordertomakethisextension,weneedonlyshowthatlimu!1VG(u)exists:First,notethateverywhereVG(u)isdened,d duG(u,VG(u))=Gu(u,VG(u))+Gv(u,VG(u))dVG(u) du. (5)Thenforallu2[0,1),sinceG(u,VG(u))=0,d duG(u,VG(u))=0.ByCondition 3 ,Gu0andGv<0,soby( 5 ),dVG du=)]TJ /F5 7.97 Tf 10.51 4.71 Td[(Gu Gv0.Thatis,VG(u)isanondecreasingfunctionofuforallu2[0,1).Furthermore,sinceGuandGvarecontinuousfunctions,Guisboundedon[0,a2][0,b2]andthereexistssomeG>0suchthatGv<)]TJ /F9 11.955 Tf 9.3 0 Td[(G<0on[0,a2][0,b2].ThenGu Gvremainsboundedon[0,a2][0,b2],sodVG duremainsboundedon[0,a2][0,b2].WeclaimthatVG(u)2[0,b2]wheneveru2[0,a2]andVG(u)exists.Supposenot.ThensinceVG(u)isanondecreasingfunctionandVG(0)=0,thereexistssomeu2(0,a2]suchthatVG(u)>b2.BytheconstructionofVG(u),G(u,VG(u))=0.ByCondition 3 ,Gv<0for(u,v)2[0,a2][0,b2].SinceGviscontinuous,wemayassumewithoutlossofgeneralitythatVG(u)iscloseenoughtob2thatGv(u,v)<0forallv2[0,VG(u)].SinceVG(u)>b2,wehavethatG(u,b2)>G(u,VG(u))=0.ByCondition 3 ,Gu>0.Thensinceua2,G(a2,b2)G(u,b2)>0,acontradictiontothatG(a2,b2)=0byCondition 1 .ThenVG(u)2[0,b2]wheneveru2[0,a2]anddVG duisboundedforu2[0,1)[0,a2].Thus,VG(u)isboundedontheinterval[0,1). 78

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SinceVG(u)isnondecreasingandboundedon[0,1),limu!1VG(u)=VG(1)exists.Then,bythecontinuityofG,G(u,VG(u))ju=1=0.Thus,VG(u)isacontinuouslydifferentiablefunctionsuchthatG(u,VG(u))=0forallu2[0,1)andG(1,VG(1))=0.ThentheImplicitFunctionTheorem(Theorem 1.3 )canbeappliedatu=1andVG(u)canbeextendedtotheinterval[0,1+2)forsome2>0.Iteratingthisuniqueconstructionandextension,wehavethatthereexistsauniquecontinuouslydifferentiablefunctionVG(u)suchthatG(u,VG(u))=0forallu2[0,a2]andVG(0)=0.WeclaimthatVG(a2)=b2.Supposenot.ThenVG(a2)>b2(orVG(a2)0),acontradictiontothedenitionofVG(u).Similarly,VG(a1)=b1.AsimilarargumentshowsthatthereexistsauniquecontinuouslydifferentiablefunctionVF(u)suchthatF(u,VF(u))=0forallu2[0,b2],VF(0)=0,VF(a1)=b1andVF(a2)=b2. NowsupposethatCondition 2 holdsandconsidertheJacobianDgofthereactionsystemg(u,v)asdenedinCondition 1 :Dg(u,v)=0B@FuFvGuGv1CA.ByCondition 2 ,botheigenvaluesofDg(E0)havenegativerealpart,sodet(Dg(E0))>0.ThenFuGv)]TJ /F3 11.955 Tf 11.95 0 Td[(FvGujE0>0FuGvjE0>FvGujE0Fu FvE0
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sinceFv>0andGv<0byCondition 3 .Wehavethat0=d duG(u,VG(u))=Gu+GvdVG du8u2[0,a2]and0=d duF(u,VF(u))=Fu+FvdVF du8u2[0,a2],so)]TJ /F5 7.97 Tf 10.51 4.71 Td[(Gu Gv=dVG duand)]TJ /F5 7.97 Tf 10.51 4.71 Td[(Fu Fv=dVF duforallu2[0,a2].Thenby( 5 ),dVG dujE0VF(u)forallu2(a1,a2). (5)ByCondition 3 ,wehavethatGv(u,v)<0andFv(u,v)>0for(u,v)2[0,a2][0,b2].Thus,wecandeterminetheregionsof[0,a2][0,b2]inwhichFandGarestrictlypositiveorstrictlynegative.TheseregionsaredisplayedinFigure 5-2 5.3.2TheWaveSpeedcNow,letusconsidertheresultwewishtoprove.Weseekamonotonetravelingwavesolutionofsystem( 5 )withlowerandupperlimitsatE0andE2,respectively.Thatis,weseekasolutiontosystem( 5 )oftheform(u(x,t),v(x,t))T=(U(),V())T,=x+ctforsomec2Rsuchthatlim!(U(),V())T=(0,0)Tandlim!1(U(),V())T=(a2,b2)T (5)where(0,0)T=E0and(a2,b2)T=E2areasdescribedinCondition 1 80

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Figure5-2. VF(u)(bluesolid)istheuniquecontinuouslydifferentiablefunctionsuchthatF(u,VF(u))=0forallu2[0,a2],VF(0)=0,VF(a1)=b1andVF(a2)=b2.VG(u)(reddashed)istheuniquecontinuouslydifferentiablefunctionsuchthatG(u,VG(u))=0forallu2[0,a2],VG(0)=0,VG(a1)=b1andVG(a2)=b2. Substituting(U(),V())Tinto( 5 ),weseethat(U(),V())Tmustsatisfy8>><>>:cU0=DU00+F(U,V)cV0=G(U,V) (5)wheretheprime(0)denotesd d.LettingW()=U0(),( 5 )isequivalentto8>>>>>><>>>>>>:U0=WcV0=G(U,V)cW=DW0+F(U,V). (5)Suppose(U(),V(),W())Tsolves( 5 )forsomec2R,U0()>0forallnite,V0()>0forallnite,lim!(U(),V(),W())T=(0,0,0)Tandlim!1(U(),V(),W())T=(a2,b2,0)T. (5)Then(U(),V())Tsatises( 5 ),( 5 )andisthereforeatravelingwavesolutionofsystem( 5 )withlowerandupperlimitsatE0andE2,respectively. 81

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SinceU0()>0forallnite,wemayreparameterizeVandWasfunctionsofU.LetV(U)=V((U))andW(U)=W((U))forU2(0,a2).By( 5 )and( 5 ),wehavecZ1(W())2d=DZ1W()W0()d+Z1F(U(),V())W()dcZ1(W())2d=D(W())2 21+Z1F(U(),V())U0()dcZ1(W())2d=Za20F(U,V(U))dU. (5)Thensincethesignoftherighthandsideof( 5 )isdeterminedbyc,Za20F(U,V(U))dU>0,c>0 (5)Za20F(U,V(U))dU=0,c=0 (5)Za20F(U,V(U))dU<0,c<0 (5)Furthermore,sinceW()=U0()>0forallnite,V0()>0forallnite,andcdV dU=cV0 U0=G(U,V(U)) W, (5)thesignofG(U,V(U))isthesameasthesignofcforallU2(0,a2).Ifc>0,thenbytheconstructionofVG(U),G(U,V(U))>0=G(U,VG(U))forallU2(0,a2).ByCondition 3 ,GV<0,sowehavethatV(U)0forall(U,V)2[0,a2][0,b2],soF(U,V(U))
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Thatis,ifc>0,thenRa20F(U,VG(U))dU>0.Similarly,ifc<0,thenZa20F(U,V(U))dU<0andG(U,V(U))<0=G(U,VG(U))forallU2(0,a2)by( 5 )and( 5 ),sobyCondition 3 ,F(U,VG(U))0forallniteandV0()>0forallnite.Then(U(),V())Tisamonotoneincreasingtravelingwavesolutionofsystem( 5 )withlowerandupperlimitsatE0andE2,respectively,andthesignofcisthesameasthesignofRa20F(U,VG(U))dUwhereVG(U)satisesG(U,VG(U))=0forallU2[0,a2]. 5.3.3Ra20F(U,VG(U))dU=0SupposerstthatRa20F(U,VG(U))dU=0.ThenbyLemma 7 ,if(U(),V(),W())Tsolves( 5 )and( 5 )forsomec2R,U0()>0forallniteandV0()>0forallnite,thenthereexistsamonotoneincreasingstationarywavesolutionto( 5 )and 83

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c=0.Ifc=0,then( 5 )isequivalentto8>><>>:U0=WW0=)]TJ /F5 7.97 Tf 10.49 5.47 Td[(F(U,VG(U)) D (5)whereVG(U)istheuniquecontinuouslydifferentiablefunctionwhichsatisesG(U,VG(U))=0forallU2[0,a2],VG(0)=0,VG(a1)=b1andVG(a2)=b2.SuchafunctionexistsbyLemma 6 .Weclaimthat(0,0)and(a2,0)aresaddlepointsof( 5 ).Indeed,theJacobianof( 5 )at(U,W)isJ=26401)]TJ /F7 7.97 Tf 11.72 4.71 Td[(1 Dd dUF(U,VG(U))0375whichhaszerotraceanddeterminantequalto1 Dd dUF(U,VG(U)).Sincethetraceiszero,theeigenvaluesofJmusteitherberealandofoppositesignorbeacomplexconjugatepairwithzerorealpart.Intheformercase,(0,0)and(a2,0)aresaddlepointsof( 5 ).Wewillexcludethelatterpossibility.SupposebywayofcontradictionthattheeigenvaluesofJareacomplexconjugatepairwithzerorealpart.Withoutlossofgenerality,wewillconsider(0,0).ThenthedeterminantofJispositive,thatis,1 Dd dUF(U,VG(U))jU=0>0.SinceD>0,thisimpliesthatFU(U,VG(U))+FV(U,VG(U))d dUVG(U)U=0=d dUF(U,VG(U))U=0>0.Byconstruction,d dUVG(U)=)]TJ /F5 7.97 Tf 10.72 5.12 Td[(GU GV.ThentheaboveinequalityandthefactthatGV(0,0)<0implythatFU(0,0)GV(0,0))]TJ /F3 11.955 Tf 12.1 0 Td[(FV(0,0)GU(0,0)<0.Ontheotherhand,byCondition 2 ,FU(0,0)GV(0,0))]TJ /F3 11.955 Tf 11.95 0 Td[(FV(0,0)GU(0,0)>0,acontradiction.Let~x=(x1x2)beaneigenvectoroftheJacobianmatrixJwitheigenvalue.Then26401)]TJ /F7 7.97 Tf 11.73 4.7 Td[(1 Dd dUF(U,VG(U))03750B@x1x21CA=0B@x1x21CA. 84

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Wehaveadependentsystemofequationsandmayassumewithoutlossofgeneralitythatx1=1.Then~x=(1).IfweconsidereigenvectorsasexistinginU,W-space,thenthisshowsthatthereexistsaneigenvectortangenttotheunstablemanifoldof(0,0)pointingnortheastfrom(0,0)andthatthereexistsaneigenvectortangenttothestablemanifoldof(a2,0)pointingnorthwestfrom(a2,0).NotethatsinceVG(U)isstrictlyincreasing,thesameanalysisshowsthatinU,V,W-spacethereexistsaneigenvectortangenttotheunstablemanifoldof(0,0,0)pointingintothepositiveorthantandthatthereexistsaneigenvectortangenttothestablemanifoldof(a2,b2,0)pointingintotheregionf(U,V,W)j00g.Considertheenergyof( 5 ):K(U,W)=1 2W2()+ZU01 DF(s,VG(s))ds. (5)K(U,W)isaconservedquantity:d dK(U,W)=WW0+U01 DF(U,VG(U))=)]TJ /F3 11.955 Tf 9.29 0 Td[(W1 DF(U,VG(U))+W1 DF(U,VG(U))=0SinceK(U,W)isconserved,anytrajectorythatstartsonalevelsetofK(U,W)mustremainonthatsamelevelsetforall.Withthisinmind,weexaminetheenergyofthepoints(0,0)and(a2,0):K(a2,0)=Za201 DF(U,VG(U))dU=0=K(0,0). (5)Now,ifthelevelsetS0=f(U,W)jK(U,W)=0,U2[0,a2]gisconnected,thensinceVG(a2)=b2,( 5 )isasufcientconditiontoshowthatatrajectorythatstartsontheunstablemanifoldof(0,0)willapproach(a2,0)ininnitetime.WeclaimthatthesetS0 85

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isconnected.Indeed,by( 5 ),(U,W)2S0ifandonlyifU2[0,a2]andW=s )]TJ /F8 11.955 Tf 9.29 0 Td[(2ZU01 DF(s,VG(s))ds. (5)WechoosethepositiverootsincewerequireU0()=W()>0forallnite.NotethatbytheanalysisinSection 5.3.1 (summarizedinFigure 5-2 ),8>>>>>><>>>>>>:F(U,VG(U))<0forU2(0,a1)F(U,VG(U))>0forU2(a1,a2)F(U,VG(U))=0forU2f0,a1,a2gThen( 5 )givesW(U)asareal,continuousfunctionofUdenedontheintervalU2[0,a2].SinceS0isthegraphofW(U)asdenedin( 5 )forU2[0,a2],andsincethecontinuousimageofaconnectedsetisconnected,S0isconnected. 5.3.4Ra20F(U,VG(U))dU>0SupposethatRa20F(U,VG(U))dU>0.ThenbyLemma 7 ,if(U(),V(),W())Tsolves( 5 )and( 5 )forsomec2R,U0()>0forallniteandV0()>0forallnite,thenthereexistsamonotonetravelingwavesolutionto( 5 )andc>0.Then( 5 )isequivalentto8>>>>>><>>>>>>:U0=WV0=G(U,V) cW0=cW)]TJ /F5 7.97 Tf 6.58 0 Td[(F(U,V) D. (5)First,wewillshowthat(0,0,0)isasaddlepointof( 5 )withaone-dimensionalunstablemanifold,denotedWu(E0),andthatWu(E0)intersectsthepositiveorthant.Todoso,wewillneedthreelemmas. 86

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Lemma8. LetEbealinearlystablesteadystateofthereactionsystem8>><>>:du dt=F(u,v)dv dt=G(u,v).Thenforanyc>0,theJacobianmatrixofthevectoreldofthesystem( 5 )at(E,0)hasonerealpositiveeigenvalueandtwoeigenvalueswithnegativerealparts. Proof. RecallthatthetraceofaJacobianmatrixisthesumofitseigenvaluesandthatthedeterminantofaJacobianmatrixistheproductofitseigenvalues.ThensinceEislinearlystable,Fu+Gv<0andFuGv)]TJ /F3 11.955 Tf 11.95 0 Td[(GuFv>0. (5)ThecharacteristicequationoftheJacobianmatrixof( 5 )isdet2666640)]TJ /F8 11.955 Tf 9.3 0 Td[(1)]TJ /F5 7.97 Tf 10.5 4.7 Td[(Gu c)]TJ /F5 7.97 Tf 13.15 4.7 Td[(Gv c0Fu DFv D)]TJ /F5 7.97 Tf 14.28 4.7 Td[(c D377775=3)]TJ /F4 11.955 Tf 11.95 16.86 Td[(Gv c+c D2+Fu+Gv D)]TJ /F3 11.955 Tf 13.15 8.09 Td[(FuGv)]TJ /F3 11.955 Tf 11.95 0 Td[(FvGu cD=0whichcanberewrittenasp()=3+22+1+0=0=()]TJ /F9 11.955 Tf 11.95 0 Td[(1)()]TJ /F9 11.955 Tf 11.96 0 Td[(2)()]TJ /F9 11.955 Tf 11.95 0 Td[(3),where1,2and3aretherootsofp().By( 5 ),Condition 3 andsincec>0,12+13+23=1<0 (5)123=)]TJ /F9 11.955 Tf 9.3 0 Td[(0>0. (5)First,observethattheremustbearealpositiverootofp().Indeed,p(0)=0<0andlim!+1p()=+1.Therefore,bytheIntermediateValueTheorem,p()hasaroot12(0,+1).Then2and3musteitherbothberealorbeacomplexconjugatepair. 87

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Since12(0,+1),( 5 )impliesthat23>0.ThenRe(2)andRe(3)musthavethesamesign.Since12(0,+1),( 5 )impliesthatnotbothofRe(2)andRe(3)arepositive.Thus,both2and3havenegativerealparts. Lemma9. IfL()satisesL0()+mL()=()wherem>0,()>0forallniteandbounded,andifL()remainsboundedforall2R,thenL()=e)]TJ /F5 7.97 Tf 6.58 0 Td[(mRems(s)ds>0. Proof. Byassumption,L0()+mL()=()em(L0()+mL())=em()L()em=L(0)+Z0ems(s)dsL()=e)]TJ /F5 7.97 Tf 6.59 0 Td[(mL(0)+Z0ems(s)ds (5)Sincelim!L()isboundedandlim!e)]TJ /F5 7.97 Tf 6.59 0 Td[(m=1,wemusthavethatlim!L(0)+Z0ems(s)ds=0.Thatis,L(0)=)]TJ /F4 11.955 Tf 11.29 16.27 Td[(Z0ems(s)dsL(0)=Z0ems(s)ds.Thenby( 5 ),L()=e)]TJ /F5 7.97 Tf 6.59 0 Td[(mZ0ems(s)ds+Z0ems(s)ds=e)]TJ /F5 7.97 Tf 6.59 0 Td[(mZems(s)ds.Sinceispositive,L()=e)]TJ /F5 7.97 Tf 6.59 0 Td[(mRems(s)ds>0forany6=. 88

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Lemma10(Lemma2.2inJinandZhao[ 47 ]). Let(U(),V(),W())Tbeasolutionof( 5 )suchthatlim!(U(),V(),W())T=(0,0,0)T. (5)Thennear=,(U(),V(),W())TsatisesU()>0,V()>0,W()=U0()>0andV0()>0. Proof. ByLemma 8 ,(0,0,0)Thasaonedimensionalunstablemanifoldcorrespondingto(c).Let~x=(x1,x2,x3)TbeaneigenvectoroftheJacobiancorrespondingto(c).Then266664001Gu(0,0) cGv(0,0) c0)]TJ /F5 7.97 Tf 10.49 5.48 Td[(Fu(0,0) D)]TJ /F5 7.97 Tf 10.49 5.48 Td[(Fv(0,0) Dc D3777750BBBB@x1x2x31CCCCA=(c)0BBBB@x1x2x31CCCCAorequivalently8>>>>>><>>>>>>:x3=(c)x1Gu(0,0) cx1+Gv(0,0) cx2=(c)x2)]TJ /F5 7.97 Tf 10.49 5.48 Td[(Fu(0,0) Dx1)]TJ /F5 7.97 Tf 13.15 5.48 Td[(Fv(0,0) Dx2+c Dx3=(c)x3Withoutlossofgenerality,wecanassumethatx1=1.Then~x=0BBBB@1Gu(0,0) c(c))]TJ /F5 7.97 Tf 6.59 0 Td[(Gv(0,0)(c)1CCCCA.Sincec>0,(c)>0andbyCondition 3 ,x20andx3>0.Notethatthesolution(U(),V(),W())Tliestangenttotheeigenvector~xattheorigin.Thennear=,(U(),V(),W())TsatisesU()>0andW()=U0()>0.IfGu(0,0)>0,thenx2>0andwehavethatV()>0andV0()>0near=.IfGu(0,0)=0,thenbyCondition 4 ,V()satisesV0()+m cV()=1 c(U(),V())= 89

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1 c()wherem>0,()>0forallniteandbounded.Thenby( 5 ),Lemma 9 andsincec>0,V()=e)]TJ /F13 5.978 Tf 7.78 3.26 Td[(m cZems c(s) cds>0andhenceV0()>0near=. ByCondition 2 andLemma 8 ,E0isasaddlepointof( 5 )withaone-dimensionalunstablemanifoldWu(E0).ByLemma 10 ,Wu(E0)intersectsthepositiveorthant.Let0BBBB@U()V()W()1CCCCAbethesolutionof( 5 )withinitialconditioncontainedinWu(E0).(U(),V(),W())Tisdenedforall2(,max)wheremaxmaybeinnite.Wenownotethatasolution(U(),V(),W())Tto( 5 )dependsontheparameterc.Wewillnotexplicitlydenoteeachsolution'sdependenceonc,buttheappropriateassociationwillbeclearbycontext.Wehaveshownthatwemaychoosetheinitialconditionof(U(),V(),W())TsuchthatitliesbothinWu(E0)andinthepositiveorthant.Wemaychoosetheinitialconditionsuchthatitliesintheregion:=f(U,V,W)j00g. (5)(SeeFigure 5-3 .)Toshowthat(U(),V(),W())Tisatravelingwavesolutionto( 5 ),weneedtoshowthatlim!1(U(),V(),W())T=(a2,b2,0)Tand(U(),V(),W())T2forall2(,1).Tothisend,weexaminethepossiblepathsof(U(),V(),W())T.First,weshowthatasolutionmaynotexitthroughthefacewhereV=0orthefacewhereU=0.Notethatsincec>0,GU>0on(0,a2](0,b2]andG(0,0)=0 90

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(byConditions 1 and 3 ),V0>0intheregionf(U,V,W)j00g.Additionally,sinceU0=W,U0>0intheregionf(U,V,W)jU=0,0Vb2,W>0g.Thus,(U(),V(),W())Tmayintersect@onlyinthefacesf(U,V,W)jU=a2,0Vb2,W0g,f(U,V,W)j0Ua2,V=b2,W0gorf(U,V,W)j0Ua2,0Vb2,W=0g.Ourstrategyistoclassifyeachc2(0,1)basedonwhichregionof@thetrajectory(U(),V(),W())Tintersects. Figure5-3. Theregion. 5.3.4.1TheSetsP1andP2Weclassifyeachc>0intothreesets:P1,P2and(0,1)n(P1[P2).P1willbethesetofparametervaluescforwhichthesolution(U(),V(),W())Texitsthroughf(U,V,W)jU=a2,0Vb2,W>0gorf(U,V,W)j0Ua2,V=b2,W>0g.P2willbethesetofparametervaluescforwhichthesolutionexitsthroughf(U,V,W)j0
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P2).Wewillthenshowthatthisspeedcadmitsaheteroclinicorbitof( 5 )connecting(0,0,0)Tand(a2,b2,0)T.DeneP1=fc>0j91<1suchthatU(1)>a2orV(1)>b2 (5)andW()>082(,1]gP2=fc>0j92<1suchthatW(2)<0,U()0forall2(,0),V0()>0forall2(,0). Proof. ByLemma 10 ,W()>0forcloseto.SinceweassumethatU0=W>0forall2(,0),wemayreparameterizeVandWasfunctionsofU.LetV(U)=V((U))andW(U)=W((U))forU2(0,U(0)).ThenVandWsatisfythefollowingequations: V0=dV dU=G(U,V) cW (5a)W0=dW dU=cW)]TJ /F3 11.955 Tf 27.41 0 Td[(F(U,V) DW (5b)forU2(0,U(0))withtheinitialconditionsV(0)=0,W(0)=0. (5)WeclaimthatV0(U)>0forallU2(0,U(0)).Indeed,supposenot.ThenbyLemma 10 thereexistssomeu2(0,U(0))suchthatV0(u)=0andV0(U)>0forallU2(0,u). 92

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SinceW>0forU2(0,U(0))andc>0,wehavethatG(U,V(U))>0forallU2(0,u)andG(u,V(u))=0.Thenitmustbetruethatd dUG(U,V(U))jU=u0.Ontheotherhand,wehavethatd dUG(U,V(U))U=u=GU(U,V(U))+GV(U,V(U))dV(U) dUU=u=GU(U,V(U))jU=u>0byCondition 3 sinceu>0,acontradiction. Lemma12. Let(U(),V(),W())Tbeasolutionof( 5 )suchthat( 5 )holds.Supposethat00forall2(,0)where0max.ThenW()isboundedforall2(,0).Furthermore,if0=max,thenmax=1andW()isboundedforall2(,1). Proof. SinceU0>0on2(,0),wecanexpressVandWasfunctionsofUforU2(0,U0)whereU0=lim!0U().ThenV(U)andW(U)satisfy( 5 )and( 5 )forU2(0,U0).SupposebywayofcontradictionthatW(U)isunboundedforU2(0,U0).ThenlimU!U0W(U)=1.Since00for(U,V)2[0,a2][0,b2],wehavethat~M:=max[0,a2][0,b2]jF(U,V)jispositive.ThenforallU2(0,U0),W0=c D)]TJ /F3 11.955 Tf 13.15 8.09 Td[(F(U,V) DWc D+~M DWSincelimU!U0W(U)=1,thereexistssomeUsuchthatforallU2(U,U0),W(U)>1andjW0j
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andhenceW(U)0forall2(,0).ThenV()2(0,b2)forall2(,0). Proof. ByLemma 10 ,thereexistssome0suchthatV()2(0,b2)forall2(,).Letbethelargestsuch.If0,thenourproofiscomplete.Supposebywayofcontradictionthat<0.ThensinceV0()>0forall2(,0)(,)byLemma 11 andthemaximalityof,V()=b2.ThenG(U(),V())=G(U(),b2)0byCondition 3 .ThenV0()j==G(U(),V()) c=<0sincec>0.ThiscontradictsthatV0()>0forall2(,0)(,). Lemma14(Step1intheproofofTheorem2.1inJinandZhao[ 47 ]). Let(U(),V(),W())Tbeasolutionof( 5 )suchthat( 5 )holds.Letbesuchthat0c DU()>0andW0()>c DU0()>0forall2(,). Proof. First,weshowthatW()>0forall2(,).Supposebywayofcontradictionthatthereexistssome0
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generality,assumethat0istherstsuch.ThenW()>0forall2(,0).NotethatbyLemma 13 ,V()2(0,b2)forall2(,0).SinceU0>0forall2(,0),wemayreparameterizethesolutionandconsiderthesystem( 5 ),( 5 )forU2(0,U0)wherelim!0U()=U0a1.ByLemma 11 ,00andc>0,G(U,V(U))>0=G(U,VG(U))forU2(0,U0). (5)ByCondition 3 ,GV<0.Thiscombinedwith( 5 )showsthatV(U)0,soF(U,V(U))c D>0 (5)forallU2(0,U0).SinceW(0)=0,thisimpliesthatW(U0)>W(0)=0,acontradiction.ToshowthatW()>c DU()>0forall2(,),weconsidertheaboveproof.Notethat( 5 )holdsforall2(,)sinceU0>0forall2(,).ThenW(U)>c DUforallU2(0,U())andhence,W()>c DU()forall2(,).Similarly,by( 5 ),W0()>c DU0()>0forall2(,). Lemma15(Step2intheproofofTheorem2.1inJinandZhao[ 47 ]). Let(U(),V(),W())Tbeasolutionof( 5 )suchthat( 5 )holds.Letc>c:=r 2MD a1 (5)where0ca1 4D>0forall2[a1,0). 95

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Proof. First,notethatsinceF(a1,b1)=0andFV>0for(U,V)2[0,a2][0,b2],Mispositive.Nowwewillprovetheclaimofthelemma.BythecontinuityofU(),sinceU()0.Supposebywayofcontradictionthatthereexistssome2[a1,0)suchthatW()=ca1 4D.Withoutlossofgenerality,supposethatistherstsuch2[a1,0).ThenW()>ca1 4Dforall2[a1,).ByLemma 14 ,U0()>0forall2(,a1),andU0()>ca1 4D>0forall2[a1,)bytheaboveassumption,sowemayreparameterizethesolutionandconsiderthesystem( 5 ),( 5 )forU2(0,U()).SinceW(U())=ca1 4D,theremustexistsome^u2(a1,U())suchthatW(^u)=ca1 2DandW(U)>ca1 2DforallU2[a1,^u).ThenforallU2[a1,^u),dW dU=cW)]TJ /F3 11.955 Tf 27.4 0 Td[(F(U,V(U)) DWcW)]TJ /F3 11.955 Tf 27.4 0 Td[(M DW=c D)]TJ /F3 11.955 Tf 18.71 8.09 Td[(M DW>c D)]TJ /F3 11.955 Tf 19.46 8.09 Td[(M Dca1 2D=c2a1)]TJ /F8 11.955 Tf 11.95 0 Td[(2MD Dca1 (5)sinceV(U)2(0,b2)byLemma 13 andwhereMdenedin( 5 )ispositive.By( 5 )and( 5 ),d dUW(U)>0,U2[a1,^u).ThereforeW(^u)>W(a1),acontradiction. Lemma16. P1asdenedin( 5 )isnonempty. Proof. Weclaimthatforallc>casdenedin( 5 )and( 5 ),c2P1.Supposebywayofcontradictionthatthereexistssome~c>csuchthat~c=2P1.Theneither (i) U()a2andV()b2forall2(,max)or (ii) thereexistssome1a2orV(1)>b2andthereexistssome1suchthatW()=0. 96

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Case1:Supposethat( i )holds.Wewillshowthatthentheremustexistsomea2,yieldingacontradiction.Weclaimthatthereexistssomea10forall2(,max)sobyLemma 13 ,V()2(0,b2)forall2(,max).Then,byLemma 12 ,max=1andwehavethatU()=ZW(s)ds>Z0W(s)dssinceW()>082(,1)>Z0W(0)dssinceW0()>082(,1)byLemma 14 =W(0).Sincemax=1,wecanndsomelargeenoughsuchthatW(0)>a1,acontradiction.Thus,thereexistssomea10forall2(,a1).WeclaimthatW()>0forall2[a1,max).Supposenot.Thenthereexistssomeleast02[a1,max)suchthatW(0)=0andW()>0forall2[a1,0).ByLemma 15 ,W()>~ca1 4Dforall2[a1,0),sobythecontinuityofW()andsince00,acontradiction.Thus,W()>0forall2[a1,max).Now,byLemma 13 ,V()2(0,b2)forall2(,max).Then,byLemma 12 ,max=1.ThenwehavethatU()=a1+Za1W(s)ds>a1+Za1~ca1 4DdsbyLemma 15 =a1+~ca1 4D()]TJ /F9 11.955 Tf 11.96 0 Td[(a1). 97

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Sincemax=1,wecanndsomelargeenoughsuchthatU()>a2,acontradiction.Thus,thereexistssomea2~ca1 4DwheneverU()2[a1,a2)byLemma 15 andsincea20.Thenthereexistssome>0smallenoughsuchthata2+a2.Thiscontradictstheassumptionthat( i )holds.Case2:Supposethat( ii )holds.Thenthereexistssome1a2orV(1)>b2andthereexistssomenite1suchthatW()=0.Withoutlossofgenerality,letbetherstsuch.ThenW()>0forall2(,).NotethatU()isincreasingforall2(,).Sinceisnite,U()6=0.If00,acontradictiontothedenitionof.Ifa1U()0,acontradictiontothedenitionof.IfU()=a2,thenbyLemma 15 ,W()>~ca1 4Dforall2[a1,).BythecontinuityofW()andsince0,acontradictiontothedenitionof.IfU()>a2,then~c2P1,acontradictiontotheassumptionthat~c=2P1.Hence,( ii )doesnothold.Sinceneither( i )nor( ii )hold,wemusthavethat(c,1)P1.Thus,P16=;. 5.3.4.3P26=;ToshowthatP2isnonempty,wewillneedanotherlemma: Lemma17. Let(U(),V(),W())Tbeasolutionof( 5 )suchthat( 5 )holds.Ifthereexistssome00forall2(,0)andU()2(0,a2)forall2(,0],thenc2P2. Proof. NotethatsinceW()>0forall2(,0)andU()2(0,a2)forall2(,0],V()2(0,b2)forall2(,0)byLemma 13 .WeclaimthatV(0)2(0,b2).Indeed,sinceV()2(0,b2)forall2(,0),V(0)2[0,b2].ByLemma 11 ,V0()>0forall2(,0),soV(0)2(0,b2].SinceV0()>0forall2(,0),V0(0)0.Then,bytheequationforV0in( 5 )andsincec>0, 98

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G(U(0),V(0))0.IfV(0)=b2,thenG(U(0),b2)=G(U(0),V(0))0=G(a2,b2). (5)ByCondition 3 ,GU>0on(0,a2](0,b2],soby( 5 ),U(0)a2.ThiscontradictstheassumptionthatU(0)2(0,a2).Hence,V()2(0,b2)forall2(,0].SinceW()>0forall2(,0)andW(0)=0,W0(0)0.IfW0(0)<0,thenthereexistssome>0smallenoughsuchthat0+0forall2(,0).ConsiderW00():W00()=cW0())]TJ /F3 11.955 Tf 11.96 0 Td[(FU(U(),V())U0())]TJ /F3 11.955 Tf 11.96 0 Td[(FV(U(),V())V0() DW00()j=0=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(FV(U(),V())V0() D=0sinceW0(0)=0andU0(0)=W(0)=0.SinceU(0)2(0,a2)andV(0)2(0,b2),FV(U(0),V(0))>0byCondition 3 .Thus,itsufcestoshowthatV0(0)>0.First,weclaimthatU(0)>a1.Ifnot,thensinceU()ismonotonicallyincreasing,U()2(0,a1)forall2(,0).ThenbyLemma 14 ,W()>c DU()>0forall2(,0).ThisimpliesthatW(0)>0,acontradictiontothatW(0)=0.Hence,U(0)>a1.Now,bytheequationforW0in( 5 )andsinceW(0)=0,wehavethatF(U(0),V(0))=0=F(U(0),VF(U(0))). (5)ByCondition 3 ,FV(U,V)>0forall(U,V)2[0,a2][0,b2],soV(0)=VF(U(0))by( 5 ).SinceU(0)2(a1,a2)andby( 5 ),VG(U(0))>VF(U(0))=V(0). 99

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ByCondition 3 ,GV<0on[0,a2][0,b2],so0=G(U(0),VG(U(0)))0. Lemma18. P2asdenedin( 5 )isnonempty. Proof. Weclaimthatthereexistssome^c>0suchthat(0,^c)P2.Supposenot.Thenthereexistssomesequencefcig1i=1suchthatci!0asi!1andci=2P2foralli.InStep1,weshowthatforallci, (I) thereexistssomeleast0(whichmaybeinnite)suchthatlim!0U()=a2andW()>0forall2(,0).InStep2,wewillusethisfacttoobtainacontradiction.Foreachci,let(Ui,Vi,Wi)Tbeasolutionof( 5 )suchthat( 5 )holds.WewillsuppressthissubscriptiforthedurationofStep1,thoughitwillbeusedexplicitlyinStep2.Step1Weclaimthat( I )holdsforallci=2P2.Foreachci,eitherW()>0forall0forall0forall0forall2(,1).SinceV()ismonotonicallyincreasingandboundedforall2(,1),lim!1V()exists.ItiseasytoseethatW0(denedin( 5 ))isbounded,sobyBarbalat'sLemma(Lemma 1 ),lim!1W()=0.SinceU,VandWconverge,Lemma 4 impliesthatthetrajectorymustapproachasteadystate.Theonlysteadystatescontainedintheclosureofasdenedin( 5 ) 100

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are(0,0,0)T,(a1,b1,0)Tand(a2,b2,0)T.Wewillshowthatlim!1U()>a1.Thisimpliesthatlim!1U()=a2.Weclaimthatthereexistssomea1<1suchthatU(a1)=a1.Ifnot,thenU()Z0W(s)dssinceW()>082(,1)>Z0W(0)dssinceW0()>082(,1)byLemma 14 =W(0).Sincemax=1,wecanndsomelargeenoughsuchthatW(0)>a1,acontradiction.Thus,thereexistssomea1a1.Sincelim!1U()2f0,a1,a2g,lim!1U()=a2.Then( I )holds.Ifthereexistssome^0forall2(,0).IfU(0)a2,thenthereexistssome<0suchthatU()=a2,U()2(0,a2)forall2(,)andW()>0forall2(,).Then( I )holds.Step2(Step3intheproofofTheorem2.1inJinandZhao[ 47 ])Wehaveshownthat( I )holdsforallci=2P2.Thatis,forallci=2P2,thereexistssomeleasti0(whichmaybeinnite)suchthatlim!i0Ui()=a2andWi()>0forall2(,i0).SinceU0i()>0forall2(,i0),wemayreparameterizethesolutionandconsiderthesystem( 5 ),( 5 )forU2(0,a2).Thenforeachi2NandforallU2(0,a2),W0i(U)=ci D)]TJ /F3 11.955 Tf 13.15 8.09 Td[(F(U,Vi(U)) DWi(U)ZU0Wi(s)W0i(s)ds=ZU0ci DWi(s))]TJ /F8 11.955 Tf 14.91 8.08 Td[(1 DF(s,Vi(s))ds 101

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(Wi(U))2 2=ZU0ci DWi(s))]TJ /F8 11.955 Tf 14.91 8.09 Td[(1 DF(s,Vi(s))ds. (5)DeneAi:=supU2[0,a2]Wi(U)foreachiandm:=min(U,V)2[0,a2][0,b2]F(U,V).WeclaimthatAiisniteforeachi.SinceWiiscontinuousandsinceWi(0)=0by( 5 ),itsufcestoshowthatlimA!A(i0)Wi(A)<1.Ifi00forall2(,i0)byLemma 10 ,Ai>0foralli2N.AlsonotethatsinceF(a1,b1)=0andFV>0for(U,V)2[0,a2][0,b2],misnegative.Step2.1WeclaimthatWi(U)isuniformlyboundedon[0,a2]forlargei.Thatis,thereexistssomeN1>0andsomeA>0suchthatWi(U)AforalliN1andallU2[0,a2].Toprovethis,wewillshowthatAiisboundedforlargei.Thatis,wewillshowthatthereexistssomeN1>0andsomeA>0suchthat00.Notethath(A):=DA2+2ma2 2a2A 102

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isanincreasingfunctionofA.Sincem<0,limA!0h(A)=.ThensincelimA!1h(A)=1,thereexistssomeuniqueAsuchthath(A)=c1andh(A)c1ifandonlyifAA. (5)Sincelimi!1ci=0,thereexistssomeN1>0suchthatcic1foralliN1.Thenby( 5 )and( 5 ),AiAforalliN1.BythedenitionofAi,00andeachU2[0,a2],thereexistssomeN2>0(whichdependsonU)suchthatjG(U,Vi(U)))]TJ /F3 11.955 Tf 11.96 0 Td[(G(U,VG(U))j
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forallU2[0,a2].First,considerRUyGV(s,Vi(s)) ciWi(s)dsforxedU,y2[0,a2],y0suchthatGV(U,V))]TJ /F9 11.955 Tf 22.9 0 Td[(forall(U,V)2[0,a2][0,b2].ByLemma 13 ,Vi(U)2(0,b2)forallU2(0,a2).Then,bycontinuity,Vi(U)2[0,b2]forallU2[0,a2]foralli.Hence,GV(U,Vi(U)))]TJ /F9 11.955 Tf 23.33 0 Td[(forallU2[0,a2].ThenZUyGV(s,Vi(s)) ciWi(s)dsZUy)]TJ /F9 11.955 Tf 9.3 0 Td[( ciWi(s)ds.ByStep2.1,00on[0,a2][0,b2],sothereexistssome>0suchthat0GU(U,V)forall(U,V)2[0,a2][0,b2].ByLemma 13 ,Vi(U)2(0,b2)forallU2(0,a2).Then,bycontinuity,Vi(U)2[0,b2]forallU2[0,a2]foralli.Hence,0GU(U,Vi(U))forallU2[0,a2]. 104

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By( 5 ),jMi(U)j=ZU0GU(y,Vi(y))expZUyGV(s,Vi(s)) ciWi(s)dsdyZU0expZUyGV(s,Vi(s)) ciWi(s)dsdyforalli2NandforeachU2[0,a2].Thenlimi!1jMi(U)jlimi!1ZU0expZUyGV(s,Vi(s)) ciWi(s)dsdy=0 (5)foreachU2[0,a2]by( 5 )andtheDominatedConvergenceTheorem(Theorem 1.4 ).Then,sinceG(U,VG(U))=0forallU2[0,a2]bydenition,andbythedenitionofMi(U),jG(U,Vi(U)))]TJ /F3 11.955 Tf 11.96 0 Td[(G(U,VG(U))j=jG(U,Vi(U))j=jMi(U)jThenby( 5 ),limi!1jG(U,Vi(U)))]TJ /F3 11.955 Tf 11.95 0 Td[(G(U,VG(U))j=limi!1jMi(U)j=0foreachU2[0,a2].Thenforall>0andeachU2[0,a2],thereexistssomeN2>0(whichdependsonU)suchthatjG(U,Vi(U)))]TJ /F3 11.955 Tf 11.96 0 Td[(G(U,VG(U))j0suchthatGV(U,V))]TJ /F9 11.955 Tf 22.37 0 Td[(forall(U,V)2[0,a2][0,b2].Thenforalli2NandforeachU2[0,a2],jVi(U))]TJ /F3 11.955 Tf 11.95 0 Td[(VG(U)j(Vi(U))]TJ /F3 11.955 Tf 11.96 0 Td[(VG(U))Z10GV(U,Vi(U)+s[VG(U))-222(Vi(U)])ds=jG(U,VG(U)))]TJ /F3 11.955 Tf 11.96 0 Td[(G(U,Vi(U))j (5) 105

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ByStep2.2,forall>0andeachU2[0,a2],thereexistssomeN2>0(whichdependsonU)suchthatjG(U,Vi(U)))]TJ /F3 11.955 Tf 12.98 0 Td[(G(U,VG(U))j0.Thenthereexistssome>0suchthatjF(U,Vi(U)))]TJ /F3 11.955 Tf 12.47 0 Td[(F(U,VG(U))j0. 5.3.4.4P1andP2areOpenandDisjointNext,weshowthatP1andP2areopenbythecontinuityofsolutionswithrespecttoparameters,timeandinitialconditions(Lemma 5 )andbythecontinuityoftheunstablemanifoldwithrespecttoparameters(Appendix C ). Lemma19. P1andP2asdenedin( 5 )and( 5 )areopen. Proof. TheproofsofopennessofP1andP2aresimilar.Here,wewillgiveonlytheproofthatP1isopen.RecallthatP1=fc>0j91<1suchthatU(1)>a2orV(1)>b2andW()>082(,1]g. 106

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Letc12P1.Wewillshowthatthereexistssome>0suchthatc2P1wheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(cj<.Let(U0c1,V0c1,W0c1)T2Wu(0,c1)denotetheinitialcondition(=0)chosenontheunstablemanifoldoftheoriginandintheopenpositiveorthant.(RecallthatsuchachoiceispossiblebyLemmas 8 and 10 .)Then,sincec12P1,thereexistssome1<1suchthatU(1)>a2orV(1)>b2andW()>0forall2(,1].Withoutlossofgenerality,supposethatU(1)>a2.Foreach2[0,1],choosesuchthatB)]TJ /F8 11.955 Tf 5.48 -9.68 Td[((U(,c1),V(,c1),W(,c1))TiscontainedintheopenpositiveorthantandsuchthatB1)]TJ /F8 11.955 Tf 5.48 -9.69 Td[((U(1,c1),V(1,c1),W(1,c1))TisadditionallycontainedintheregionfU>a2g.NotethatthisprecludesthepossibilitythatanysteadystatesarecontainedinB)]TJ /F8 11.955 Tf 5.48 -9.68 Td[((U(,c1),V(,c1),W(,c1))T.Bythecontinuityofsolutionswithrespecttoinitialconditions,time,andparameters(Lemma 5 ),foreach2[0,1],thereexist,,andsuchthat(U(s,c),V(s,c),W(s,c))T2B)]TJ /F8 11.955 Tf 5.48 -9.68 Td[((U(,c1),V(,c1),W(,c1))Twhenever(U0c1,V0c1,W0c1)T)]TJ /F8 11.955 Tf 11.96 0 Td[((U0c,V0c,W0c)T<,jc1)]TJ /F3 11.955 Tf 12.17 0 Td[(cj0suchthatB)]TJ /F8 11.955 Tf 5.47 -9.69 Td[((U0c1,V0c1,W0c1)T\Wu(0,c)6=;whenever 107

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jc1)]TJ /F3 11.955 Tf 13.51 0 Td[(cj<.Let=minf,g.Thenforanyjc1)]TJ /F3 11.955 Tf 13.5 0 Td[(cj<,wemaychoose(U0c,V0c,W0c)T2Wu(0,c)suchthat(U0c1,V0c1,W0c1)T)]TJ /F8 11.955 Tf 11.95 0 Td[((U0c,V0c,W0c)T<.Thenby( 5 ),(U(s,c),V(s,c),W(s,c))Tiscontainedintheopenpositiveorthantfors2[0,1]and(U(1,c),V(1,c),W(1,c))TisadditionallycontainedinthesetfU>a2g.ThenbyLemma 10 ,c2P1.Thus,P1isopen. Lastly,P1\P2=;bydenition.Thus,P1andP2arenonempty,disjoint,andopen.Now,aswearguedinSection 5.3.4.1 ,thereexistssomec2(0,1)n(P1[P2). 5.3.4.5TheExistenceofaHeteroclinicConnection Lemma20. Letc2(0,1)n(P1[P2).Thenforall2(,max),U()
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Case1:a0.Thenthereexistssome>0smallenoughsuchthatU(a+)>a2andW()>0forall2(,a+].Thatis,c2P1,acontradiction.Case2:b0on(0,a2](0,b2]andhenceG(U(b),V(b))=G(U(b),b2)<0=G(a2,b2).Thensincec>0,V0(b)=G(U(b),V(b)) c<0,acontradiction.Case3:00forall<0.ThenitmustbetruethatW0(0)0andsinceW0=cW)]TJ /F5 7.97 Tf 6.59 0 Td[(F(U,V) D,wehavethatF(U(0),V(0))0.IfF(U(0),V(0))>0,thenW0(0)=cW(0))]TJ /F5 7.97 Tf 6.58 0 Td[(F(U(0),V(0)) D=)]TJ /F5 7.97 Tf 10.49 5.47 Td[(F(U(0),V(0)) D<0.Thenthereexistssome>0suchthatforall2(0,0+),W()0.ByLemma 11 ,V0(0)0.IfV0(0)>0,thenW00(0)<0,inwhichcasethereexistssome<0suchthatW()<0,acontradictiontothedenitionof0.IfV0(0)=0,thensinceV0=G(U,V) c,G(U(0),V(0))=0.RecallthatF(U(0),V(0))=0.Then(U(0),V(0),W(0))Tisasteadystateofsystem( 5 ).Thiscontradictstheuniquenessofsolutions.Case4:a=b<0.DeneLI=(U,V,W)TjU=a2,V=b2,W>0. 109

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(SeeFigure 5-3 .)Then(U(a),V(a),W(a))T2LI.ConsiderV00(a)=1 c(GU(U(a),V(a))U0(a)+GV(U(a),V(a))V0(a)).ByCondition 3 ,GU>0on(0,a2](0,b2].ByCondition 1 ,G(a2,b2)=0,soV0(a)=0.Byassumption,U0(a)=W(a)>0andc>0.Hence,V00(a)>0.Then,sinceV(a)=b2,V0(a)=0andV00(a)>0,thereexistssome0for(U,V)2[0,a2][0,b2],sobythecontinuityofFV,V(a)b2.Thenba,acontradiction.Case6:b=00,G(U(b),V(b))=G(U(b),b2)0=G(a2,b2).ByCondition 3 ,GU>0for(U,V)2R2+nf(0,0)g,sowemusthavethatU(b)a2.Thenab,acontradiction.Case7:a=b=0.Here,a,band0mustallbelessthanmaxandthereforenite.Then(U(a),V(a),W(a))T=(a2,b2,0)T,anequilibriumpointof( 5 ).Thiscontradictstheuniquenessofsolutions. 110

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Wenowhaveallofthelemmasnecessarytoshowthatc2R+n(P1[P2)admitsaheteroclinicorbitof( 5 )connecting(0,0,0)Tand(a2,b2,0)T. Lemma21. Let(U(),V(),W())Tbeasolutionof( 5 )suchthat( 5 )holds.Supposethatforall2(,max),00forall,theUandVcomponentsofthetrajectoryaremonotonicallyincreasingforall.Thus,U()andV()havelimitsas!1.ByBarbalat'sLemma(Lemma 1 ),ifU0=Wisuniformlycontinuous,thenlim!1W()=0.Indeed,sinceU,V,andWarebounded,FiscontinuousandW0=cW)]TJ /F5 7.97 Tf 6.59 0 Td[(F(U,V) D,wehavethatW0isboundedandhenceWisuniformlycontinuous.SinceU,VandWconverge,Lemma 4 impliesthatthetrajectorymustapproachasteadystate.Theonlysteadystatescontainedintheclosureofasdenedin( 5 )are(0,0,0)T,(a1,b1,0)Tand(a2,b2,0)T.Weclaimthat(U(),V(),W())Tcannotconvergeto(0,0,0)Torto(a1,b1,0)T.Indeed,sinceUandVaremonotonicallyincreasingforall2(,1)byLemmas 11 and 20 ,(U(),V(),W())Tcannotconvergeto(0,0,0)T.Supposebywayofcontradictionthat(U(),V(),W())Tconvergesto(a1,b1,0)T.ThenU()0forall2(,1).Thenlim!1W()>0.SinceWdoesnotapproachzeroas!1,(U(),V(),W())Tcannotconvergeto(a1,b1,0)T.Since(U(),V(),W())Tcannotconvergeto(0,0,0)Torto(a1,b1,0)T,thesolutionmustconvergeto(a2,b2,0)T. 5.3.4.6TheExistenceofaTravelingWaveByRemark 2 (afterLemma 20 )andLemmas 20 21 and 11 ,(U(),V(),W())Tsolves( 5 )and( 5 )forc2R+n(P1[P2),U0()>0forallniteandV0()>0 111

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forallnite.Then(U(),V())Tsatises( 5 ),( 5 )andisthereforeatravelingwavesolutionofsystem( 5 )withlowerandupperlimitsatE0andE2,respectively. 5.3.5Ra20F(U,VG(U))dU<0(Step4intheproofofTheorem2.1inJinandZhao[ 47 ])SupposethatRa20F(U,VG(U))dU<0.Bythechangeofvariablesu=a2)]TJ /F3 11.955 Tf 11.95 0 Td[(uandv=b2)]TJ /F3 11.955 Tf 11.95 0 Td[(v,( 5 )becomes8>><>>:@u @t=D@2u @x2+F(u,v)@v @t=G(u,v) (5)whereF(u,v)=)]TJ /F3 11.955 Tf 9.3 0 Td[(F(a2)]TJ /F8 11.955 Tf 12.2 0 Td[(u,b2)]TJ /F8 11.955 Tf 12.25 0 Td[(v)andG(u,v)=)]TJ /F3 11.955 Tf 9.3 0 Td[(G(a2)]TJ /F8 11.955 Tf 12.2 0 Td[(u,b2)]TJ /F8 11.955 Tf 12.24 0 Td[(v).DenevG(u):=b2)]TJ /F3 11.955 Tf 11.95 0 Td[(VG(a2)]TJ /F8 11.955 Tf 12.2 0 Td[(u).ThenG(u,vG(u))=)]TJ /F3 11.955 Tf 9.3 0 Td[(G(u,VG(u))=0forallu2[0,a2]bytheconstructionofVG(u)inSection 5.3.1 .BythedenitionsofFandG,g(u,v):=(F(u,v),G(u,v))ThasonlythreezerosE0=(0,0),E1=(a2)]TJ /F3 11.955 Tf 11.97 0 Td[(a1,b2)]TJ /F3 11.955 Tf 11.96 0 Td[(b1)andE2=(a2,b2)intheorderinterval[E0,E2].SinceFandGsatisfyConditions 1 4 for( 5 ),itiseasytoseethatZa20F(u,vG(u))du=)]TJ /F4 11.955 Tf 11.29 16.27 Td[(Za20F(U,VG(U))dU>0 112

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andthatFandGsatisfyConditions 1 4 for( 5 ).ThenbytheseriesofproofsinSection 5.3.4 ,thereexistssomec2R+suchthat(U(x+ct),V(x+ct))Tisamonotoneincreasingtravelingwavesolutionof( 5 )suchthatlim!(U(),V())T=(0,0)Tandlim!1(U(),V())T=(a2,b2)Twhere=x+ct.DeneU()=a2)]TJ /F8 11.955 Tf 13.42 2.66 Td[(U()]TJ /F9 11.955 Tf 9.3 0 Td[()andV()=b2)]TJ /F8 11.955 Tf 13.82 2.66 Td[(V()]TJ /F9 11.955 Tf 9.3 0 Td[()forall2R.Thenlim!(U(),V())T=(0,0)Tandlim!1(U(),V())T=(a2,b2)T.Itfollowsthat(U(),V())Tisamonotoneincreasingtravelingwavesolutionofsystem( 5 )withlowerandupperlimitsatE0andE2,respectively.ByLemma 7 ,thiswavehasnegativespeed,thatis,c<0. 5.3.6StatementofExistenceTheoremWesummarizetheresultsofthischapterinatheorem: Theorem5.2. Suppose( 5 )satisesConditions 1 4 .Thenthereexistssomec2Rsuchthat(U(x+ct),V(x+ct))=(U(),V())isabistablemonotoneincreasingtravel-ingwavesolutionto( 5 )withlim!(U(),V())=E0andlim!1(U(),V())=E2.Furthermore,thewavespeedchasthesamesignastheintegralRa20F(U,VG(U))dUwhereVG(U)satisesG(U,VG(U))=0forallU2[0,a2]. 113

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CHAPTER6FUTUREWORKThereareseveraldirectionsinwhichwemayextendthiswork.OneobviousextensionisdeterminingconditionsunderwhichthetravelingwavesolutionfoundinSection 5.3 isuniqueuptotranslation.WeexpectthatsucharesultwouldfollowfrommonotonicitypropertiesofthefunctionsFandGsimilarto,butpossiblystrongerthan,thoselistedinCondition 3 .AsufcientproofwouldshowthatthesetsP1andP2(denedin( 5 )and( 5 ))areconnectedandthatthelowerboundofP1isequaltotheupperboundofP2,orpossiblythatasc>0increases,thevalueofUatwhichthetrajectory(U(),V(),W())TintersectstheU,V-planeincreasesmonotonically.Anotherextensiontoourexistenceproofisananalysisofthestabilityofourtravelingwavesolution.Inaseriesofarticles,Evans(1972,1975)provesthatforacertainclassofreaction-diffusionsystems(thatincludesourmodel),atravelingwavesolutionisstableifandonlyifthecorrespondinglinearizationaboutthissolutionisstable.[ 28 31 ]Weexpectthattheseresultscouldbereadilyappliedtoourtravelingwavesolution.Finally,weplantoexperimentallyparameterizeourmodelofQSup-regulationinAliivibrioscheri,describedinSection 5.1 .TheparameterizationwillrequireincorporatingcellgrowthdependenceintherateofproductionofLuxIasafunctionofAHLconcentration.Thisdependencewillthenneedtobeappropriatelyremoved,perhapsbyapproximatingthetime-dependentcellgrowthratebytheinstantaneouscellgrowthrateatanappropriatetime.AnothercomplicationinourparameterizationliesinthatwewillnotbeabletodirectlymeasuretheAHLconcentrationA.WithoutatimeseriesrepresentationofA(t),wewouldnotbeabletodetermine,theAHLproductionrateperunitofLuxI.TodeducetheAHLconcentrationAinacultureofA.scheriattimet,weextractasampleofthecultureandremovethesupernatant.WethenintroducethesensorstrainE.coli+pJBA132,whoseresponsetoAHL 114

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iswell-characterized,andmeasuretheuorescenceresponse.ThisallowsustoapproximatetheAHLconcentrationA(t).OncewehaveparameterizedthemodelinSection 5.1 ,wewillbeabletodetermineparameterrangesunderwhichatravelingwavesolutionexistsandtonumericallyapproximatethespeedofsignalpropagationinaspatiallyextendedcolonyofA.scheri. 115

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APPENDIXAPROOFOFTHEOREM2.2InthisAppendixweshallproveTheorem 2.2 .FirstwesetIi=Rl0n(x)dxandIo=R1ln(x)dx.Usingthedenitionofn(x)in( 2 )and( 2 ),wehave:Ii=Zl0n(x)dx=Zl0ccosh(ix)+R idx=c isinh(il)+Rl i=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(R 1 i)]TJ /F8 11.955 Tf 16.28 8.09 Td[(1 oDoo Dii1 isinh(il)sinh(o(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l))+Rl iwhereisgivenby( 2 ),andsimilarlyforIo:Io=Z1ln(x)dx=Z1ldcosh(o(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x))+R odx=d osinh(o(1)]TJ /F3 11.955 Tf 11.95 0 Td[(l))+R(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l) o=R 1 i)]TJ /F8 11.955 Tf 16.28 8.09 Td[(1 o1 osinh(il)sinh(o(1)]TJ /F3 11.955 Tf 11.95 0 Td[(l))+R(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l) oRecallingthedenitionsofiandoin( 2 ),andusing( 2 ),itfollowsthat:Ii=)]TJ /F4 11.955 Tf 11.29 16.86 Td[(R1 i)]TJ /F8 11.955 Tf 16.28 8.09 Td[(1 op o is D sinhp il1 p Dsinhp o(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l)q D +Rl iIo=R1 i)]TJ /F8 11.955 Tf 16.28 8.09 Td[(1 o1 p os D sinhp il1 p Dsinhp o(1)]TJ /F3 11.955 Tf 11.95 0 Td[(l)q D +R(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l) oand=r o i1 p coshp il1 p Dsinh p o(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l)r D! 116

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+sinhp il1 p Dcosh p o(1)]TJ /F3 11.955 Tf 11.95 0 Td[(l)r D!Deningi=R1 i)]TJ /F8 11.955 Tf 16.28 8.09 Td[(1 op o i,o=R1 i)]TJ /F8 11.955 Tf 16.28 8.09 Td[(1 o1 p o,i=p ilo=p o(1)]TJ /F3 11.955 Tf 11.95 0 Td[(l),wendthat: Ii=)]TJ /F9 11.955 Tf 9.3 0 Td[(is D sinhi1 p Dsinhoq D +Rl i (A) Io=os D sinhi1 p Dsinhoq D +R(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l) o (A) and =r o i1 p coshi1 p Dsinh or D!+sinhi1 p Dcosh or D!(A)TheYieldY,TotalAbundanceAandLogRatioLasdenedin( 2 ),( 2 )and( 2 )respectively,canbewrittenmorecompactlyintermsofIiandIo:Y=(o)]TJ /F9 11.955 Tf 11.95 0 Td[(i)Io,A=Ii+Io,andL=ln(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l)Ii lIo.WestartbyexaminingthesignsofthederivativesdIi dDanddIo dD,andwillthenusethisinformationtodeterminethesignsofdY dD,dA dDanddL dD.Fact: dIi dD0anddIo dD0. 117

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Beforeprovingthis,weintroducemorenotation.Weset~o=op ,=q o i,~i=i=p ,andwelety=1 p D.Thenwecanrewritein( A )asfollows: =sinh(~oy)cosh(iy)+sinh(iy)cosh(~oy),(A)and( A )becomes:Ii=)]TJ /F8 11.955 Tf 9.91 0 Td[(~isinh(~oy)sinh(iy) y+Rl i,andthendIi dD=dy dDdIi dy=)]TJ /F8 11.955 Tf 9.92 0 Td[(~iy3 )]TJ /F8 11.955 Tf 9.3 0 Td[(2d dysinh(~oy)sinh(iy) y=~iy3 21 (y)2h(~ocosh(~oy)sinh(iy)+isinh(~oy)cosh(iy))(y))]TJ /F8 11.955 Tf 13.15 0 Td[(sinh(~oy)sinh(iy)yd dy+=~iy 22f(y),whereweused( A )inthelaststepandintroducedf(y):=sinh2(iy)[~oy)]TJ /F8 11.955 Tf 11.95 0 Td[(sinh(~oy)cosh(~oy)]+sinh2(~oy)[iy)]TJ /F8 11.955 Tf 11.96 0 Td[(sinh(iy)cosh(iy)]. (A)Thus,thesignofdIi dDisequaltothesignoff(y).Todeterminethis,weexaminethefunctiona)]TJ /F8 11.955 Tf 13.19 0 Td[(sinh(a)cosh(a)=a)]TJ /F7 7.97 Tf 14.38 4.7 Td[(1 2sinh(2a)fora0.Notethatwhena=0,1 2sinh(2a)=0andthatd da(a)=1cosh(2a)=d da)]TJ /F7 7.97 Tf 6.68 -4.98 Td[(1 2sinh(2a)fora0.Thus,a1 2sinh(2a)foralla0andconsequently( A )isnonpositivefory>0.ThisshowsthatdIi dD0.WeuseasimilarcalculationforIo.Setting~o=o=p ,werewriteIoasIo=~osinh(~oy)sinh(iy) y+R(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l) o. 118

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ThendIo dD=dy dDdIo dy=~oy3 )]TJ /F8 11.955 Tf 9.3 0 Td[(2d dysinh(~oy)sinh(iy) y=)]TJ /F8 11.955 Tf 9.91 0 Td[(~oy 22f(y)0forally>0.ProofofTheorem 2.2 1. dIo dD0.Thishasalreadybeenshown. 2. dY dD0.Indeed,sinceY=(o)]TJ /F9 11.955 Tf 11.95 0 Td[(i)IoanddIo dD0,itfollowsthatdY dD=2(o)]TJ /F9 11.955 Tf 11.96 0 Td[(i)dIo dD0. 3. dA dD0.SinceA=Ii+Io,itfollowsthatdA dD=dIi dD+dIo dD=~i1 2p D2f1=p D)]TJ /F8 11.955 Tf 12.57 0 Td[(~o1 2p D2f1=p D=1 2p D2(i)]TJ /F9 11.955 Tf 11.96 0 Td[(o)fp DRecallfrom( A )thatf(y)0ify>0.Moreover,~i)]TJ /F8 11.955 Tf 12.57 0 Td[(~o=R p 1 i)]TJ /F8 11.955 Tf 16.28 8.09 Td[(1 oo)]TJ /F9 11.955 Tf 11.96 0 Td[(i ip o>0,andthereforedA dD0. 4. dL dD0.NoticethatL=ln(1)]TJ /F3 11.955 Tf 11.95 0 Td[(l)Ii lIo=ln1)]TJ /F3 11.955 Tf 11.95 0 Td[(l l+lnIi Io. 119

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Moreover,Ii Io=Rl0n(x)dx R1ln(x)dx=R10n(x)dx R1ln(x)dx)]TJ /F8 11.955 Tf 11.96 0 Td[(1=A Y=(0)]TJ /F9 11.955 Tf 11.96 0 Td[(i))]TJ /F8 11.955 Tf 11.95 0 Td[(1SinceAisnonincreasingandYisnondecreasingwithD,itfollowsthatIi=IoandhenceLisnonincreasingwithD. 120

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APPENDIXBSKETCHOFTHEPROOFOFTHEOREM2.4HereweonlysketchpartoftheproofofTheorem 2.4 ,asitissimilartothatofTheorem 2.2 .SinceIi=Rl0n(x)dx,andn(x)isgivenby( 2 ),with( 2 ),thereholdsthatIi=)]TJ /F3 11.955 Tf 13.28 8.09 Td[(Doo Di2isinh(o(1)]TJ /F3 11.955 Tf 11.96 0 Td[(l))sinh(il)R(z)+Rl iwhereR(z)isgivenby( 2 ),andby( 2 ).By( 2 ),recalling( 2 )andsetting~o=p o(1)]TJ /F3 11.955 Tf 11.95 0 Td[(l)p andi=p il,itfollowsthatIi=)]TJ 12.98 16.14 Td[(p o ip sinh(~oy)sinh(iy) yR(z)+Rl iwhereasbefore,wehavesety=1=p D,andwhere:=sinh(~oy)cosh(iy)+1)]TJ /F3 11.955 Tf 11.96 0 Td[(z 1+zp cosh(~oy)sinh(iy)with=p o=(i).Therefore,dIi dD=dy dDdIi dy=+p o ip y3 2R(z)d dysinh(~oy)sinh(iy) yProceedingsimilarlyasintheproofofTheorem 2.2 ,butbeingcautiousbecausenowthefactor(1)]TJ /F3 11.955 Tf 11.95 0 Td[(z)=(1+z)appearsinthesecondtermofhere,itfollowsthat:dIi dD=+p o ip y 22R(z)sinh2(~oy)g(iy)+1)]TJ /F3 11.955 Tf 11.95 0 Td[(z 1+zp sinh2(iy)g(~oy),whereg(x)=x)]TJ /F8 11.955 Tf 11.96 0 Td[(sinh(x)cosh(x).Sinceg(x)0ifx0,itfollowsthatthesignofdIi=dDisoppositetothesignofR(z)(R(z)isdenedin( 2 )).Asthelatterisnegativeifz
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ifz>z,itfollowsthatdIi dD8>>>>>><>>>>>>:<0,ifz>z(weakbias)=0,ifz=z(criticalbias)>0,ifz
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APPENDIXCCONTINUITYOFTHESTABLEMANIFOLDWITHRESPECTTOPARAMETERSInthisappendix,weprovethatthestablemanifoldislocallycontinuouswithrespecttoparameters.ThisproofisanextensionofthatfoundinCoddingtonandLevinson'sclassictext,TheoryofOrdinaryDifferentialEquations.[ 17 ]Section C.5 closelyfollowstheprooffoundinCoddingtonandLevinson,butdiffersinthatCoddingtonandLevinsonuseaPicardIterationtoshowtheexistenceofthestablemanifoldwhileweusetheContractionMappingTheorem. C.1AssumptionsConsiderthesystemx0=A(c)x+f(t,x,c) (C)wherex2Rn,t2R,theprime)]TJ /F16 7.97 Tf 5.48 -5.35 Td[(0=d dtdenotesthetimederivative,c2RisaparameterandA(c)isareal-valuedmatrixwhoseentriesdependcontinuouslyonc,butareindependentoft.Supposethatf(t,x,c)dependscontinuouslyon(t,x,c)forx2,t0,andc2whereRn,~02int()andR.Supposethatf(t,0,c)=0forallt0,c2.Moreover,supposethatgivenany>0,thereexistsome>0andsomeT>0suchthatforanytT,kxk,k~xkandallc2,kf(t,~x,c))]TJ /F3 11.955 Tf 11.95 0 Td[(f(t,x,c)kk~x)]TJ /F3 11.955 Tf 11.95 0 Td[(xk. (C)ThisconditionisstrongerthantheusualassumptionthatfislocallyLipschitz. C.2OutlineFixc2andsupposethatA(c)haskcharacteristicrootswithnegativerealpartsandn)]TJ /F3 11.955 Tf 12.44 0 Td[(kcharacteristicrootswithpositiverealparts.Wewillshowthatthereexistsak)]TJ /F1 11.955 Tf 9.3 0 Td[(dimensionalmanifold~S(c)2Rn,somekcurvilinearcoordinatesy1,y2,...ykand 123

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somen)]TJ /F3 11.955 Tf 11.96 0 Td[(krealcontinuousfunctions j(y1,y2,...,yk,c),k+1jn,suchthat0BBBBBBBBBBBBBB@y1...yk k+1(y1,y2,...,yk,c)... n(y1,y2,...,yk,c)1CCCCCCCCCCCCCCAdenes~S(c)forsufcientlysmalljyij,1ik.WewillalsoshowthatthereexistsarealnonsingularmatrixP(c)whoseentriesdependcontinuouslyoncsuchthaty=P(c)xandx=P(c))]TJ /F7 7.97 Tf 6.59 0 Td[(10BBBBBBBBBBBBBB@y1...yk k+1(y1,y2,...,yk,c)... n(y1,y2,...,yk,c)1CCCCCCCCCCCCCCAdenesthesought-afterstablemanifoldS(c)intermsofthekcurvilinearcoordinatesy1,...,yk.Lastly,wewillshowthatifwexc1,thenwhenevercissufcientlyclosetoc1,S(c)isclosetoS(c1).Moreprecisely,weshowthatthereexistssomecompactset)]TJ /F1 11.955 Tf -437.63 -23.91 Td[((whichwillbedenedlater)containing(~0,c1)suchthatgivenany>0thereexists 124

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some>0suchthat0BBBBBBBBBBBBBB@y1...yk k+1(y1,y2,...,yk,c)... n(y1,y2,...,yk,c)1CCCCCCCCCCCCCCA)]TJ /F4 11.955 Tf 11.95 74.24 Td[(0BBBBBBBBBBBBBB@y1...yk k+1(y1,y2,...,yk,c1)... n(y1,y2,...,yk,c1)1CCCCCCCCCCCCCCA0,thereexistsa>0sothatifr(z)=zn+b1zn)]TJ /F7 7.97 Tf 6.59 0 Td[(1++bn,jai)]TJ /F3 11.955 Tf 11.96 0 Td[(bij
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planeandsuchthatBi(i)\Bj(j)=;foralli6=j.Let=min1iqi.ThenbyLemma 22 ,thereexistssome>0suchthatifr(z)=zn+b1zn)]TJ /F7 7.97 Tf 6.59 0 Td[(1++bn,jai)]TJ /F3 11.955 Tf 11.96 0 Td[(bij0suchthatjai)]TJ /F3 11.955 Tf 12.53 0 Td[(bij
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SinceA(c)variescontinuouslywithrespecttoc,andthereforeR(,c)does,eachPi(c)variescontinuouslywithrespecttocsolongastheconditionsunderwhichweconstructedPi(c)stillhold:Fixc1andletcbeclosetoc1.ThenPi(c)variesfromPi(c1)continuouslywithrespecttocsolongasmc1icharacteristicrootsofA(c)(includingmultiplicity)areenclosedby)]TJ /F5 7.97 Tf 6.78 5.3 Td[(c1iand)]TJ /F5 7.97 Tf 6.77 5.3 Td[(c1idoesnotintersectthespectrumofA(c).Leteach)]TJ /F5 7.97 Tf 6.78 5.29 Td[(c1ibethecircleofradiuscenteredatc1i,whereischosensothat)]TJ /F5 7.97 Tf 6.78 5.29 Td[(c1idoesnotintersecttherealaxisofthecomplexplaneandnotwo)]TJ /F5 7.97 Tf 6.77 5.29 Td[(c1iintersect.ThenbyLemma 22 andtheabovediscussion,Pi(c)variescontinuouslywithrespecttocwheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(cj<1,where1isthesameaswasdeterminedinSection C.3 .Recallthatsincejc1)]TJ /F3 11.955 Tf 13.13 0 Td[(cj<1,A(c)hasnoeigenvalueswithzerorealpart.SupposethatA(c)hasq)]TJ /F5 7.97 Tf -.59 -7.29 Td[(cdistincteigenvalueswithnegativerealpart(ci,1iq)]TJ /F5 7.97 Tf -.59 -7.29 Td[(c),q+c=qc)]TJ /F3 11.955 Tf 11.96 0 Td[(q)]TJ /F5 7.97 Tf -.59 -7.3 Td[(cdistincteigenvalueswithpositiverealpart(ci,q)]TJ /F5 7.97 Tf -.59 -7.3 Td[(c+1iqc).DeneP(c):=)]TJ /F8 11.955 Tf 16.94 8.08 Td[(1 2iZ)]TJ /F27 5.978 Tf 4.83 3.58 Td[(cR(,c)dwhere)]TJ /F16 7.97 Tf 6.77 4.34 Td[()]TJ /F5 7.97 Tf 0 -7.29 Td[(c()]TJ /F7 7.97 Tf 6.78 4.34 Td[(+c)isasimpleclosedcontourinthecomplexplanethatenclosesallofthecharacteristicrootsofA(c)withnegative(positive)realpart,each)]TJ /F5 7.97 Tf 6.78 4.34 Td[(ci,1iq)]TJ /F5 7.97 Tf -.59 -7.29 Td[(c(q)]TJ /F5 7.97 Tf -.58 -7.3 Td[(c+1iqc),andthatdoesnotintersectthespectrumofA(c).ThenbyCauchy'sResidueTheorem(Theorem 1.2 ),P)]TJ /F8 11.955 Tf 7.09 1.79 Td[((c)(P+(c))isthesumoftheprojectionsontoallgeneralizedeigenspacescorrespondingtoeigenvaluesenclosedby)]TJ /F16 7.97 Tf 6.77 4.34 Td[()]TJ /F5 7.97 Tf 0 -7.29 Td[(c()]TJ /F7 7.97 Tf 6.78 4.34 Td[(+c):P)]TJ /F8 11.955 Tf 7.08 1.79 Td[((c)=)]TJ /F8 11.955 Tf 16.94 8.09 Td[(1 2iZ)]TJ /F27 5.978 Tf 4.82 3.57 Td[()]TJ /F13 5.978 Tf 0 -5.09 Td[(cR(,c)d=)]TJ /F5 7.97 Tf 14.42 15.4 Td[(q)]TJ /F13 5.978 Tf -.38 -5.08 Td[(cXi=1Res=ciR(,c)=)]TJ /F5 7.97 Tf 14.42 15.39 Td[(q)]TJ /F13 5.978 Tf -.38 -5.09 Td[(cXi=11 2iZ)]TJ /F13 5.978 Tf 4.82 2.74 Td[(ciR(,c)d=q)]TJ /F13 5.978 Tf -.38 -5.09 Td[(cXi=1Pi(c). 127

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Similarly,P+(c)=qcXi=q)]TJ /F13 5.978 Tf -.38 -5.08 Td[(c+1Pi(c).Furthermore,bythePrimaryDecompositionTheorem(Theorem 1.1 )andsinceeachPi(c)isalinearoperator(andthereforeP)]TJ /F8 11.955 Tf 7.08 1.79 Td[((c)andP+(c)are),E)]TJ /F8 11.955 Tf 7.09 1.8 Td[((c):=P)]TJ /F8 11.955 Tf 7.08 1.8 Td[((c)X=q)]TJ /F13 5.978 Tf -.39 -5.09 Td[(ci=1~EciandE+(c):=P+(c)X=qci=q)]TJ /F13 5.978 Tf -.38 -5.09 Td[(c+1~Eci.Inparticular,P)]TJ /F8 11.955 Tf 7.09 1.79 Td[((c)mapsthedomainX=RnontothegeneralizedstableeigenspaceE)]TJ /F8 11.955 Tf 7.09 1.8 Td[((c)andP+(c)mapsthedomainX=RnontothegeneralizedunstableeigenspaceE+(c).SinceeachPi(c)variescontinuouslywithrespecttocwheneverjc1)]TJ /F3 11.955 Tf 12.39 0 Td[(cj<1,P(c)variescontinuouslywithrespecttocwheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(cj<1.SupposethatA(c1)haskeigenvalueswithnegativerealpartandn)]TJ /F3 11.955 Tf 11.99 0 Td[(keigenvalueswithpositiverealpart.Thendim(E)]TJ /F8 11.955 Tf 7.08 1.79 Td[((c1))=kanddim(E+(c1))=n)]TJ /F3 11.955 Tf 11 0 Td[(k.Letfv1,v2,...,vkgbeabasisforE)]TJ /F8 11.955 Tf 7.09 1.8 Td[((c1)andfvk+1,...,vngbeabasisforE+(c1).Thenthereexistssome2>0suchthatfP)]TJ /F8 11.955 Tf 7.09 1.79 Td[((c)v1,P)]TJ /F8 11.955 Tf 7.09 1.79 Td[((c)v2,...,P)]TJ /F8 11.955 Tf 7.09 1.79 Td[((c)vkgisabasisforE)]TJ /F8 11.955 Tf 7.08 1.79 Td[((c)andfP+(c)vk+1,...,P+(c)vngisabasisforE+(c)wheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(cj<2.Let=minf1,2ganddeneP(c):=[P)]TJ /F8 11.955 Tf 7.09 1.79 Td[((c)v1,P)]TJ /F8 11.955 Tf 7.09 1.79 Td[((c)v2,...,P)]TJ /F8 11.955 Tf 7.09 1.79 Td[((c)vk,P+(c)vk+1,...,P+(c)vn] (C)forjc1)]TJ /F3 11.955 Tf 12.58 0 Td[(cj<.ThensinceP)]TJ /F8 11.955 Tf 7.08 1.8 Td[((c)andP+(c)varycontinuouslywithrespecttoc,sodoesP(c).Recallthatforjc1)]TJ /F3 11.955 Tf 12.08 0 Td[(cj<1,A(c)haskeigenvalueswithnegativerealpartandn)]TJ /F3 11.955 Tf 12.51 0 Td[(keigenvalueswithpositiverealpart.BytheaboveconstructionofP(c),wehavethatP(c))]TJ /F7 7.97 Tf 6.59 0 Td[(1A(c)P(c)=0B@B1(c)00B2(c)1CA=B(c) (C) 128

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forjc1)]TJ /F3 11.955 Tf 13.57 0 Td[(cj<,whereB1(c)isamatrixofkrowsandcolumnshavingallitscharacteristicrootswithnegativerealpartsandB2(c)isamatrixofn)]TJ /F3 11.955 Tf 12.78 0 Td[(krowsandcolumnshavingallitscharacteristicrootswithpositiverealparts.Additionally,sinceP(c)andA(c)varycontinuouslywithrespecttoc,B1(c)andB2(c)varycontinuouslywithrespecttoc. C.5ConstructionoftheStableManifold C.5.1PreliminariesFixsomec2suchthatjc1)]TJ /F3 11.955 Tf 12.04 0 Td[(cj0,thereexist>0,T>0,notnecessarilyequaltothosefrom( C ),suchthatjg(t,~y,c))]TJ /F3 11.955 Tf 11.95 0 Td[(g(t,y,c)jj~y)]TJ /F3 11.955 Tf 11.96 0 Td[(yj (C)forj~yj,jyj,tTandallc2.LetU1(t,c)=0B@etB1(c)0001CAandU2(t,c)=0B@000etB2(c)1CA. 129

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ThenetB(c)=U1(t,c)+U2(t,c)and@ @tUj(c)=B(c)Uj(c)forj=1,2.Furthermore,asadirectconsequenceofTheorem 1.5 ,weobtainthefollowingboundsonU1(t,c)andU2(t,c): Lemma23. Let00bechosensothat2Kc c<1 2 (C)andlet,Tbeasin( C )forthechosen.Fixt0Tandlettt0.Considertheintegralequation(t,a,c)=U1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(t0,c)a+Ztt0U1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c)g(s,(s,a,c),c)ds)]TJ /F4 11.955 Tf 11.96 16.27 Td[(Z1tU2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c)g(s,(s,a,c),c)ds (C)wherea2Rnisaconstantvector.Wewillshowthatforaxedc,thereexistsauniquecontinuoussolution(t,a,c)to( C )onthespace[T,1) B 2Kcfcg,whereB 2Kcistheopenn)]TJ /F1 11.955 Tf 9.3 0 Td[(dimensionalballofradius 2Kccenteredattheoriginand B 2Kcdenotesitsclosure.LetusdenotebyC(Y)thesetofcontinuousfunctionsonametricspaceY.ConsiderthecompletemetricspaceoffunctionsCb[T,1) B 2Kcfcg=n~(t,a,c)2C[T,1) B 2Kcfcgsuchthat (C) 130

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~(t,0,c)=0,~(t,a,c)10.Wewillshowthatthereexistssome>0suchthatk()(t2,a2,c))]TJ /F9 11.955 Tf 11.95 0 Td[(()(t1,a1,c)k
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kU1(t2)]TJ /F3 11.955 Tf 11.95 0 Td[(t0,c)a2)]TJ /F3 11.955 Tf 11.96 0 Td[(U1(t1)]TJ /F3 11.955 Tf 11.96 0 Td[(t0,c)a1k (C)+Zt2t0U1(t2)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c)g(s,(s,a2,c),c)ds)]TJ /F4 11.955 Tf 11.95 16.27 Td[(Zt1t0U1(t1)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c)g(s,(s,a1,c),c)ds (C)+Z1t2U2(t2)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c)g(s,(s,a2,c),c)ds)]TJ /F4 11.955 Tf 11.95 16.27 Td[(Z1t1U2(t1)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c)g(s,(s,a1,c),c)ds (C)First,weconsider( C ).Bydenition,kU1(t2)]TJ /F3 11.955 Tf 11.96 0 Td[(t0,c)a2)]TJ /F3 11.955 Tf 11.95 0 Td[(U1(t1)]TJ /F3 11.955 Tf 11.95 0 Td[(t0,c)a1kkU1(t2)]TJ /F3 11.955 Tf 11.96 0 Td[(t0,c)kka2)]TJ /F3 11.955 Tf 11.95 0 Td[(a1k+ka1kkU1(t2)]TJ /F3 11.955 Tf 11.95 0 Td[(t0,c))]TJ /F3 11.955 Tf 11.96 0 Td[(U1(t1)]TJ /F3 11.955 Tf 11.96 0 Td[(t0,c)k=e(t2)]TJ /F5 7.97 Tf 6.59 0 Td[(t0)B1(c)ka2)]TJ /F3 11.955 Tf 11.96 0 Td[(a1k+ka1ke(t2)]TJ /F5 7.97 Tf 6.58 0 Td[(t0)B1(c))]TJ /F3 11.955 Tf 11.95 0 Td[(e(t1)]TJ /F5 7.97 Tf 6.59 0 Td[(t0)B1(c).Sincethematrixexponentialiscontinuous,andsincecisxed,thereexistssome1>0suchthate(t2)]TJ /F5 7.97 Tf 6.59 0 Td[(t0)B1(c))]TJ /F3 11.955 Tf 11.96 0 Td[(e(t1)]TJ /F5 7.97 Tf 6.58 0 Td[(t0)B1(c)
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Next,consider( C ).Firstnotethatby( C )andsinceiscontinuous,wemayconsiderg(t,(t,a,c),c)asafunctionthatiscontinuousin(t,a).LetG(t,a):=g(t,(t,a,c),c). (C)ThenG(t,a)iscontinuousin(t,a).Supposewithoutlossofgeneralitythatt2>t1.ThenbythedenitionofU1andrewritinggasin( C ),Zt2t0U1(t2)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c)G(s,a2)ds)]TJ /F4 11.955 Tf 11.96 16.27 Td[(Zt1t0U1(t1)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c)G(s,a1)dsZt1t0U1(t2)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c)G(s,a2))]TJ /F3 11.955 Tf 11.95 0 Td[(U1(t1)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c)G(s,a1)ds+Zt2t1U1(t2)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c)G(s,a2)ds=Zt1t0e(t2)]TJ /F5 7.97 Tf 6.59 0 Td[(s)B1(c)G(s,a2))]TJ /F3 11.955 Tf 11.95 0 Td[(e(t1)]TJ /F5 7.97 Tf 6.59 0 Td[(s)B1(c)G(s,a1)ds (C)+Zt2t1e(t2)]TJ /F5 7.97 Tf 6.59 0 Td[(s)B1(c)G(s,a2)ds (C)Consider( C )alone.SinceGiscontinuousandbythecontinuityofthematrixexponential,Zt1t0e(t2)]TJ /F5 7.97 Tf 6.59 0 Td[(s)B1(c)G(s,a2))]TJ /F3 11.955 Tf 11.96 0 Td[(e(t1)]TJ /F5 7.97 Tf 6.58 0 Td[(s)B1(c)G(s,a1)dsZt1t0e(t2)]TJ /F5 7.97 Tf 6.58 0 Td[(s)B1(c))]TJ /F3 11.955 Tf 11.95 0 Td[(e(t1)]TJ /F5 7.97 Tf 6.59 0 Td[(s)B1(c)kG(s,a2)kds+Zt1t0e(t1)]TJ /F5 7.97 Tf 6.59 0 Td[(s)B1(c)kG(s,a1))]TJ /F3 11.955 Tf 11.96 0 Td[(G(s,a2)kdsZt1t0e(t2)]TJ /F5 7.97 Tf 6.58 0 Td[(s)B1(c))]TJ /F3 11.955 Tf 11.95 0 Td[(e(t1)]TJ /F5 7.97 Tf 6.59 0 Td[(s)B1(c)M1ds+Zt1t0M2kG(s,a1))]TJ /F3 11.955 Tf 11.95 0 Td[(G(s,a2)kdse(t2)]TJ /F5 7.97 Tf 6.58 0 Td[(t1)B1(c))]TJ /F8 11.955 Tf 11.95 0 Td[(1Zt1t0e(t1)]TJ /F5 7.97 Tf 6.58 0 Td[(s)B1(c)M1ds+Zt1t0M2kG(s,a1))]TJ /F3 11.955 Tf 11.96 0 Td[(G(s,a2)kdse(t2)]TJ /F5 7.97 Tf 6.58 0 Td[(t1)B1(c))]TJ /F8 11.955 Tf 11.95 0 Td[(1M3+Zt1t0M2kG(s,a1))]TJ /F3 11.955 Tf 11.95 0 Td[(G(s,a2)kds (C) 133

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whereM1=max(s,a)2[T,t1] B 2KckG(s,a)kM2=maxs2[T,t1]e(t1)]TJ /F5 7.97 Tf 6.58 0 Td[(s)B1(c)andM3=M2(t1)]TJ /F3 11.955 Tf 12.24 0 Td[(T).NotethatallthreeofM1,M2andM3arenonnegativeandnite.Bythecontinuityofthematrixexponential,thereexistssome2>0suchthate(t2)]TJ /F5 7.97 Tf 6.59 0 Td[(t1)B1(c))]TJ /F8 11.955 Tf 11.96 0 Td[(1M3< 12wheneverjt2)]TJ /F3 11.955 Tf 12.49 0 Td[(t1j<2.SinceGiscontinuousin(t,a),Gisuniformlycontinuousfor(t,a)2[t0,t1] B 2Kc.Thenthereexistssome3>0suchthatkG(s,a1))]TJ /F3 11.955 Tf 11.96 0 Td[(G(s,a2)k< 12M2(t1)]TJ /F3 11.955 Tf 11.96 0 Td[(T)wheneverka2)]TJ /F3 11.955 Tf 11.96 0 Td[(a1k<3foralls2[T,t1].Thenby( C ),Zt1t0e(t2)]TJ /F5 7.97 Tf 6.59 0 Td[(s)B1(c)G(s,a2))]TJ /F3 11.955 Tf 11.96 0 Td[(e(t1)]TJ /F5 7.97 Tf 6.58 0 Td[(s)B1(c)G(s,a1)ds< 12+ 12= 6 (C)wheneverk(t2,a2,c))]TJ /F8 11.955 Tf 11.96 0 Td[((t1,a1,c)k
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Thenfort22t1,Zt2t1e(t2)]TJ /F5 7.97 Tf 6.58 0 Td[(s)B1(c)G(s,a2)dsjt2)]TJ /F3 11.955 Tf 11.96 0 Td[(t1jM4M5.Thenthereexistssome0<40suchthatZ1T1U2(t2)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c)g(s,(s,a2,c),c)ds)]TJ /F4 11.955 Tf 11.95 16.27 Td[(Z1T1U2(t1)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c)g(s,(s,a1,c),c)ds< 6.Thenbyananalysissimilartothatfor( C ),thereexistssome5suchthatZT1t2U2(t2)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c)g(s,(s,a2,c),c)ds)]TJ /F4 11.955 Tf 11.95 16.28 Td[(ZT1t1U2(t1)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c)g(s,(s,a1,c),c)ds< 6wheneverk(t2,a2,c))]TJ /F8 11.955 Tf 11.96 0 Td[((t1,a1,c)k<5.Now,bytheabovetwoequations,( C )and( C ),k()(t1,a1,c))]TJ /F9 11.955 Tf 11.95 0 Td[(()(t2,a2,c)k
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By( C ),Lemma 23 andby( C )sincek(t,a,c)k1<,k()(t,a,c)k1Kce)]TJ /F7 7.97 Tf 6.59 0 Td[((c+c)(t)]TJ /F5 7.97 Tf 6.59 0 Td[(t0)kak+Ztt0Kce)]TJ /F7 7.97 Tf 6.59 0 Td[((c+c)(t)]TJ /F5 7.97 Tf 6.59 0 Td[(s)ds+Z1tKcec(t)]TJ /F5 7.97 Tf 6.58 0 Td[(s)dst0,c>0,kak 2Kcandby( C ).Thustheclaimisproven. C.5.2.3isaContractionNext,weclaimthatisacontraction.Let1,22Cb.ThenbyLemma 23 ,( C ),( C ),and( C ),k(1)(t,a,c))]TJ /F9 11.955 Tf 11.96 0 Td[((2)(t,a,c)k1Ztt0Kce)]TJ /F7 7.97 Tf 6.58 0 Td[((c+c)(t)]TJ /F5 7.97 Tf 6.58 0 Td[(s)k1(s,a,c))]TJ /F9 11.955 Tf 11.96 0 Td[(2(s,a,c)k1ds+Z1tKcec(t)]TJ /F5 7.97 Tf 6.59 0 Td[(s)k1(s,a,c))]TJ /F9 11.955 Tf 11.96 0 Td[(2(s,a,c)k1dsk1(s,a,c))]TJ /F9 11.955 Tf 11.96 0 Td[(2(s,a,c)k1Kc c+c)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F7 7.97 Tf 6.58 0 Td[((c+c)(t)]TJ /F5 7.97 Tf 6.58 0 Td[(t0)+Kc c)]TJ /F8 11.955 Tf 5.47 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(ectk1(s,a,c))]TJ /F9 11.955 Tf 11.96 0 Td[(2(s,a,c)k12Kc c
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From( C )itfollowsthattherstkcomponentsofj(t0,a,c)arej(t0,a,c)=aj1jkandthelatercomponentsaregivenbyj(t0,a,c)=)]TJ /F4 11.955 Tf 11.29 16.85 Td[(Z1t0U2(t0)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c)g(s,(s,a,c),c)dsjk+1jnwhere()jdenotesthejthcomponent.Ifthefunctions jaredenedby j(a1,...,ak,c)=)]TJ /F4 11.955 Tf 11.29 16.86 Td[(Z1t0U2(t0)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c)g(s,(s,a,c),c)dsjfork+1jn,thenclearlytheinitialvaluesyj=j(t0,a,c)satisfytheequationsyj= j(y1,...,yk,c)k+1jninyspace,whichdeneamanifold~S(c)inyspace.Weclaimthatnosolutionpof( C )withp(t0)2 B 2Kcandp(t0)noton~Scansatisfykp(t)kforalltt0,whereisthesameaswasdenedin( C ).Indeed,supposebywayofcontradictionthatthereexistssomesolutionpof( C )withp(t0)2 B 2Kcsuchthatkp(t)kforalltt0.Thenbythevariationofparametersformula,p(t)=e(t)]TJ /F5 7.97 Tf 6.59 0 Td[(t0)B(c)p(t0)+Ztt0e(t)]TJ /F5 7.97 Tf 6.59 0 Td[(s)B(c)g(s,p(s),c)ds.BythedenitionsofU1andU2,wehavethatp(t)=(U1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(t0)+U2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(t0))p(t0)+Ztt0(U1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s)+U2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s))g(s,p(s),c)dsp(t)=U1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(t0)p(t0)+U2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(t0)b+Ztt0U1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s)g(s,p(s),c)ds)]TJ /F4 11.955 Tf 11.96 16.27 Td[(Z1tU2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s)g(s,p(s),c)ds (C)whereb=Z1t0U2(t0)]TJ /F3 11.955 Tf 11.95 0 Td[(s)g(s,p(s),c)ds+p(t0) 137

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isaniteconstantvectorbyLemma 23 andby( C ).Sincethelefthandsideof( C )isboundedast!1byassumption,therighthandsidemustbealso.Eachtermontherighthandsideof( C ),exceptU2(t)]TJ /F3 11.955 Tf 12.35 0 Td[(t0)b,isboundedast!1byLemma 23 andby( C ).ThenitmustbetruethatU2(t)]TJ /F3 11.955 Tf 12.14 0 Td[(t0)bisboundedast!1.ByLemma 2 andsincealleigenvaluesofB2(c)havepositiverealparts,eachnonzerocomponentofU2(t)]TJ /F3 11.955 Tf 10.98 0 Td[(t0)growsunboundedast!1.Thenitmustbetruethatbj=0fork+1jn.Thus,by( C ),p(t)=U1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(t0)p(t0)+Ztt0U1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s)g(s,p(s),c)ds)]TJ /F4 11.955 Tf 11.95 16.28 Td[(Z1tU2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s)g(s,p(s),c)ds.Thenp(t)solves( C ),sop(t)ison~S,acontradiction.Weclaimthatx=P(c))]TJ /F7 7.97 Tf 6.58 0 Td[(10BBBBBBBBBBBBBB@y1...yk k+1(y1,...,yk,c)... n(y1,...,yk,c)1CCCCCCCCCCCCCCAdenesthestablemanifoldS(c)intermsofkcurvilinearcoordinatesy1,...,yk.Toprovethisclaim,weneedonlyshowthatforaxeda2 B 2Kc(andaxedc),limt!1(t,a,c)=0.Indeed,notethatsincesolves( C ),()=.Recallthatisacontractionwithcontractioncoefcient1 2.Thenforanyxedt2[t0,1),a2 B 2Kcandc2,k()(t,a,c))]TJ /F9 11.955 Tf 11.96 0 Td[((0)k1 2k(t,a,c)kk(t,a,c))]TJ /F3 11.955 Tf 11.96 0 Td[(U1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(t0,a)ak1 2k(t,a,c)kk(t,a,c)k2kakkU1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(t0,a)kk(t,a,c)k2kakKce)]TJ /F7 7.97 Tf 6.59 0 Td[((c+c)(t)]TJ /F5 7.97 Tf 6.59 0 Td[(t0) (C) 138

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by( C )andbyLemma 23 .Sincekak 2Kc,therighthandsideof( C )goestozeroastapproachesinnity.Thenlimt!1k(t,a,c)k=0andlimt!1(t,a,c)=0.Thustheclaimisproven. C.6(t,a,c)isContinuouswithRespecttocWenowshowthat(t,a,c)iscontinuouswithrespecttoc.First,weprovealemmaandestablishthedomainofexistenceof(t,a,c)forarangeofcvalues. Lemma24. Foraxedc1,thereexistsome,,Kand1allpositivesuchthatkU1(t,c)kKe)]TJ /F7 7.97 Tf 6.59 0 Td[((+)tforallt0kU2(t,c)kKetforallt0foralljc1)]TJ /F3 11.955 Tf 11.96 0 Td[(cj<1. Proof. WewillshowthatthereexistssomeK>0andsome1>0suchthatetB1(c)wKe)]TJ /F14 7.97 Tf 6.59 0 Td[(tkwk (C)forallw2Rk,t0,andc2[c1)]TJ /F9 11.955 Tf 11.98 0 Td[(1,c1+1]whereischosensuchthattherealpartsofalleigenvaluesofB1(c1)arelessthan)]TJ /F8 11.955 Tf 9.3 0 Td[(2.ThisgivesusananalogtoTheorem 1.5 thatholdsuniformlyforallc2[c1)]TJ /F9 11.955 Tf 12.44 0 Td[(1,c1+1].Thenbyasimilarproofandresultfor)]TJ /F3 11.955 Tf 9.3 0 Td[(B2(c),Lemma 24 holds.Leti,1ikbetheeigenvaluesofB1(c1).RecallthatalleigenvaluesofB1(c1)havenegativerealpart.Choose>0suchthatRe(i)<)]TJ /F8 11.955 Tf 9.3 0 Td[(2forall1ik.ThenbyTheorem 1.5 ,thereexistssomeK>0suchthatetB1(c1)w1Ke)]TJ /F7 7.97 Tf 6.59 0 Td[(2tkwk1 (C)wherekk1denotestheL1norm,denedasfollows.Letw2Rk,w=(w1,w2,...,wk)T.Thenkwk1:=kXi=1jwij. 139

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LetAbeakkconstantmatrixwithentriesaij.ThentheinducednormonAkAk1:=max1jkkXi=1jaijjisthemaximumoftheabsolutecolumnsumsofA.Fortheremainderofthisproof,letkk=kk1.WeclaimthatforanyconstantmatrixAsuchthatKkAk<,e(B1(c1)+A)tKe)]TJ /F14 7.97 Tf 6.59 0 Td[(t. (C)Indeed,considertheinitialvalueproblemdx dt=B1(c1)x+Ax,x(0)=x0.Bythevariationofparametersformula,andby( C ),wehavethatx(t)=eB1(c1)tx0+Zt0eB1(c1)(t)]TJ /F5 7.97 Tf 6.59 0 Td[(s)Ax(s)dskx(t)keB1(c1)tx0+Zt0eB1(c1)(t)]TJ /F5 7.97 Tf 6.58 0 Td[(s)kAkkx(s)kdsKe)]TJ /F7 7.97 Tf 6.59 0 Td[(2tkx0k+Zt0Ke)]TJ /F7 7.97 Tf 6.59 0 Td[(2(t)]TJ /F5 7.97 Tf 6.58 0 Td[(s)kAkkx(s)kdse2tkx(t)kKkx0k+Zt0KkAke2skx(s)kds.Letz(t):=e2tkx(t)k.Thentheaboveinequalitybecomesz(t)Kz(0)+Zt0KkAkz(s)ds.ByGronwall'sInequality(Theorem 1.7 ),z(t)Kz(0)eKkAkt 140

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e2tkx(t)kKkx0keKkAktkx(t)kKkx0ke()]TJ /F7 7.97 Tf 6.59 0 Td[(2+KkAk)t. (C)Now,sincee(B1(c1)+A)tistheprincipalfundamentalmatrixsolutionofdx dt=(B1(c1)+A)x,columniofe(B1(c1)+A)tisthesolutiontotheinitialvalueproblemdx dt=(B1(c1)+A)x,x(0)=eiwhereeidenotestheithstandardbasisvector.Letxi(t)bethissolution.Thenby( C ),kxi(t)kKe()]TJ /F7 7.97 Tf 6.59 0 Td[(2+KkAk)t.Thenthenormofeachcolumnofe(B1(c1)+A)tisboundedbythesamefunction.Thus,e(B1(c1)+A)tKe()]TJ /F7 7.97 Tf 6.59 0 Td[(2+KkAk)t.ChooseAsuchthatKkAk<.Thentheclaim( C )holds.Letc22andletA=B1(c2))]TJ /F3 11.955 Tf 12.91 0 Td[(B1(c1).ThensincetheentriesofB1(c)varycontinuouslywithrespecttocforjc1)]TJ /F3 11.955 Tf 13 0 Td[(cj<(whereisasinSection C.4 ),andsincethenormisacontinuousfunction,thereexistssome1
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Let)]TJ /F2 11.955 Tf 10.91 0 Td[(Rbeacompactsetsuchthat)]TJ /F2 11.955 Tf 10.91 0 Td[((c1)]TJ /F9 11.955 Tf 12.28 0 Td[(1,c1+1)andc1liesintheinteriorof)]TJ /F1 11.955 Tf 6.78 0 Td[(.Let1>0bechosensuchthat21K <1 2 (C)whereKandareasinLemma 24 .By( C ),thereexistsome>0,T>0suchthatkg(t,~y,c))]TJ /F3 11.955 Tf 11.95 0 Td[(g(t,y,c)k<1k~y)]TJ /F3 11.955 Tf 11.96 0 Td[(yk (C)forkyk<,k~yk<,tTandc2)]TJ /F2 11.955 Tf 11.14 0 Td[(.Thenbytheproofoftheexistenceofthestablemanifold(Section C.5 ),foreachc2)]TJ /F1 11.955 Tf 6.77 0 Td[(,thereexistsaunique(t,a,c)2Cb[T,1) B 2Kfcg,whereCbisdenedin( C ),thatsolves( C ).Notethatk(t,a,c)ke)]TJ /F7 7.97 Tf 6.59 0 Td[((+)(t)]TJ /F5 7.97 Tf 6.58 0 Td[(t0) (C)forall(t,a,c)2[T,1) B 2K)]TJ /F1 11.955 Tf 6.77 0 Td[(,tt0Tby( C ),wheretheestimate( C )holdsforallc2)]TJ /F1 11.955 Tf 10.1 0 Td[(sinceLemma 24 holdsforallc2)]TJ /F1 11.955 Tf 10.1 0 Td[(andsince1ischosen(andandTarefound)uniformlyforallc2)]TJ /F1 11.955 Tf 6.77 0 Td[(.Wenowshowthatforanyt0T,(t,a,c)variescontinuouslywithrespecttocfor(t,a,c)2[t0,1) B 2K)]TJ /F1 11.955 Tf 6.77 0 Td[(.Let>0andxsomet0T,tt0,a2 B 2K.Recallthatc1isxed.Weshowthatthereexistssome>0suchthatk(t,a,c1))]TJ /F9 11.955 Tf 11.96 0 Td[((t,a,c2)k
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forallt^T.If^T=t0,ourproofiscomplete.Otherwise,lett2[t0,^T].By( C ),k(t,a,c1))]TJ /F9 11.955 Tf 11.95 0 Td[((t,a,c2)kkU1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(t0,c1))]TJ /F3 11.955 Tf 11.95 0 Td[(U1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(t0,c2)kkak (C)+Ztt0kU1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c1))]TJ /F3 11.955 Tf 11.96 0 Td[(U1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c2)kkg(s,(s,a,c1),c1)kds (C)+Ztt0kU1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c2)kkg(s,(s,a,c1),c1))]TJ /F3 11.955 Tf 11.96 0 Td[(g(s,(s,a,c2),c1)kds (C)+Ztt0kU1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c2)kkg(s,(s,a,c2),c1))]TJ /F3 11.955 Tf 11.96 0 Td[(g(s,(s,a,c2),c2)kds (C)+Z1tkU2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c1))]TJ /F3 11.955 Tf 11.95 0 Td[(U2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c2)kkg(s,(s,a,c1),c1)kds (C)+Z1tkU2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c2)kkg(s,(s,a,c1),c1))]TJ /F3 11.955 Tf 11.96 0 Td[(g(s,(s,a,c2),c1)kds (C)+Z1tkU2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c2)kkg(s,(s,a,c2),c1))]TJ /F3 11.955 Tf 11.96 0 Td[(g(s,(s,a,c2),c2)kds (C)Firstconsider( C ).Notethatkak 2K.SinceU1(t,c)iscontinuouswithrespectto(t,c)for(t,c)2R)]TJ /F1 11.955 Tf 6.77 0 Td[(,U1(t,c)isuniformlycontinuouswithrespectto(t,c)for(t,c)2[T,^T])]TJ /F1 11.955 Tf 6.77 0 Td[(.Thenthereexistssome1>0suchthatkU1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(t0,c1))]TJ /F3 11.955 Tf 11.96 0 Td[(U1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(t0,c2)k
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Consider( C ).Since2Cb,k(t,a,c)k0suchthatkU1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c1))]TJ /F3 11.955 Tf 11.96 0 Td[(U1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c2)k< 10^T1wheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2j<2.ThenZtt0kU1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c1))]TJ /F3 11.955 Tf 11.96 0 Td[(U1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c2)kkg(s,(s,a,c1),c1)kds< 10 (C)wheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2j<2.Consider( C ).Sinceg(s,x,c)isuniformlycontinuousfor(s,x,c)2[T,^T][)]TJ /F9 11.955 Tf 9.3 0 Td[(,])]TJ /F1 11.955 Tf 10.09 0 Td[(andkk<,thereexistssome3>0suchthatkg(s,(s,a,c2),c1))]TJ /F3 11.955 Tf 11.95 0 Td[(g(s,(s,a,c2),c2)k<(+) 10Kwheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2j<3.ThenbytheaboveandbyLemma 24 ,Ztt0kU1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c2)kkg(s,(s,a,c2),c1))]TJ /F3 11.955 Tf 11.96 0 Td[(g(s,(s,a,c2),c2)kds< 10 (C)wheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2j<3.Consider( C ).By( C )andLemma 24 ,theintegral( C )isnite.Sincetheintegrandisnonnegative,thereexistssomeT1t0suchthatZ1T1kU2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c1))]TJ /F3 11.955 Tf 11.95 0 Td[(U2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c2)kkg(s,(s,a,c1),c1)kds< 20.NotethatsincegisboundeduniformlyandsinceLemma 24 givesauniformboundforallc2)]TJ /F1 11.955 Tf 6.77 0 Td[(,T1isindependentofc1,c22)]TJ /F1 11.955 Tf 6.77 0 Td[(.IftT1,thisboundisenough.Otherwise, 144

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considerZT1tkU2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c1))]TJ /F3 11.955 Tf 11.96 0 Td[(U2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c2)kkg(s,(s,a,c1),c1)kds.Asweshowedforequation( C ),thereexistssome4>0suchthatZT1tkU2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c1))]TJ /F3 11.955 Tf 11.95 0 Td[(U2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c2)kkg(s,(s,a,c1),c1)kds< 20wheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2j<4.ThenZ1tkU2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c1))]TJ /F3 11.955 Tf 11.95 0 Td[(U2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c2)kkg(s,(s,a,c1),c1)kds< 10 (C)wheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2j<4.Consider( C ).By( C )andLemma 24 ,theintegral( C )isnite.Sincetheintegrandisnonnegative,thereexistssomeT2t0suchthatZ1T2kU2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c2)kkg(s,(s,a,c2),c1))]TJ /F3 11.955 Tf 11.95 0 Td[(g(s,(s,a,c2),c2)kds< 20.SimilarlyasforT1,T2isindependentofc1,c22)]TJ /F1 11.955 Tf 6.78 0 Td[(.IftT2,thisboundisenough.Otherwise,considerZT2tkU2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c2)kkg(s,(s,a,c2),c1))]TJ /F3 11.955 Tf 11.96 0 Td[(g(s,(s,a,c2),c2)kds.Asweshowedforequation( C ),thereexistssome5>0suchthatZT2tkU2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c2)kkg(s,(s,a,c2),c1))]TJ /F3 11.955 Tf 11.95 0 Td[(g(s,(s,a,c2),c2)kds< 20wheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2j<5.ThenZ1tkU2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c2)kkg(s,(s,a,c2),c1))]TJ /F3 11.955 Tf 11.96 0 Td[(g(s,(s,a,c2),c2)kds< 10 (C)wheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2j<5. 145

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Combiningtheestimates( C ),( C ),( C ),( C )and( C )andbyapplying( C )to( C )and( C ),wehavethatk(t,a,c1))]TJ /F9 11.955 Tf 11.95 0 Td[((t,a,c2)k< 2+1Ztt0kU1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s,c2)kk(s,a,c1))]TJ /F9 11.955 Tf 11.96 0 Td[((s,a,c2)kds+1Z1tkU2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s,c2)kk(s,a,c1))]TJ /F9 11.955 Tf 11.96 0 Td[((s,a,c2)kds (C)wheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2j
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C.7TheStableManifoldisContinuouswithRespecttocSinceP(c)iscontinuouswithrespecttocandinvertible,andsince(t,a,c)iscontinuouswithrespecttoc,thecompositionP)]TJ /F7 7.97 Tf 6.59 0 Td[(1(c)(t,a,c)iscontinuouswithrespecttoc.Sincethisdenesthestablemanifold,wehaveshownthatthestablemanifoldiscontinuouswithrespecttoc.Thatis,givenc1,>0,anda2 B 2K,thereexistssome>0,T>0suchthatsupt2[T,1)P)]TJ /F7 7.97 Tf 6.59 0 Td[(1(c1)(t,a,c1))]TJ /F3 11.955 Tf 11.96 0 Td[(P)]TJ /F7 7.97 Tf 6.58 0 Td[(1(c2)(t,a,c2)< (C)wheneverjc1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2j<. Remark3. Asimilarstatementholdsfortheunstablemanifold.Theprooffollowsbymakingthesubstitutiont=)]TJ /F3 11.955 Tf 9.3 0 Td[(tin( C ). Remark4. Notethatif( C )isanautonomoussystem,thenwemaychooseTarbitrarilyin( C ). 147

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BIOGRAPHICALSKETCH JessicagraduatedfromtheUniversityofFloridain2008withaB.S.inmathematicswithminorsinphysicsandstatistics.Jessicacontinuedstudyingmathematics,inparticularmathematicalbiology,attheUniversityofFloridaandreceivedherPh.D.inmathematicsinAugust2013. 156