Stability and bifurcation analysis of reaction-diffusion systems with

Stability and bifurcation analysis of reaction-diffusion systems with

delays

by

@Rui Hu,B.S, M.S.

DISSERTATION Submitt~totheSchoolofGraduateStudiesof

MemorialUniversityof1Newfoundland

fortheDegreeof DOCTOR OF PHILOSOPHY

e art entof at e aticaand Statistics e oria University of e fo nd and

October,2009 St.John's, Newfoundland,Canada

Acknowledgements

Iwould like toexpressmy heartfelt gratitude to allthosewho gave me the possibility tocomplete this thesis.

Iamgreatlyindebtedtomysllpervisor,ProfessorYuanYuaD,forberhelp, stimulating suggestionsandencouragementduring my Ph.Dprogram,which was supportedbytheDepartment of Mathematics and Statistics, School of Graduate Studies, and Dr. Yuan's research grants, at Memorial University.

Ihavefurthermore to thank Professors Edgar Goodaire, ChunhuaOu, Jie Xiao,XiaoqiangZhaowhoeithertaughtmecoursesorgavemanyvaluablehelps formy studying and living.

Ialsoowe mysincere gratitudetomy friends and myfellowclassmates for theirhelp,supportand valuable hints: FangFang,MinChen,YijunLou,Qiong

Especially,Iarnobligedtomybelovedfarnily: mymother,father,parents- in-law and my brother.Theirconsistentsupporthas providedme the strength tokeepgoing.

Mylastanddeepestgratitudegoestomyhusband,ZhichunZhai,forhislove and greatconfidencein me all through these years.

Abstract

Theworkfocuseson the stability of steady state andlocal bifurcationanaIysis in partial differentialequationswith differentdelays.Especially,aneural network modelwithdiscretedelayanddiffusionisproposedinthefirstpart; a diffusive competition model with uniformlydistributeddelay is studied in part 2. An extendedreaction-diffusionsystemwithgeneraldistributed delay is treatedin part 3.Inthelast part, a Nicholson'sblowflies modelwithnonlocaI delayand

For adiffusiveneuralnetworkmodelwithdiscrete delay,by analyzingthe distributionsoftheeigenvalues ofthesystem and applying the centermanifold theoryandnormal formcomputation,weshow that,regardingthe connection coefficients as theperturbationparameter, thesystem,withdifferentboundary conditions,undergoessomebifurcations includingtranscritical bifurcation,Hopf bifurcationandHopf-zerobifurcation.The normal formsare givento determine

Insomecases,themodelwithdistributeddelayismoreaccuratethanthat withdiscrete delay. Westudy a competitiondiffusionsystemwith uniformly distributeddelay. Thecomplete analysis ofthecharacteristic equation is given

Andvia theanalysis,the stability of the constructedpositive spatiallynon- homogeneous steady state solutionis obtained.Moreover,theoccurrence of Hopfbifurcationnearthe steady state solutionis provedbyusingthe implicit function theorem with time delayasthebifurcationparameter.Finally, the forrnuladeterminingthe stability of the periodic solutionsisgiven

Theuniformly distributed kernel isonlyoneofthewidelyusedtimekernel Itis natural todiscllSS moregeneral timekernels.Weconsider a classof reaction- diffusion system with generalkernelfunctions.Thestability of the constructed positivespatiallynon-homogeneoussteady statesolutionisohtainedundergen- era!kernelsbyusing the similar methodin part 2.Moreover, taking minimal time delayasthebifurcationparameter, wecannotonlyshowtheexistenceof Hopfbifurcationsnearthesteadystatesolution,but alsoprovethattheHopf bifurcation isforwardand thebifurcated periodicsolutionsarestableundercer- taincondition.ThegeneraIresults are applied tocompetitiveandcooperative systems withweak kernelfunction

Inmany applicationmodels,ifindividualsmove,itismorereasonable to modeldelayanddiffusionsimultaneously, whichinducesnonlocaldelay by em- ployingBritton's randomwalkmethod. Westudy the stability of theuniform steady statesandHopfbifurcationofdiffusive Nicholson's blowfliesequation withnonlocaldelay.Byusingtheupper-andlower-solutionsmethod,wehave obtainedthe globalstabilityconditionsattheconstant steady states,anddis- cussedthelocalstability.Moreover,for a specialkernel,wehaveproved the occurrenceofHopfbifurcationnearthesteadystatesolutionandgivenformula indeterrniningstability of bifurcatedperiodicsolutions.

Contents

Acknowledgements

List ofSymbols 1 Introductionand preliminary

2.1 Neumannboundarycondition 2.1.1 Localstability

2.2 Dirichletboundarycondition 3Competitionsystem

3.1Existence o£ positive steady state solution and correspondingeigen-

3.2 Stability of thepositiveequilibrium 3.3 The existence ofHopf bifurcation 3.4 Stabilityofperiodicsolutions

3.5 Example andnumericalsimulation 4Areaction-diffusionsystem

4.1 Existenceofp05itivesteadystatesolution 4.2 Stabilityof thepositiveequilibrium 4.3 Hopfbifurcation.

4.4Examples andnumericalsimulation 5 Nicholson'sblowfliesequation

5.1Positivityandboundednessofsolution 5.2Globalasymptoticbehavioroftheuniformequilibria 5.3Linearizedstabilityofconstantsteady state

5.4Hopfbifurcationfromthenon-zerouniformstatewithstrongkernelll2 5.5 Numericalsimulations

Bibliography

List of Symbols

fieldof realnumbers n dimensional EuclideanSpace fieldofcomplexnumbers

X aHilbertspaceof functionsfrom

n

tol withinnerproductC,}, ( 0);r,X)(r>0)withtbe supremumnorm Zc Z iZ={XI+ix,lxj,x,EZ}

u,(O) u(I+O)

Chapter 1

Introduction and preliminary

Nonlinear dynamical systems areubiquitousinbiology,chemistry,engineeriog, ecology,economics,andevensociology. Thereis a vastliteratureon theap- plicationofnonlinear dynamicson th05edisciplines (seee.g.[19,20,21,29, 33,36,39,40,49,62,63,66,73,74,75,76)).The mathematicalanalysis of the dynamicalmodelsinscienceandengineeringmakesthesystematic study of complexinteractionbetweenfactors available, and deeperunderstandingof theentirety ofprocessesthat happeninsystemsis thereforepossible.Indeed, nonlineardynarnics andothersciencefieldshasbroughtgreatbenefit foreach other. Whiledynamicsplaysacrucialrolebyprovidingmethodsandtools,many developmentsin mathematical theoryis also stimulated by its application([88)) Thequalitative analysisfor modelsinnaturescienceincludesaspects ofstabil- ityandcomplexbifurcationbehavior, which are two of thefundamental tasksin dynamical theory. Studying thestabilitydetermines whether the systemsetUes down to equilibrium orkeepsrepeatingincycles.A dynamicalsystemusually hasseveralindependentparameters. With the varyingofsystem parameters

the stability canbe lost,then thequalitative properties havesignificant change

Bifurcationtheory is amain theme of dynamics.Inapplications,thebifur- cation theory attempts to explain various phenomena that have been discQvered and described in natural science. The principal theoriesfordealing with local bi- furcation analysis at fixed point are the center manifold and normalform.Both of them are fundamental and rigorollsmathematical techniques, whichingen- eralareusedtoreducethe dimensionality ofthesystem without changing the dynamical behaviors

Before the time of Volterra [72], inmostapplications, one assumed the system under consideration was independent of the past states andWBSonly determined by the present.However,itis getting apparent that theprincipleof setting modelsin the formof ordinary or partial differential equations, is often only a first approximationtothe consideredrealsystem ([28]). Andinsomecases,itis more realistic to includesome ofthe paststates ofthesesystems,i.e. a system shouldbe modeled by differentialequationswith time delays. Indeed,aftereffect arisesfrom variouscauses such aspresenceoftime delaysinactuatio andin informationtransmissionandprocessingof controlled signals,hatching period of species,durationof gestation, and slowreplacementoffoodsupplies (see e.g. [8,37,38,61,77]),anditcanberegardedasauniversalpropertyofthe surroundingworld.Sincelastcenturyl the study and application ofthe aftereffect havedeveloped and spread to a remarkable extentin biological1ecological and controlmodels,etc. (see e.g. [5,50,51,89]).Insome cases,itturns out that only certain past events influence onfutureones,forwhich the discrete delay canbeused to describe the hereditary systems with selective memory.For

example,discretedelay isa good approximationincontrol theory, whenitmodels afeedbacksignal transmitted as anerve impulse[78].But inotherdisciplines, discretedelaymay not bea good choice and spread of thedelayaround some meanvalue,i.e.adistributed.delay,ismorereasonable.Forexample,pollution of an environment bydeadorganismsisa cumulative effect126]

Inmanydisciplines,thedynamical modelsareintheformofreactiondiffusion equations sinceindividuals underconsideration are allowed todiffusespatially

(l52,60]}.Forinstance,iobiologicalorecologicalsystems,itiswellknownthat

mostspecieshavethe tendency thatmigratetowardsregionsoflower population density([14]).Theseeminglyrandommovement ofparticlessuspendedinafluid (i.e.aliquidorgas),knownasthefamous<lBrownianmotion",is described by areaction diffusionsysteminparticletheory(l4])

Incorporatingbothtemporaldelaysandspatialdiffusionintoamodelisvery natural to make the modelclosertothe reality, andthepartial functiona1differ- entialequations (PFDE)is applicable.In population ecology,thelogisticequa- tionwith delay and diffusion is proposed to describea single speciesdistributed uniformlyinanisolatedenvironment([54]);a three-compartment model with diffusionanddelay inone spacedimensionarisesin modelinggeneticrepression ([47J)

Inmostoftheexisting literature, investigatorssimplyaddadiffusion term tothecorresponding ordinarydifferentialequations.Recently,someresearchers pointedoutthat diffusion and timedelays arenot independentof each ather, sinceindividuals may movearound and shouldbeatdifferent pointsatdifferent times(l26]).Britton[3J is the firstoneto modeldelay and diffusion simul- taneouslyviarandom walkmethod foraFisherequation on aninfinitespatial

domain1inwhicha so-called spatiotemporaldelay ornonlocaldelay isintroduced (see, e.g. [22,25,70,80,81])

fusionsystemwithdelayhasbeenextensivelystudiedby many investigators(see [67,77]andreferencestherein).Theabstractform ofreactiondiffusion equations with time delay is

~=dD'u(t)+(.,u,)+F(.,u,) (1.0.1) whereu= d )= ( eisapara eter,dL >01D²isthe Laplacianoperator,dom(D2) CX X isaHilbert spaceoffunctions, andLisa linear operator and Fanonlinearfunction. Without loss ofgenerality,one can assume thatF(.,O)=0andDF(.,O)=0,i.e.thereis an equilibriumpointat the origin. Furthermore,F(e,·)has the Taylor expansion neartrivialequiJibrium F(.,u)=Fn(.,u) o(llull n~2, (1.0.2) whereFnisannmultilinearmapping

In[71],the existence and stabilitypropertiesof solutionsto(1.0. 1) are in- vestigated.lntheworkofI48],stableandunstablemanifoldsnearahyperbolic equilibrium of (1.0.1) wereconsidered. Basedon this work,Lin,So and Wu144!

developed acentermanifold theoryfor(1.0.1).Later,Faria derived a method toobtaintheexplicit normal form ofPFDE (1.0.1) by relating the PFDEtoa correspondingfunctionaldifferential equations (FDE).In117j withthefollowing hypothesis(H1)-(H4) (see also [44],[48jand177]),

(HI) dD' generates a Co semigroupT(t)'0onXwithIT(t)15MeW'forM~1, wERandt~O,andT(t)isacompactoperatorfort>O;

(H2)theeigenfunctions{lJ,}~ofdD2,withcorrespondingeigenvalues{o;)~, form an orthonormal basis forXand

o

^-ooask-00;

(H3)thesubspacesB,ofC, ,:=span(v('),i ,)i ,lvEC)satisfyL(B,)c span i ,

(H4)Lcanbe extendedto aboundedlinear operator fromBCtoX,where BC=(,p[-T,O]~X ,pis continuouson 0),3T,Iim, o,p( )EX), withthesuperemum norm,

thenormal form is proved to coincidewith the normal formforaFDEassociated withthegivenPFDE,uptoacertainorderofterms

In116],amoregeneralcaseisconsidered,i.e.,Ldoesnotsatisfy(H3),but thereexistbiocksofeigenfunctionsofdD²forminggeneralizedeigenspacessuch thatL ,for E,dom(L),canbe expressedas sHoearcombinationofthe generalized eigenfunctions.Forthis case, the assumptions (H2) and (H3) can be replaced by

(H2')let{o~': E ,=I""p,}betheeigenvalues of dD2andiJ~'be eigenfunctions corresponding to{O~·},suchthat{13~·:k E ,i=I, .,Pk}

(H3') thesubspaces ,ofC, ,:=span(v('),i')i ' vEC,i,=I,,,,,p,}

satisfyL(B,)cspan{iJ~," i :'

Withhypotheses(HI),(H2'),(H3'),(H4),theauthor showed thedecompo- sitionof thecharacteristicequation, which isapplicablefor thelocalstability analysis ofconstantsteady state solutions.The characteristic equation ofthe linearized systemofPFDE (1.0.1),

-dD -, =0

CHAPTER1.INTRODUCTiON AND PRELIMINARY forsomenonzeroyEdorn(D2),isequivalent to the sequenceof equations detLl..( )=O,(kEN),here

satisfying

forop=(opl, ,op )E ,,= ( O ,, ) L

the linearizedequation

~U(t) = dD2u(t)+L(',U')

isequivalentto the FDEitt)=M.z(t)+L.("z,)onCpo.Let .= E :is.asolution of detLl..( )=0 with =O}

and =

uf ,

^.,forsomeNE1 1.One canassume 0. Otherwise,there exists onlyastable and unstable manifold, andthe dynamical propertiesare

quiteclear.Thendecomposing as = where = an d

and<I>kistheeigenfunctionspaceoftheFDEonCp",correspondingtoAkoThus, thephasespacenfPFDE(1.0.1)canbedecomposedbyaprojection7r:C~p, P= ,Q= n and ,EC,

(1 1•• 1 .). [.(.•.).heing thehilinear form([30J).

Accordingto116. Theorem 4.1], if anotherhypothesis(HS')holds (H5')(DF,(u)(<p/3?),/3~")= . uEp. pEC{[-r,0];IR) for1~n~N,1in~Pnj>Nand1 ij Pj,then thenormalforms of thePFDE(1.0.1)and its associatedFDEarethesame,upto atleastthethird order terms onthecentermanifold. TheassociatedFDEis defined as

x(t)=R«.x,)+G«.x,) (1.0.3) where x(t)=(x.(t)):~lwith x. E IRP'.andR«).G«): CJ IR^JwithJ=

1 k^are

forp=... pNp CJ, p.= •...,.·p ···Cp.. k=l.···,N Faria's methodis veryuseful for theoreticalanalysis ofmanykinds of bi- furcations,including the importantHopfbifurcationwhichismarkedby the appearance ofa small periodic orbitnearthe steady state.Besidesusing Faria's approach,weeanalso employ themethodin131]for Hopfbifurcation, which needs toobtaina centermanifold first. Suppose thatwhenparameterf= the characteristic equation of thelinearequation of (1.0.1)hasa pair ofpurely imaginaryeigenvaluesiwoand-iwowithcorrespondingeigenfunctionsqandq

CHAPTER1.INTRODUCTIONAND PRELIMINARY respectively, ir The adjointeigenfunction isq·,thenonlinearfunctionF hasTaylorexpansion as(1.0.2)withn=2.Itis well known that

has a decomposition asXc=X'e^{X' whereX' ={zq}+ ^{lzE C}and} X' = uEXcI(q",u) =0).Thenu canbe written in theform

whereWEX'.Accordingtothe decomposition,the system (1.0.1)becomes

~~iwoz (q",F( o,zq Ui w»), w~^{L( o)l ow}+H(z,z,w),

H(z,z,w)=F(Eo,zq Ui w) (q",F(Eo,zq Uiw))q(q',F(EO, zq Ui w))q

w=W20Z'+WIlZZ+woii'+O(lzl'), H(z,z,w)=H20z'+Hllzi+Hoii'+O(lzl') Thenthe system (1.0.1) onthecentermanifoldis

(1.0.4)

T

⁼(q",F,(q,q,AO»),gil=(q",F,(q,q,Ao) F,(q,q,Ao»),

=(q",F,(q,q,Ao», =(q",F,(WIl,q,Ao) F,(W20,q,Ao) F,(q,q,Ao»

CHAPTER1.INTRODUCTION AND PRELIMINARY

L(>'o)lxow+H=

~

=

~~

+

~~

L(>.ollxo(w,oz'+WllZZ+ 02z') +H2OZ'+HlI H 2Z'+h.o.t+

= ( ,o + lI (+(q",F( .oo

+(WlIZ+ 02z)( iwoz+(,F( .o" , (1.0.5)

bycomparingthecoeflicientofz'andzzon bothsidesof(1.0.5),wehave

whereH2O.andHu are definedas

H2O=F,(q, q, >'0) (q",F,(q, q, 'o q( t,F,(q,q, 'o q,

HlI=F,(q, q, >'0)+F,(q, q, >'0) (q",F,(q, q, ' +F,(q,q, 'o q(qO,F,(q, q, >'0)+F,(q, q, 'o q WithW20,Wll,W02determinedas above,the flow onthecenter manifold (1.0.4)isobtained.Onecanfind a transformation

under which (1.0.4)canbe transformed into the Poincare form

c,(O)=

~[9209"

-219,,1' -

J

⁺

^T'

The bifurcationdirectionand the stability of thebifurcating periodicsolutions aredeterminedbYIVz=-~Re(c,(>.o))andRe(cl(>.o))respectively.The bifurcationissupercritical (subcritical)ifl z>0«0);thebifurcatingperiodic solutionsarestable (unstable) when e(c,(>.0))<0(>0).

Inthepresentwork,westudymodelsofneural network and populationdy- narnics in the formof (1.0.1). Inthe following we will describethemodels

InChapter 2, weconsideramodelincludingapairofneurons with time- delayedconnections betweenthe neuronsandtime delayedfeedbackfromeaeh

¥u

=d,D'u-u(t)+a/(u(t-r)) +b/(v(t-r)),

=d,D'v- v(t)+a/(v(t-r)+b/(u(t -r)) (1.0.6) The recurrent neuralnetworkssuchascellularneural networks(CNNs)and delayed cellular neuralnetworks(DeNNs)are widely used insomeimage pro cessing,quadratic optimizationand pattern recognition problems ([12],[131,[58]).

Becauseofthefiniteprocessingspeedofinformation,timedelaysareinevitably involvedinthemodelingofthe biological neuron networksorartificialneural networks. Sincetimedelaysmaylead to bifurcation,oscilJation, divergence or instability,thestudy of dynamicphenomenonofdelayedproblemisimportant

for high quality neural networks. In[86], by considering aneural networkof four identical neurons with time-delayedconnections, Yuan and Weigavesorne parameterregionsforglobal,localstabilityandsynchronization,anddiscussed the occurrence ofpitchforkbifurcation,HopfandequivariantHopfbifurcations.

Formore study of dynamics of delayedneutralnetworksystems,see184,86]and

Most previous work did not consider the effect of diffusion inneural net.- works.However, with themovementof neuronsthediffusionis unavoidable.

Forexarnple,inman-madeneuralnetworks,diffusioneffectsshouldbeinvolved whenelectrons arernovinginasymmetric electromagneticfields. Thestability of neuralnetworks with diffusion terms, but withoutdelay,have been consideredin literature (see e.g.[91.111J,[151.132]).Recently, the problem of delayed neural net- workswithdiffusion terrnsisattractingsome experts'attention.In[43LCaoand Lianggavenew sufficientconditionsforexistence,uniqueness and globalexpo- nentialstabilityoftheequilibriumpointofaclassofreaction-diffusionrecurrent neural networks withtime-varying delays.Byconstructing suitableLyapunov functionalsand utilizingsomeinequalitytechniques,Lu145] analyzed the global exponentialstabilityand periodicityfora classof reaction-diffusiondelayedre- currentneural networks withDirichletboundary conditions.

The model (1.0.6) is based onthemodel in[59Jwithout diffusion.In1591, Campbell and Shayerconsidered a model with multiple parameters forapair ofneuronswith time-delayed connections between theneuronsandtimedelayed feedback fromeach neurontoitself.Theyshowed conditionsforthestability of the trivialsolution.Moreover,they analyzedpossible bifurcations that may occurattrivialfixedpointsuchaspitchforkbifurcation,Hopfbifurcation,orone

CHAPTER1.INTRODUCTION AND PRELIMINARY ofthree types ofcodimension-two bifurcations.

Inourwork,weinvestigatethestability ofthefixedpoints andbifurcations in(1.0.6)under dilferent boundaryconditionby computing thenormalformand tryingtofindouttheelfect of diffusion onthemodelby comparing with the resultin [59]. We cancheckthatwith dilferentboundary conditions,system (1.0.6) satisfiesthegeneral assumptionsin[161.Sowe willfollowthe work of 116!todothecomputationandanalysis

In Chapter3,weconsideroneofthemostinterestingand applicable popula- tionmodels,the competitiondiffusionmodelwithdelays inthe following form

~=d u ( ,- K(O) (t-O, )dli-- , K(O)v(t-O,x)dO)

~ ~d, v v( ,- ,K(O)v(t-O, )d - K(O)u(t-O,x)dO)(l.O.7) whereu,varethepopulationdensitiesofthetwospecies,allthecoefficients,

, C,(i=l,2)arepositive,andthekernel functionKsatisfies

.I

With different kindsofkernel functions,especiaUy thedelta functionwhich correspondsto adiscrete delay,system(1.0.7) hasbeeninvestigated and many interestingdynamical results have beenobtained(see, e.g. [771andreferences therein).Althoughthediffusioneffectisconcerned,whendealingwiththelocaI stability andbifurcation problem,most of the research focusedona spatiallyho- mogeneoussteady state solution.Whenconsideringaconstant steady state solu- tionofsystem (l.O.7),byfollowingaroutinecalculation,one candecomposethe characteristicequationintoasetofalgebraictlcharacteristicequations"(see,e.g [17,341).Butasforthespatiallynonhomogeneoussteady state solution,there

are only afew worksin literature because thedecompositionofthecharacteris- tic equationisunavailable,whicbmakesthe analysismucbmoredifficult([61).

Byusingthe implicit functiontheorem andtechnicalconstruction,Busenberg andHuangin16)skillfully overcome the obstacle oftheanalysisofcharacteristic equation andinvestigatedthe existence anddirectionofHopfbifurcationnear a spatiallynon-homogeneoussteady state solution of thediffusiveHutchinson equation.Motivatedby themethod in[6j,someresearchers investigatedthedy- namicalbehaviorforsome particular systemsneara spatiallynonhomogeneous steady state solution.Forexample,in[90j,for a coupled competition diffusion system,not only theOccurrenceand the directionofHopfbifurcation,but also the stability of the periodic solutionwereobtained;in[68],apopulationequation based on[6jwitha generaltime-delayedgrowthrate functionis discussed; and in[42],the authors showedtheexistence andpropertiesofHopfbifurcationfor a cooperationsystem.Itisnoticeable that allthemodelsin[6],[42],[68]and [90] arediscussed withdiscretedelay.To ourbestknowledge,thereislittledis- cussion([2j) aboutthebifurcation behavior nearthespatially non-homogeneous steadystatesolutionofmodelswithdistributeddelaywhichisfoundtobemore realisticandaccurateinsomecases ( [7,10,231).In[2],the authorsshowthe existenceofHopf bifurcationnear a spatiallynonhomogeneous steady stateofa kindof reaction-diffusionequation withuniformlydistributed delaybyusing the techniquesin16].

Inthis chapter we considerthe dynamical propertiesnear a spatiallyn00- homogeneoussteady state solution of system (1.0.7) with a simple but widely used kernel function-uniform distribution,i.e. thekernel functionintheforrn

(O)={

~, T~O~6+T,

(T20,6>0) O,otherwise.

ThehomogeneousDirichlet boundary conditionisimposedin the system (1.0.7), whichmeanstbat the exteriorenvironmentis hostileand the species cannot survive onthe boundary or outside of the domain. Let~=d,l1i=a (i=1,2)for simplicity.After re-scaling,system (1.0.7) becomes

~

= u ( ((t O, ) (t , )) )

f

= v ( v(tO, )(,,u(t O, )) O),t>O, u(t,O)=u(t,,,)=v(t,O)=v(t,,,)~O,t O, (u,v)~( 11, 1 2),-(T+O)<t~O,O x Wekeep the assumption

(1.0.8)

toenSUfethat thespatiallynon-homogeneoussteady state solution('U~,v.B)con- structedispositiveandstable =•.

We employthe methodanalogueto thatin[2,6,42,68,90].Existence of positivesteady stateandHopfbifurcation areaddressed. Theanalysis of the distributeddelaymodelsis notjust simpleand paralleltothatofthediscrete delayonesbecauseofthe complex calculationand toughanalysisof stability of thespatiallynon-homogeneous steady state solution.

InChapter4,aclassofreaction-diffusionsystemwithageneraldistributed

delay is proposed.Weconsider a model in the followingform

= dD'u13u(x,t)

1+~

(O) .(u(x,tO),v(x,t O))dD,

= dD'v+13v(x,t)

1+~

(O) ,(u(x,t O),v(x,t O»)dD, u(t,O)=u(t, ')=v(t,0)=v(t, ')= 0,t^~0,

(u,v)= ( I lo(t,x)2),E(-00,0] x 0,.1 (1.0.9) Inthis chapter, we will investigate the stability ofthe spatiallynonhomoge- neous positive steadystatesolutions of(1.0.9)andHopfbifurcationwhenthe stabilityislostwith the varying of the minimum time delayT.WecallaHopf bifurcationIIforward"ifthereexistperiodicsolutions forparameter Tsatisfying Denote~(O)=: I.u., =:Ji,.,j;;~\U}.=:]u., : f=: and:

~(O)=: (i=1,2).We studyEq.(1.0.9)underassumptions (G,) (O)u ,(O)2:0, i,j=I,2, ' (G,)( 'u. u.)( ,v. ,..»O

Assumption isimposedsinceitguarantees the simplicity ofpure imaginary eigenvalue and issatisfiedfor many population hiologicalmodels.(C2)isrequired tomakesuretheexistence ofpairsOfp05itivesteady state solutions. Especially, we consider thefollowing foursubcases of(G,)

CHAPTER1.INTRODUCTION AND PRELIMINARY (C f,u. -) u.>0'/2,. -f".>0 and .,u.- ,. u.>o.

We mainly discussthefirsttwocaseshy following the basic frameworkofl6j and [90].The lasttwocases canbestudied in the same way and similar results

[nChapterS,we studydiffusive icholson's blowflies modelwithnonlocal delay.Gurney[27] modified Nicholson's model andmade it more realistic,which is later referredas the"Nicholson's blowfliesequ&tion",

~=-dD2u(t)+pu(t-T)exp[-au(t-T)], (1.0.10) wherepisthemaximumpercapitadailyeggproductionrate,l/aisthesizeat which theblowfly population reproducesatits maximum rate, is thepercapita dailyadultdeath fateandTis thegenerationtime. Toexplaininteractions among organismsIthe diffusioneffectwas introducedin[65, the,authors extended(1.0.10)toadiffusive form and via a rescaling

u= u,t=Tt, = T,( =p ,

=du( ,t)-Tu(x,t) ( T (x,t- )exp -u(x,t-(1.0.11)) The globalstabilityof the equilibriumof (1.0.11)with homogeneousDirichlet boundaryconditionisstudiedin 6 ]and the existenceHopfbifurcation and its properties underNeumannboundaryconditionis addressed in[831.Especially, the occurrence of steady statebifurcation andHopfbifurcationatpositiveequi- libriumareinvestigatedinl67j.Basedon (1.0.11),adistributeddelayisusedby RuanandGourley[23jinthe equation

=d-T ( ,t) T(- s)u(x,s)ds)exp(-,u

-

s)u(x,s)ds)(1.0.12)

(t

for(x,t) Eflx [0,00),whereflis either alloflRn orsomefinitedomain, and thekernelsatisfiesf(t) O,

J.~f(t)dt=l,J.~tf(t)dt=1. (1.0.13) Intheir paper,the globaland localstability ofuniformsteadystatesaremainly studied.Especially,forthe globalstability, energy methodsand acomparison principle fordelay equationsareemployed.

By usingtherandomwalk method[3,25),onecanincorporatetimedelayand spatialdiffusionsimultaneously.Inthe presentchapter,weconsider themodified

au~;x)=dD'u(t,x)-ru(t,x)+{3r(g.u)(t,x)exp[-(g.u)(t,x)] (1.0.14) for(t,x) E[0,00)x 10, ..withinitial condition

u(s,x)= (s,x) (s,x),E (-00,OJx[0,..

andhomogeneousNeumannboundarycondition

~=O,

t>O, x=O,",

(g.u)(t,x)=

1~ J.·

( +

~ ~e-dn'('-')cos(nx)cos(ny))

f(t-s)u(y,s)dyds, f(t)satisfiesconditionsin(1.0.13)anditiseasytoseethatJ~~Jo· g(s,x,y)dyds= 1

As faraswe know,themaintopicin mostoftheliteratureabout (1.0.14)is abouttravelingwave.Forexample,in[41J,theexistence of travellingwRve-front

CHAPTER1.INTRODUCTlON AND PRELIMINARY solutions of (1.0.14) is established. [79]provesthe existence ofnon-monotone traveling wavesfromthe trivial solution to the positive equilibriurn of (1.0.14) Works about thedynamicalbehavior aroundtheuniform steady state solutions are few.So,our mainpurposeis to investigate the stability of two constant steady states and possibleHopfbifurcationwhen the stability is lost.

Chapter 2

Stability, bifurcation analysis in a neural network model with delay and diffusion

lncorporatingtheeffectofdiffusionandtimedelay,weconsidern modelincluding apairofneurons with time-delayedconnectionsbetweentheneuronsandtime delayedfeedbackfromeachneuron toitself,

=d,D'u-u(t) +al(u(t-r))+bl(v(t-r)),

=d,D'v-v(t)+al(v(t-r))+bl(u(t-r)), (2.0.1) wherea,b denotes thefeedback andconnection strengthrespectively,Tisthe timedelaY,d1andd2arediffusioncoefficients,thenonlinearfeedbackfunction ,IR~IRissmoothenoughwith1(0)=0and without loss of generality, 1'(0)=1,1"(0) O. Moreover, we denote =( ,, E,C(1-r,OJ,IR'), ( EC,kEN,,) nE NU{O)=No, =(1,lf,V,=(1,-lfand

T:=(; :),fixthevalueofthedelayrandchoosethedilfusioncoefficients d,= =1.Thework of this chapteristhe main content of[341which will appearin Expandedvolume ofDiscreteContin.Dyn.Syst.

2.1 Neumann boundary condition First,we consider (2.0.1) with eumannboundaryconditionin

x

={(u,v): u,vEW '(a,,du/dx=) dv/dx=a atx= a, f, with theinner product inducedbythatof theSobolevspace W '(a,f) SettingW(t) = (u(t),v(t)jTandusingTayiorexpansionatthetrivia1equilibrium point, (2.a.l) canbegiven,inabstractform inC=C([-r,al;X) as

dW/dt=D'W(t)+L(W)+F(W), (2.1.1) L(~)=-~(a)+^Tb~(-r)

F(~)=Tb ~?(-r)f(j)(a)/j!.

Theeigenvalues ofthe aplacian onXareo~·=-(k_I)'=: =1,2, witheigenfunctionsf3~= b.,ajT andf3~=(a,'l'.)T,respectively,for

'l'.(x)=II

c:~~ik_-l~~~!"'''

(Hl)-(H4) holdwith p.=2,since thelinearpart of (2.1.1) satisfies

2.1.1 Localstability

Thecharacteristicequationofthelinearizedequationof(2.1.1)isequivalentto det ,(,l,)= +(k-I)'+I-(a-b)e- T I)'+I-(a+b)e- T)+(k- = O.(k E ) ,l,E is aneigenvalueif andonlyif forsomek

c(,l,)=,l,+ (k-I)'+1-Ce- T=0 withC = a -borC =a+b. We firstanalyze thedistributionof zeros ofp~

withzeroreal parts. et,l, it,tE Bycomparingthe imaginary and real partsofP~(it),wegeta parametricsystemas

C(t) = (I +(k-I)')cos(tr)-tsin(tr)andImp~(,)(it)=O.

Tosolvethissystem,weconsideritscorresponding curverkdeterminedby 8(t)=C(t) andT(t)=(I+(k-I)')sin(tr)+tcos(tr).Letott)=T(t)/8(t) ThenO'(t)>oforalltElRsatisfying8(t) Thusr, moves counterclockwise aroundthe origin inthe (8,T)-plane.Itis easy toseethatat a sequenceof critical values{t~}::'=owitht~=O,t~E«2n-1) . (2r),mr r),r,intersects with8-axisat(C~,O).Since

8'(t)+T'(t) = (I +(k-I)')'+t', C~= (-I) (I+(k-I)')'+(t~)' and{IC~llnENois an increasing sequence.ObviouslyC=1+(k-I)' and (-l)nc~>0,hence the following result holds

Lemma2.1.1(seeFigure 1) Forsomek,(i)P~(,l,)hasa simplepairof purely imaginary roots±it~if and only ifC=C~forn 0;p ,l,) has asimple zero root,l, ifandonlyifC C

(ii)Pg~)onlyhasrootswith negative ealpartsifC~<C<C~;2{1+1)roots with positive realpartsifC~+3 C<C~+I;21+1rootswith positive ealparts ifC~, C<C~l+"IENo·

Proof.(i)FromP~('\)=0and(P~('\))'=1+ e-'P'( it )0 and P'(O) ' (i) is obvious fromtheprocesstoformC~

(ii)First, isa continuousfunctionofC according to the implicitfunetion theorem. IfC=0,P~{'\)=0has only one root ,=-{I+(k -I)')<0 Moreover, differentiatingP~(),)=0 withrespecttoC,wehave

e

=I+ e-'

Bycomputation,sign(Re~lc=c~)=sign(C~).Hence, as CincreasestoC~=

I+{k-I)'>0, only one rootofP~=0is,ero whilethe othershave negative realparts; whenC lies betweenC~and

C ,

^p~hasone zerowith positive real part while the others have negative realparts.AsCreachesC;,apairofcomplex rootsofP~=Ohavezerorealpartandonehaspositiverealpartwhiletheothers have negativerealparts; whenCcrossesC~,p~hasthreezeroswith positive realparts while the others have negative real parts. Similarly, we can finish the remaining proof.0

CHAPTER2. NEURAL NETWORKMODEL

Inordertostudythedynamical behavior in (2.1.1), weneedtodiscussthe distribution ofrootsindet .( )=O.P~(it~)=0givesus-t~/(I+(k_I)')= tan(t~T),t~<t~+l,andIC~I<IC~+lI.Thuswehave,

Theorem 2.1.2(SeeFigure2.) For the ehameteristie equation of the linearized equationof(2.1.1)with Neumann boundary condition, (i)alleigenvalueshave negative real partsifand onlyifC:<a+b< =1 andCi<a-b<I, whichimpliesthat, when (a,b)E ((a,b): C: <a+b<

I,

Ci

<a-b<I},the trivial solution 01system(2.0.1) isasymptotically stable;

(ii)ifa+b=C: andCi<a-b<1ora-b= Ci andC:<a+b<I, then all eigenvaluesbut>.= ithave strictiy negative reaiparts,where±it~isa pair of purelyimaginaryroots 01 det :.,( )=0whenC=Ci

(iii)ifa+b=Cf,a-b=1ora+b=1,a-b=

at

then all eigenvalues,except

> = itfand>.=Ojhavestrictly negative real parts

nextC Ci 0 nextC C

Figure 2

Combining theresults inLemma 2.1.1 andTheorem 2.1.2,weknow thatif the parameterCisbeyond C there exists at, least oneeigenvaluewith positive realpart and the trivial solution may lose stability and bifurcationoccur

Theoccurrenceof bifurcation impliesaqualitativechangein thesolutioos.The study of such changesis important,especially when the systempossessesonly a center manifold and astablemanifold nearthe trivial solution, we are able to determine thewhole dynamicalbehavior of the system.Inthis subsection,we will study all generic bifurcations at the trivial solution of (2.0.1) withNeumann boundary condition.Weare onlyinterested inthe bifurcations at theboundary critical values,implyingdetl~'1=0 hasonly eigenvalues with zeroreal part,so N=lin(1.0.3). More precisely, thepotentialbifurcationsincludesteady state bifurcationwithsimple zero eigenvalue at0 = =I,Hopf bifurcationwith a simple pair ofpurely imaginaryeigenvalues±it~atC = and Hopf-zero bifurcation, the interactionofthe twocodimeosion-onebifurcations

To discuss thecodimension-onebifurcation, wefixbandperturbtheparam- eter0at the critical value as 0= ,,+1 ,1EJR.Then in(2.1.1) = andF(iji)=1'~iji(-r)+1'bLj~2ijii(-r)f(j)(0)/j!.In(1.0.3),

= I since 01=0 andM I=0,

=1'~<p(-r)+Lj~21"';(-r)<;j/j! (2.1.3) where = Therefore, withI ( 1, I)= (I JiTl -l,the FOEassociated with (2.1.1)byAatthetrivialequilibriumpointis

wherex(t)=( (t), ,(t)EC([-r,O!;IR'),xi(t)=(x{(t),~(t)jT.Denote

~bethesecond-ordertermofthenonlineartermsinthis855OCiatedFDE, we have

,( ,) 2=T~ip(-r)+ b" "(O)-r)/2.'( (2.1.5) Casel.Transcritical bifurcationWe firstconsiderthe simplest bifurca- tionoccurringin(2.1.1).When thecriticalvalue aosatisfies (i)ao+b=C = I and o-bE(CLI),or(ii) o-bCandao+bE (CLI),Theorem??implies A={O}.Itsufficestodiscussthecase(i).The phasespaceCofthe linearized equationof(2.1.1)can be decomposed asC=P Qwithrespectto :C~P, ,,( =)(1 [ I (Ifi, ip( ),.a ))1:) 1)1, (2.1.6) with 1 =V"1 V(2 2C r)-1",Vt "1 1= (1.,fiT,O Tand1 1=(0,1.,fiT T

FollowingTheorem4.1 in[171, the normalforms ofPFDE and its associated FDEare the sameforthefirst-andsecond-orderterms. Bycomputation,wecan obtainthenormal formof the associatedFDE(2.1.4)up to the second order, with respect toA={O)as

i=2 ,(/"+CJ~,z2/2)+h.o.t (2.1.7) Thus the bifurcationat aoistranscriticalsince~,=f"(O)/.,fiT¥O

Case2.Hopfbifurcationlfaosatisfies(i)ao+b=CI andao-bE (CLI), or(ii) o-b C!and o bE(CLI),thesystemundergoes aHopfbifurcation ata=aosince thetransversalityconditionisconfirmed in theproofofLemma 2.1,A={-itLitn.Weconsider(i)only,(ii)canbetreatedin asimilarway The phase space of the linearizedsystemof(2.1.1)C=p Qwith respect to

CHAPTER2. NEURAL NETWORKMODEL rrdefined by(2.1.6)for

I= le":'Vi,e-"I'ViJ:=( )= co!(v."I 1e-":',1I '11e"I'),(2.1.8)

IIEP=span{(/3:,/3~)¢" ({31.1J~)th)' II=p(/3:,/3~)¢.+q(/3I.1J~)th:=(' "' 2f/. rr forp,qEIR.(H5')holdssincefrom(2.1.5),fork~2,,pI,¢,EC(I-r,OI;R)and

~DIF2(ll,,..)(,p./3~+¢'/3n { .. .( K2 .( P).( ) 2(-r)J}/3~) ( +(,..,p2(-r}K2[b>p.(-r),pI(-r)+ao'P2(-r)¢,(-r)J}/3~ (2.1.9) Hence we canderivethe normalformof(2.1.4)in polarcoordinates,up to thethirdorder, as

{

.=Re(e-"I'. +.. +h.o.t (2.1.10)

~= t h.o.t

wherewith,,=!,"(O)/rr,and

XI~-2Re(D.)+2i~:r'~I~.+~)_~e-<t:'(D.+.::t~Dl)J} ,

=

4{1~2Re(DI(-I-~+~))I(I-Cn-·+2Re(D:;: n.

^(2.1.11)

CHAPTER2.NEURAL NETWORK MODEL

Case 3.Hopf-zero bifurcationTo discuss thecodimension-two bifurca.

tion,weperturb the parameters a,bat thecritical valuesao and boasa=ao+l',

= o+v, ',vE .Then in (2.1.1), (i= - ( )+Tl (-T)and ( )= T~$(-T)+TrLmf(j)(O)$i(-T)Jj!.Whenthe critical values ao,bo satisfy(i) ao-bo=C and o+ o=C .or (ii) o- o=C/and ao+bo=C , = itl,It. issufficienttoconsider the case(i).The phase spaceC relatedto(2.1.1) canbe decomposed by rrsimi arlyas(2.1.6)with 4 = and,,op,Wsatisfying

4 (8)=e<t:8Ve-<t18V, ,V,], ( )= olev,TD,e-<ti',v{D,e<ti', V,TD, ,(2.1.12) where T8 0 T,D,andD,aredefined in(2.1.6)and(2.1.8) respectively.

TheFDE associated with(2.1.1) by has asimilarform as(2.1.4)with T andT~replaced by Tl andT~respectively.It is easy to verify that(H5') holds Sincein the normalform of theassociated.FDE the coefficients ofthe second- orderterms are zeroduetothe structure in(2.0.1),thehigher-order termshave qualitative effects andweneed to compute the normal form upto the thirdorder Therefore,thenormalformof(2.1.1),uptothethirdorder,canbeobtainedin cylindricalcoordinate as

I

^=(I'+v)Re(D,e-<tiT)p+Re(K,)p3+Re(K3)pz'+h.o.t

8=-tl+h.o.t (2.1.13)

Z=2D,(1' - v)z+K,zp'+K4Z3+h.o.t, with Ktandh,givenin(2.1.11),and

K2=2D2(2CJRe(4iC;2Cfe-itlrDdt~+eitITh3+~)/2+2( 2C , K3=D,<,C/{2i(, C (D,e-'<tiT- ,)-2e- ITD,C l/t:+h3)⁺(3D,Cle-"iT, K4=2D,(,C Re(2i(,C/e-<tiTD,/tD+(3D,C 3, h3=4D,C <,e-1':T{i/t:-e;118~/2+C/i/tU(itl+I-C e- IT)-t

2.2 Dirichlet boundary condition Inthis section we study(2.0.1)withDirichlet boundaryconditionu(t,O)= v(t,O)=u(t,)=v(t, )=O.Under this condition, (H5') doesnot holdand we willsetup the relationshipbetween the normal formsof thePFDEand its associated FDE.DefineX=((u,v):u,vEW"'(O,):u(O)=v(O)= u(..) = v(..)=O}withtheinnerproduct(.,-)inducedbythatofL'(O, ..).Itis easy tosee thatzeroisnolongeraneigenvalueofLaplacianinthiscase.lnfact,eigenvalues inXofD²are~..=_k²=:kik=1,2 withcorrespondingnormalized eigenfunctions1 =(oyk,O)T,13~=(O,' k)respectively,' k(X)=sin(k )fiF Similarly(Hl)-(H4)hold withpk = 2 and at the trivial equilibrium point,(2.0.1) canbe transformedinto (2.1.1)in C=C([-r,O];X).Denote

p~=. k'+I-C -,C~=(-I)nJ(1+k')'+(t~)'.Lemma2.1.1 holdsand thedistributionoftheeigenvaluesofthecharacteristicequationisthe sameas

For the same reason as that in the previous section,weonly need toconsider theregion flo=((a,b):

CI

C=a±b CnwhereC= 2 andC= +Atthe boundary criticalvaluesof the systemhaspossible bifurcations includingsteady state(simplezero) atC=C ,Hopfbifurcation at C=Cfandtheinteractionofthesetwoone-codimensionbifurcations

To discussthe codimension-one bifurcation, wefix bandlet E Then in(2.1.1)

( ) - (O)+ :O (-r)

( p)=T~rp(-r)+Tg~rpi{-r)fU)(O)fj!.

Corresponding to(1.0.3), ,=-I,M,= diag(-I, -I).In theassociatedFDE of (2.1.1), ( is defined in) (2.1.3)withour choice of1 1and1 1,R{<p)= M,<p{O)+L,(<p)whereL,(-)is definedin(2.1.2).Parallelto thediscussion in Section2.2,wehave

Case1.Transcritical bifurcation Leta;, satisfy a , = C = 2 and a ,- E ( /,2),thenA= {O}.The phasespaceCcanbe decomposedsimilarly withrespecttorr as (2.1.6) with4=V" 111=V D"The normal fonnof the associatedFDEwith respecttoA has thesarne form as(2.1.7)withC=2and (,=4(2/rr)'/'f"{0)/3.ltisclearthatthebifurcationata;,=2-bistranscritical

Case2.HopfbifurcationLeta;, aatisfy a , = and a ,- (C/,C ), thenA={ i The phasespaceCcanbe decomposedasbefore byrr as (2.1.6) associatedwithA,and4 andlllhave theaame formas thatin(2.1.8).But(HS') fails.In fact, foralluEP, = ( {<p,,<p,) ,for 2, ",,, / 2EC( -r,O ;I ), 1/2D,F,{u,,,, " ,1 i 1/ i Dhasthe sameformas (2.1.9), whereas

({1/2)D,F,{u, ",)(" ,1 i+!/J213~)'P;)~~,

= {<p,( ) ( ),<p,( ) ,( (O)

with"k ={,,(k"f,,"(,)=Oifkiseven,or-4(2/rr)'/'/ k(k'-4)ifkisodd.

Since (H5')doesnothold,we cannotobtain theinformationdirectly from thenormalform ofthe associated FDE.However, wecan still make use of the relationshipbetween thenormal formsofPFDE(2.1.1) anditsassociatedFDE tostudytheHopfbifurcation.By the decompositionofC,(2.1.1) canbetrans-

I I

I

CHAPTER2. NEURALNETWORKMODEL

whereB=diag itL-itn,z = l, )E C',Y EC n, CJ:={iOEC:~EC,iO(O)Edom(D')},

liO=~+Xo[ (iO)+ 'iO O)-~(O)], f](z,y)= 0)({Pi((PUml<I>zl+y),P;»:~I

fJ(z,y)={I- rr)XoFi((PUn[ I zJ y) Sincethecharacteristicequationoftheassociated FOE,detl'l, =Oonlyhas apairofeigenvalueswith zerorealparts,Le.±t~iwhichcorrespondto theeigen- functionspace<I>,thenwecandecompose the phasespaceC,=C([-r,0]; I ')of theassociated FOEasC,= span{ IEDQLetx,=Iz(t) +y,withz(t) EC' and here,different from that in PFOE,yEQndom(AOl),

oI'l'=P+o R( - ( IEdom(AodcC,

fJ)z,y) = 0)({Pi((PUn<I>z+y),P:»:~I' f )z,y)={I-rr)Xo (Pi(( l<I>z+y )n ,P~»):~"

i:=Bz+~fJi(Z,y)Jj! dY/dt=AOIY+~fgi(Z,y)Jj!

CHAPTER 2. NEURAL NETWORKMODEL Letz =Bz (z,O,")/2... and z=Bz ,,(z,O,")/2 forms in complexcoordinates on the centermanifold atzero for PFDE(2.1.1) anditsassociatedFDErespectively,wehavethefollows' Theorem2.2.1The norma/form ofPFDE(2.1.1) i

i=Bz+g~(z,O,I")f2!+ ,O(z,")/+ h.o.t

=Bz+ .,(z,,I )2+,,(z, 0, 1 ) 1+(K.z1zK.~zdT+ h.o.t

Ks= +Cl,keitlT)Dll ,C ,i=k₂+2~~ ^I

C", (2tli+k'+ii )e I_ withii.= "(O)C . Proof.From theproofofl17,Theorem4,1)andbecauseoftheoccurrence of HopfbifurcationinassociatedFDE, ,,(z,O)= (z,O)=Oanfory=0 7~(z,0,1 )=7i,,(z, 0, 1 )+~[D,li(z,y, " ,=oUi(z, ")-D,/ ,,(z,y,II,=o ,,(z,1 )

~7i,,(z,0'1")+3'!'(O)(~(D.F,«PI.P1)[<I'z],I"J x(f (hiPk+h%p~)(z,I")),pf))I=. (2.2.1) whereh(z, 1 ):=Ui(Z,I")=L'~I(hiPk+^{h f)is the}uniquesolutionof

(Mih)(z, =D,h(z, ")) Bz-A.(h(z, "=fi(z, ,1") (see[18)).Forhi(z) =hi(z,)(i=I,2),

m

^{DF,((l.l 1)lz )(Lk (hiPk+}h~pm(z),PDJI=.

=(1' " "(O)(eil1 z.+e-ili z,)')'.Lk.' .(hi(z),h (zW,

)

1)

=.

=Lk .(e-iti' z.+^eiti'z,) . "(O)1' "(hi(z)(-T),h (z)(-TW

Now, we need to computeh1{z),h~{z)in(2.2.1) bysolving (M~h)(z,O)=mz,O,O),

f~(z,0,0)=XoF2((I :,l I )-( l nI (O)({F2((I3L13~)I ,O)'~)):=I]

Ontheotherhand, since(Mih)(z,0)=D.h(z)Bz-A1h(z),then fork>I,i=

1,2,

{

~~:~:1~~~:;~HO)+h~(0)={e-"ITzl+e"ITz2)'iik

^(2.2.2)

whereii = I (O), (=(I , ), l)and)=

h~(z)(O)=1eh~{z)(O)I.=o.Starting fromthelowest order, weset

Solving(2.2.2),wehave

h,k=2iik 2+k^1-C:=CO,k,

After obtainingh =1,2)andsubstitutingh(z,1 )=L:k~1(h11 1+h~I3~)into (2.2.1),then

(z,0)

=7~,3{Z,0)+ 2:., "(O) kT:"(h1,h )T(e": z+e-":'z2)(-T)]

= "(z,0)+61~r(0)<>kCI((CO'ke-":T+Cl,ke":T)z~Z2 +Co,ke"IT+C\,ke-":T)~ZI)I(DI'1 1)T+h.D.t.

NEURAL NETWORK MODEL

andwe completed the proof.0

In fact, thenormalform ofthe associatedFDE in polarcoordinatehas the sameformas (2.1.10)withcorrespondingvalue oftl.Thenthenormalformof PFDEinpolarcoordinateis

{ ^=Re(e-it

T l+Re(Kp I+K,)p3+h.G.t

~=-tl+h.G.t.

Case 3.Hopf-zero bifurcationLeta=a+ b=bo+.a,bosatisfy a,,+b = anda,,-bo=C .ThenA= it ,and in(2.1.1),

L -( )+-r)

MI O)+LI with L) I (·)defined in (2.1.2), MI=-1 and is defined) in (2.1.3)with i replacedbyY~.Withthe sameprocedureasin Case 2, it iseasy to verify that (H5')fails.As for the relationship betweenthenormalformsof PFDE (2.1.1)anditsassociatedFDE,similarto Theorem2.2.1,we have the following result,the proof issimilarto that ofTheorem2.2.1andweomitit

i=Bz+ i(z,0, )+. andi=Bz+ .,(z,0, )+

NEURALNETWORKMODEL

Z=(Zt,Z"Z3)EC3• B=diag{itL-itLO).

benonnaljorms in complex coordinates on the centermanijold at zerojorPFDE (2.1.I}and its associated FDErespectively. Then, the nonnaljonn ojPFDEis

whereKS1 o andCI,1eare given inTheorem2.2.1,and K,= akDt(Co,ke- t ,j2+C"k). K,=~ "(O)CtkD,C 'CO,k. K,= "(OPkD,C '(2Re(C',k)+CO,k),C"k=2 "(O)Ctk (tli+k'+ )e'i"C '-)

Accordingtothe result in Section 2. the normal formoftheassociatedFOE isas (2.1.13)incylindricalcoordinates, Thus, the normal form of PFOE in cylindrical coordinatesis

p.'=(I+v)Re(D e- ti,)p+Re(K,+K,)p3+Re(K3+K6)z'p+h.o.t B=-tl+h.o.t

z=2D,( '-v)z+(K,+K,)zp'+(K,+K,)z3+h.o.t.

Chapter 3

Dynamics in a competition diffusion system with uniformly distributed delay

Weconsideracompetitionsystemwith uniformly distributed delayanddiffusion,

8 ~

8v=

u(t,D)=u(t.1r)~v(t.D)=,,(t.1r)'~D,

(u, )=( ,, )( ) tD,D ,

with initial functions EC«-(T+),D),Y).[nthe present chapter,X = H²HwhereH2, isthe standa.rd notationforthereal-valuedSobolev spaces Thework in this chapteris the main content of 135j whichhas beensubmitted

CHAPTER 3. COMPETITIONSYSTEM

3.1 Existence of positive steady state solution and corresponding eigenvalues ThesteadystateequationofEq.(3.0.1)is

{dD'u+l3u(l-b1u-c,v)=0.

(3.1.1) dD'v+l3v(I- ,v-c,u)=O Similar to [6),wehavethefollowingdecomposition

L'(O,lT)=N(dD' +13.)ffR(dD'+I3.) where13.=d, N(dD'+(3.)andR(dD'+ (3.) arenulland rangespaces of the operatordD'+{3.withtheform

N(dD'+{3.)=span{sinx},R(dD'+{3.)={UEL'(O,lT):(sinx,u)=0), respectively. Let

{up( )= ({3 {3.)ol(sinx+({3 {3.)6(x) (3.1.2) vp( )=((3-{3.)o,(sinx+({3-{3.)~,(x», where(~"sinx)=0 (i=I,2).Substituting(3.1.2)into (3.1.1) yields

(dD'+i '){ +sinx+(ii.){ i (sinx+(i .){,)i xlb,ol(sinx+(iJ-iJ.){,)+c,o,(sinx+(iJ-iJ.)6)J~O, (dD'+i.) +in+(i .)6 i (si inx+ (i .){,)i x in,o,(s+ (i .)6)+c,ol(sinxi +(i .){,i )=0 Next,wearegoingtousetheimplicitfunctiontheoremtoverifytheexistenee of the solution(up,vp)of(3.1.3)for{3near{3•.At{3={3..(3.1.3)becomes

{ ~:~::;:~;:: : :~~::: :::::~ : ~:

^(3.1.4)

~~I

h in'xdx ('hsin3 dx)=3tr/(8p.) =ao.

Forminginnerproductwith sinx on both sides of (3.1.4),afteran algebraic ca!culationwehaveat.,Q2.as

at.=

b,~ = ~:c,

ao> 0, a=

b,~ =

^{c,ao> 0, (3.1.5)}

under the given condition ( d.From(3.1.4) and (3.1.5),~"and .are well definedwhich solveEq.(3.1.4). Wehavethefollowingtheorem.

Theorem 3.1.1{gO,Theoreme.1Thereare a small enough constant p'>p.

and a continuouslydifferentiablemapping 13 (~,p,~,p,a,p,a,p)from 113.,13'1 toX'xJR' suchthat(3.1.1) holdsand(~,p,sinx)=0 (i=1,2)

Wewillomit some similarproofsand only emphasize theoneswhich are different from thatin[901.

Accordingto Theorem3.J.1,itiseasy toseethat (up,vp)definedin(3.1.2) satisfiesthe steady state equation(3.1.1).Consequently,thefollowingcorollary

Corollary 3.1.2 ForeverypE ,8.,13',(3. .1)hasapositive solution (up,vp) withtheasymptotic expression(3.1.e)

LetpE ,8., ,O .p«l,and(up(x),vp(x))bethepositivespatially nonhomogeneous equilibriumof system (3.0.1) expressed as (3.1.2).Definethe operatorA(p) (A(p)) y'as

A(p)=(dD'+p)I 13(b' + 'vp 0 ),

o b,vp c,up

CHAPTER3.COMPETITION SYSTEM withdomain (A(, =X'. Then the linearized systemof (3.0.1) is

(3.1.6)

with beinga2x2matrixand each element Tinthe space ofbounded variationBV([-(r+ ^),0];Y). ThenA(, generates a compactCosemigroup 156J.LetAr(, be theinfinitesimalgenerator ofthesemigroup induced by the solutions of (3.1.6) with

Ar(, (

~

⁾⁼

( ~ )

^,-(r+

e

^0,

and ( r(, ))being theset of all

satisfying

(~ )

Ec(f-(r+o),O),Y') ( );_;EC([-(r+o),O),Y'),

( 1(0))

₍₀₎^EX',

CHAPTER3.COMPETITION SYSTEM

Thereforethecharacteristicequation of (3.0.1)is

["(>"fJ'T)=A(fJ)_fJ IT+6~e-"dO ( b'uP

^ClUP)_>.I

TO I

EigenvaluesofA,.(fJ) withzeroreal partsplaykeyrolesforthe analysis of stability of steady state solution.We firstanalyze the existence of thezero eigenvalue.

Lemma3.1.3 IfT<':O,thenOisnotaneigenvalueofAT(fJ)forfJ.<fJ~fJ' Proof. IfOisaneigenvalue,then(3.1.8)holdsforsome(Yl,Y2)T",(O,o Tand

>'(fJ)=0,ie.

YI=nlsinx+(fJ-fJ.)~I.(~l,sinx)=0, Y2=n,sinx+(fJ-p,)'I2,(I ,sinx)=0, (3.1.10)

CHAPTER3.COMPETITION SYSTEM wheren Then. substitutingYl,Y2into(3.1.9),wehave

D2 3

(2b'U~+CIV~

3

C1U~

))(

nlsinx+({3-{3 ')~l

)=0.(3.1.11) C2v~ 2b,v~+C2u~ n2 l x+( 3- 3.) 12 Bya simple calculation, (3.1.11)isequivalenttothesystem

(dD2+{3')~1 +(nlsinx+({3-{3.)~,)-{31(2bl<>l(sinx+({3-{3.)(tl + 2( in +( 3- 3.) )(n,sinx+ ( 3.3, (dD²:

;~~~s:(::s~:x-:~(~;.~:)S~;I::(::~)::~~

.) ) (3.1.12)

x(n,sinx+({3-{3)~,)+(2b,<>2(sinx+({3-{3.)6) +c2<>,(sinx+({3-{3.)6))x(n2sinx+({3-{3')'I2)1~0 WithasimilarprocessasbeforewecanverifythatTJi,1l.j(i=1,2),arecontinuous withrespecttofJ.Wecan expand (i=l,2) as

=i

~~Y)({3 -I3.)j-l,ni

=

~nY)(I3-I3.)j-1,

~ij)

= i

~i

i-

L~(~ :i;i?_~

3.) - ,

= i i-

L~(~~l;)~~

3.) -1 When13=13.,(3.1.l2)becnmes

(3.1.13) Withoutloss ofgenerality,wefirst assume thatbothn n~')), o.Then(3.1.13) becomes

{ ~:~::::~~j:;~j:; = ;::~ = :::::::ni,~~~:)~~:~::::::::~,

^(31.14)

CHAPTER3.COMPETITION SYSTEM where~••is defined in(3.1.4).If

then~I)In~')= bI/el=-l ,le,which contradictsthecondition CSIfone or both of

is/arenonzero,(3.1.14)does not holdsincesin'x

tt

n(dD'+{3.).From the above.discussion,we h.,veni=

°

^{and tl,en~i'"=O(i=1.,2)With thesame}

with thesameanalysis. The proof iseompleted.0

InthefoJlowing,we considertheexistenceof purely imaginaryeigenvalues It isobvious that AT({3)hasanimaginaryeigenvalue=i' t' i'0)if and0nly ifthat thefollowingequationissolvablefor( i i'(O,O)Ti )T

(3.1.15)

=

where yT=ro+2n-n-,n=0,1,2, .. n Denote Tn=( +2 r)(n=0,1,2,...) We have thefollowinglemmas

Lemma 3.1.4{gO, e .ljlj ,i ) ol ei ,.( .1.1 ) i (O,0)Ti' ( i , i )T, en'=O({3-{3.),' ({3-{3.) = is unijonnlyboundedjor

fJE (fJ..fJ·j,and')'(II!pdl~c+llthll~c)ise ato

-1m

U:

^fJe-i].'

~e-i7'dO(up(b,!P'

^{+c,th) ,+ ( ,+ ,th) ,)d, ).}

Lemma 3.1.5(6,Lemma2.Ifz EXcand (sin(x),z)=0, then 1«dD'+fJ.)z,z)1~3fJ.llzlI~c

NowforfJE (fJ.,fJ ,assumethat( y,r ], p th)isa solutionof (3.1.15) with ( p th)Tf(, )T. fweignoreascalarfactor,( canth)be representedas

,=sinx+(fJ-fJ.)~,(x), (sinx,~,)=O, th = (N+iM)in +(fJ-fJ.) ) ( ),(sinx, )2)=0 (3.1.16) Toshow the existenceof~,,!)2,M and NforfJE(fJ.,fJ ,substituting ( ,in(3.1.2),) ( th)in(3.1.16)and')'=(fJ-fJ.)hinto(3.1.15) yieldsthe

gl(~"!)2,h,,,,,M,N,fJ)

= (dD'+fJ.)~,+(I-ih)(sinx+(fJ-fJ.)~,)

(3.1.17) X[bl(sinx+(fJ-fJ.)~,)+c,((N+iM)sinx+(fJ-fJ.)!)2»)=O,

!/2(~l,!)2,h,r;],M,N,fJ)

=(dD'+fJ.) I-ih)(N+iM)+( inz+(fJ-fJ.) ) fJlb,o,p(sin +(fJ-fJ.) , )+ 2o,p(sin+ (fJ-fJ.), )

«N+iM) in +(fJ-fJ.) o )-fJ( in +(fJ-fJ.) , ) e-iJgiei"d9[b,«N+iM)sinx+(fJ-fJ.)~,) +c,(sinz+(fJ-fJ.)T/2»)~O.

(3.1.18)

CHAPTER3.COMPETITIONSYSTEM Notice that

'''d e-' --

1as/3-- /3.,we cao choose

~h=(I-ih.)M.~,.,'72.=(I-ih.)N'~h, with~hdefinedin(3.1.4),M.=0, .'=,,/2, andh.,N.E satisfying

hc,N (b,.-b,, .)N,, -o, .,= 0, ,.(b, c,N.)=h, o, i.e.N.=~>

°

^{from(CS,)andh.=1.}Then it is easytosee that

gi(~h,'72"h.. '.. M.,N.,/3.)=0,i=1,2 DefiningG=(g)I•••,96):X~x X , with

(3.1.19)

(3.1.20) g3(~','72,h,,,,,M,N,/3)=Re(sinx'~l)=0, g'(~I,'72,h,,,,,M,N,/3)=lm(sinx'~l)=0, g5(~','72,h,,,,,M,N,/3)= e(sinx,'72)=0, g6(~','72,h,,,,,M,N,/3)=lm(sinx,~,)=0, it isobviousthatG(~h,'72..h....M..N../3.)=' 0.Toobtaintheexistence ol roots in(3.1.17), (3.1.18)and(3.1.20)by usingtheimplicit lunctiontheorem, we need to prove that the operator

is bijective,whereJ=D('71m,h,w,M,N)G(7JI.,Tf2.,h.,ro.,M.,N.,/3.) Theorem 3.1.6/90,Theorem 9.1/ There is a continuously differentiable map-

/3--(~,~,Tf2t ,h~, "'~, M~, N~) from1/3../3')toX'X ' such that(~I~,Tf2tJ,h~,,,,~,M~,N~)solves (9.1.17). (9.1.18) and(9.1.20).More- over,ij/3E(/3••/3·j.the solution mapping is uni ue.

Theorem 3.1.6showstheexistenceofgeometricsimplepurelyimaginary eigenvaJuei"(anditseigenfunction(w..,p.,)T (,jTfor{3EI{3.,{3·]·

Thefollowingcorollary can be obtained immediately fromtheabove theorem Corollary3.1.7Foreach{3E({3.,{3·]theeigenvalueproblem

ha.sasolution( (,T, .. i!andonlyi!

(p=({3 -{3.)hp, T=Tn=( +pmr),n=p0,1,.·

( )=C(

:i~P~:~):~

^{({3 _{3.)rnp), (3.121)}

whereCisanarbitrarynonzeroconstant,an , ,h , iare de·, , scribedinTheorem ..

3.2 Stabilityof thepositive equilibrium Inthissectionwe studythestabilityofthe positiveequilibrium(up,vp)with{3E ({3.,{3']fixed,andthedelayTas a parameterpassingthroughTn,n=0,1,··

First,weneedto findthe eigenfunctions ofthe adjointoperator ofthelinear operatorof (3.0.1)bysolvingtheadjointof(3.18),

6(')(i"(~AT) ( )

=

(A({3)-i"(p-{3e-'~

_J

r'~e-;.y.dO

_o ^(b_{lupb ,}^c,vp))

(wl~)

_wi) ^=0.(3.2.1)

Similarly, let

Thenthere is acontinuouslydifferentiablemapping{3~(ry~;'>,

r(

N~o), M~o»),

,

from{3.,{30to xlR'suchthat(3.2.2)satisfies(3.2.1),andat{3={3.

ry~:)= (I-ih,)~,o, ryl )= (I -iho)N!o)~lo, =~No, =0

T Wecan choose a basis ofeigenspaceinC([-(T+ 6),O],IR') of thelinear operatorof (3.0.1) as(cl>~,~~)wherecl>~=(,p,~,",,~)Te;>6', ,p;~(i =1,2)isgiven in(3.1.21),for-(Tn+6) O 0

andtheinner productof1 ,as

= -

,p(~

-

O)dry(O)1'(~)~dx

C ([-(T andryis defined in(3.1.7) LetSfJ,.denotetheinnerproductofwpl(f.,erelatedtoTn8S

~('Pp,~~)

~ 'Pp(O)cl>~(O)dx- I~(T"H)J~ -O)ldry(8)J~~({)d{dx

= I;(,pl;'>"',~H~;'>~)dx

whereL(·)is defined in(3.1.7).Thenwehavethefollowinglemroawhichwill be useful in theproofof the algebraic simplicity of the purelyimaginaryeigenvalue i {pinLemma3.2.2.

Lemma 3.2.1Foreach{3E({3.,{3·j, ' 'l.

Proof. otingthat yp= ({3 {3.)and

l.

~ i{3.(~+2mr)(I,N!·»

^(b

₀

^l""

_{. '} C _b,,,,. '''I') ( _N.

^1)Jo'sin3xdx

J (I N· N. sin'xdx' 'Oas{3~{3.

whereOi_,i=1,2,N!·),N",areallpositive.O

Lemma 3.2.2(90,Lemma .2 Foreach{3 E ({3.,{3'j,A=i {pis a simple eigenvalueoj A({3),n=O,I,··

SinceA=i"YfJisasimpleeigenvalueofATnlitisnotdifficulttoshowthat there are aneighborhood of " in l.x x l.C IRxCxX~anda continuouslydifferentiablemapping .. -. x.X~suchthatforeachTE Op...

theonlyeigenvalueof .({3)inCpisA(randitscorresponding eigenfunction is (,p,(r,{3),,p,(r,{3)fwith

In the following,we discuss thesignof Re '(rn) which will be for ana

CHAPTER3.COMPETITION SYSTEM lyzing thepropertyofthesteadystate(u~,v~).Differentiating

( h( {3

( ( {3 =0,TEO~"

( {3

ithrespectto at .,multiplying by

( i l, l

and integrating

)

on (O,,,.), e have

C1U~ ) ( Wl~ )

dx

b,v~ W,~

Therefore,

x

J;(Wi;]Wl~+wl;]w,~)dx

Lemma 3.2.3 oeach{3 E ({3.,{3'j, e ( >0 (n=O,l,"')' Proof.Since'Y~=O({3 - (3.) and =~+O({3-(3.),itis easyto see that

e ='Y~6+0(({3_{3.)3),d

CHAPTER3.COMPETITION SYSTEM

J;(w~i,w~}

^{( ) ( )dx}

= (I l)(bo, , b,^lPc1alP) (P N.1)J;Sin3Xdx)(P-P. (P-P.)'.

ThenR.e11is equal to

",pO(I+N!·)N.~

^(I,N!'))(b_{o, ,}^lPc al_b,P)_{P N.}

(I)

_Jo^{' in3Xdx)( J- 3.) .(33.}

To checkthe stability ofthe nonconstant steady state solution(u,O,vp)with delays , nda O, firstwhenT=Otheeigenvalueproblern is reduced to

YI=sinx+(P-P.), Y,=ppsin+ (P-P.)' wherepp andpp-- pas --•. SubstitutingYI,Y' into thecharacteristic equation,multiplyingbothsidesbysinxandintegratingitfromOto7r,yields

~ao=-(bl+cIP)al.

J

^{! - 'd+(P-P.)}

~aop=-(b, o, .

J

,

^e-

^{dO+ (P-P.) (3.2.3)}

DenotingOIo/({3 {3.) = eA d = ,(, ),(3.2.3)implies p=-

'\M~:~~:.~,('\)

F( =),\'+~'\M,(,\)+ (b''' 'ha,.,la ( )

=: +B, ),(+B, ( ) =0,

where according to condition(CS,),B.,B,>O.Toanalyze the zeros of ( ), wefollowthe method in[23]and[24).Fromageneral resultincomplex variable

theory,thenumberofrootsofF('\)=Oin the righthalfof the complexpIane

willbe givenby

A~2k ( ) , ~d'\ ~

^0,

sinceF('\)isanalyticforRe'\>O. erey(R) is takenas theclosedsemicircular contour centeredat the originand containedinRe'\~O

FromAppendixA,wehave

It followsthat the numberof eigenvalues ofAo({3)with positive realpartsisO Thenwehavethefollowing lemma

Lemma 3.2.4 ForanyiJ>0andT=0,the steadystate solution (up,vp)is stable

Remark3.2.1Forthe case=0,i.e. theuniformlydistributed delay becomes discrete delay,itiswellprovedin(Sthat all the eigenvalues ofA,({3) have negativerealpartsatT=O

The followingtheoremholdssinceReX(Tn )>0 from Lemma3.2.3 Theorem3.2.5FOTany{JE({J.,{J'],0<(J'-{J.«I,there exist2(n+l) eigenvaluesof theinfinitesimalgenerntorAr(.B)withpositiverealpart whenrE (Tn,Tn+d,n=O,I,··

3.3 The existence of Hopf bifurcation Inthissectionwe willstudytheHopfbifurcation fromthe positive equilibrium (Up,Vp)asthe timedelayTcrossesro. A sirnilardiscussioncanbecarriedout for all otherTn,n=1,2"" .Forfixed{JE({J..{J'!andT=To+',denote

U(t,·)=u(t,.)-up,V(t,·)=u(t,·)-up SubstitutingU,V into(3.0.1),wehaveasystemequivalentto(3.0.1),

d^t

(u)

^{V =A ,(iJ,')(U(t»V(t))}+^{g(U.,\It,,) (3.3.1)}

A_'"((J,,)= A({J)(

u

_v

)_

^{J(b'Uc,up lJ.,up^{c'U) (J}J:::"+6^+'

^~U(t

~V(t-9)d9^{' - ))} -{J(b'Uc'U)

( J:::"+'~U(t-9)d9 )

c,V J.,V J ~V(t+-)

andg:C<l-(ro+o),O],X')~Y'isanonlinearoperatordefinedby

Letwp=2 andWI(t)=U(I+u)t),w,(t)=V(I+u)t).Then(U(t), V(t»

isanwp(l u) periodic solutionof(3.3.1) if andonlyif(w,(t),w,(t))isan wp periodicsolutionof

( w w, ' ) (w

^{( )W,}

)

^{- ,p}

(b U

,

^cu

,pb,

)(.J:'b+6~d8)

f::~d8+ u t- )+G«,u,wt),(3.3.2) whereG«,u,w,)is equalto

setting(J= andfforconvenienceomitting the tilde. Similar to[611 weuse thefollowing notations

(I)~(8)=(~p(O),~p(o»),-(TO ) 0, ( )

=

( ~~(S)/~p,, ) ,

^{0 TO+O,}

( ) [ 1(1)( )

(= 11(1)(0),1I )()=~(O)H, 1( )= = - ),(

1 1 )( )

H= ~

₂

(I

_I

-i

_i

),

^{( (( (}₍

and 1(1)( )=( 1 I)( ), 1 I)( ),i=I,2.

(2)LetA be theeigenspace ofA ,( )corresponding to eigenvalues i yp

(3)Let beaBanachspacedefinedas

w,={ (

~~

^{)E (, ,f,(t)}+wp) = /;(t),i =1,2,tE I . (4) p=(pl, 2) ,p,w,- ,i= ,2,aredefinedby

p;/= ['/. ( +( )fl( ) ( )f,( ))dxds,i= ,2 Itis easy toseethat~is areal basisofA,and isareal basisof the eigenfunction subspace of theformal adjointoperator.Byadirectcalculation, itisclearthat('l!,~)=I

Westate thefollowing lemmaabout the existence ofaperiodicsolution(see [6]and[77])

Lemma 3.3.1ForjEPw",theequation

(3.3.3) has anwp periodicsolution ifand onlyiff EN(p),thatisp;/=O,i= 1,2 Hence thereisalinear operator from N(p) tow,suchthat foreach fixed fEN(p), fisthewp eriodicsolution of ( . . )satisfying( llg=0,i.e ( ( ilo), =0,where( ilois defined by ( l O)=( l(O),OE[-(ro+o),O) ComparingwithEq. (3.3.3), weknow Eq.(3.3.2)hasanwp periodicsolution w(t) if and onlyif there isa constant csuch that

pC(f,q,W)=0, (3.3.4)

wet)=~(ll(t)+[ (f,q,w)j(t),tE , (3.3.5)

Followingtheprocedurein[6], weintroduceachangeofvarisblcsf=ce,a =

wet) = cI( )(t)+cW(t)], tE ,Wet)Ew" (1 1,(W)o) =0 Then(3.3.4)and(3.3.5) are equivalent to

. (c,e,(,W)=

J.

^{w (I}( ), (c,e,(,W( )))d , (3.3.6)

W=.l (c,e,(,W)=.1( ) (3.3.7)

whereNIisequalto

(d +{3-{3(b, + ))( t)+CWI(t))-{3(1+ ()[1 ,)(t)+cW,(t)]

xJ +l olb,( ,)+cWI)(t- g)+,( +cW,)(t -) g)]dB +{3 Jo +lb,Jo)(t B c)+cl )(tIB c )]d dB

{3 J+ lb,(o w,(t-g) -WI(t-B))+c,(W,(t-g) -W,(t-B )]dB {3( Jobl( I+ l)+cWIl(t-g)+ ( )+cW,)(t - g)ldB, and2is equalto

(d +{3-{3(c, +b, )( )(t)+cW,(t))-{3(1+()[ 1 )(t)+cW,(t)!

xJ + c,( I+cW,)(t- g)) + ,( )+cW,)(t-g)JdB

+{3 oJ + Jo lc, )(t Bt B c )+b,c )d dB I)(

{3 J o+ lc,(W,(t-g)-W,(t -B)+b,(W,(t-g)- W,(t - B)) dB {3( Jo[c,+( 1+cW,)(t -) g)+ ,( +cW,)(t -) g)JdB,