A review of traveling wave solutions of one-dimens

AReviewofTravellingWaveSolutionsofOne-dimensionalReaction-DiusionEquationswithNon-LinearDiusionTermFaustinoSanchez-Gardu~no1,PhilipK.Maini2andE.Kappos31DepartamentodeMatematicas,FacultaddeCiencias,UNAM,CircuitoExterior,C.U.,Mexico04510,D.F.,MEXICO.2CentreforMathematicalBiology,MathematicalInstitute,UniversityofOxford,24-29St.Giles’,OxfordOX13LB,U.K.3AppliedMathematicsSection,SchoolofMathematicsandStatistics,UniversityofSheeld,SheeldS102UN,UK.May22,1997AbstractInthispaperwereviewtheexistenceofdierenttypesoftravellingwavesolutionsu(x;t)=(xct)ofdegeneratenon-linearreaction-diusionequationsoftheformut=[D(u)ux]x+g(u)fordierentdensity-dependentdiusioncoecientsDandkineticpartg.Thesein-cludethenon-lineardegenerategeneralizedFisher-KPPandtheNagumoequations.Also,weconsideranequationwhosediusioncoecientchangessignasthediusivesubstanceincreases.Thisdescribesadiusive-aggregativeprocess.Inthiscasethetravellingwavesolutionsareexploredandtheill-posednessoftwoboundary-valueproblemsas-sociatedwiththeaboveequationisstated.1IntroductionSincetheclassicalworksbyFisher,1937andKolmogorovetal.,1937,whointroducedthemodelut=Duxx+g(u)todescribethepropagationofanadvantageousgenewithinanone-dimensionalhabitat,agreatdealofworkhasbeencarriedouttoextendtheirmodeltotakeintoaccountotherbi-ological,chemicalorphysicalfactors.Oneoftheseextensionsconsiders1non-lineardiusionterms,whichcanbeseenasanon-linearFick’sdiu-sionlaw.Thenon-linearitycanariseintermsofaspace,timeordensitydependentdiusioncoecient.If,inthelattercase,thediusioncoecientanditsderivativevanishatcertainvaluesofthediusivesubstance,thenthecorrespondingreaction-diusionequationdegeneratesintoanODEequationatthesevalues.Thisdegeneracyhastwomaineectsonthequalitativefeaturesofitssolutions.Firstly,theydonotpropagatethroughspacewithinnitespeed,asforthecasewhentheequationhaspositiveconstantdiusioncoecientD.Infact,thereisanitespeedofpropagation.Inthecaseofconstantdif-fusioncoecient,forsuitableinitialconditions,thecorrespondingboundaryvalueproblemhassmoothsolutionsonitsdomain.Thisisnotthesituationforthedegeneratecase;normally,wedonotexpectsmoothsolutions(someofthemcanbeofsharptype),anditisnecessary,instead,tointroduceasuitableweaksolutionconcept.Theclassicalapproachtotheinvestigationoftheexistenceoftravellingwavesolutions(t.w.s.),u(x;t)=(xct),oflinearandnonlinearreaction-diusionsystemswasintroducedbyKolmogorovetal.,1937.Itconsistsofre-statingtheoriginalinitialandboundaryvalueproblem(whichisinaninnitedimensionalspace),intermsofsearchingforasetofparameters(inwhichthespeedcisincluded)forwhichanitesystemofODEshastrajectoriesconnectingpairsofequilibriumpoints(heteroclinicand/orho-moclinic).Theboundaryconditionsforthet.w.s.arere-statedintermsoftheasymptoticbehaviouroftheheteroclinictrajectoriesastimettendsto1andto+1.ThisODEsystemisobtainedbyre-statingtheproblemintheappropriatetravellingwavevariables.AwiderangeofmethodshavebeendevelopedtosearchfortravellingwavesolutionsusingKolmogorov’smethodinmanymodelsinbiology,ecol-ogy,physiology,chemistry,etc.Arstclassofmethodsinvolvesadirect,adhocexaminationofthedynamicsofthespecicODEsystemineachappli-cation,see,forexample,Fife,1979;Smoller,1983;Britton,1986;Murray,1989;Grindrod,1991;SwinneyandKrinsky,1992;Sanchez-Gardu~noandMaini,1994a,1995a.AnotherapproachistousetheConleyIndex.Thisisamoretopologicalapproachandhasalsobeenusedtoinvestigatetheexistenceofheteroclinic2connectionsandglobalbifurcationsforODEsystems(Smoller,1983;Kap-posetal.,1991;seeKappos1995foranaccessiblepresentationoftheCon-leyindexmethod.).Shootingargumentshavealsobeenused(seeDunbar,1984;Sanchez-Gardu~noetal.,1995c)toprovetheexistenceofheteroclinictrajectoriescorrespondingtot.w.s.ofcertainnon-linearreaction-diusionequations.Incaseswhereitisknownthatheteroclinictrajectoriesexist,onecananalysethemusingnumericaloranalyticaltools.InthisdirectiondierentnumericalmethodshavebeendevelopedtoapproximatetheappropriatetrajectoriesforthecorrespondingODEsystem,(seeDoedelandFriedman,1989;Beyn,1990)aswellassolvingthePDEmodel.Forcertainspecialcases,perturbationmethodscanbeusedtoderiveanalyticapproximationstothet.w.s.(Sanchez-Gardu~noandMaini,1994b).Inthispaperwereviewresultsontheexistenceofdierenttypesoft.w.s.forseveraltypesofone-dimensionaldensity-dependentreaction-diusionequations.Thepaperisstructuredasfollows:InSection2wepresentanoverviewofthet.w.s.dynamicsforthedegenerateFisher-KPPequation.Section3containsasimilarreviewforthegeneralizednon-lineardiusionNagumoequation.InSection4weconsideradensity-dependentreaction-diusionequationinwhichthediusivetermchangessign.Thenegativediusivetermisassociatedwithanaggregativeprocess.Theaggregativetravellingwavedynamicsisexploredandtheill-posednessofacoupleofinitialandboundaryvalueproblemsisstated.Inthispaperwewillomitthetechnicalproofoftheresults,andgive,instead,therelevantreferences.2Non-lineartravellingwavesinthedegenerateFisher-KKPequationsDensity-dependentdispersalhasbeenobservedinmanybiologicalpopula-tions,forexample,sq