Transport, reaction, and delay in mathematical biology, and the inverse problem for traveling fronts

HomeTransport,reaction,anddelayinmathematicalbiology,andtheinverseproblemfortravelingfronts
TITLETransport,reaction,anddelayinmathematicalbiology,andtheinverseproblemfortravelingfronts
ABSTRACTReaction-diffusionequationsarethestandardmodelsforreactingandmovingparticlesinecology,cellbiology,andotherfieldsofbiology.Inmanysituations,amoredetaileddescriptionofthemovementsofparticlesorindividualsisrequired.Thenreaction-transportsystems,reactionCattaneosystems,andKramers-Langevinapproachescanbeused.Thetypicallimitsolutionsinunboundeddomainsaretravelingfrontswhichposenewmathematicalproblemsinthecaseoftransportequations.Theinverseproblemfortravelingfrontsisanovelproblem,whichisinvestigatedhereingreatdetail.Additionalfeaturesaredelays,whichtypicallyleadtooscillations,andquiescentphases,whichcanbeshowntostabilizeagainsttheonsetofoscillations.Inparticular,neutraldelayequationscanberigorouslyderivedfromfirst-orderhyperbolicequationswithappropriateboundaryconditionsmodelingagestructure.Multi-speciessystemsleadtostudyingvariousphenomenasuchasTuringinstability,interactionofdiffusionanddelay,andcrossdiffusion.
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