Table Of ContentLecture Notes in Mathematics 2232
Mathematical Biosciences Subseries
Raluca Eftimie
Hyperbolic and
Kinetic Models
for Self-organised
Biological
Aggregations
A Modelling and Pattern Formation
Approach
Lecture Notes in Mathematics 2232
Editors-in-Chief:
Jean-MichelMorel,Cachan
BernardTeissier,Paris
AdvisoryBoard:
MichelBrion,Grenoble
CamilloDeLellis,Princeton
AlessioFigalli,Zurich
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
GáborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,NewYork
AnnaWienhard,Heidelberg
In recentyears, the role of mathematicsin the life sciences has evolved a long
wayfromtheroleitplayedinthe1970’s,intheearlydaysof“biomathematics”,and
isasomewhatdifferentonenow,anditsperceptionbythemathematicalcommunity
is also different. We feel it is important for the Lecture Notes in Mathematics to
reflect this and thus underline the immense significance of the life sciences as a
fieldofapplicationandinteractionformathematicsinthe21stcentury.
Weareparticularlyinterestedingoingfarbeyondthetraditionalareasinwhich
mathematicswasappliedtoecology,suchaspopulationdynamics,andwouldlike
to attract publicationsin areas such as cell growth, proteinstructures, physiology,
vision,shaperecognition&gestalttheory,neuraldynamics,genomics,perhapsalso
somestatisticalaspects(thislistisnon-exhaustive).
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Raluca Eftimie
Hyperbolic and Kinetic
Models for Self-organised
Biological Aggregations
A Modelling and Pattern Formation Approach
123
RalucaEftimie
DivisionofMathematics
UniversityofDundee
Dundee,UK
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISSN2524-6771 ISSN2524-678X (electronic)
MathematicalBiosciencesSubseries
ISBN978-3-030-02585-4 ISBN978-3-030-02586-1 (eBook)
https://doi.org/10.1007/978-3-030-02586-1
LibraryofCongressControlNumber:2018963056
MathematicsSubjectClassification(2010):92-01,92-02,92C15,92D50,35C07,35Lxx,35Q20,35Q92,
35R09,35R60,37G40,58J55,65Nxx
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To Georgi
Preface
Self-organisedbiologicalaggregations(i.e.,aggregationsthatformintheabsenceof
aleaderorexternalfactors)havecaughttheinterestandimaginationofscientistsfor
thousandsofyears.Forexample,PlinytheElder,inhisbookTheNaturalHistory
(publishedaround77–79AD)[1],discussedthemigrationandmovementofvarious
flocks of birds, from thrushes and blackbirds to starlings and swallows. About
starlings,henotedthat“itisapeculiarityofthestarlingtoflyintroops,asitwere,
andthentowheelroundinaglobularmasslikeaball,thecentraltroopactingasa
pivotfortherest”.Inregardtoswallows,PlinytheElderremarkedthattheyare“the
onlybirdsthathavea sinuousflightofremarkablevelocity”[1].Thesecomments
emphasise humanity’s long-term fascination with the spatial and spatio-temporal
patternsdisplayedbyvariousanimalaggregations(includingflocksofbirds).
Over the last 60 years or so, researchers have used mathematical models to
identify the biologicalmechanismsthat could explain the formationand structure
of these animal grouppatterns. One of the first studies in this area was published
by Breder [2], who used algebraic equations for the repulsive-attractive forces
amongindividuals,tounderstand“thebasicnatureofinfluencesatworkinaschool
of fishes as well as in other less compact aggregations”. In his pioneering work,
Brederproposedtheideathatthereareattractionandrepulsionforcesbetweenfish,
which vary with the distance between them and those forces are likely mediated
bydifferentsensorymechanisms.Breder[2]identifiedvisiontobethemainfactor
involved in the attraction of fish towards each other and suggested that repulsion
couldbecausednotonlybyvisionbutalsobywatermovementorsound.
Currently,mostofmathematicalmodelsforself-organisedaggregationsassume,
one way or another, that the attractive-repulsive interactions are the basic mech-
anism behind the formation and persistence of biological aggregations. (Here,
I define self-organised aggregations as being those biological aggregations that
form in the absence of a leader or some external stimulus.) In general, these
social interactions are nonlocal, with repulsion acting on short distances and
attraction acting on large spatial distances. Thus, many mathematical models for
self-organisedbiologicalaggregationsarenonlocal.Nevertheless,theincorporation
into the mathematicalmodels of these social interactionsalone cannotexplain all
vii
viii Preface
complex patterns observed empirically in cell, bacterial and animal aggregations.
Thisled researchersto consideranothersocial interaction:alignment/polarisation.
However, alignment behaviours cannot be properly described with the help of
parabolic-typepartialdifferentialequations(which focuson randommovements).
A more natural approach for modelling alignment behaviours sees the use of
hyperbolicandkinetictransportmodels.
Thepurposeofthismonographistointroducethisresearcharea,ofmathematical
approachesfortheinvestigationofspatialandspatio-temporalpatternsdisplayedby
self-organisedbiologicalaggregations,tostudentsandresearchersnotfamiliarwith
the topic. To this end, I consider a step-by-step approach to describe various 1D
and 2D local hyperbolic and kinetic models (where interactions depend only on
thelocaldensityofneighbours),aswellasnonlocalmodels(whereindividualscan
perceive, via different sensory mechanisms, their conspecifics positioned further
away). I discuss the patterns obtained numerically with these models, as well as
other patterns that have been shown to exist or not with the help of analytical
methods. For completeness, I also give a brief overview of the most common
analytical approaches used to investigate the dynamics of hyperbolic and kinetic
models.In addition,I discuss brieflya varietyof numericalschemesdevelopedto
approximatethesolutionsofdifferenthyperbolicandkineticmodels(mainlyrelated
toproblemsinphysics,butwhichcanbeconsideredalsoinbiologicalcontexts).
The complexity and variety of these hyperbolic and kinetic models makes
it difficult to include here all types of models existent in the literature (and
all analytical and numerical approaches developed to investigate these models).
Moreover, because the investigation of collective aggregations and movement is
currentlyoneofthe mostactiveresearchareasinmathematicalbiology,moreand
more models are developed every month. Since one needs to stop somewhere, I
tried to focus on models that either introduced a new idea in terms of modelling
self-organisedaggregationsorused particularanalyticalandnumericaltechniques
to investigate the formation of patterns. However, there is a feeling that many
modelling/analytical/numericalaspectsshouldhavebeenpresentedinmoredetail.
Ultimately, I hope that this monograph will offer a first overview into the
use of kinetic and hyperbolic models to reproduce and investigate stationary and
moving biological aggregations. Moreover, I hope that researchers interested in
analytical and numericalapproachesfor hyperbolicand kinetic models (that have
been applied so far mainly to problems in physics and engineering) will become
awareofthecomplexityofphenomenainbiologyandthenumerousopenanalytical
and numerical problems associated with the models for self-organised biological
aggregations.
This monographis the result of multiple research collaborations(overthe past
15years)onvarioustopicsrelatedtopatternformationinecologicalandbiological
systemsanddiscussionswithcolleaguesandmentors.Iamparticularlygratefulto
myPhD supervisors,Prof.MarkA. LewisandProf.GerdadeVries, aswellasto
Prof. Frithjof Lutscher (with whom I collaborated at the beginning of my PhD),
who introducedme to the use of hyperbolicsystems to describe 1D movementin
biological/ecologicalaggregations and guided my first steps in the analytical and
Preface ix
numericalinvestigationofthesemathematicalmodels.Prof.MarkLewiswasvery
supportive to extend a review article on hyperbolic and kinetic models for self-
organisedbiologicalaggregations(publishedin2012intheJournalofMathematical
Biology) into a book. I am also very grateful to Prof. Pietro-Luciano Buono who
introduced me to the fascinating field of equivariant bifurcation theory: without
our collaboration on classifying the various spatial and spatio-temporal patterns
exhibited by 1D nonlocal hyperbolic systems, this monograph would not have
come to light. Prof. Thomas Hillen opened my eyes to the theory of hyperbolic
conservation laws (through the postgraduate courses he taught at the University
of Alberta while I was a PhD student). I must also thank Prof. Jose Carrillo de
la Plata and Prof. Nicola Bellomo who introduced me to their research on higher
dimensionalkineticequations(whichledtoourcurrentcollaborations).Manymore
colleagues and collaborators, among which I mention Prof. Razvan Fetecau and
Prof. Kees Weijer, have influenced over the past years my research on pattern
formationinbiologicalsystems,whichwasthestartingpointofthismonograph.
Finally,IwouldliketothanktheeditorialstaffatSpringer,inparticularDr.Eva
Hiripi,fortheirapproachabilityandhelpwiththisbook.
Dundee,UK RalucaEftimie
October2018
References
1. P.theElder,Birdswhichtaketheirdeparturefromus,andwhithertheygo;The
thrush, the blackbird, and the starling – Birds which lose their feathers during
theirretirement–Theturtle-doveandthering-dove–Theflightofstarlingsand
swallows,inTheNaturalHistory(TaylorandFrancis,London,1855).Translated
byJohnBostock,M.D.,F.R.S.,andH.T.Riley,Esq.B.A.
2. C.M.Breder,Ecology35(3),361(1954)
Contents
1 Introduction .................................................................. 1
1.1 ModellingSelf-organisedAggregationandMovement.............. 1
1.1.1 Individual-BasedModels ..................................... 4
1.1.2 KineticandMacroscopicModels............................. 17
1.2 The Importanceof Communicationfor Self-organised
BiologicalBehaviours................................................. 20
1.3 Overview............................................................... 23
References..................................................................... 32
2 AShortIntroductiontoOne-DimensionalConservationLaws......... 37
2.1 Introduction............................................................ 37
2.2 FundamentalResultsforSystemsofConservationLaws............ 40
2.2.1 Travelling Waves, Rarefaction Waves, Shocks
andContactDiscontinuities................................... 45
2.2.2 TheRankine-HugoniotJumpCondition ..................... 47
2.2.3 Admissibility Conditions: Entropy,Vanishing
ViscosityandSpeedStability................................. 48
References..................................................................... 52
3 One-EquationLocalHyperbolicModels.................................. 55
3.1 Introduction............................................................ 55
3.2 First-OrderTrafficModels............................................ 57
3.3 Second-OrderTrafficModels......................................... 62
3.4 Third-OrderTrafficModels........................................... 68
3.5 TrafficModelsthatIncludeReactionTerms ......................... 69
3.6 Advection-ReactionEquationsforAnimalPopulation
Dynamics............................................................... 72
3.7 AnalyticalApproachesforthe InvestigationofPatterns:
SpeedofTravellingWaves............................................ 74
3.8 NumericalApproaches................................................ 77
References..................................................................... 78
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