Nonautonomous Dynamical Systems in the Life Sciences

326 Pages·2013·7.56 MB·English

by  Peter E. Kloeden

Lecture Notes in Mathematics 2102
Mathematical Biosciences Subseries
Peter E. Kloeden
Christian Pötzsche Editors
Nonautonomous
Dynamical Systems
in the Life Sciences
Lecture Notes in Mathematics 2102
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
CamilloDeLellis(Zürich)
MariodiBernardo(Bristol)
AlessioFigalli(Austin)
DavarKhoshnevisan(SaltLakeCity)
IoannisKontoyiannis(Athens)
GaborLugosi(Barcelona)
MarkPodolskij(Heidelberg)
SylviaSerfaty(ParisandNY)
CatharinaStroppel(Bonn)
AnnaWienhard(Heidelberg)
EditorsMathematicalBiosciencesSubseries:
P.K.Maini,Oxford
Forfurthervolumes:
http://www.springer.com/series/304

Peter E. Kloeden (cid:129) Christian P oRtzsche
Editors
Nonautonomous Dynamical
Systems in the Life Sciences
123
Editors
PeterE.Kloeden ChristianPoRtzsche
InstitutfürMathematik InstitutfürMathematik
Goethe-UniversitätFrankfurt Alpen-AdriaUniversitätKlagenfurt
FrankfurtamMain,Germany KlagenfurtamWörthersee,Austria
ISBN978-3-319-03079-1 ISBN978-3-319-03080-7(eBook)
DOI10.1007/978-3-319-03080-7
SpringerChamHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2013956227
MathematicsSubjectClassification(2010):37B55,92XX,34C23,34C45,37HXX
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Preface
Thetheoryofdynamicalsystemsisawell-developedandsuccessfulmathematical
frameworktodescribetime-varyingphenomena.Itsapplicationsinthelifesciences
range from simple predator–preymodels to complicated signal transduction path-
ways in biological cells, in physics from the motion of a pendulum to complex
climate models,and beyondthatto furtherfieldsas diverseas chemistry(reaction
kinetics),economics,engineering,sociology,demography,andbiosciences.Indeed,
Systems Biology relies heavily on methods from Dynamical Systems. Moreover,
these diverse applications have provided a significant impact on the theory of
dynamicalsystemsitself andisoneofthemainreasonsforits popularityoverthe
lastdecades.
As a general principle, before abstract mathematical tools can be applied to
real-world phenomena from the above areas, one needs corresponding models in
terms of some kind of evolutionarydifference or differentialequation. Their goal
is to provide a realistic and tractable picture for the actual behavior of, e.g., a
biodynamicalsystem.Athoroughunderstandinghelpstooptimizetime-consuming
andcostlyexperiments,likethedevelopmentofharvestingordosingstrategiesand
mightevenenablefieldstudiestobeavoided.
Fromaconceptionallevel,indevelopingsuchmodelsonedistinguishesanactual
dynamicalsystem fromits surroundingenvironment.The system is givenin terms
of physical or internal feedback laws that yield an evolutionary equation. The
parametersinthisequationdescribethecurrentstateoftheenvironment.Thelatter
mayormaynotvaryintime,butisassumedtobeunaffectedbythesystem.
For autonomous dynamical systems the basic law of evolution is static in
the sense that the environment does not change with time. However, in many
applications such a static approach is too restrictive and a temporally fluctuating
environmentisrequired:
– Parametersinreal-worldsituationsandparticularlyinthelifesciencesarerarely
constant over time. This has various reasons, like absence of lab conditions,
adaptionprocesses,seasonaleffectsondifferenttimescales,changesinnutrient
supply,oranintrinsic“backgroundnoise.”
v
vi Preface
– On the other hand, sometimes it is desirable to include regulation or control
strategiesintoamodel(e.g.,harvestingandfishing,dosingofdrugsorradiation,
stimulatingchemicalsorcatalyticsubmissions),aswellasextrinsicnoise,andto
studytheireffects.Inparticular,biochemicalsignalingwithinandintocellsisa
nonautonomousprocess.
– Several problems can be decomposed into coupled subsystems. Provided the
influence of some of them is understood, without the requirement to precisely
knowtheirexplicitform,theycanbeseenasanon-constanttime-varyinginput.
Thesetemporalfluctuationsmightbedeterministicorrandomandin thefirstcase
oftenmorethan just periodicin time. The evidenceof time-dependentparameters
canalsobeverifiedstatistically,whenitcomestotheproblemoffittingparameters
toactualmeasureddataorinphenomenalikecardiovascularageing.
Consequently,inreasonablemodelsthatareadaptedtoandwellsuitedforprob-
lems in temporally fluctuating environments, the resulting evolutionary equations
have to depend explicitly on time. In order to study such realistic problems, the
classicaltheoryofdynamicalsystemshastobeextended.Thefieldofnonautono-
mous and random dynamicalsystems has thus received a wide attraction over the
recent 10–15 years and is expected to develop to further maturity. Both the fields
ofnonautonomousandofrandomdynamicalsystemsareparalleltheoriesfeaturing
verysimilarconcepts.
In the area of biomathematics, for example, the corresponding contributions
deal with nonautonomous equations and have provided major progress in our
understandingof classical boundedness,global stability, persistence, permanence,
or positivity aspects. Nevertheless, often more subtle questions are crucial. For
instance, it is of utmost importance to identify “key players” in biodynamical
processes,i.e.,thevariablesorparameterscruciallyaffectingthelong-termbehavior
ofasystem.Knowingthesequantitiesenablesresearcherstoreducethedimension
ofasystemsignificantlyandthusmakesitamenableforanalyticaltools,asopposed
to sometimes problematic numerical methods and simulations. Such questions
clearlyfitintotheframeworkofbifurcationtheorydescribingqualitativechanges.
However,whendealingwithnonautonomousandrandomequationsnewideasand
concepts are required:For instance, the classical notions of invariance,attraction,
hyperbolicity,andinvariantmanifoldshavehadtobeextended.
Despite being well motivated, one rarely finds biological processes modeled
using nonautonomous equations. A cause seems to be the problem that classical
methods from autonomous dynamical systems theory do not apply to them,
while more recent tools tailor-made for time-dependentproblems still need to be
popularized.
For these reasons, the contemporary fields of nonautonomous and random
dynamical systems on the one side, and biodynamics on the other side, strongly
benefitfromeachother.Actually,itisessentialto
– illustrate such a modern theory using successful and convincing real-world
applications,
Preface vii
– gaininputfromreal-lifeapplicationinducingafurtherdevelopmentofthetheory
intodirectionsofabroaderinterest.
Ontheotherhand,researchersinterestedinmathematicalapproachestolifesciences
will
– find suitable mathematical methods promising and fruitful in various applica-
tions,
– obtainasolidtoolboxforanunderstandinganditerativerefinementofmodelsin
fluctuatingenvironments,
– loweraninhibitionthresholdtousenonautonomousmodelsfromthebeginning.
In conclusion, the main motivation for this book is to bring readers’ attention
tovariousrecentdevelopmentsandmethodsfromthefieldofnonautonomousand
randomdynamicalsystemsandpromisingapplicationsoriginatinginlifesciences.
For this reason we collected three articles (Chaps. 1–3) illustrating theoretical
aspects, as well as six further papers (Chaps. 4–8) focussing on more concrete
applications,wherethesenewideasandtoolscouldbeused:
1. The introductorycontribution of the editors shares the title with this book and
surveys several key concepts from the mathematical theory of deterministic
nonautonomousdynamicalsystems. Theyareexemplifiedusingvarioussimple
modelsfromthelifesciences.
2. ThechapteronRandomDynamicalSystemswithInputsbyMichaelMarcondes
deFreitasandEduardoD.Sontagdescribestheconceptofarandomdynamical
system and extends it to problems with inputs and outputs. Applications to
feedbackconnectionsaregiven.
3. Multiple time scales are an important feature of physiological systems and
functions. Martin Wechselberger, John Mitry, and John Rinzel give a modern
introduction to the basic geometrical singular perturbationtheory and use it in
theirchapterCanardTheoryandExcitabilitytotacklerelatedproblemsandtheir
transientbehavior.
4. Stimulus-ResponseReliability of BiologicalNetworks by Kevin K. Lin reviews
some basic concepts and results from the ergodic theory of random dynamical
systems and explains how these ideas can be used (partly in combination with
numericalsimulations)tostudythereliabilityofnetworks,i.e.,thereproducibil-
ityofanetwork’sresponsewhenrepeatedlypresentedwithagivenstimulus.
5. The “Lancaster group” Philip Clemson, Spase Petkoski, Tomislav Stankovski,
andAnetaStefanovskaexplainhownetworksofnonautonomousself-sustained
oscillators can model a virtual physiological human. Their chapter Coupled
Nonautonomous Oscillators includes novel methods suitable to reconstruct
nonautonomousdynamicsusingdatafromareallivingsystembystudyingtime-
dependentcouplingbetweencardiacandrespiratoryrhythms.
6. GermánEnciso’scontributionMultisite MechanismsforUltrasensitivity inSig-
nalTransductiongivesa mathematicalreviewof the mostimportantmolecular
modelsfeaturingultrasensitivebehavior.
viii Preface
7. ThechapterMathematicalConceptsinPharmacokineticsandPharmacodynam-
ics with Application to Tumor Growth by Gilbert Koch and Johannes Schropp
describes corresponding models and their applications in the pharmaceutical
industry.Moreover,amodelfortumorgrowthandanticancereffectsisdeveloped
anddiscussed.
8. InViralKineticModelingofChronicHepatitisCandBInfection,EvaHerrmann
andYusukeAsaidemonstratetheinterplaybetweenmathematicalandstatistical
analysis of compartment ODE models for hepatitis B and C. They give an
account of clinical use of models in treatment. Moreover, the most relevant
modelsforsuchinfectionsaresurveyed.
9. Finally, Christina and Nicolae Surulescu study Some Classes of Stochastic
Differential Equations as an Alternative Modeling Approach to Biomedical
Problems. In detail, models for an intracellular signaling pathway, a radio-
oncologicaltreatment,andcelldispersalarepresentedandstudied.
Finally,wecordiallythankallthecontributorstothisvolumeandhopeto have
contributedtobuildingabridgebetweennonautonomous/randomdynamicsandthe
lifesciences.
FrankfurtamMain,Germany PeterE.Kloeden
Klagenfurt,Austria ChristianPötzsche
September2013
Acknowledgments
Theideaofthisbookaroseduringdiscussionsataworkshop
NonautonomousandRandomDynamicalSystemsinLifeSciences
heldin Inzell, GermanyfromAugust1 to 5, 2011andorganizedby the editorsin
collaborationwithProf.Dr.RupertLasser(HelmholtzCentreMunichandMunich
UniversityofTechnology).TheworkshopwasfundedbytheVolkswagenStiftung
and the Institute for Biomathematics and Biometrics at the Helmholtz Centre
Munich. We are grateful for their support. We also thank Prof. Dr. R. Lasser for
hisassistanceinorganizingtheworkshop.
We are gratefulto the anonymousreferees—theirsuggestionsand constructive
criticismledtoamorecoherentandaccessiblepresentation.
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