Oscillation and Stability of Delay Models in Biology

347 Pages·2014·2.49 MB·English

by  Ravi P. Agarwal

Ravi P. Agarwal · Donal O'Regan
Samir H. Saker
Oscillation and
Stability of
Delay Models
in Biology
Oscillation and Stability of Delay Models
in Biology

Ravi P. Agarwal • Donal O’Regan (cid:129) Samir H. Saker
Oscillation and Stability
of Delay Models in Biology
123
RaviP.Agarwal DonalO’Regan
DepartmentofMathematics SchoolofMathematics,Statistics
TexasA&MUniversity-Kingsville andAppliedMathematics
Kingsville,TX,USA NationalUniversityofIreland
Galway,Ireland
SamirH.Saker
DepartmentofMathematics
MansouraUniversity
Mansoura,Egypt
ISBN978-3-319-06556-4 ISBN978-3-319-06557-1(eBook)
DOI10.1007/978-3-319-06557-1
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014939117
MathematicsSubjectClassification(2010):34K11,34C10,34C20
©SpringerInternationalPublishingSwitzerland2014
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RaviP.Agarwal:ToSadhna, Sheba,and
Danah
DonalO’Regan:ToAlice,Aoife,Lorna,
Daniel,andNiamh
SamirH.Saker:ToMona,Meran,Maryam,
Menah,andAhmed

Preface
Inmathematicsyoudonotunderstandthings.Youjustgetused
tothem.
JohnL.VonNeumann(1903–1957).
Weareservantsratherthanmastersinmathematics.
CharlesHermit(1822–1901).
A mathematical model is an equation or a system of equations used to describe a
naturalphenomenon.Manyresearchersstudythequalitativebehaviorofnonlinear
delay mathematical models in a single species and also species with interactions.
The qualitative analysis of delay models with constant coefficients (autonomous
models)hasbeenstudiedextensively.Weknowthatthevariationoftheenvironment
plays animportantroleinmany biologicalandecological dynamical systems.For
example, physical environment conditions such as temperature and humidity and
availabilityoffood,water,andotherresourcesusuallyvaryintimewithseasonalor
dailyvariation.Therefore,morerealisticmodelswouldbenonautonomoussystems.
One of the purposes of our book is to study oscillation and global stability of
specific types of nonautonomous delay models in biology. In particular, our book
presentsrecentresearchresultsonthequalitativebehaviorofmathematicalmodels
inbiology.
Thebookconsistsofsixchaptersandisorganizedasfollows:
In Chap.1, we discuss the derivation and extensions of logistic models and
some of their applications. This chapter also contains some useful results from
mathematicalanalysiswhichareneededthroughoutthebook.
In Chap.2, we are concerned with oscillation and nonoscillation of different
typesofdelaylogisticmodelsandtheirmodifiedforms.Inparticular,westudythe
oscillation of models of Hutchinson type, models with delayed feedback, ˛-delay
logisticmodels,˛-delaymodelswithseveraldelays,modelswithnonlineardelays,
hyperlogistic models, delay models with harvesting, and models with varying
capacity.
vii
viii Preface
In Chap.3, we discuss the local and global stability of different types of
delay logistic models. In particular, we are concerned with the local and global
stabilityofautonomouslogisticmodelsandtheuniformand3/2globalstabilityof
nonautonomousdelaylogisticmodels.Alsowediscussageneralizedlogisticmodel
andmodelswithimpulses.
In Chap.4, we discuss autonomous and nonautonomous logistic models with
piecewisearguments.
InChap.5,wediscusstheoscillationofautonomousandnonautonomous“food-
limited” population models with delay times and impulsive effects as well as the
existenceofperiodicsolutions.Alsowestudythe3/2globalstabilityoftheclassical
modelandthe3/2uniformstabilityofamodelwithaparameterl.Inaddition,we
discusstheglobalstabilityofmodelswithimpulsesandmoregeneralizedmodels,
“food-limited” population models with periodic coefficients, and the existence of
periodicsolutions.
In Chap.6, we are concerned with oscillation, global stability, and periodicity
of some diffusive logistic models. In particular, we present oscillation results of a
diffusive Malthus model with several delays, oscillation results of an autonomous
diffusive logistic model with a Neumann boundary condition (flux conditions),
oscillationresultsofanonautonomousdiffusivelogisticmodelwithseveraldelays
and a Neumann boundary condition (flux conditions), global stability of the delay
logistic diffusion model with a Neumann boundary condition, and periodicity and
stability of a periodic diffusive logistic model of Volterra-type with instantaneous
anddelayeffects.
Wewishtoexpressourthankstoourfamiliesandfriends.
Kingsville,TX,USA RaviP.Agarwal
Galway,Ireland DonalO’Regan
Mansoura,Egypt SamirH.Saker
Contents
1 LogisticModels............................................................... 1
1.1 TheLogisticModels.................................................... 2
1.2 ExtendedLogisticModels.............................................. 3
1.3 DelayLogisticModels.................................................. 4
1.4 SomeResultsfromAnalysis ........................................... 4
2 OscillationofDelayLogisticModels....................................... 9
2.1 ModelsofHutchinsonType ............................................ 9
2.2 ModelswithDelayedFeedback........................................ 12
2.3 ˛-DelayModels......................................................... 15
2.4 ˛-ModelswithSeveralDelays ......................................... 26
2.5 ModelswithHarvesting ................................................ 33
2.6 ModelswithNonlinearDelays......................................... 42
2.7 HyperlogisticModels................................................... 53
2.8 ModelswithaVaryingCapacity ....................................... 72
3 StabilityofDelayLogisticModels ......................................... 79
3.1 AutonomousModelsofHutchinsonType ............................. 80
3.1.1 LocalStability .................................................. 80
3.1.2 3-GlobalStability............................................... 84
2
3.1.3 GlobalExponentialStability................................... 91
3.2 ANonautonomousHutchinsonModel................................. 96
3.2.1 3(cid:2)UniformStability............................................ 96
2
3.2.2 3-GlobalStability............................................... 100
2
3.2.3 GlobalExponentialStability................................... 112
3.3 AGeneralizedLogisticModel ......................................... 116
3.4 ModelswithImpulses .................................................. 120
4 LogisticModelswithPiecewiseArguments............................... 127
4.1 OscillationofAutonomousModels.................................... 128
4.2 StabilityofAutonomousModels....................................... 130
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