Explorations of Mathematical Models in Biology with Maple

307 Pages·2014·2.131 MB·English

by  Mazen Shahin

EXPLORATIONS OF
MATHEMATICAL
MODELS IN BIOLOGY
WITH MAPLE™

EXPLORATIONS OF
MATHEMATICAL
MODELS IN BIOLOGY
WITH MAPLE™
MAZEN SHAHIN
Department of Mathematical Sciences
Delaware State University
Dover, DE, USA
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LibraryofCongressCataloging-in-PublicationData:
Shahin,Mazen,1947–author.
ExplorationsofmathematicalmodelsinbiologywithMaple™/MazenShahin,Departmentof
MathematicalSciences,DelawareStateUniversity,Dover,DE.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-1-118-03211-4(hardback)
1. Biology–Mathematicalmodels. 2. Biology–Dataprocessing. 3. Maple(Computerfile) I. Title.
QH323.5.S4642014
570.1051–dc23
2014011432
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
CONTENTS
Preface vii
Acknowledgments xi
1 Overview of Discrete Dynamical Modeling and Maple™ 1
1.1 Introduction to Modeling and DifferenceEquations, 1
1.2 The Modeling Process, 8
1.3 Getting Started withMaple, 12
2 Modeling with First-Order Difference Equations 37
2.1 Modeling with First-Order Linear Homogeneous Difference
Equations with Constant Coefficients, 37
2.2 Modeling with Nonhomogeneous First-Order Linear
Difference Equations, 53
2.3 Modeling with Nonlinear DifferenceEquations: Logistic
Growth Models, 72
2.4 Logistic Equations and Chaos, 89
3 Modeling with Matrices 100
3.1 Systems of Linear Equations Having UniqueSolutions, 100
3.2 The Gauss–Jordan Elimination Method withModels, 114
3.3 Introduction to Matrices, 136
3.4 Determinants and Systems of LinearEquations, 164
3.5 Eigenvalues and Eigenvectors, 176
3.6 Eigenvalues and Stability of Linear Models, 199
v
vi CONTENTS
4 Modeling with Systems of LinearDifferenceEquations 209
4.1 Modeling with Markov Chains, 209
4.2 Age-Structured Population Models, 230
4.3 Modeling with Second-Order Linear DifferenceEquations, 240
5 Modeling with Nonlinear Systems of Difference Equations 258
5.1 Modeling of InteractingSpecies, 258
5.2 The SIR Model of InfectiousDiseases, 272
5.3 Modeling with Second-Order Nonlinear Difference Equations, 280
Bibliography 286
Index 288
vi
PREFACE
MAIN GOALS
ThemaingoalofExplorationsofMathematicalModelsinBiologywithMaple™isto
offer students a textbook to help them explore and discover mathematical concepts
andutilizetheseconceptsinbuildingandanalyzingmathematicalmodelsoflifesci-
ence disciplines such as biology, ecology, and environmental sciences. The main
mathematical tools utilized in this text are difference equations and matrices. The
useofthemathematicssoftwareMapleisanintegralpartofexploringandanalyzing
themodels.Itisimportanttostressthatthistextisneitheratextondifferenceequa-
tions noratext on how to use Maple.
THE NEED FOR ATEXTBOOK IN ELEMENTARY MATHEMATICAL
MODELING AND MATRIX ALGEBRA
TheMathematicalAssociationofAmerica(MAA)publishedMath&Bio2010:Link-
ing Undergraduate Disciplines in 2005. Math & Bio 2010 envisions a new educa-
tional paradigm in which the disciplines of mathematics and biology, currently
quite separate, will be productively linked in the undergraduate science programs
ofthetwenty-firstcentury.Asascience,biologydependsincreasinglyondata,algo-
rithms,andmodels;invirtuallyeveryrespect,itisbecomingmorequantitative,more
computational, and more mathematical.
One of the recommendations of the Curriculum Renewal Across the First Two
Years(CRAFTY)oftheMAApublishedin2004wastheinclusionofmathematical
vii
viii PREFACE
modeling,discretemathematics,andmatrixalgebrainthemathematicscurriculumfor
biologymajors.
Following recommendations of these professional organizations, we believe that
thereisaneedforacourseinelementarymathematicalmodelingandmatrixalgebra
forbiologyandlifesciencemajors.Traditionally,mathematicalmodelsutilizediffer-
entialequations.AdifferentialequationscourseisusuallyofferedaftercalculusIand
calculusII.Consequently,thereisagreatneedtoofferacourseinmathematicalmod-
eling that utilizes difference equations rather than differential equations. Difference
equationsandmatrixalgebraaresimple,yetpowerful,toolsinmodelingdiscretetime
dynamicalsystems.ThesetoolsareaccessibletostudentswithhighschoolalgebraII
or college algebra anddo not require calculus and differential equations.
Sinceoneofthemainobjectivesinwritingthisbookisprovidingstudentswitha
self-contained textbook, we included the background materials necessary to under-
standthemaintopicsinthetext.Forexample,wepresentednecessarymaterialsfrom
linear algebra.
APPROACH
Inthistext,weintroducethemodelingofreal-lifesituationswithdifferenceequations
andmatricesusingMaple.Weemphasizetheuseofgraphicalandnumericaltechni-
ques, rather than theoretical techniques, to investigate and analyze the behavior of
solutionsofthemathematicalmodels.Wealsoinvestigateinterestinglinearandnon-
linearmodelsfromdiverselifesciencedisciplinessuchasbiology,ecology,andenvi-
ronmental sciences.
We utilize a discovery pedagogical approach. To introduce a concept, first we
investigateamodelnumericallyand/orgraphicallyandrecognizeapatternorcertain
propertiesthatcharacterizethatconcept.Thenwegiveadefinitionoftheconceptwith
examplesandapplications.Forexample,tointroducetheeigenvaluesandcorrespond-
ingeigenvectorsofasquarematrix,weinvestigateanage-structurepopulationmodel
withdifferent initial population vectors that lead to a visualization of an eigenvalue
andcorrespondingeigenvectors.Thenthedefinitionoftheeigenvaluesandeigenvec-
tors isintroduced and some properties are discussed.
WHY MODELING WITH DIFFERENCE EQUATIONS AND
MATRICES?
Differenceequationsrepresentaverysophisticatedandpowerfulmathematicaltoolto
modelawiderangeofreal-lifediscretetimesituationsindiverseareas,includingthe
life sciences. And matrices provide an excellent tool in modeling linear problems.
Moreover, these powerful tools do not require a sophisticated mathematics back-
ground, being accessible to anyone who has successfully completed high school
algebra or college algebra.