Mathematical Models In Biology

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  • Format: Paperback
  • Copyright: 2005-02-28
  • Publisher: Society for Industrial & Applied

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Mathematical Models in Biology is an introductory book for readers interested in biological applications of mathematics and modeling in biology. Connections are made between diverse biological examples linked by common mathematical themes, exploring a variety of discrete and continuous ordinary and partial differential equation models. Although great advances have taken place in many of the topics covered, the simple lessons contained in Mathematical Models in Biology are still important and informative. Shortly after the first publication of Mathematical Models in Biology, the genomics revolution turned Mathematical Biology into a prominent area of interdisciplinary research. In this new millennium, biologists have discovered that mathematics is not only useful, but indispensable! As a result, there has been much resurgent interest in, and a huge expansion of, the fields collectively called mathematical biology. This book serves as a basic introduction to concepts in deterministic biological modeling.

Author Biography

Leah Edelstein-Keshet is a member of the Mathematics Department at the University of British Columbia and past president of the Society for Mathematical Biology

Table of Contents

Preface to the Classics Editionp. xv
Prefacep. xxiii
Acknowledgmentsp. xxvii
Erratap. xxxi
Discrete Process in Biologyp. 1
The Theory of Linear Difference Equations Applied to Population Growthp. 3
Biological Models Using Difference Equationsp. 6
Cell Divisionp. 6
An Insect Populationp. 7
Propagation of Annual Plantsp. 8
Statement of the Problemp. 8
Definitions and Assumptionsp. 9
The Equationsp. 10
Condensing the Equationsp. 10
Checkp. 11
Systems of Linear Difference Equationsp. 12
A Linear Algebra Reviewp. 13
Will Plants Be Successful?p. 16
Qualitative Behavior of Solutions to Linear Difference Equationsp. 19
The Golden Mean Revisitedp. 22
Complex Eigenvalues in Solutions to Difference Equationsp. 22
Related Applications to Similar Problemsp. 25
Growth of Segmental Organismsp. 26
A Schematic Model of Red Blood Cell Productionp. 27
Ventilation Volume and Blood CO[subscript 2] Levelsp. 27
For Further Study: Linear Difference Equations in Demographyp. 28
Problemsp. 29
Referencesp. 36
Nonlinear Difference Equationsp. 39
Recognizing a Nonlinear Difference Equationp. 40
Steady States, Stability, and Critical Parametersp. 40
The Logistic Difference Equationp. 44
Beyond r = 3p. 46
Graphical Methods for First-Order Equationsp. 49
A Word about the Computerp. 55
Systems of Nonlinear Difference Equationsp. 55
Stability Criteria for Second-Order Equationsp. 57
Stability Criteria for Higher-Order Systemsp. 58
For Further Study: Physiological Applicationsp. 60
Problemsp. 61
Referencesp. 67
Appendix to Chapter 2: Taylor Seriesp. 68
Functions of One Variablep. 68
Functions of Two Variablesp. 70
Applications of Nonlinear Difference Equations to Population Biologyp. 72
Density Dependence in Single-Species Populationsp. 74
Two-Species Interactions: Host-Parasitoid Systemsp. 78
The Nicholson-Bailey Modelp. 79
Modifications of the Nicholson-Bailey Modelp. 83
Density Dependence in the Host Populationp. 83
Other Stabilizing Factorsp. 86
A Model for Plant-Herbivore Interactionsp. 89
Outlining the Problemp. 89
Rescaling the Equationsp. 91
Further Assumptions and Stability Calculationsp. 92
Deciphering the Conditions for Stabilityp. 96
Comments and Extensionsp. 98
For Further Study: Population Geneticsp. 99
Problemsp. 102
Projectsp. 109
Referencesp. 110
Continuous Processes and Ordinary Differential Equationsp. 113
An Introduction to Continuous Modelsp. 115
Warmup Examples: Growth of Microorganismsp. 116
Bacterial Growth in a Chemostatp. 121
Formulating a Model
First Attemptp. 122
Corrected Versionp. 123
A Saturating Nutrient Consumption Ratep. 125
Dimensional Analysis of the Equationsp. 126
Steady-State Solutionsp. 128
Stability and Linearizationp. 129
Linear Ordinary Differential Equations: A Brief Reviewp. 130
First-Order ODEsp. 132
Second-Order ODEsp. 132
A System of Two First-Order Equations (Elimination Method)p. 133
A System of Two First-Order Equations (Eigenvalue-Eigenvector Method)p. 134
When Is a Steady State Stable?p. 141
Stability of Steady States in the Chemostatp. 143
Applications to Related Problemsp. 145
Delivery of Drugs by Continuous Infusionp. 145
Modeling of Glucose-Insulin Kineticsp. 147
Compartmental Analysisp. 149
Problemsp. 152
Referencesp. 162
Phase-Plane Methods and Qualitative Solutionsp. 164
First-Order ODEs: A Geometric Meaningp. 165
Systems of Two First-Order ODEsp. 171
Curves in the Planep. 172
The Direction Fieldp. 175
Nullclines: A More Systematic Approachp. 178
Close to the Steady Statesp. 181
Phase-Plane Diagrams of Linear Systemsp. 184
Real Eigenvaluesp. 185
Complex Eigenvaluesp. 186
Classifying Stability Characteristicsp. 186
Global Behavior from Local Informationp. 191
Constructing a Phase-Plane Diagram for the Chemostatp. 193
The Nullclinesp. 194
Steady Statesp. 196
Close to Steady Statesp. 196
Interpreting the Solutionsp. 197
Higher-Order Equationsp. 199
Problemsp. 200
Referencesp. 209
Applications of Continuous Models to Population Dynamicsp. 210
Models for Single-Species Populationsp. 212
Malthus Modelp. 214
Logistic Growthp. 214
Allee Effectp. 215
Other Assumptions; Gompertz Growth in Tumorsp. 217
Predator-Prey Systems and the Lotka-Volterra Equationsp. 218
Populations in Competitionp. 224
Multiple-Species Communities and the Routh-Hurwitz Criteriap. 231
Qualitative Stabilityp. 236
The Population Biology of Infectious Diseasesp. 242
For Further Study: Vaccination Policiesp. 254
Eradicating a Diseasep. 254
Average Age of Acquiring a Diseasep. 256
Models for Molecular Eventsp. 271
Michaelis-Menten Kineticsp. 272
The Quasi-Steady-State Assumptionp. 275
A Quick, Easy Derivation of Sigmoidal Kineticsp. 279
Cooperative Reactions and the Sigmoidal Responsep. 280
A Molecular Model for Threshold-Governed Cellular Developmentp. 283
Species Competition in a Chemical Settingp. 287
A Bimolecular Switchp. 294
Stability of Activator-Inhibitor and Positive Feedback Systemsp. 295
The Activator-Inhibitor Systemp. 296
Positive Feedbackp. 298
Some Extensions and Suggestions for Further Studyp. 299
Limit Cycles, Oscillations, and Excitable Systemsp. 311
Nerve Conduction, the Action Potential, and the Hodgkin-Huxley Equationsp. 314
Fitzhugh's Analysis of the Hodgkin-Huxley Equationsp. 323
The Poincare-Bendixson Theoryp. 327
The Case of the Cubic Nullclinesp. 330
The Fitzhugh-Nagumo Model for Neural Impulsesp. 337
The Hopf Bifurcationp. 341
Oscillations in Population-Based Modelsp. 346
Oscillations in Chemical Systemsp. 352
Criteria for Oscillations in a Chemical Systemp. 354
For Further Study: Physiological and Circadian Rhythmsp. 360
Appendix to Chapter 8. Some Basic Topological Notionsp. 375
Appendix to Chapter 8. More about the Poincare-Bendixson Theoryp. 379
Spatially Distributed Systems and Partial Differential Equation Modelsp. 381
An Introduction to Partial Differential Equations and Diffusion in Biological Settingsp. 383
Functions of Several Variables: A Reviewp. 385
A Quick Derivation of the Conservation Equationp. 393
Other Versions of the Conservation Equationp. 395
Tubular Flowp. 395
Flows in Two and Three Dimensionsp. 397
Convection, Diffusion, and Attractionp. 403
Convectionp. 403
Attraction or Repulsionp. 403
Random Motion and the Diffusion Equationp. 404
The Diffusion Equation and Some of Its Consequencesp. 406
Transit Times for Diffusionp. 410
Can Macrophages Find Bacteria by Random Motion Alone?p. 412
Other Observations about the Diffusion Equationp. 413
An Application of Diffusion to Mutagen Bioassaysp. 416
Appendix to Chapter 9. Solutions to the One-Dimensional Diffusion Equationp. 426
Partial Differential Equation Models in Biologyp. 436
Population Dispersal Models Based on Diffusionp. 437
Random and Chemotactic Motion of Microorganismsp. 441
Density-Dependent Dispersalp. 443
Apical Growth in Branching Networksp. 445
Simple Solutions: Steady States and Traveling Wavesp. 447
Nonuniform Steady Statesp. 447
Homogeneous (Spatially Uniform) Steady Statesp. 448
Traveling-Wave Solutionsp. 450
Traveling Waves in Microorganisms and in the Spread of Genesp. 452
Fisher's Equation: The Spread of Genes in a Populationp. 452
Spreading Colonies of Microorganismsp. 456
Some Perspectives and Commentsp. 460
Transport of Biological Substances Inside the Axonp. 461
Conservation Laws in Other Settings: Age Distributions and the Cell Cyclep. 463
The Cell Cyclep. 463
Analogies with Particle Motionp. 466
A Topic for Further Study: Applications to Chemotherapyp. 469
Summaryp. 469
A Do-It-Yourself Model of Tissue Culturep. 470
A Statement of the Biological Problemp. 470
A Simple Casep. 471
A Slightly More Realistic Casep. 472
Writing the Equationsp. 473
The Final Stepp. 475
Discussionp. 476
For Further Study: Other Examples of Conservation Laws in Biological Systemsp. 477
Models for Development and Pattern Formation in Biological Systemsp. 496
Cellular Slime Moldsp. 498
Homogeneous Steady States and Inhomogeneous Perturbationsp. 502
Interpreting the Aggregation Conditionp. 506
A Chemical Basis for Morphogenesisp. 509
Conditions for Diffusive Instabilityp. 512
A Physical Explanationp. 516
Extension to Higher Dimensions and Finite Domainsp. 520
Applications to Morphogenesisp. 528
For Further Studyp. 535
Patterns in Ecologyp. 535
Evidence for Chemical Morphogens in Developmental Systemsp. 537
A Broader View of Pattern Formation in Biologyp. 539
Selected Answersp. 556
Author Indexp. 571
Subject Indexp. 575
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