Mathematical Models In Biology

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ISBN: 9780898715545 | 0898715547
Cover: Paperback
Copyright: 2/28/2005

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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.

Preface to the Classics Edition	p. xv
Preface	p. xxiii
Acknowledgments	p. xxvii
Errata	p. xxxi
Discrete Process in Biology	p. 1
The Theory of Linear Difference Equations Applied to Population Growth	p. 3
Biological Models Using Difference Equations	p. 6
Cell Division	p. 6
An Insect Population	p. 7
Propagation of Annual Plants	p. 8
Statement of the Problem	p. 8
Definitions and Assumptions	p. 9
The Equations	p. 10
Condensing the Equations	p. 10
Check	p. 11
Systems of Linear Difference Equations	p. 12
A Linear Algebra Review	p. 13
Will Plants Be Successful?	p. 16
Qualitative Behavior of Solutions to Linear Difference Equations	p. 19
The Golden Mean Revisited	p. 22
Complex Eigenvalues in Solutions to Difference Equations	p. 22
Related Applications to Similar Problems	p. 25
Growth of Segmental Organisms	p. 26
A Schematic Model of Red Blood Cell Production	p. 27
Ventilation Volume and Blood CO[subscript 2] Levels	p. 27
For Further Study: Linear Difference Equations in Demography	p. 28
Problems	p. 29
References	p. 36
Nonlinear Difference Equations	p. 39
Recognizing a Nonlinear Difference Equation	p. 40
Steady States, Stability, and Critical Parameters	p. 40
The Logistic Difference Equation	p. 44
Beyond r = 3	p. 46
Graphical Methods for First-Order Equations	p. 49
A Word about the Computer	p. 55
Systems of Nonlinear Difference Equations	p. 55
Stability Criteria for Second-Order Equations	p. 57
Stability Criteria for Higher-Order Systems	p. 58
For Further Study: Physiological Applications	p. 60
Problems	p. 61
References	p. 67
Appendix to Chapter 2: Taylor Series	p. 68
Functions of One Variable	p. 68
Functions of Two Variables	p. 70
Applications of Nonlinear Difference Equations to Population Biology	p. 72
Density Dependence in Single-Species Populations	p. 74
Two-Species Interactions: Host-Parasitoid Systems	p. 78
The Nicholson-Bailey Model	p. 79
Modifications of the Nicholson-Bailey Model	p. 83
Density Dependence in the Host Population	p. 83
Other Stabilizing Factors	p. 86
A Model for Plant-Herbivore Interactions	p. 89
Outlining the Problem	p. 89
Rescaling the Equations	p. 91
Further Assumptions and Stability Calculations	p. 92
Deciphering the Conditions for Stability	p. 96
Comments and Extensions	p. 98
For Further Study: Population Genetics	p. 99
Problems	p. 102
Projects	p. 109
References	p. 110
Continuous Processes and Ordinary Differential Equations	p. 113
An Introduction to Continuous Models	p. 115
Warmup Examples: Growth of Microorganisms	p. 116
Bacterial Growth in a Chemostat	p. 121
Formulating a Model
First Attempt	p. 122
Corrected Version	p. 123
A Saturating Nutrient Consumption Rate	p. 125
Dimensional Analysis of the Equations	p. 126
Steady-State Solutions	p. 128
Stability and Linearization	p. 129
Linear Ordinary Differential Equations: A Brief Review	p. 130
First-Order ODEs	p. 132
Second-Order ODEs	p. 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 Chemostat	p. 143
Applications to Related Problems	p. 145
Delivery of Drugs by Continuous Infusion	p. 145
Modeling of Glucose-Insulin Kinetics	p. 147
Compartmental Analysis	p. 149
Problems	p. 152
References	p. 162
Phase-Plane Methods and Qualitative Solutions	p. 164
First-Order ODEs: A Geometric Meaning	p. 165
Systems of Two First-Order ODEs	p. 171
Curves in the Plane	p. 172
The Direction Field	p. 175
Nullclines: A More Systematic Approach	p. 178
Close to the Steady States	p. 181
Phase-Plane Diagrams of Linear Systems	p. 184
Real Eigenvalues	p. 185
Complex Eigenvalues	p. 186
Classifying Stability Characteristics	p. 186
Global Behavior from Local Information	p. 191
Constructing a Phase-Plane Diagram for the Chemostat	p. 193
The Nullclines	p. 194
Steady States	p. 196
Close to Steady States	p. 196
Interpreting the Solutions	p. 197
Higher-Order Equations	p. 199
Problems	p. 200
References	p. 209
Applications of Continuous Models to Population Dynamics	p. 210
Models for Single-Species Populations	p. 212
Malthus Model	p. 214
Logistic Growth	p. 214
Allee Effect	p. 215
Other Assumptions; Gompertz Growth in Tumors	p. 217
Predator-Prey Systems and the Lotka-Volterra Equations	p. 218
Populations in Competition	p. 224
Multiple-Species Communities and the Routh-Hurwitz Criteria	p. 231
Qualitative Stability	p. 236
The Population Biology of Infectious Diseases	p. 242
For Further Study: Vaccination Policies	p. 254
Eradicating a Disease	p. 254
Average Age of Acquiring a Disease	p. 256
Models for Molecular Events	p. 271
Michaelis-Menten Kinetics	p. 272
The Quasi-Steady-State Assumption	p. 275
A Quick, Easy Derivation of Sigmoidal Kinetics	p. 279
Cooperative Reactions and the Sigmoidal Response	p. 280
A Molecular Model for Threshold-Governed Cellular Development	p. 283
Species Competition in a Chemical Setting	p. 287
A Bimolecular Switch	p. 294
Stability of Activator-Inhibitor and Positive Feedback Systems	p. 295
The Activator-Inhibitor System	p. 296
Positive Feedback	p. 298
Some Extensions and Suggestions for Further Study	p. 299
Limit Cycles, Oscillations, and Excitable Systems	p. 311
Nerve Conduction, the Action Potential, and the Hodgkin-Huxley Equations	p. 314
Fitzhugh's Analysis of the Hodgkin-Huxley Equations	p. 323
The Poincare-Bendixson Theory	p. 327
The Case of the Cubic Nullclines	p. 330
The Fitzhugh-Nagumo Model for Neural Impulses	p. 337
The Hopf Bifurcation	p. 341
Oscillations in Population-Based Models	p. 346
Oscillations in Chemical Systems	p. 352
Criteria for Oscillations in a Chemical System	p. 354
For Further Study: Physiological and Circadian Rhythms	p. 360
Appendix to Chapter 8. Some Basic Topological Notions	p. 375
Appendix to Chapter 8. More about the Poincare-Bendixson Theory	p. 379
Spatially Distributed Systems and Partial Differential Equation Models	p. 381
An Introduction to Partial Differential Equations and Diffusion in Biological Settings	p. 383
Functions of Several Variables: A Review	p. 385
A Quick Derivation of the Conservation Equation	p. 393
Other Versions of the Conservation Equation	p. 395
Tubular Flow	p. 395
Flows in Two and Three Dimensions	p. 397
Convection, Diffusion, and Attraction	p. 403
Convection	p. 403
Attraction or Repulsion	p. 403
Random Motion and the Diffusion Equation	p. 404
The Diffusion Equation and Some of Its Consequences	p. 406
Transit Times for Diffusion	p. 410
Can Macrophages Find Bacteria by Random Motion Alone?	p. 412
Other Observations about the Diffusion Equation	p. 413
An Application of Diffusion to Mutagen Bioassays	p. 416
Appendix to Chapter 9. Solutions to the One-Dimensional Diffusion Equation	p. 426
Partial Differential Equation Models in Biology	p. 436
Population Dispersal Models Based on Diffusion	p. 437
Random and Chemotactic Motion of Microorganisms	p. 441
Density-Dependent Dispersal	p. 443
Apical Growth in Branching Networks	p. 445
Simple Solutions: Steady States and Traveling Waves	p. 447
Nonuniform Steady States	p. 447
Homogeneous (Spatially Uniform) Steady States	p. 448
Traveling-Wave Solutions	p. 450
Traveling Waves in Microorganisms and in the Spread of Genes	p. 452
Fisher's Equation: The Spread of Genes in a Population	p. 452
Spreading Colonies of Microorganisms	p. 456
Some Perspectives and Comments	p. 460
Transport of Biological Substances Inside the Axon	p. 461
Conservation Laws in Other Settings: Age Distributions and the Cell Cycle	p. 463
The Cell Cycle	p. 463
Analogies with Particle Motion	p. 466
A Topic for Further Study: Applications to Chemotherapy	p. 469
Summary	p. 469
A Do-It-Yourself Model of Tissue Culture	p. 470
A Statement of the Biological Problem	p. 470
A Simple Case	p. 471
A Slightly More Realistic Case	p. 472
Writing the Equations	p. 473
The Final Step	p. 475
Discussion	p. 476
For Further Study: Other Examples of Conservation Laws in Biological Systems	p. 477
Models for Development and Pattern Formation in Biological Systems	p. 496
Cellular Slime Molds	p. 498
Homogeneous Steady States and Inhomogeneous Perturbations	p. 502
Interpreting the Aggregation Condition	p. 506
A Chemical Basis for Morphogenesis	p. 509
Conditions for Diffusive Instability	p. 512
A Physical Explanation	p. 516
Extension to Higher Dimensions and Finite Domains	p. 520
Applications to Morphogenesis	p. 528
For Further Study	p. 535
Patterns in Ecology	p. 535
Evidence for Chemical Morphogens in Developmental Systems	p. 537
A Broader View of Pattern Formation in Biology	p. 539
Selected Answers	p. 556
Author Index	p. 571
Subject Index	p. 575
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Mathematical Models In Biology

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