# Mathematical Models In Biology

, by Edelstein-Keshet, Leah**Note:**Supplemental materials are not guaranteed with Rental or Used book purchases.

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

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.

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

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 |

Table of Contents provided by Ingram. All Rights Reserved. |