- Preface ix
- Chapter 1: Mathematical Modeling in Biology
- 1 1.1 Introduction
- 1 1.2 HIV
- 2 1.3 Models of HIV/AIDS
- 5 1.4 Concluding Message
- 14
- Chapter 2: How to Construct a Model
- 17 2.1 Introduction
- 17 2.2 Formulate the Question
- 19 2.3 Determine the Basic Ingredients
- 19 2.4 Qualitatively Describe the Biological System
- 26 2.5 Quantitatively Describe the Biological System
- 33 2.6 Analyze the Equations
- 39 2.7 Checks and Balances
- 47 2.8 Relate the Results Back to the Question
- 50 2.9 Concluding Message
- 51
- Chapter 3: Deriving Classic Models in Ecology and Evolutionary Biology
- 54 3.1 Introduction
- 54 3.2 Exponential and Logistic Models of Population Growth
- 54 3.3 Haploid and Diploid Models of Natural Selection
- 62 3.4 Models of Interactions among Species
- 72 3.5 Epidemiological Models of Disease Spread
- 77 3.6 Working Backward--Interpreting Equations in Terms of the Biology
- 79 3.7 Concluding Message
- 82
- Primer
- 1: Functions and Approximations
- 89 P1.1 Functions and Their Forms
- 89 P1.2 Linear Approximations
- 96 P1.3 The Taylor Series
- 100
- Chapter 4: Numerical and Graphical Techniques--Developing a Feeling for Your Model
- 110 4.1 Introduction
- 110 4.2 Plots of Variables Over Time
- 111 4.3 Plots of Variables as a Function of the Variables Themselves
- 124 4.4 Multiple Variables and Phase-Plane Diagrams
- 133 4.5 Concluding Message
- 145
- Chapter 5: Equilibria and Stability Analyses--One-Variable Models
- 151 5.1 Introduction
- 151 5.2 Finding an Equilibrium
- 152 5.3 Determining Stability
- 163 5.4 Approximations
- 176 5.5 Concluding Message
- 184
- Chapter 6: General Solutions and Transformations--One-Variable Models
- 191 6.1 Introduction
- 191 6.2 Transformations
- 192 6.3 Linear Models in Discrete Time
- 193 6.4 Nonlinear Models in Discrete Time
- 195 6.5 Linear Models in Continuous Time
- 198 6.6 Nonlinear Models in Continuous Time
- 202 6.7 Concluding Message
- 207
- Primer
- 2: Linear Algebra
- 214 P2.1 An Introduction to Vectors and Matrices
- 214 P2.2 Vector and Matrix Addition
- 219 P2.3 Multiplication by a Scalar
- 222 P2.4 Multiplication of Vectors and Matrices
- 224 P2.5 The Trace and Determinant of a Square Matrix
- 228 P2.6 The Inverse
- 233 P2.7 Solving Systems of Equations
- 235 P2.8 The Eigenvalues of a Matrix
- 237 P2.9 The Eigenvectors of a Matrix
- 243
- Chapter 7: Equilibria and Stability Analyses--Linear Models with Multiple Variables
- 254 7.1 Introduction
- 254 7.2 Models with More than One Dynamic Variable
- 255 7.3 Linear Multivariable Models
- 260 7.4 Equilibria and Stability for Linear Discrete-Time Models
- 279 7.5 Concluding Message
- 289
- Chapter 8: Equilibria and Stability Analyses--Nonlinear Models with Multiple Variables
- 294 8.1 Introduction
- 294 8.2 Nonlinear Multiple-Variable Models
- 294 8.3 Equilibria and Stability for Nonlinear Discrete-Time Models
- 316 8.4 Perturbation Techniques for Approximating Eigenvalues
- 330 8.5 Concluding Message
- 337
- Chapter 9: General Solutions and Tranformations--Models with Multiple Variables
- 347 9.1 Introduction
- 347 9.2 Linear Models Involving Multiple Variables
- 347 9.3 Nonlinear Models Involving Multiple Variables
- 365 9.4 Concluding Message
- 381
- Chapter 10: Dynamics of Class-Structured Populations
- 386 10.1 Introduction
- 386 10.2 Constructing Class-Structured Models
- 388 10.3 Analyzing Class-Structured Models
- 393 10.4 Reproductive Value and Left Eigenvectors
- 398 10.5 The Effect of Parameters on the Long-Term Growth Rate
- 400 10.6 Age-Structured Models--The Leslie Matrix
- 403 10.7 Concluding Message
- 418
- Chapter 11: Techniques for Analyzing Models with Periodic Behavior
- 423 11.1 Introduction
- 423 11.2 What Are Periodic Dynamics?
- 423 11.3 Composite Mappings
- 425 11.4 Hopf Bifurcations
- 428 11.5 Constants of Motion
- 436 11.6 Concluding Message
- 449
- Chapter 12: Evolutionary Invasion Analysis
- 454 12.1 Introduction
- 454 12.2 Two Introductory Examples
- 455 12.3 The General Technique of Evolutionary Invasion Analysis
- 465 12.4 Determining How the ESS Changes as a Function of Parameters
- 478 12.5 Evolutionary Invasion Analyses in Class-Structured Populations
- 485 12.6 Concluding Message
- 502
- Primer
- 3: Probability Theory
- 513 P3.1 An Introduction to Probability
- 513 P3.2 Conditional Probabilities and Bayes' Theorem
- 518 P3.3 Discrete Probability Distributions
- 521 P3.4 Continuous Probability Distributions
- 536 P3.5 The (Insert Your Name Here) Distribution
- 553
- Chapter 13: Probabilistic Models
- 567 13.1 Introduction
- 567 13.2 Models of Population Growth
- 568 13.3 Birth-Death Models
- 573 13.4 Wright-Fisher Model of Allele Frequency Change
- 576 13.5 Moran Model of Allele Frequency Change
- 581 13.6 Cancer Development
- 584 13.7 Cellular Automata--A Model of Extinction and Recolonization
- 591 13.8 Looking Backward in Time--Coalescent Theory
- 594 13.9 Concluding Message
- 602
- Chapter 14: Analyzing Discrete Stochastic Models
- 608 14.1 Introduction
- 608 14.2 Two-State Markov Models
- 608 14.3 Multistate Markov Models
- 614 14.4 Birth-Death Models
- 631 14.5 Branching Processes
- 639 14.6 Concluding Message
- 644
- Chapter 15: Analyzing Continuous Stochastic Models--Diffusion in Time and Space
- 649 15.1 Introduction
- 649 15.2 Constructing Diffusion Models
- 649 15.3 Analyzing the Diffusion Equation with Drift
- 664 15.4 Modeling Populations in Space Using the Diffusion Equation
- 684 15.5 Concluding Message
- 687 Epilogue: The Art of Mathematical Modeling in Biology
- 692
- Appendix 1: Commonly Used Mathematical Rules
- 695 A1.1 Rules for Algebraic Functions
- 695 A1.2 Rules for Logarithmic and Exponential Functions
- 695 A1.3 Some Important Sums
- 696 A1.4 Some Important Products
- 696 A1.5 Inequalities
- 697
- Appendix 2: Some Important Rules from Calculus
- 699 A2.1 Concepts
- 699 A2.2 Derivatives
- 701 A2.3 Integrals
- 703 A2.4 Limits
- 704
- Appendix 3: The Perron-Frobenius Theorem
- 709 A3.1: Definitions
- 709 A3.2: The Perron-Frobenius Theorem
- 710
- Appendix 4: Finding Maxima and Minima of Functions
- 713 A4.1 Functions with One Variable
- 713 A4.2 Functions with Multiple Variables
- 714
- Appendix 5: Moment-Generating Functions
- 717
- Index of Definitions, Recipes, and Rules
- 725 General Index 727.
- (source: Nielsen Book Data)
Thirty years ago, biologists could get by with a rudimentary grasp of mathematics and modeling. It is not so today. In seeking to answer fundamental questions about how biological systems function and change over time, the modern biologist is as likely to rely on sophisticated mathematical and computer-based models as traditional fieldwork. In this book, Sarah Otto and Troy Day provide biology students with the tools necessary to both interpret models and to build their own. The book starts at an elementary level of mathematical modeling, assuming that the reader has had high school mathematics and first-year calculus. Otto and Day then gradually build in depth and complexity, from classic models in ecology and evolution to more intricate class-structured and probabilistic models. The authors provide primers with instructive exercises to introduce readers to the more advanced subjects of linear algebra and probability theory. Through examples, they describe how models have been used to understand such topics as the spread of HIV, chaos, the age structure of a country, speciation, and extinction. Ecologists and evolutionary biologists today need enough mathematical training to be able to assess the power and limits of biological models and to develop theories and models themselves. This innovative book will be an indispensable guide to the world of mathematical models for the next generation of biologists. It is a how-to guide for developing new mathematical models in biology. It provides step-by-step recipes for constructing and analyzing models. It covers interesting biological applications. It explores classical models in ecology and evolution. There are questions at the end of every chapter. Primers cover important mathematical topics. Exercises are provided with answers. Appendixes summarize useful rules. Labs and advanced material are available.
(source: Nielsen Book Data)