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Thirty years ago, biologists could get by with a rudimentary grasp of mathematics and modeling. 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.

- A how-to guide for developing new mathematical models in biology

- Provides step-by-step recipes for constructing and analyzing models

- Interesting biological applications

- Explores classical models in ecology and evolution

- Questions at the end of every chapter

- Primers cover important mathematical topics

- Exercises with answers

- Appendixes summarize useful rules

- Labs and advanced material available

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
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