Modeling the Dynamics of Life CALCULUS AND PROBABILITY FOR LIFE SCIENTISTS

Frederick R. Adler University of Utah

Brooks/Cole Publishing Company

l ( J / ) j r An International Thomson Publishing Company

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Contents

Chapter 1 Introduction to Discrete-Time Dynamical Systems 1

1.1 Biology and Dynamics 2

1.2 Updating Functions: Describing Growth 6

1.3 Units and Dimensions of Measurements and Functions 14

1.4 Linear Functions and TheirGraphs 22

1.5 Finding Solutions: Describing the Dynamics 32

1.6 Combining and Manipulating Functions 39

1.7 Expressing Solutions with Exponential Functions 49

1.8 Power Functions and Allometry 59

1.9 Oscillations and Trigonometry 69

1.10 Modeling and Graphical Analysis of Updating Functions 78

1.11 Equilibria 86

1.12 An Example of Nonlinear Dynamics 95

1.13 Excitable Systems: The Heart 106

Chapter 2 Limits and Derivatives 118

2.1 Introduction to Derivatives 119

2.2 Limits 129

2.3 Continuity 138

2.4 Computing Derivatives: Linear and Quadratic Functions 148

2.5 Derivatives of Sums, Powers, and Polynomials 156

2.6 Derivatives of Products and Quotients 168

2.7 The Second Derivative, Curvature, and Acceleration 176

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2.8 Derivatives of Exponential and Logarithmic Functions 184

2.9 Derivatives of Trigonometrie Functions 190

2.10 The Chain Rule 199

Chapter 3 Applications of Derivatives and Dynamical Systems 212

3.1 Stability and the Derivative 213

3.2 More Complex Dynamics 224

3.3 Maximization 237

3.4 Reasoning About Functions 248

3.5 Limitsat Inf inity 257

3.6 Leading Behavior and L'Höpital's Rule 268

3.7 Approximating Functions with Lines and Polynomials 280

3.8 Newton's Method 289

3.9 Panting and Deep Breathing 299

Chapter 4 Differential Equations, Integrals, and Their Applications 310

4.1 Differential Equations 311

4.2 Basic Differential Equations 322

4.3 Solving Pure-Time Differential Equations 330

4.4 Integration of Special Functions

and Integration by Substitution 340

4.5 Integrals and Sums 348

4.6 Definite and Indefinite Integrals 356

4.7 Applications of Integrals 366

4.8 Improper Integrals 374

Chapter 5 Analysis of Differential Equations 386

5.1 Equilibria and Display of

Autonomous Differential Equations 387

5.2 Stable and Unstable Equilibria 395

5.3 Solving Autonomous Differential Equations 402

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5.4 Two-Dimensional Differential Equations 409

5.5 The Phase Plane 417

5.6 Solutions in the Phase Plane 425

5.7 The Dynamics ofa Neuron 435

Chapter 6 Probability Theory and Descriptive Statistics 447

6.1 Introduction to Probabilistic Models 448

6.2 Stochastic Models of Diffusion 454

6.3 Stochastic Models ofGenetics 462

6.4 Probability Theory 468

6.5 Conditional Probability 474

6.6 Independence and Markov Chains 480

6.7 Displaying Probabilities 486

6.8 Random Variables 497

6.9 Descriptive Statistics 506

6.10 Descriptive Statistics forSpread 515

Chapter 7 Probability Models 529

7.1 Joint Distributions 530

7.2 Covariance and Correlation 538

7.3 Sums and Products of Random Variables 547

7.4 The Binomial Distribution 557

7.5 Applications of the Binomial Distribution 567

7.6 Waiting Times: Geometrie and Exponential Distributions 574

7.7 The Poisson Distribution 584

7.8 The Normal Distribution 592

7.9 Applying the Normal Approximation 602

Chapter 8 Introduction to Statistical Reasoning 618

8.1 Statistics: Estimating Parameters 619

8.2 Confidence Limits 627

8.3 Estimating the Mean 636

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8.4 Hypothesis Testing 647

8.5 Hypothesis Testing: Normal Theory 654

8.6 Comparing Experiments: Unpaired Data 663

8.7 Regression 671

Answers 685

Index 777