Part I Calculus and Modeling
1 A Brief Summary of Calculus ............................... 5
1.1 Working with Parameters .................................... 7
1.1.1 Scaling Parameters .................................. 8
1.1.2 Nonlinear Parameters ............................... 10
1.1.3 Bifurcations ....................................... 12
i.2 Rates of Change and the Derivative ........................ 18
1.2.1 Rate of Change for a Function of Discrete Time ..... 18
1.2.2 Rate of Change for a Function of Continuous Time ... 19
1.2.3 The Derivative ..................................... 21
1.2.4 Slope of a Tangent to a Graph ...................... 22
1.3 Computing Derivatives ..................................... 26
1.3.1 Two Notations ...................................... 26
1.3.2 Elementary Derivative Formulas ..................... 27
1.3.3 General Derivative Rules ........................... 28
1.3.4 Partial Derivatives ................................ 31
1.4 Local Behavior and Linear Approximation ................... 34
1.4.1 Tangent Lines ...................................... 34
1.4.2 Local Extrema ...................................... 35
1.4.3 Linear Approximation ............................... 39
1.5 Optimization .............................................. 42
1.5.1 The Marginal Value Theorem ......................... 44
1.6 Related Rates ............................................. 51
1.6.1 Differential Equations ............................. 52
1.6.2 The Chain Rule ..................................... 54
1.7 Accumulation and the Definite Integral .................... 59
1.7.1 Estimating Total Volume from Flow Rate Data ........ 59
1.7.2 The Definite Integral .............................. 61
1.7.3 Applications of the Definite Integral .............. 63
1.8 Properties of the Definite Integral ....................... 68
1.8.1 The Fundamental Theorem of Calculus ................ 69
1.8.2 Computing Definite Integrals with the Fundamental
Theorem ............................................ 71
1.9 Computing Antiderivatives and Definite Integrals .......... 74
1.9.1 Substitution ....................................... 75
1.9.2 Constructing Antiderivatives with the Fundamental
Theorem ............................................ 77
1.9.3 Obtaining a Graph of ƒ from a Graph of ƒ' .......... 78
References ................................................ 81
2 Mathematical Modeling ..................................... 83
2.1 Mathematics in Biology .................................... 85
2.1.1 Biological Data .................................... 85
2.1.2 Overall Patterns in a Random World ................. 86
2.1.3 Determining Relationships .......................... 88
2.2 Basic Concepts of Modeling ................................ 89
2.2.1 Mechanistic and Empirical Modeling ................. 91
2.2.2 Aims of Mathematical Modeling ...................... 93
2.2.3 The Narrow and Broad Views of Mathematical Models .. 94
2.2.4 Accuracy, Precision, and Interpretation of
Results ............................................ 95
2.3 Empirical Modeling I: Fitting Linear Models to Data ....... 99
2.3.1 The Basic Linear Least Squares Method (y = mx) .... 100
2.3.2 Adapting the Method to the General Linear Model ... 103
2.3.3 Implied Assumptions of Least Squares .............. 105
2.4 Empirical Modeling II: Fitting Semilinear Models to
Data ..................................................... 108
2.4.1 Fitting the Exponential Model by Linear Least
Squares ........................................... 108
2.4.2 Linear Least Squares Fit for the Power Function
Model у = AxP ..................................... 110
2.4.3 Semilinear Least Squares .......................... 111
2.5 Mechanistic Modeling I: Creating Models from Biological
Principles ............................................... 117
2.5.1 Constructing Mechanistic Models ................... 117
2.5.2 Dimensional Analysis .............................. 120
2.5.3 A Mechanistic Model for Resource Consumption ...... 121
2.5.4 A More Sophisticated Model for Food Consumption ... 122
2.5.5 A Compartment Model for Pollution in a Lake ....... 123
2.5.6 "Let the Buyer Beware" ............................ 125
2.6 Mechanistic Modeling II: Equivalent Forms ................ 127
2.6.1 Notation .......................................... 127
2.6.2 Algebraic Equivalence ............................. 128
2.6.3 Different Parameters .............................. 129
2.6.4 Visualizing Models with Graphs .................... 130
2.6.5 Dimensionless Variables ........................... 131
2.6.6 Dimensionless Forms ............................... 131
2.7 Empirical Modeling III: Choosing Among Models ............ 135
2.7.1 Quantitative Accuracy ............................. 135
2.7.2 Complexity ........................................ 136
2.7.3 The Akaike Information Criterion .................. 138
2.7.4 Choosing Among Models ............................. 139
2.7.5 Some Recommendations .............................. 140
References ............................................... 143
Part II Probability
3 Probability Distributions ................................ 149
3.1 Characterizing Data ...................................... 150
3.1.1 Types of Data ..................................... 150
3.1.2 Displaying Data ................................... 151
3.1.3 Measures of Central Tendency ...................... 153
3.1.4 Measuring Variability ............................. 154
3.2 Concepts of Probability .................................. 158
3.2.1 Experiments, Outcomes, and Random Variables ....... 159
3.2.2 Probability Distributions ......................... 161
3.2.3 Sequences and Complements ......................... 162
3.3 Discrete Probability Distributions ....................... 165
3.3.1 Distribution Functions ............................ 167
3.3.2 Expected Value, Mean, and Standard Deviation ...... 169
3.4 The Binomial Distribution ................................ 172
3.4.1 Bernoulli Trials and Binomial Distributions ....... 174
3.5 Continuous Probability Distributions ..................... 179
3.5.1 Cumulative Distribution Functions ................. 180
3.5.2 Probability Density Functions ..................... 183
3.5.3 Expected Value, Mean, and Variance ................ 185
3.6 The Normal Distribution .................................. 188
3.6.1 The Standard Normal Distribution .................. 189
3.6.2 Standard Probability Intervals .................... 193
3.7 The Poisson and Exponential Distributions ................ 195
3.7.1 The Poisson Distribution .......................... 196
3.7.2 The Exponential Distribution ...................... 198
3.7.3 Memory in Probability Distributions ............... 200
References ............................................... 205
4 Working with Probability ................................. 207
4.1 An Introduction to Statistical Inference ................. 207
4.2 Tests on Probability Distributions ....................... 214
4.2.1 A Graphical Test for Distribution Type ............ 214
4.2.2 The Cramer-von Mises Test ......................... 216
4.2.3 Outliers .......................................... 217
4.3 Probability Distributions of Samples ..................... 222
4.3.1 Sums and Means of Two Random Variables ............ 222
4.3.2 General Characteristics of Samples ................ 223
4.3.3 Means and Standard Deviations of Sample Sums and
Means ............................................. 224
4.3.4 Distributions of Sample Means ..................... 225
4.4 Inferences About Populations ............................. 229
4.4.1 Inferences About Sample Means ..................... 229
4.4.2 Inferences About Sample Proportions ............... 232
4.5 Estimating Parameters for Distributions .................. 236
4.5.1 Confidence Intervals .............................. 236
4.5.2 Estimating Success Probability with a Likelihood
Function .......................................... 237
4.5.3 Estimating Population Size by Mark and Recapture .. 240
4.5.4 A Final Observation ............................... 241
4.6 Conditional Probability .................................. 244
4.6.1 The Concept of Conditional Probability ............ 246
4.6.2 Formulas for Conditional Probability .............. 246
4.6.3 The Partition Rule ................................ 248
4.7 Conditional Probability Applied to Diagnostic Tests ...... 250
4.7.1 The Standard Test Interpretation Problem .......... 252
References ............................................... 256
Part III Dynamical Systems
5 Dynamics of Single Populations ........................... 261
5.1 Discrete Population Models ............................... 262
5.1.1 Discrete Exponential Growth ....................... 263
5.1.2 The Discrete Logistic Model ....................... 265
5.1.3 Simulations ....................................... 266
5.1.4 Fixed Points ...................................... 267
5.2 Cobweb Analysis .......................................... 270
5.2.1 Cobweb Diagrams ................................... 271
5.2.2 Stability Analysis ................................ 273
5.3 Continuous Population Models ............................. 275
5.3.1 Exponential Growth ................................ 276
5.3.2 Logistic Growth ................................... 276
5.3.3 Equilibrium Solutions ............................. 278
5.3.4 A Renewable Resource Model ........................ 278
5.3.5 Equilibria of the Renewable Resource Model ........ 280
5.3.6 A Graphical Method ................................ 281
5.4 Phase Line Analysis ...................................... 286
5.4.1 The Phase Line .................................... 287
5.4.2 The Phase Line for the Holling Type III
Renewable Resource Model .......................... 288
5.4.3 Comparison of Graphical Methods ................... 289
5.5 Linearized Stability Analysis ............................ 291
5.5.1 Stability Analysis for Discrete Models:
A Motivating Example .............................. 291
5.5.2 Stability Analysis for Discrete Models: The
General Case ...................................... 293
5.5.3 Stability Analysis for Continuous Models .......... 295
5.5.4 Similarities and Differences ...................... 296
References ............................................... 299
6 Discrete Dynamical Systems ............................... 301
6.1 Discrete Linear Systems .................................. 301
6.1.1 Simple Structured Models .......................... 302
6.1.2 Finding the Growth Rate and Stable Distribution ... 304
6.1.3 General Properties of Discrete Linear Structured
Models ............................................ 305
6.2 A Matrix Algebra Primer .................................. 310
6.2.1 Matrices and Vectors .............................. 311
6.2.2 Population Models in Matrix Notation .............. 313
6.2.3 The Central Problem of Matrix Algebra ............. 313
6.2.4 The Determinant ................................... 314
6.2.5 The Equation Ax = 0 ............................... 315
6.3 Long-Term Behavior of Linear Models ...................... 317
6.3.1 Eigenvalues and Eigenvectors ...................... 318
6.3.2 Solutions of xt+1 = Mxt ........................... 320
6.3.3 Long-Term Behavior ................................ 321
References ............................................... 326
7 Continuous Dynamical Systems ............................. 327
7.1 Pharmacokinetics and Compartment Models .................. 328
7.1.1 Compartment Models ................................ 328
7.1.2 A Simplified Model ................................ 330
7.1.3 The Dimensionless Model ........................... 332
7.2 Enzyme Kinetics .......................................... 336
7.2.1 Nondimensionalization ............................. 338
7.2.2 Simulation ........................................ 339
7.2.3 The Briggs-Haldane Approximation .................. 340
7.3 Phase Plane Analysis ..................................... 344
7.3.1 Equilibrium Solutions ............................. 345
7.3.2 Solution Curves in the Phase Plane ................ 345
7.3.3 Nullclines ........................................ 346
7.3.4 Nullcline Analysis ................................ 347
7.3.5 Using Small Parameters ............................ 349
7.3.6 Nullcline Analysis in General ..................... 350
7.4 Stability in Linear Systems .............................. 357
7.4.1 Linear Systems .................................... 357
7.4.2 Eigenvalues and Stability ......................... 359
7.4.3 The Routh-Hurwitz Conditions ...................... 360
7.4.4 The Routh-Hurwitz Conditions for a Three-
Dimensional System ................................ 362
7.5 Stability in Nonlinear Systems ........................... 365
7.5.1 Approximating a Nonlinear System at an
Equilibrium Point ................................. 365
7.5.2 The Jacobian and Stability ........................ 367
7.6 Primary HIV Infection ................................... 372
7.6.1 Nondimensionalization ............................. 373
7.6.2 Reduction to Two Variables ........................ 374
7.6.3 Equilibria and Stability .......................... 375
7.6.4 Phase Plane Analysis .............................. 376
References ............................................... 379
Appendix A. Additional Topics in Discrete Dynamical Systems ... 381
A.l Discrete Nonlinear Systems ............................... 381
A.1.1 Linearization for Discrete Nonlinear Systems ...... 383
A.1.2 A Structured Population Model with One Nonlinearity 384
A.2 Markov Chains ............................................ 392
A.2.1 Some Scientific Background ........................ 393
A.2.2 A Model for DNA Change ............................ 394
A.2.3 Equilibrium Analysis of Markov Chain Models ....... 395
A.2.4 Analysis of the DNA Change Model .................. 396
A.3 Boolean Algebra Models ................................... 400
A.3.1 Boolean Algebra ................................... 400
A.3.2 Boolean Functions and Boolean Networks ............ 402
A.3.3 Using a Boolean Network for a Consistency Check ... 403
References ............................................... 405
Appendix В The Definite Integral via Riemann Sums ............. 407
Appendix С A Runge-Kutta Method for Numerical Solution of
Differential Equations ................................... 409
Hints and Answers to Selected Problems ........................ 411
Index ......................................................... 429
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