Table of Contents
Preface xi
Ways to use this book xv
Acknowledgments xix
1 Review 1
1.1 Rules for differentiation 1
1.2 Interpretations of the derivative 2
1.3 Intermediate and mean value theorems 3
1.4 The fundamental theorem of calculus 4
1.5 Discrete and continuous models 4
2 Discrete dynamics 6
2.1 Logistic map dynamics 6
2.2 Fixed points and their stability 13
2.3 Cycles and their stability 18
2.4 Chaos 24
2.5 Basic cardiac dynamics 36
2.6 A simple cardiac model 37
2.7 Arnold's model 44
3 Differential equations models 52
3.1 Some types of models 53
3.2 Examples: diffusion, Gompertz, SIR 55
3.3 Laird's growth logic 67
3.4 The Norton-Simon model 75
4 Single-variable differential equations 89
4.1 Simple integration 90
4.2 Integration by substitution 94
4.3 Integration by parts 97
4.4 Integral tables and computer integration 103
4.5 Fixed point analysis 107
4.6 Separation of variables 113
4.7 Integration by partial fractions 118
5 Definite integrals and improper integrals 125
5.1 Definite integrals by substitution 126
5.2 Volume by integration 129
5.3 Lengths of curves 131
5.4 Surface area by integration 133
5.5 Improper integrals 137
5.6 Improper integral comparison tests 140
5.7 Stress testing growth models 145
6 Power laws 149
6.1 The circumference of a circle 149
6.2 Scaling of coastlines, log-log plots 152
6.3 Allometric scaling 156
6.4 Power laws and dimensions 160
6.5 Some biological examples 167
7 Differential equations in the plane 173
7.1 Vector fields and trajectories 174
7.2 Differential equations software 181
7.3 Predator-prey equations 182
7.4 Competing populations 187
7.5 Nullcline analysis 191
7.6 The Fitzhugh-Nagumo equations 195
8 Linear systems and stability 203
8.1 Superposition of solutions 204
8.2 Types of fixed points 205
8.3 The matrix formalism 211
8.4 Eigenvalues and eigenvectors 213
8.5 Eigenvalues at fixed points 219
8.6 The trace-determinant plane 227
9 Nonlinear systems and stability 232
9.1 Partial derivatives 232
9.2 The Hartman-Grobman theorem 236
9.3 The pendulum as guide 247
9.4 Liapunov functions 250
9.5 Fixed points not at the origin 261
9.6 Limit cycles 266
9.7 The Poincaré-Bendixson theorem 272
9.8 Lienard's and Bendixson's theorems 283
10 Infinite series and power series 292
10.1 The Integral Test 295
10.2 The Comparison Test 297
10.3 Alternating series 300
10.4 The Root and Ratio Tests 307
10.5 Numerical series practice 310
10.6 Power series 312
10.7 Radius and interval of convergence 315
10.8 Taylor's theorem, series manipulation 320
10.9 Power series solutions 327
11 Some probability 333
11.1 Discrete variables 334
11.2 Continuous variables 338
11.3 Some combinatorial rules 343
11.4 Simpson's paradox 350
11.5 Causality calculus 356
11.6 Expected value and variance 365
11.7 The binomial distribution 377
11.8 The Poisson distribution 386
11.9 The normal distribution 399
11.10 Infinite moments 408
11.11 Classical hypothesis tests 418
11.12 Bayesian Inference 426
12 Why this matters 435
Appendix A Technical notes 444
A.1 Integrating factors 444
A.2 Existence and uniqueness 447
A.3 The threshold theorem 452
A.4 Vokerra's trick 454
A.5 Euler's formula 459
A.6 Proof of the second-derivative test 460
A.7 Some linear algebra 462
A.8 A proof of Liapunov's theorem 471
A.9 A proof of the Hartman-Grobman theorem 473
A.10 A proof of the Poincaré-Bendixson theorem 477
A.11 The Law of Large Numbers 482
Appendix B Some Mathematica code 485
B.1 Cycles and their stability 486
B.2 Chaos 487
B.3 A simple cardiac model 488
B.4 The Norton-Simon model 488
B.5 Log-log plots 489
B.6 Vector fields and trajectories 490
B.7 Differential equations software 490
B.8 Predator-prey models 491
B.9 The Fitzhugh-Nagumo equations 492
B.10 Eigenvalues and eigenvectors 493
Appendix C Some useful integrals and hints 494
Figure credits 498
References 499
Index 508
