Dynamic Models in Biology

Stephen P. Ellner is Professor of Ecology and Evolutionary Biology at Cornell University. He has published numerous papers on subjects from measles epidemics to bumblebee behavior, in publications including "Science" and "Nature". John Guckenheimer is Professor of Mathematics at Cornell University. He is the coauthor of "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields".

"What is remarkable about Dynamic Models in Biology is that it truly speaks to students of biological sciences. It puts biology first, and then tries to explain how mathematical tools can explain biological phenomena. Nothing else I've seen does this anywhere near as well. The authors have combined their experience to produce and excellent textbook."--Bill Satzer, MAA Reviews "This is a great book and I expect that it will play an important role in the teaching of mathematical biology and the development of the next generation of mathematical biologists for many years to come."--Marc Mangel, SIAM Review "Dynamic Models in Biology stands apart from existing textbooks in mathematical biology largely because of its interdisciplinary approach and its hands-on, project-oriented case studies and computer laboratories. In an effort to explore biology in more detail, the authors bravely chose a style that differs from the classical biomath texts ... whose focus is more on formal mathematics."--Lewi Stone, BioScience "The book begins with a stellar overview of the purpose of modeling, contrasting statistical with dynamical models, and theoretical with practical models both clearly and even-handedly...[E]ngaging the full breadth and depth of this book could be an education for both instructors and students alike."--Frederick R. Adler, Mathematical Biosciences "[S]tudents from both biology and mathematics can gain much from this book. Dynamic Models in Biology would be appropriate for use in a semester or two-quarter course; however, with judicious selection of topics, it can be used in a quarter. My students included undergraduates in biology with knowledge only of calculus, undergraduates in mathematics, and graduate students and academic staff in biology, all enrolled on a ten-week course... Overall, Dynamic Models in Biology fills an important niche in the biological modeling canon. It occupies a place on my shelf next to Edelstein-Keshet (1988) and Murray (1989), and like them, will become a well-thumbed reference."--Carole L. Hom, Environmental Conservation

• List of Figures ix List of Tables xiv Preface xvi Chapter 1: What Are Dynamic Models? 1 1.1 Descriptive versus Mechanistic Models 2 1.2 Chinook Salmon 4 1.3 Bathtub Models 6 1.4 Many Bathtubs: Compartment Models 7 1.4.1 Enzyme Kinetics 8 1.4.2 The Modeling Process 11 1.4.3 Pharmacokinetic Models 13 1.5 Physics Models: Running and Hopping 16 1.6 Optimization Models 20 1.7 Why Bother? 21 1.8 Theoretical versus Practical Models 24 1.9 What's Next? 26 1.10 References 28 Chapter 2: Matrix Models and Structured Population Dynamics 31 2.1 The Population Balance Law 32 2.2 Age-Structured Models 33 2.2.1 The Leslie Matrix 34 2.2.2 Warning: Prebreeding versus Postbreeding Models 37 2.3 Matrix Models Based on Stage Classes 38 2.4 Matrices and Matrix Operations 42 2.4.1 Review of Matrix Operations 43 2.4.2 Solution of the Matrix Model 44 2.5 Eigenvalues and a Second Solution of the Model 44 2.5.1 Left Eigenvectors 48 2.6 Some Applications of Matrix Models 49 2.6.1 Why Do We Age? 49 2.6.2 Elasticity Analysis and Conservation Biology 52 2.6.3 How Much Should We Trust These Models? 58 2.7 Generalizing the Matrix Model 59 2.7.1 Stochastic Matrix Models 59 2.7.2 Density-Dependent Matrix Models 61 2.7.3 Continuous Size Distributions 63 2.8 Summary and Conclusions 66 2.9 Appendix 67 2.9.1 Existence and Number of Eigenvalues 67 2.9.2 Reproductive Value 67 2.10 References 68 Chapter 3: Membrane Channels and Action Potentials 71 3.1 Membrane Currents 72 3.1.1 Channel Gating and Conformational States 74 3.2 Markov Chains 77 3.2.1 Coin Tossing 78 3.2.2 Markov Chains 82 3.2.3 The Neuromuscular Junction 86 3.3 Voltage-Gated Channels 90 3.4 Membranes as Electrical Circuits 92 3.4.1 Reversal Potential 94 3.4.2 Action Potentials 95 3.5 Summary 103 3.6 Appendix: The Central Limit Theorem 104 3.7 References 106 Chapter 4: Cellular Dynamics: Pathways of Gene Expression 107 4.1 Biological Background 108 4.2 A Gene Network That Acts as a Clock 110 4.2.1 Formulating a Model 111 4.2.2 Model Predictions 113 4.3 Networks That Act as a Switch 119 4.4 Systems Biology 125 4.4.1 Complex versus Simple Models 129 4.5 Summary 131 4.6 References 132 Chapter 5: Dynamical Systems 135 5.1 Geometry of a Single Differential Equation 136 5.2 Mathematical Foundations: A Fundamental Theorem 138 5.3 Linearization and Linear Systems 141 5.3.1 Equilibrium Points 141 5.3.2 Linearization at Equilibria 142 5.3.3 Solving Linear Systems of Differential Equations 144 5.3.4 Invariant Manifolds 149 5.3.5 Periodic Orbits 150 5.4 Phase Planes 151 5.5 An Example: The Morris-Lecar Model 154 5.6 Bifurcations 160 5.7 Numerical Methods 175 5.8 Summary 181 5.9 References 181 Chapter 6: Differential Equation Models for Infectious Disease 183 6.1 Sir Ronald Ross and the Epidemic Curve 183 6.2 Rescaling the Model 187 6.3 Endemic Diseases and Oscillations 191 6.3.1 Analysis of the SIR Model with Births 193 6.3.2 Summing Up 197 6.4 Gonorrhea Dynamics and Control 200 6.4.1 A Simple Model and a Paradox 200 6.4.2 The Core Group 201 6.4.3 Implications for Control 203 6.5 Drug Resistance 206 6.6 Within-Host Dynamics of HIV 209 6.7 Conclusions 213 6.8 References 214 Chapter 7: Spatial Patterns in Biology 217 7.1 Reaction-Diffusion Models 218 7.2 The Turing Mechanism 223 7.3 Pattern Selection: Steady Patterns 226 7.4 Moving Patterns: Chemical Waves and Heartbeats 232 7.5 References 241 Chapter 8: Agent-Based and Other Computational Models for Complex Systems 243 8.1 Individual-Based Models in Ecology 245 8.1.1 Size-Dependent Predation 245 8.1.2 Swarm 247 8.1.3 Individual-Based Modeling of Extinction Risk 248 8.2 Artificial Life 252 8.2.1 Tierra 253 8.2.2 Microbes in Tierra 255 8.2.3 Avida 257 8.3 The Immune System and the Flu 259 8.4 What Can We Learn from Agent-Based Models? 260 8.5 Sensitivity Analysis 261 8.5.1 Correlation Methods 264 8.5.2 Variance Decomposition 266 8.6 Simplifying Computational Models 269 8.6.1 Separation of Time Scales 269 8.6.2 Simplifying Spatial Models 272 8.6.3 Improving the Mean Field Approximation 276 8.7 Conclusions 277 8.8 Appendix: Derivation of Pair Approximation 278 8.9 References 279 Chapter 9: Building Dynamic Models 283 9.1 Setting the Objective 284 9.2 Building an Initial Model 285 9.2.1 Conceptual Model and Diagram 286 9.3 Developing Equations for Process Rates 291 9.3.1 Linear Rates: When and Why? 291 9.3.2 Nonlinear Rates from "First Principles" 293 9.3.3 Nonlinear Rates from Data: Fitting Parametric Models 294 9.3.4 Nonlinear Rates from Data: Selecting a Parametric Model 298 9.4 Nonlinear Rates from Data: Nonparametric Models 302 9.4.1 Multivariate Rate Equations 304 9.5 Stochastic Models 306 9.5.1 Individual-Level Stochasticity 306 9.5.2 Parameter Drift and Exogenous Shocks 309 9.6 Fitting Rate Equations by Calibration 311 9.7 Three Commandments for Modelers 314 9.8 Evaluating a Model 315 9.8.1 Comparing Models 317 9.9 References 320 Index 323