From controlling disease outbreaks to predicting heart attacks, dynamic models are crucial for understanding biological processes. This book teaches how to use dynamic models in biology. It is organized around biological applications, with mathematics and computing developed through case studies at the molecular, cellular, and population levels.

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

"This book is written with the reality of biology students and their apprehension about mathematics in mind. The applications of mathematical models to real biological problems are not contrived, as they are in a number of other texts. And the biology examples are taken from the current literature—a wonderful help to those who will be teaching with this book."—Jim Keener, University of Utah, author of Principles of Applied Mathematics and Mathematical Physiology"Dynamic Models in Biology is a new and significant contribution to the field. Very well written and clearly presented, it fulfills its goal of bringing dynamic models into the undergraduate biology curriculum. Indeed it puts biology first, and then seeks to show how biological phenomena can be explained in mathematical terms."—Martin Henry H. Stevens, Miami University"This excellent book is a major contribution to the literature. Strong biologically and mathematically, well-organized, and engagingly written, it introduces the subject of dynamical models in biology in as coherent a way as I have seen anywhere. Few authors could approach this topic as authoritatively as do Ellner and Guckenheimer."—Simon Levin, Princeton University, author of The Importance of Species and The Encyclopedia of Biodiversity

This book is written with the reality of biology students and their apprehension about mathematics in mind. The applications of mathematical models to real biological problems are not contrived, as they are in a number of other texts. And the biology examples are taken from the current literature--a wonderful help to those who will be teaching with this book. -- Jim Keener, University of Utah, author of "Principles of Applied Mathematics" and "Mathematical Physiology" Dynamic Models in Biology is a new and significant contribution to the field. Very well written and clearly presented, it fulfills its goal of bringing dynamic models into the undergraduate biology curriculum. Indeed it puts biology first, and then seeks to show how biological phenomena can be explained in mathematical terms. -- Martin Henry H. Stevens, Miami University This excellent book is a major contribution to the literature. Strong biologically and mathematically, well-organized, and engagingly written, it introduces the subject of dynamical models in biology in as coherent a way as I have seen anywhere. Few authors could approach this topic as authoritatively as do Ellner and Guckenheimer. -- Simon Levin, Princeton University, author of "The Importance of Species" and "The Encyclopedia of Biodiversity"