Course in Mathematical Biology : Quantitative Modeling With Mathematical And Computational Methods

Preface ix I Theoretical Modeling Tools 1(198) Introduction 3(6) The Modeling Process 3(1) Probabilities and Rates 4(3) Model Classes 7(1) Exercises for Modeling 8(1) Discrete-Time Models 9(46) Introduction to Discrete-Time Models 9(1) Scalar Discrete-Time Models 10(26) Growth of a Population and the Discrete Logistic Equation 10(5) Cobwebbing, Fixed Points, and Linear Stability Analysis 15(3) Analysis of the Discrete Logistic Equation 18(7) Alternatives to the Discrete Logistic Equation 25(4) Models in Population Genetics 29(7) Systems of Discrete-Time Equations 36(12) Love Affairs: Introduction 36(2) Fixed Points and Linear Stability Analysis for Systems of Discrete-Time Equations 38(4) Love Affairs: Model Analysis 42(2) Host-Parasitoid Models 44(4) Exercises for Discrete-Time Models 48(7) Ordinary Differential Equations 55(42) Introduction to ODEs 55(1) Scalar Equations 56(4) The Picard-Lindelof Theorem 59(1) Systems of Equations 60(6) Reaction Kinetics 60(2) A General Interaction Model for Two Populations 62(2) A Basic Epidemic Model 64(1) Nondimensionalization 65(1) Qualitative Analysis of 2 x 2 Systems 66(14) Phase-Plane Analysis: Linear Systems 67(7) Nonlinear Systems and Linearization 74(2) Qualitative Analysis of the General Population Interaction Model 76(2) Qualitative Analysis of the Epidemic Model 78(2) General Systems of Three or More Equations 80(1) Discrete-Time Models from Continuous-Time Models 81(2) Numerical Methods 81(1) The Time-One Map 81(2) Elementary Bifurcations 83(6) Saddle-Node Bifurcation 84(1) Transcritical Bifurcation 84(1) Pitchfork Bifurcation 85(1) Hopf Bifurcation 86(2) The Spruce Budworm Model 88(1) Further Reading 89(1) Exercises for ODEs 90(7) Partial Differential Equations 97(24) Partial Derivatives 97(1) An Age-Structured Model 98(4) Derivation 98(2) Solution 100(2) Reaction-Diffusion Equations 102(13) Derivation of Reaction-Diffusion Equations 102(2) The Fundamental Solution 104(2) Critical Domain Size 106(5) Traveling Waves 111(4) Further Reading 115(1) Exercises for PDEs 116(5) Stochastic Models 121(34) Introduction 121(1) Markov Chains 122(4) A Two-Tree Forest Ecosystem 122(2) Markov Chain Theory 124(1) The Princeton Forest Ecosystem 124(2) Working with Random Variables 126(6) Probability Density 126(1) Probability Mass 127(2) Descriptive Statistics 129(1) The Generating Function 130(2) Diffusion Processes 132(4) Random Motion in One Dimension 133(2) Diffusion Equation 135(1) Branching Processes 136(5) Galton--Watson Process 136(3) Polymerase Chain Reaction 139(2) Linear Birth and Death Process 141(6) Pure Birth Process 141(3) Birth and Death Process 144(3) Nonlinear Birth-Death Process 147(4) A Model for the Common Cold in Households 148(1) Embedded Time-Discrete Markov Process and Final Size Distribution 149(2) Exercises for Stochastic Models 151(4) Cellular Automata and Related Models 155(20) Introduction to Cellular Automata 155(6) Wolfram's Classification 157(2) The Game of Life 159(1) Some Theoretical Results on Cellular Automata 160(1) Greenberg-Hastings Automata 161(4) Relation to an SIR Model 164(1) Generalized Cellular Automata 165(5) Automata with Stochastic Rules 165(2) Grid Modifications 167(1) Asynchronous Dynamics 168(2) Related Models 170(1) Further Reading 171(1) Exercises for Cellular Automata 172(3) Estimating Parameters 175(24) Introduction 175(1) The Likelihood Function 176(11) Stochastic Models without Measurement Error 176(5) Deterministic Models 181(6) Model Comparison 187(6) Akaike Information Criterion 187(2) Likelihood Ratio Test for Nested Models 189(3) Cross Validation 192(1) Optimization Algorithms 193(2) Algorithms 193(1) Positivity 194(1) What Did We Learn? 195(1) Further Reading 196(1) Exercises for Parameter Estimation 196(3) II Self-Guided Computer Tutorial 199(38) Maple Course 201(36) First Steps in Maple 201(11) Constants and Functions 201(4) Working with Data Sets 205(2) Linear Regression 207(5) Discrete Dynamical Systems: The Ricker Model 212(8) Procedures in Maple 215(2) Feigenbaum Diagram and Bifurcation Analysis 217(2) Application of the Ricker Model to Vespula vulgaris 219(1) Stochastic Models with Maple 220(3) ODEs: Applications to an Epidemic Model and a Predator--Prey Model 223(5) The SIR Model of Kermack and McKendrick 223(2) A Predator--Prey Model 225(3) PDEs: An Age-Structured Model 228(4) Stochastic Models: Common Colds in Households 232(5) Application to Data 234(3) III Projects 237(46) Project Descriptions 239(24) Epidemic Models 239(6) Population Dynamics 245(4) Models for Spatial Spread 249(5) Physiology 254(9) Solved Projects 263(20) Cell Competition 263(11) Paramecium caudatum in Isolation 263(3) The Two Populations in Competition 266(3) Phase-Plane Analysis of the Competition Model 269(4) Model Prediction 273(1) An Alternative Hypothesis 273(1) The Chemotactic Paradox 274(9) A Resolution of the Chemotactic Paradox 274(4) Discussion 278(5) Appendix: Further Reading 283(4) Bibliography 287(10) Author Index 297(4) Index 301