Serves as a how-to guide for developing mathematical models in biology. Starting at an elementary level of mathematical modeling, this title gradually builds from classic models in ecology and evolution to more intricate class-structured and probabilistic models. It provides primers with instructive exercises. Preface ix Mathematical Modeling in Biology 1(16) Introduction 1(1) HIV 2(3) Models of HIV/AIDS 5(9) Concluding Message 14(3) How to Construct a Model 17(37) Introduction 17(2) Formulate the Question 19(1) Determine the Basic Ingredients 19(7) Qualitatively Describe the Biological System 26(7) Quantitatively Describe the Biological System 33(6) Analyze the Equations 39(8) Checks and Balances 47(3) Relate the Results Back to the Question 50(1) Concluding Message 51(3) Deriving Classic Models in Ecology and Evolutionary Biology 54(56) Introduction 54(1) Exponential and Logistic Models of Population Growth 54(8) Haploid and Diploid Models of Natural Selection 62(10) Models of Interactions among Species 72(5) Epidemiological Models of Disease Spread 77(2) Working Backward---Interpreting Equations in Terms of the Biology 79(3) Concluding Message 82(7) Primer 1: Functions and Approximations 89(1) Functions and Their Forms 89(7) Linear Approximations 96(4) The Taylor Series 100(10) Numerical and Graphical Techniques---Developing a Feeling for Your Model 110(41) Introduction 110(1) Plots of Variables Over Time 111(13) Plots of Variables as a Function of the Variables Themselves 124(9) Multiple Variables and Phase-Plane Diagrams 133(12) Concluding Message 145(6) Equilibria and Stability Analyses---One-Variable Models 151(40) Introduction 151(1) Finding an Equilibrium 152(11) Determining Stability 163(13) Approximations 176(8) Concluding Message 184(7) General Solutions and Transformations-One-Variable Models 191(63) Introduction 191(1) Transformations 192(1) Linear Models in Discrete Time 193(2) Nonlinear Models in Discrete Time 195(3) Linear Models in Continuous Time 198(4) Nonlinear Models in Continuous Time 202(5) Concluding Message 207(7) Primer 2: Linear Algebra 214(1) An Introduction to Vectors and Matrices 214(5) Vector and Matrix Addition 219(3) Multiplication by a Scalar 222(2) Multiplication of Vectors and Matrices 224(4) The Trace and Determinant of a Square Matrix 228(5) The Inverse 233(2) Solving Systems of Equations 235(2) The Eigenvalues of a Matrix 237(6) The Eigenvectors of a Matrix 243(11) Equilibria and Stability Analyses---Linear Models with Multiple Variables 254(40) Introduction 254(1) Models with More than One Dynamic Variable 255(5) Linear Multivariable Models 260(19) Equilibria and Stability for Linear Discrete-Time Models 279(10) Concluding Message 289(5) Equilibria and Stability Analyses---Nonlinear Models with Multiple Variables 294(53) Introduction 294(1) Nonlinear Multiple-Variable Models 294(22) Equilibria and Stability for Nonlinear Discrete-Time Models 316(14) Perturbation Techniques for Approximating Eigenvalues 330(7) Concluding Message 337(10) General Solutions and Tranformations---Models with Multiple Variables 347(39) Introduction 347(1) Linear Models Involving Multiple Variables 347(18) Nonlinear Models Involving Multiple Variables 365(16) Concluding Message 381(5) Dynamics of Class-Structured Populations 386(37) Introduction 386(2) Constructing Class-Structured Models 388(5) Analyzing Class-Structured Models 393(5) Reproductive Value and Left Eigenvectors 398(2) The Effect of Parameters on the Long-Term Growth Rate 400(3) Age-Structured Models---The Leslie Matrix 403(15) Concluding Message 418(5) Techniques for Analyzing Models with Periodic Behavior 423(31) Introduction 423(1) What Are Periodic Dynamics? 423(2) Composite Mappings 425(3) Hopf Bifurcations 428(8) Constants of Motion 436(13) Concluding Message 449(5) Evolutionary Invasion Analysis 454(113) Introduction 454(1) Two Introductory Examples 455(10) The General Technique of Evolutionary Invasion Analysis 465(13) Determining How the ESS Changes as a Function of Parameters 478(7) Evolutionary Invasion Analyses in Class-Structured Populations 485(17) Concluding Message 502(11) Primer 3: Probability Theory 513(1) An Introduction to Probability 513(5) Conditional Probabilities and Bayes' Theorem 518(3) Discrete Probability Distributions 521(15) Continuous Probability Distributions 536(17) The (Insert Your Name Here) Distribution 553(14) Probabilistic Models 567(41) Introduction 567(1) Models of Population Growth 568(5) Birth-Death Models 573(3) Wright-Fisher Model of Allele Frequency Change 576(5) Moran Model of Allele Frequency Change 581(3) Cancer Development 584(7) Cellular Automata---A Model of Extinction and Recolonization 591(3) Looking Backward in Time---Coalescent Theory 594(8) Concluding Message 602(6) Analyzing Discrete Stochastic Models 608(41) Introduction 608(1) Two-State Markov Models 608(6) Multistate Markov Models 614(17) Birth-Death Models 631(8) Branching Processes 639(5) Concluding Message 644(5) Analyzing Continuous Stochastic Models---Diffusion in Time and Space 649(43) Introduction 649(1) Constructing Diffusion Models 649(15) Analyzing the Diffusion Equation with Drift 664(20) Modeling Populations in Space Using the Diffusion Equation 684(3) Concluding Message 687(5) Epilogue: The Art of Mathematical Modeling in Biology 692(3) Appendix 1: Commonly Used Mathematical Rules 695(4) A1.1 Rules for Algebraic Functions 695(1) A1.2 Rules for Logarithmic and Exponential Functions 695(1) A1.3 Some Important Sums 696(1) A1.4 Some Important Products 696(1) A1.5 Inequalities 697(2) Appendix 2: Some Important Rules from Calculus 699(10) A2.1 Concepts 699(2) A2.2 Derivatives 701(2) A2.3 Integrals 703(1) A2.4 Limits 704(5) Appendix 3: The Perron-Frobenius Theorem 709(4) A3.1 Definitions 709(1) A3.2 The Perron-Frobenius Theorem 710(3) Appendix 4: Finding Maxima and Minima of Functions 713(4) A4.1 Functions with One Variable 713(1) A4.2 Functions with Multiple Variables 714(3) Appendix 5: Moment-Generating Functions 717(8) Index of Definitions, Recipes, and Rules 725(2) General Index 727