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