# A Biologist's Guide to Mathematical Modeling in Ecology and Evolution

, by Otto, Sarah P.**Note:**Supplemental materials are not guaranteed with Rental or Used book purchases.

- ISBN: 9780691123448 | 0691123446
- Cover: Hardcover
- Copyright: 2/20/2007

A how-to guide for developing new mathematical models in biology, Provides step-by-step recipes for constructing and analyzing models, Interesting biological applications, Exploration of classical models in ecology and evolution, Questions at the end of every chapter, Primers cover important mathematical topics, Exercises with answers, Appendices summarize useful rules, Labs and advanced material available through the book's Web site at http://press.princeton.edu/titles/8458.html.

Sarah P. Otto is Professor of Zoology at the University of British Columbia. Troy Day is Associate Professor of Mathematics and Biology at Queen's University

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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