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

by Otto, Sarah P.

• ISBN13: 9780691123448
• ISBN10: 0691123446
• Format: Hardcover
• Copyright: 2007-02-20
• Publisher: Princeton Univ Pr

Summary

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.

Author Biography

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

Table of Contents

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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