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

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  • Format: Hardcover
  • Copyright: 2007-02-20
  • Publisher: Princeton Univ Pr

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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

Prefacep. ix
Mathematical Modeling in Biologyp. 1
Introductionp. 1
HIVp. 2
Models of HIV/AIDSp. 5
Concluding Messagep. 14
How to Construct a Modelp. 17
Introductionp. 17
Formulate the Questionp. 19
Determine the Basic Ingredientsp. 19
Qualitatively Describe the Biological Systemp. 26
Quantitatively Describe the Biological Systemp. 33
Analyze the Equationsp. 39
Checks and Balancesp. 47
Relate the Results Back to the Questionp. 50
Concluding Messagep. 51
Deriving Classic Models in Ecology and Evolutionary Biologyp. 54
Introductionp. 54
Exponential and Logistic Models of Population Growthp. 54
Haploid and Diploid Models of Natural Selectionp. 62
Models of Interactions among Speciesp. 72
Epidemiological Models of Disease Spreadp. 77
Working Backward-Interpreting Equations in Terms of the Biologyp. 79
Concluding Messagep. 82
Functions and Approximationsp. 89
Functions and Their Formsp. 89
Linear Approximationsp. 96
The Taylor Seriesp. 100
Numerical and Graphical Techniques-Developing a Feeling for Your Modelp. 110
Introductionp. 110
Plots of Variables Over Timep. 111
Plots of Variables as a Function of the Variables Themselvesp. 124
Multiple Variables and Phase-Plane Diagramsp. 133
Concluding Messagep. 145
Equilibria and Stability Analyses-One-Variable Modelsp. 151
Introductionp. 151
Finding an Equilibriump. 152
Determining Stabilityp. 163
Approximationsp. 176
Concluding Messagep. 184
General Solutions and Transformations-One-Variable Modelsp. 191
Introductionp. 191
Transformationsp. 192
Linear Models in Discrete Timep. 193
Nonlinear Models in Discrete Timep. 195
Linear Models in Continuous Timep. 198
Nonlinear Models in Continuous Timep. 202
Concluding Messagep. 207
Linear Algebrap. 214
An Introduction to Vectors and Matricesp. 214
Vector and Matrix Additionp. 219
Multiplication by a Scalarp. 222
Multiplication of Vectors and Matricesp. 224
The Trace and Determinant of a Square Matrixp. 228
The Inversep. 233
Solving Systems of Equationsp. 235
The Eigenvalues of a Matrixp. 237
The Eigenvectors of a Matrixp. 243
Equilibria and Stability Analyses-Linear Models with Multiple Variablesp. 254
Introductionp. 254
Models with More than One Dynamic Variablep. 255
Linear Multivariable Modelsp. 260
Equilibria and Stability for Linear Discrete-Time Modelsp. 279
Concluding Messagep. 289
Equilibria and Stability Analyses-Nonlinear Models with Multiple Variablesp. 294
Introductionp. 294
Nonlinear Multiple-Variable Modelsp. 294
Equilibria and Stability for Nonlinear Discrete-Time Modelsp. 316
Perturbation Techniques for Approximating Eigenvaluesp. 330
Concluding Messagep. 337
General Solutions and Tranformations-Models with Multiple Variablesp. 347
Introductionp. 347
Linear Models Involving Multiple Variablesp. 347
Nonlinear Models Involving Multiple Variablesp. 365
Concluding Messagep. 381
Dynamics of Class-Structured Populationsp. 386
Introductionp. 386
Constructing Class-Structured Modelsp. 388
Analyzing Class-Structured Modelsp. 393
Reproductive Value and Left Eigenvectorsp. 398
The Effect of Parameters on the Long-Term Growth Ratep. 400
Age-Structured Models-The Leslie Matrixp. 403
Concluding Messagep. 418
Techniques for Analyzing Models with Periodic Behaviorp. 423
Introductionp. 423
What Are Periodic Dynamics?p. 423
Composite Mappingsp. 425
Hopf Bifurcationsp. 428
Constants of Motionp. 436
Concluding Messagep. 449
Evolutionary Invasion Analysisp. 454
Introductionp. 454
Two Introductory Examplesp. 455
The General Technique of Evolutionary Invasion Analysisp. 465
Determining How the ESS Changes as a Function of Parametersp. 478
Evolutionary Invasion Analyses in Class-Structured Populationsp. 485
Concluding Messagep. 502
Probability Theoryp. 513
An Introduction to Probabilityp. 513
Conditional Probabilities and Bayes' Theoremp. 518
Discrete Probability Distributionsp. 521
Continuous Probability Distributionsp. 536
The (Insert Your Name Here) Distributionp. 553
Probabilistic Modelsp. 567
Introductionp. 567
Models of Population Growthp. 568
Birth-Death Modelsp. 573
Wright-Fisher Model of Allele Frequency Changep. 576
Moran Model of Allele Frequency Changep. 581
Cancer Developmentp. 584
Cellular Automata-A Model of Extinction and Recolonizationp. 591
Looking Backward in Time-Coalescent Theoryp. 594
Concluding Messagep. 602
Analyzing Discrete Stochastic Modelsp. 608
Introductionp. 608
Two-State Markov Modelsp. 608
Multistate Markov Modelsp. 614
Birth-Death Modelsp. 631
Branching Processesp. 639
Concluding Messagep. 644
Analyzing Continuous Stochastic Models-Diffusion in Time and Spacep. 649
Introductionp. 649
Constructing Diffusion Modelsp. 649
Analyzing the Diffusion Equation with Driftp. 664
Modeling Populations in Space Using the Diffusion Equationp. 684
Concluding Messagep. 687
Epilogue: The Art of Mathematical Modeling in Biologyp. 692
Commonly Used Mathematical Rulesp. 695
Rules for Algebraic Functionsp. 695
Rules for Logarithmic and Exponential Functionsp. 695
Some Important Sumsp. 696
Some Important Productsp. 696
Inequalitiesp. 697
Some Important Rules from Calculusp. 699
Conceptsp. 699
Derivativesp. 701
Integralsp. 703
Limitsp. 704
The Perron-Frobenius Theoremp. 709
Definitionsp. 709
The Perron-Frobenius Theoremp. 710
Finding Maxima and Minima of Functionsp. 713
Functions with One Variablep. 713
Functions with Multiple Variablesp. 714
Moment-Generating Functionsp. 717
Index of Definitions, Recipes, and Rulesp. 725
General Indexp. 727
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