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

by and

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

 

Synopses & Reviews

Publisher Comments:


Thirty years ago, biologists could get by with a rudimentary grasp of mathematics and modeling. Not so today. In seeking to answer fundamental questions about how biological systems function and change over time, the modern biologist is as likely to rely on sophisticated mathematical and computer-based models as traditional fieldwork. In this book, Sarah Otto and Troy Day provide biology students with the tools necessary to both interpret models and to build their own.

The book starts at an elementary level of mathematical modeling, assuming that the reader has had high school mathematics and first-year calculus. Otto and Day then gradually build in depth and complexity, from classic models in ecology and evolution to more intricate class-structured and probabilistic models. The authors provide primers with instructive exercises to introduce readers to the more advanced subjects of linear algebra and probability theory. Through examples, they describe how models have been used to understand such topics as the spread of HIV, chaos, the age structure of a country, speciation, and extinction.

Ecologists and evolutionary biologists today need enough mathematical training to be able to assess the power and limits of biological models and to develop theories and models themselves. This innovative book will be an indispensable guide to the world of mathematical models for the next generation of biologists.

  • A how-to guide for developing new mathematical models in biology
  • Provides step-by-step recipes for constructing and analyzing models
  • Interesting biological applications
  • Explores classical models in ecology and evolution
  • Questions at the end of every chapter
  • Primers cover importantmathematical topics
  • Exercises with answers
  • Appendixes summarize useful rules
  • Labs and advanced material available

Synopsis:

"A wonderfully pedagogical introduction to mathematical modeling in population biology: an ideal first course for biologists."--Simon A. Levin, Princeton University

"This book is an amazing teaching resource for developing a comprehensive understanding of the methods and importance of biological modeling. But more than that, this book should be read by every student of evolutionary biology and ecology so that they can come to a deeper appreciation of the fundamental ideas and models that underlie these fields."--Patrick C. Phillips, University of Oregon

"There is an increasing use of mathematics throughout the biological sciences, yet the training of most biologists still woefully lacks crucial mathematical tools. Sally Otto and Troy Day are themselves two masters at the deft use of theoretical models to crystallize conceptual insights about ecological and evolutionary problems, and in this wonderful book they make accessible to a broad audience the essential mathematical tool kit biologists need, both to read the literature and to craft and analyze models themselves."--Robert D. Holt, University of Florida

"I am often asked by biologists to recommend a book on mathematical modeling, but I must tell them that there is no single good book that will guide them through the difficult first stages of learning to make models. Otto and Day's book fills the gap. The quality is high throughout, the scholarship is sound, the book is comprehensive. The authors are both first-rate scientists. I think this will be a classic."--Steven A. Frank, author of Immunology and Evolution of Infectious Disease

"This book provides a general introduction to mathematical modeling--in particular, to population modeling--in the biological sciences. This past year I taught a 400-level course in mathematical modeling of biological systems, and I had to do so without a textbook because no adequate text existed. Otto and Day's book would have met my needs beautifully. This book is an important addition to the field."--Carl Bergstrom, University of Washington

"This book has the ambitious and worthy goal of teaching biologists enough about modeling and about mathematical methods to be both intelligent consumers of models and competent creators of their own models. Its concentration on the process of building rather than analyzing models is its strongest point."--Frederick R. Adler, author of Modeling the Dynamics of Life: Calculus and Probability for Life Scientists

Synopsis:

Thirty years ago, biologists could get by with a rudimentary grasp of mathematics and modeling. Not so today. In seeking to answer fundamental questions about how biological systems function and change over time, the modern biologist is as likely to rely on sophisticated mathematical and computer-based models as traditional fieldwork. In this book, Sarah Otto and Troy Day provide biology students with the tools necessary to both interpret models and to build their own.

The book starts at an elementary level of mathematical modeling, assuming that the reader has had high school mathematics and first-year calculus. Otto and Day then gradually build in depth and complexity, from classic models in ecology and evolution to more intricate class-structured and probabilistic models. The authors provide primers with instructive exercises to introduce readers to the more advanced subjects of linear algebra and probability theory. Through examples, they describe how models have been used to understand such topics as the spread of HIV, chaos, the age structure of a country, speciation, and extinction.

Ecologists and evolutionary biologists today need enough mathematical training to be able to assess the power and limits of biological models and to develop theories and models themselves. This innovative book will be an indispensable guide to the world of mathematical models for the next generation of biologists.

  • A how-to guide for developing new mathematical models in biology
  • Provides step-by-step recipes for constructing and analyzing models
  • Interesting biological applications
  • Explores classical models in ecology and evolution
  • Questions at the end of every chapter
  • Primers cover important mathematical topics
  • Exercises with answers
  • Appendixes summarize useful rules
  • Labs and advanced material available

About the Author

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 ix

Chapter 1: Mathematical Modeling in Biology 1

1.1 Introduction 1

1.2 HIV 2

1.3 Models of HIV/AIDS 5

1.4 Concluding Message 14

Chapter 2: How to Construct a Model 17

2.1 Introduction 17

2.2 Formulate the Question 19

2.3 Determine the Basic Ingredients 19

2.4 Qualitatively Describe the Biological System 26

2.5 Quantitatively Describe the Biological System 33

2.6 Analyze the Equations 39

2.7 Checks and Balances 47

2.8 Relate the Results Back to the Question 50

2.9 Concluding Message 51

Chapter 3: Deriving Classic Models in Ecology and Evolutionary Biology 54

3.1 Introduction 54

3.2 Exponential and Logistic Models of Population Growth 54

3.3 Haploid and Diploid Models of Natural Selection 62

3.4 Models of Interactions among Species 72

3.5 Epidemiological Models of Disease Spread 77

3.6 Working Backward--Interpreting Equations in Terms of the Biology 79

3.7 Concluding Message 82

Primer 1: Functions and Approximations 89

P1.1 Functions and Their Forms 89

P1.2 Linear Approximations 96

P1.3 The Taylor Series 100

Chapter 4: Numerical and Graphical Techniques--Developing a Feeling for Your Model 110

4.1 Introduction 110

4.2 Plots of Variables Over Time 111

4.3 Plots of Variables as a Function of the Variables Themselves 124

4.4 Multiple Variables and Phase-Plane Diagrams 133

4.5 Concluding Message 145

Chapter 5: Equilibria and Stability Analyses--One-Variable Models 151

5.1 Introduction 151

5.2 Finding an Equilibrium 152

5.3 Determining Stability 163

5.4 Approximations 176

5.5 Concluding Message 184

Chapter 6: General Solutions and Transformations--One-Variable Models 191

6.1 Introduction 191

6.2 Transformations 192

6.3 Linear Models in Discrete Time 193

6.4 Nonlinear Models in Discrete Time 195

6.5 Linear Models in Continuous Time 198

6.6 Nonlinear Models in Continuous Time 202

6.7 Concluding Message 207

Primer 2: Linear Algebra 214

P2.1 An Introduction to Vectors and Matrices 214

P2.2 Vector and Matrix Addition 219

P2.3 Multiplication by a Scalar 222

P2.4 Multiplication of Vectors and Matrices 224

P2.5 The Trace and Determinant of a Square Matrix 228

P2.6 The Inverse 233

P2.7 Solving Systems of Equations 235

P2.8 The Eigenvalues of a Matrix 237

P2.9 The Eigenvectors of a Matrix 243

Chapter 7: Equilibria and Stability Analyses--Linear Models with Multiple Variables 254

7.1 Introduction 254

7.2 Models with More than One Dynamic Variable 255

7.3 Linear Multivariable Models 260

7.4 Equilibria and Stability for Linear Discrete-Time Models 279

7.5 Concluding Message 289

Chapter 8: Equilibria and Stability Analyses--Nonlinear Models with Multiple Variables 294

8.1 Introduction 294

8.2 Nonlinear Multiple-Variable Models 294

8.3 Equilibria and Stability for Nonlinear Discrete-Time Models 316

8.4 Perturbation Techniques for Approximating Eigenvalues 330

8.5 Concluding Message 337

Chapter 9: General Solutions and Tranformations--Models with Multiple Variables 347

9.1 Introduction 347

9.2 Linear Models Involving Multiple Variables 347

9.3 Nonlinear Models Involving Multiple Variables 365

9.4 Concluding Message 381

Chapter 10: Dynamics of Class-Structured Populations 386

10.1 Introduction 386

10.2 Constructing Class-Structured Models 388

10.3 Analyzing Class-Structured Models 393

10.4 Reproductive Value and Left Eigenvectors 398

10.5 The Effect of Parameters on the Long-Term Growth Rate 400

10.6 Age-Structured Models--The Leslie Matrix 403

10.7 Concluding Message 418

Chapter 11: Techniques for Analyzing Models with Periodic Behavior 423

11.1 Introduction 423

11.2 What Are Periodic Dynamics? 423

11.3 Composite Mappings 425

11.4 Hopf Bifurcations 428

11.5 Constants of Motion 436

11.6 Concluding Message 449

Chapter 12: Evolutionary Invasion Analysis 454

12.1 Introduction 454

12.2 Two Introductory Examples 455

12.3 The General Technique of Evolutionary Invasion Analysis 465

12.4 Determining How the ESS Changes as a Function of Parameters 478

12.5 Evolutionary Invasion Analyses in Class-Structured Populations 485

12.6 Concluding Message 502

Primer 3: Probability Theory 513

P3.1 An Introduction to Probability 513

P3.2 Conditional Probabilities and Bayes’ Theorem 518

P3.3 Discrete Probability Distributions 521

P3.4 Continuous Probability Distributions 536

P3.5 The (Insert Your Name Here) Distribution 553

Chapter 13: Probabilistic Models 567

13.1 Introduction 567

13.2 Models of Population Growth 568

13.3 Birth-Death Models 573

13.4 Wright-Fisher Model of Allele Frequency Change 576

13.5 Moran Model of Allele Frequency Change 581

13.6 Cancer Development 584

13.7 Cellular Automata--A Model of Extinction and Recolonization 591

13.8 Looking Backward in Time--Coalescent Theory 594

13.9 Concluding Message 602

Chapter 14: Analyzing Discrete Stochastic Models 608

14.1 Introduction 608

14.2 Two-State Markov Models 608

14.3 Multistate Markov Models 614

14.4 Birth-Death Models 631

14.5 Branching Processes 639

14.6 Concluding Message 644

Chapter 15: Analyzing Continuous Stochastic Models--Diffusion in Time and Space 649

15.1 Introduction 649

15.2 Constructing Diffusion Models 649

15.3 Analyzing the Diffusion Equation with Drift 664

15.4 Modeling Populations in Space Using the Diffusion Equation 684

15.5 Concluding Message 687

Epilogue: The Art of Mathematical Modeling in Biology 692

Appendix 1: Commonly Used Mathematical Rules 695

A1.1 Rules for Algebraic Functions 695

A1.2 Rules for Logarithmic and Exponential Functions 695

A1.3 Some Important Sums 696

A1.4 Some Important Products 696

A1.5 Inequalities 697

Appendix 2: Some Important Rules from Calculus 699

A2.1 Concepts 699

A2.2 Derivatives 701

A2.3 Integrals 703

A2.4 Limits 704

Appendix 3: The Perron-Frobenius Theorem 709

A3.1: Definitions 709

A3.2: The Perron-Frobenius Theorem 710

Appendix 4: Finding Maxima and Minima of Functions 713

A4.1 Functions with One Variable 713

A4.2 Functions with Multiple Variables 714

Appendix 5: Moment-Generating Functions 717

Index of Definitions, Recipes, and Rules 725

General Index 727

Product Details

ISBN:
9780691123448
Author:
Sarah P. Otto and Troy Day
Publisher:
Princeton University Press
Author:
Otto, Sarah P.
Author:
Day, Troy
Location:
Princeton
Subject:
General
Subject:
Ecology
Subject:
Evolution (Biology)
Subject:
Life Sciences - Ecology
Subject:
Life Sciences - Biology - General
Subject:
Biology
Subject:
Biological Sciences.
Subject:
Mathematics
Subject:
Ecology -- Mathematical models.
Subject:
Evolution (Biology) -- Mathematical models.
Subject:
Mathema
Subject:
tics
Subject:
Environmental Studies-General
Copyright:
Publication Date:
February 2007
Binding:
HARDCOVER
Grade Level:
College/higher education:
Language:
English
Illustrations:
207 line illus. 22 tables.
Pages:
744
Dimensions:
10 x 8 in

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» Science and Mathematics » Mathematics » General
» Textbooks » General

A Biologist's Guide to Mathematical Modeling in Ecology and Evolution New Hardcover
0 stars - 0 reviews
$98.95 In Stock
Product details 744 pages Princeton University Press - English 9780691123448 Reviews:
"Synopsis" by ,

"A wonderfully pedagogical introduction to mathematical modeling in population biology: an ideal first course for biologists."--Simon A. Levin, Princeton University

"This book is an amazing teaching resource for developing a comprehensive understanding of the methods and importance of biological modeling. But more than that, this book should be read by every student of evolutionary biology and ecology so that they can come to a deeper appreciation of the fundamental ideas and models that underlie these fields."--Patrick C. Phillips, University of Oregon

"There is an increasing use of mathematics throughout the biological sciences, yet the training of most biologists still woefully lacks crucial mathematical tools. Sally Otto and Troy Day are themselves two masters at the deft use of theoretical models to crystallize conceptual insights about ecological and evolutionary problems, and in this wonderful book they make accessible to a broad audience the essential mathematical tool kit biologists need, both to read the literature and to craft and analyze models themselves."--Robert D. Holt, University of Florida

"I am often asked by biologists to recommend a book on mathematical modeling, but I must tell them that there is no single good book that will guide them through the difficult first stages of learning to make models. Otto and Day's book fills the gap. The quality is high throughout, the scholarship is sound, the book is comprehensive. The authors are both first-rate scientists. I think this will be a classic."--Steven A. Frank, author of Immunology and Evolution of Infectious Disease

"This book provides a general introduction to mathematical modeling--in particular, to population modeling--in the biological sciences. This past year I taught a 400-level course in mathematical modeling of biological systems, and I had to do so without a textbook because no adequate text existed. Otto and Day's book would have met my needs beautifully. This book is an important addition to the field."--Carl Bergstrom, University of Washington

"This book has the ambitious and worthy goal of teaching biologists enough about modeling and about mathematical methods to be both intelligent consumers of models and competent creators of their own models. Its concentration on the process of building rather than analyzing models is its strongest point."--Frederick R. Adler, author of Modeling the Dynamics of Life: Calculus and Probability for Life Scientists

"Synopsis" by ,

Thirty years ago, biologists could get by with a rudimentary grasp of mathematics and modeling. Not so today. In seeking to answer fundamental questions about how biological systems function and change over time, the modern biologist is as likely to rely on sophisticated mathematical and computer-based models as traditional fieldwork. In this book, Sarah Otto and Troy Day provide biology students with the tools necessary to both interpret models and to build their own.

The book starts at an elementary level of mathematical modeling, assuming that the reader has had high school mathematics and first-year calculus. Otto and Day then gradually build in depth and complexity, from classic models in ecology and evolution to more intricate class-structured and probabilistic models. The authors provide primers with instructive exercises to introduce readers to the more advanced subjects of linear algebra and probability theory. Through examples, they describe how models have been used to understand such topics as the spread of HIV, chaos, the age structure of a country, speciation, and extinction.

Ecologists and evolutionary biologists today need enough mathematical training to be able to assess the power and limits of biological models and to develop theories and models themselves. This innovative book will be an indispensable guide to the world of mathematical models for the next generation of biologists.

  • A how-to guide for developing new mathematical models in biology
  • Provides step-by-step recipes for constructing and analyzing models
  • Interesting biological applications
  • Explores classical models in ecology and evolution
  • Questions at the end of every chapter
  • Primers cover important mathematical topics
  • Exercises with answers
  • Appendixes summarize useful rules
  • Labs and advanced material available

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