Synopses & Reviews
"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
Review
A gentle but thorough introduction to the mathematical techniques employed in ecological and evolutionary theory. Readers who . . . finish this well-written book will be prepared to read and understand a sizeable fraction of the current literature. -- Donald L. DeAngelis, Quarterly Review of Biology At long last, Sally Otto and Troy Day have provided relief for biologists and epidemiologists in search of an easily read, practical, and thorough starting point from which to learn mathematical modeling. . . . We would recommend this book over shorter texts that are labeled as 'introductory'. . . . The depth and detail that Otto and Day have included in this text arc appealing rather than intimidating, and the structure of the text is empowering rather than didactic or formulaic. -- Sanjay Basu and Alison P. Galvani, Siam Review [T]he great value of the Otto/Day book is that it attempts pedagogical soundness, and so is useful for teaching. Besides being perfectly readable, it engages and impresses the reader quickly not only with the subject matter, but also with the quality of printing and layout which have to be seen to be believed. These praises may sound lavish by many a reader of these columns but first see the book or better still buy the volume and you will see our passion and rage for going all out in praise of this volume. -- Current Engineering Practice I highly recommend this book for every university biology department because it provides both a unique, and often uplifting, introduction and a comprehensive reference of techniques for building and analysing mathematical models. -- Volker Grimm, Basic and Applied Ecology
Review
"A gentle but thorough introduction to the mathematical techniques employed in ecological and evolutionary theory. Readers who . . . finish this well-written book will be prepared to read and understand a sizeable fraction of the current literature."--Donald L. DeAngelis, Quarterly Review of Biology
Review
"At long last, Sally Otto and Troy Day have provided relief for biologists and epidemiologists in search of an easily read, practical, and thorough starting point from which to learn mathematical modeling. . . . We would recommend this book over shorter texts that are labeled as 'introductory'. . . . The depth and detail that Otto and Day have included in this text arc appealing rather than intimidating, and the structure of the text is empowering rather than didactic or formulaic."--Sanjay Basu and Alison P. Galvani, Siam Review
Review
"[T]he great value of the Otto/Day book is that it attempts pedagogical soundness, and so is useful for teaching. Besides being perfectly readable, it engages and impresses the reader quickly not only with the subject matter, but also with the quality of printing and layout which have to be seen to be believed. These praises may sound lavish by many a reader of these columns but first see the book or better still buy the volume and you will see our passion and rage for going all out in praise of this volume."--Current Engineering Practice
Review
"I highly recommend this book for every university biology department because it provides both a unique, and often uplifting, introduction and a comprehensive reference of techniques for building and analysing mathematical models."--Volker Grimm, Basic and Applied Ecology
Review
Honorable Mention for the 2007 Best Professional/Scholarly Book in Biological Sciences, Association of American Publishers
Review
"I cannot help but think that future textbook authors will want to have Otto and Day front and center on the work desk, for this is a valuable source of material. . . . This book stands out, and its contribution is quite apparent. In sum, this book is a valuable contribution to the literature, and one to which I expect to refer regularly in connection with my teaching and writing duties."--Steven G. Krantz, UMAP Journal
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 importantmathematical topics
- Exercises with answers
- Appendixes summarize useful rules
- Labs and advanced material available
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
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
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
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