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

Sarah P. Otto and Troy Day
Princeton University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Jason M. Graham
, on

There is a long history of mathematical approaches to the study of population dynamics. A particularly early example is that of the Fibonacci sequence as a model for reproducing rabbits. However contrived Fibonacci’s rabbit problem may be, the second order linear recurrence relation that generates the Fibonacci sequence still provides a nice introduction to the usefulness of linear difference equations as discrete-time models for the dynamics of structured populations. Euler is known to have examined exponential growth, Daniel Bernoulli and d’Alembert both proposed models to study smallpox inoculation. More recently, work by Hardy and Weinberg, Lotka, May, and Leslie have served to develop mathematical population dynamics into a mature field. Furthermore, applications of the mathematical study of populations has lead to notable progress in epidemiology and the spread of infectious disease.

The recent book A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution by Sarah P. Otto and Troy Day is a testament to the value of mathematics in population dynamics, and even in biology more broadly. The book is certainly strong evidence of a general recognition that there is a desire (need?) for practicing biologists to gain greater exposure to mathematical modeling. As it turns out, if you are a biologist and it is your desire (need?) to gain greater exposure to mathematical model, then this is a great place to begin.

As the authors point out, for a biologist first constructing their own mathematical models the greatest challenge is often the initial step of putting pencil to paper. How does a beginner know where or how to start? This is a legitimate question that Otto and Day do a great job in addressing. Specifically, they describe seven steps for constructing a dynamical model, either discrete or continuous, that one should follow to obtain a mathematical model for a biological problem. I believe that their approach to explaining the modeling process is highly effective and useful for wide variety of readers. The authors follow up their discussion of model construction with examples and methods common in mathematical ecology and a few other areas of mathematical biology.

Most of the rest of the text is concerned with common techniques for deriving information from a mathematical model. The reader will learn many standard mathematical techniques, some of which date back to the early days of population dynamics alluded to before, for solving problems that arise in the biological sciences. Specifically, the authors cover stability analysis in one and several variables, with the several variable case treated after a brief primer in linear algebra, and a few other analytical techniques. I think that their treatment is highly accessible and they take great care to develop only as much mathematics as is necessary, and then only at the point when it will be used. Their approach is well motivated. Students, especially those coming from outside of mathematics, will appreciate the efforts of the authors.

A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution has some overlap with other texts such as Quantifying Life by Dmitry Kondrashov and Mathematics for the Life Sciences by Bodine, Lenhart and Gross. The three listed books are similar in spirit in that they seek to teach relevant mathematical concepts to biologists, rather than mathematical biology to mathematics students. There is enough of a difference, however, to make A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution a unique text and worth studying even if the reader has familiarity with some other text on mathematical modeling in biology.

I want to point out that this book does not make use of computing technology in problem solving. Whether this is a shortcoming of the book is of course a matter of opinion, but I would argue that developing skills in programming and computing to implement mathematical models or develop model-based simulations is as important for today’s biologists as is developing the skills that are well-covered here.

There are several features that make A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution a great textbook, such as a masterful use of figures and illustrations and exercises that are both interesting and provide the reader with valuable practice in constructing models and implementing related mathematical techniques. I certainly recommend this text and can attest to its usefulness for budding researchers in the biological sciences.

Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.

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