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More About This Textbook
Overview
"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 firstrate 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 modelingin particular, to population modelingin the biological sciences. This past year I taught a 400level 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
Editorial Reviews
Quarterly Review of Biology  Donald L. DeAngelis
A gentle but thorough introduction to the mathematical techniques employed in ecological and evolutionary theory. Readers who . . . finish this wellwritten book will be prepared to read and understand a sizeable fraction of the current literature.Siam Review  Sanjay Basu and Alison P. Galvani
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.Basic and Applied Ecology  Volker Grimm
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.From the Publisher
Honorable Mention for the 2007 Best Professional/Scholarly Book in Biological Sciences, Association of American Publishers
"A gentle but thorough introduction to the mathematical techniques employed in ecological and evolutionary theory. Readers who . . . finish this wellwritten 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
"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
Quarterly Review of Biology
A gentle but thorough introduction to the mathematical techniques employed in ecological and evolutionary theory. Readers who . . . finish this wellwritten book will be prepared to read and understand a sizeable fraction of the current literature.— Donald L. DeAngelis
Siam 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
Current Engineering Practice
[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.Basic and Applied Ecology
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
Product Details
Related Subjects
Meet 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
Read an Excerpt
Biologist's Guide to Mathematical Modeling in Ecology and Evolution
By Sarah P. Otto Troy Day
Princeton University Press
Copyright © 2007 Princeton University PressAll right reserved.
ISBN: 9780691123448
Chapter One
Mathematical Modeling in Biology1.1 Introduction
Mathematics permeates biology. Unfortunately, this is far from obvious to most students of biology. While many biology courses cover results and insights from mathematical models, they rarely describe how these results were obtained. Typically, it is only when biologists start reading research articles that they come to appreciate just how common mathematical modeling is in biology. For many students, this realization comes long after they have chosen the majority of their courses, making it difficult to build the mathematical background needed to appreciate and feel comfortable with the mathematics that they encounter. This book is a guide to help any student develop this appreciation and comfort. To motivate learning more mathematics, we devote this first chapter to emphasizing just how common mathematical models are in biology and to highlighting some of the important ways in which mathematics has shaped our understanding of biology.
Let's begin with some numbers. According to BIOSIS, 886,101 articles published in biological journals contain the keyword "math" (including math, mathematical,mathematics, etc.) as of April 2006. Some of these articles are in specialized journals in mathematical biology, such as the Bulletin of Mathematical Biology, the Journal of Mathematical Biology, Mathematical Biosciences, and Theoretical Population Biology. Many others, however, are published in the most prestigious journals in science, including Nature and Science. Such a coarse survey, however, misses a large fraction of articles describing theoretical models without using "math" as a keyword.
We performed a more indepth survey of all of the articles published in one year within some popular ecology and evolution journals (Table 1.1). Given that virtually every statistical analysis is based on an underlying mathematical model, nearly all articles relied on mathematics to some extent. With a stricter definition that excludes papers whose only use of mathematics is through statistical analyses, 35% of Evolution and Ecology articles and nearly 60% of American Naturalist articles reported predictions or results obtained using mathematical models. The extent of mathematical analysis varied greatly, but mathematical equations appeared in almost all of these articles. Furthermore, many of the articles used computer simulations to describe changes that occur over time in the populations under study. Such simulations can be incredibly helpful, allowing the reader to "see" what the equations predict and allowing authors to obtain results from even the most complicated models.
An important motivation for learning mathematical biology is that mathematical equations typically "say" more than the surrounding text. Given the space constraints of many journals, authors often leave out intermediate steps or fail to state every assumption that they have made. Being able to read and interpret mathematical equations is therefore extremely important, both to verify the conclusions of an author and to evaluate the limitations of unstated assumptions.
To describe all of the biological insights that have come from mathematical models would be an impossible task. Therefore, we focus the rest of this chapter on the insights obtained from mathematical models in one tiny, but critically important, area of biology: the ecology and epidemiology of the human immunodeficiency virus (HIV). As we shall see, mathematical models have allowed biologists to understand otherwise hidden aspects of HIV, they have produced testable predictions about how HIV replicates and spreads, and they have generated forecasts that improve the efficacy of prevention and health care programs.
1.2 HIV
On June 5, 1981, the Morbidity and Mortality Weekly Report of the Centers for Disease Control reported the deaths of five males in Los Angeles, all of whom had died from pneumocystis, a form of pneumonia that rarely causes death in individuals with healthy immune systems. Since this first report, acquired immunodeficiency syndrome (AIDS), as the disease has come to be known, has reached epidemic proportions, having caused more than 20 million deaths worldwide (Joint United Nations Programme on HIV/AIDS 2004b). AIDS results from the deterioration of the immune system, which then fails to ward off various cancers (e.g., Karposi's sarcoma) and infectious agents (e.g., the protozoa that cause pneumocystis, the viruses that cause retinitis, and the bacteria that cause tuberculosis). The collapse of the immune system is caused by infection with the human immunodeficiency virus (Figure 1.1). HIV is transmitted from infected to susceptible individuals by the exchange of bodily fluids, primarily through sexual intercourse without condoms, sharing of unsterilized needles, or transfusion with infected blood supplies (although routine testing for HIV in donated blood has reduced the risk of infection through blood transfusion from 1 in 2500 to 1 in 250,000 [Revelle 1995]).
Once inside the body, HIV particles infect white blood cells by attaching to the CD4 protein embedded in the cell membranes of helper T cells, macrophages, and dendritic cells. The genome of the virus, which is made up of RNA, then enters these cells and is reverse transcribed into DNA, which is subsequently incorporated into the genome of the host. (The fact that normal transcription from DNA to RNA is reversed is why HIV is called a retrovirus.) The virus may then remain latent within the genome of the host cell or become activated, in which case it is transcribed to produce both the proteins necessary to replicate and daughter RNA particles (Figure 1.2). When actively replicating, HIV can produce hundreds of daughter viruses per day per host cell (Dimitrov et al. 1993), often killing the host cell in the process. These virus particles (or virions) then go on to infect other CD4bearing cells, repeating the process. Eventually, without treatment, the population of CD4+ helper T cells declines dramatically from about 1000 cells per cubic millimeter of blood to about 200 cells, signaling the onset of AIDS (Figure 1.3).
Normally, CD4+ helper T cells function in the cellular immune response by binding to fragments of viruses and other foreign proteins presented on the surface of other immune cells. This binding activates the helper T cells to release chemicals (cytokines), which stimulate both killer T cells to attack the infected cells and B cells to manufacture antibodies against the foreign particles. What makes HIV particularly harmful to the immune system is that the virus preferentially attacks activated helper T cells; by destroying such cells, HIV can eliminate the very cells that recognize and fight other infections.
Early on in the epidemic, the median period between infection with HIV1 (the strain most common in North America) and the onset of AIDS was about ten years (Bacchetti and Moss 1989). The median survival time following the onset of an AIDSassociated condition (e.g., Karposi's sarcoma or pneumocystis) was just under one year (Bacchetti et al. 1988). Survival statistics have improved dramatically with the development of effective antiretroviral therapies, such as protease inhibitors, which first became available in 1995, and with the advent of combination drug therapy, which uses multiple drugs to target different steps in the replication cycle of HIV. In San Francisco, the median survival after diagnosis with an AIDSrelated opportunistic infection rose from 17 months between 1990 and 1994 to 59 months between 1995 and 1998 (San Francisco Department of Public Health 2000). Unfortunately, modern drug therapies are extremely expensive (typically over US$10,000 per patient per year) and cannot be afforded by the majority of individuals infected with HIV worldwide. Until effective therapy or vaccines become freely available, HIV will continue to take a devastating toll (Figure 1.4; Joint United Nations Programme on HIV/AIDS 2004a).
1.3 Models of HIV/AIDS
Mathematical modeling has been a very important tool in HIV/AIDS research. Every aspect of the natural history, treatment, and prevention of HIV has been the subject of mathematical models, from the thermodynamic characteristics of HIV (e.g., Hansson and Aqvist 1995; Kroeger Smith et al. 1995; Markgren et al. 2001) to its replication rate both within and among individuals (e.g., Funk et al. 2001; Jacquez et al. 1994; Koopman et al. 1997; Levin et al. 1996; Lloyd 2001; Phillips 1996). In the following sections, we describe four of these models in more detail. These models were chosen because of their implications for our understanding of HIV, but they also illustrate the sorts of techniques that are described in the rest of this book.
1.3.1 Dynamics of HIV after Initial Infection After an individual is infected by HIV, the number of virions within the bloodstream skyrockets and then plummets again (Figure 1.3). This period of primary HIV infection is known as the acute phase; it lasts approximately 100 days and often leads to the onset of flulike symptoms (Perrin and Yerly 1997; Schacker et al. 1996). The rapid rise in virus particles reflects the infection of CD4+ cells and the replication of HIV within actively infected host cells. But what causes the decline in virus particles? The most obvious answer is that the immune system acts to recognize and suppress the viral infection (Koup et al. 1994). Phillips (1996), however, suggested an alternative explanation: the number of virions might decline because most of the susceptible CD4+ cells have already been infected and thus there are fewer host cells to infect. Phillips developed a model to assess whether this alternative explanation could mimic the observed rise and fall of virions in the blood stream over the right time frame. In his model, there are four variables (i.e., four quantities that change over time): R, L, E, and V. R represents the number of activated but uninfected CD4+ cells, L represents the number of latently infected cells, E represents the number of actively infected cells, and V represents the number of virions in the blood stream. The dynamics of each variable (i.e., how the variable changes over time) depend on the values of the remaining variables. For example, the number of viruses changes over time in a manner that depends on the number of cells infected with actively replicating HIV. In the next chapter, we describe the steps involved in building models such as this one (see Chapter 2, Box 2.4).
Phillips' model contains several parameters, which are quantities that are constant over time (see Chapter 2, Box 2.4). In particular, the death rate of actively infected cells ([delta]) and the death rate of viruses ([sigma]) are parameters in the model and are not allowed to change over time. ([delta] and [sigma] are the lowercase Greek letters "delta" and "sigma." Greek letters are often used in models, especially for terms that remain constant ("parameters"). See Table 2.1 for a complete list of Greek letters.) Thus, Phillips built into his model the crucial assumption that the body does not get better at eliminating infected cells or virus particles over time, under the null hypothesis that the immune system does not mount a defense against HIV during the acute phase. To model the progression of HIV within the body, Phillips then needed values for each of the parameters in the model. Unfortunately, few data existed at the time for many of them. To proceed, Phillips chose plausible values for each parameter and numerically ran the model (a technique that we will describe in Chapter 4). The numerical solution for the number of virus particles, V, predicted from Phillips' model is plotted in Figure 1.5 (compare to Figure 1.3). Phillips then showed that similar patterns are observed under a variety of different parameter values. In particular, he observed that the number of virus particles typically rose and then fell by several orders of magnitude over a period of a few days to weeks. (An order of magnitude refers to a factor of ten. The number 100 is two orders of magnitude larger than one.)
Phillips thus came to the counterintuitive conclusion that "the reduction in virus concentration during acute infection may not reflect the ability of the HIVspecific immune response to control the virus replication" (p. 497, Phillips 1996). The wording of this conclusion is critical and insightful. Phillips did not use his model to prove that the immune system plays no role in viral dynamics during primary infection. In fact, his model cannot say one way or the other whether there is a relevant HIVspecific immune response during this time period. What Phillips can say is that an immune response is not necessary to explain the observed data. This result illustrates an important principle in modeling: the principle of parsimony. The principle of parsimony states that one should prefer models containing as few variables and parameters as possible to describe the essential attributes of a system. Paraphrasing Albert Einstein, a model should be as simple as possible, but no simpler. In Phillips' case, he could have added more variables describing an immune response during acute infection, but his results showed that adding such complexity was unnecessary. A simpler hypothesis can explain the rise and fall of HIV in the bloodstream: as infection proceeds, a decline in susceptible host cells reduces the rate at which virus is produced. Without having a good reason to invoke a more complex model, the principle of parsimony encourages us to stick with simple hypotheses.
Phillips' model accomplished a number of important things. First, it changed our view of what was possible. Without such a model, it would seem unlikely that a dramatic viral peak and decline could be caused by the dynamics of a CD4+ cell population without an immune response. Second, it produced testable predictions. One prediction noted by Phillips is that the viral peak and decline should be observed even in individuals that do not mount an immune response (i.e., do not produce antiHIV antibodies) over this time period. Indeed, this prediction has been confirmed in several patients (Koup et al. 1994; Phillips 1996). Employing a more quantitative test, Stafford et al. (2000) fitted a version of Phillips' model to data on the viral load in ten patients from several time points during primary HIV infection; they found a good fit to the data within the first 100 days following infection. Third, Phillips' model generated a useful null hypothesis: viral dynamics do not reflect an immune response. This null hypothesis might be wrong, but at least it can be tested.
Phillips acknowledged that this null hypothesis can be rejected as a description of the longerterm dynamics of HIV. His model predicts that the viral load should reach an equilibrium (as described in Chapter 8), but observations indicate that the viral load slowly increases over the long term as the immune system weakens (the chronic phase in Figure 1.3). Furthermore, Schmitz et al. (1999) directly tested Phillips' hypothesis by examining the role of the immune system in rhesus monkeys infected with the simian immunodeficiency virus (SIV), the equivalent of HIV in monkeys. By injecting a particular antibody, Schmitz et al. were able to eliminate most CD8+ lymphocytes, which are the killer T cells thought to prevent the replication of HIV and SIV. Compared to control monkeys, the experimentally treated monkeys showed a much more shallow decline in virus load following the peak. This proves that, at least in monkeys, an immune response does play some role in the viral dynamics observed during primary infection. Nevertheless, the peak viral load was observed at similar levels in antibodytreated and untreated monkeys. Thus, an immune response was not responsible for stalling viral growth during the acute phase, which is best explained, instead, by a decline in the number of uninfected CD4+ cells (the targets of HIV and SIV).
(Continues...)
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 BackwardInterpreting 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 TechniquesDeveloping 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 PhasePlane Diagrams 133
4.5 Concluding Message 145
Chapter 5: Equilibria and Stability AnalysesOneVariable 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 TransformationsOneVariable 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 AnalysesLinear 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 DiscreteTime Models 279
7.5 Concluding Message 289
Chapter 8: Equilibria and Stability AnalysesNonlinear Models with Multiple Variables 294
8.1 Introduction 294
8.2 Nonlinear MultipleVariable Models 294
8.3 Equilibria and Stability for Nonlinear DiscreteTime Models 316
8.4 Perturbation Techniques for Approximating Eigenvalues 330
8.5 Concluding Message 337
Chapter 9: General Solutions and TranformationsModels 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 ClassStructured Populations 386
10.1 Introduction 386
10.2 Constructing ClassStructured Models 388
10.3 Analyzing ClassStructured Models 393
10.4 Reproductive Value and Left Eigenvectors 398
10.5 The Effect of Parameters on the LongTerm Growth Rate 400
10.6 AgeStructured ModelsThe 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 ClassStructured 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 BirthDeath Models 573
13.4 WrightFisher Model of Allele Frequency Change 576
13.5 Moran Model of Allele Frequency Change 581
13.6 Cancer Development 584
13.7 Cellular AutomataA Model of Extinction and Recolonization 591
13.8 Looking Backward in TimeCoalescent Theory 594
13.9 Concluding Message 602
Chapter 14: Analyzing Discrete Stochastic Models 608
14.1 Introduction 608
14.2 TwoState Markov Models 608
14.3 Multistate Markov Models 614
14.4 BirthDeath Models 631
14.5 Branching Processes 639
14.6 Concluding Message 644
Chapter 15: Analyzing Continuous Stochastic ModelsDiffusion 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 PerronFrobenius Theorem 709
A3.1: Definitions 709
A3.2: The PerronFrobenius 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: MomentGenerating Functions 717
Index of Definitions, Recipes, and Rules 725
General Index 727