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Preface | |
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Mathematical Modeling in Biology | |
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Introduction | |
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HIV | |
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Models of HIV/AIDS | |
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Concluding Message | |
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How to Construct a Model | |
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Introduction | |
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Formulate the Question | |
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Determine the Basic Ingredients | |
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Qualitatively Describe the Biological System | |
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Quantitatively Describe the Biological System | |
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Analyze the Equations | |
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Checks and Balances | |
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Relate the Results Back to the Question | |
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Concluding Message | |
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Deriving Classic Models in Ecology and Evolutionary Biology | |
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Introduction | |
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Exponential and Logistic Models of Population Growth | |
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Haploid and Diploid Models of Natural Selection | |
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Models of Interactions among Species | |
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Epidemiological Models of Disease Spread | |
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Working Backward-Interpreting Equations in Terms of the Biology | |
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Concluding Message | |
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Functions and Approximations | |
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Functions and Their Forms | |
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Linear Approximations | |
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The Taylor Series | |
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Numerical and Graphical Techniques-Developing a Feeling for Your Model | |
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Introduction | |
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Plots of Variables Over Time | |
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Plots of Variables as a Function of the Variables Themselves | |
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Multiple Variables and Phase-Plane Diagrams | |
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Concluding Message | |
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Equilibria and Stability Analyses-One-Variable Models | |
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Introduction | |
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Finding an Equilibrium | |
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Determining Stability | |
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Approximations | |
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Concluding Message | |
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General Solutions and Transformations-One-Variable Models | |
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Introduction | |
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Transformations | |
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Linear Models in Discrete Time | |
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Nonlinear Models in Discrete Time | |
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Linear Models in Continuous Time | |
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Nonlinear Models in Continuous Time | |
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Concluding Message | |
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Linear Algebra | |
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An Introduction to Vectors and Matrices | |
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Vector and Matrix Addition | |
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Multiplication by a Scalar | |
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Multiplication of Vectors and Matrices | |
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The Trace and Determinant of a Square Matrix | |
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The Inverse | |
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Solving Systems of Equations | |
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The Eigenvalues of a Matrix | |
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The Eigenvectors of a Matrix | |
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Equilibria and Stability Analyses-Linear Models with Multiple Variables | |
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Introduction | |
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Models with More than One Dynamic Variable | |
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Linear Multivariable Models | |
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Equilibria and Stability for Linear Discrete-Time Models | |
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Concluding Message | |
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Equilibria and Stability Analyses-Nonlinear Models with Multiple Variables | |
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Introduction | |
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Nonlinear Multiple-Variable Models | |
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Equilibria and Stability for Nonlinear Discrete-Time Models | |
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Perturbation Techniques for Approximating Eigenvalues | |
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Concluding Message | |
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General Solutions and Tranformations-Models with Multiple Variables | |
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Introduction | |
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Linear Models Involving Multiple Variables | |
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Nonlinear Models Involving Multiple Variables | |
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Concluding Message | |
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Dynamics of Class-Structured Populations | |
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Introduction | |
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Constructing Class-Structured Models | |
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Analyzing Class-Structured Models | |
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Reproductive Value and Left Eigenvectors | |
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The Effect of Parameters on the Long-Term Growth Rate | |
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Age-Structured Models-The Leslie Matrix | |
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Concluding Message | |
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Techniques for Analyzing Models with Periodic Behavior | |
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Introduction | |
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What Are Periodic Dynamics? | |
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Composite Mappings | |
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Hopf Bifurcations | |
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Constants of Motion | |
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Concluding Message | |
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Evolutionary Invasion Analysis | |
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Introduction | |
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Two Introductory Examples | |
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The General Technique of Evolutionary Invasion Analysis | |
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Determining How the ESS Changes as a Function of Parameters | |
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Evolutionary Invasion Analyses in Class-Structured Populations | |
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Concluding Message | |
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Probability Theory | |
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An Introduction to Probability | |
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Conditional Probabilities and Bayes' Theorem | |
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Discrete Probability Distributions | |
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Continuous Probability Distributions | |
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The (Insert Your Name Here) Distribution | |
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Probabilistic Models | |
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Introduction | |
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Models of Population Growth | |
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Birth-Death Models | |
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Wright-Fisher Model of Allele Frequency Change | |
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Moran Model of Allele Frequency Change | |
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Cancer Development | |
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Cellular Automata-A Model of Extinction and Recolonization | |
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Looking Backward in Time-Coalescent Theory | |
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Concluding Message | |
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Analyzing Discrete Stochastic Models | |
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Introduction | |
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Two-State Markov Models | |
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Multistate Markov Models | |
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Birth-Death Models | |
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Branching Processes | |
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Concluding Message | |
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Analyzing Continuous Stochastic Models-Diffusion in Time and Space | |
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Introduction | |
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Constructing Diffusion Models | |
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Analyzing the Diffusion Equation with Drift | |
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Modeling Populations in Space Using the Diffusion Equation | |
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Concluding Message | |
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Epilogue: The Art of Mathematical Modeling in Biology | |
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Commonly Used Mathematical Rules | |
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Rules for Algebraic Functions | |
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Rules for Logarithmic and Exponential Functions | |
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Some Important Sums | |
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Some Important Products | |
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Inequalities | |
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Some Important Rules from Calculus | |
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Concepts | |
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Derivatives | |
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Integrals | |
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Limits | |
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The Perron-Frobenius Theorem | |
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Definitions | |
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The Perron-Frobenius Theorem | |
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Finding Maxima and Minima of Functions | |
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Functions with One Variable | |
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Functions with Multiple Variables | |
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Moment-Generating Functions | |
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Index of Definitions, Recipes, and Rules | |
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General Index | |