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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 | |