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Discrete-Time Dynamical Systems | |

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Biology and Dynamics | |

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Growth: Models of Malaria | |

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Maintenance: Models of Neurons | |

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Replication: Models of Genetics | |

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Types of Dynamical Systems | |

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Variables and Functions | |

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Describing Measurements with Variables, Parameters and Graphs | |

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Describing Relations Between Measurements with Functions | |

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Combining Functions | |

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Units and Dimensions | |

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Converting Between Units | |

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Translating Between Dimensions | |

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Checking: Dimensions and Estimation | |

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Linear Functions and their Graphs | |

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Proportional Relations | |

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Linear Functions and the Equation of a Line | |

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Finding Equations and Graphing Lines | |

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Solving Equations Involving Lines | |

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Finding Inverse Functions | |

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Discrete-Time Dynamical Systems | |

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Discrete-time Dynamical Systems and Updating Functions | |

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Manipulating Updating Functions | |

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Discrete-Time Dynamical Systems: Units and Dimensions | |

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Solutions | |

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Analysis of Discrete-Tie Dynamical Systems | |

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Cobwebbing: A Graphical Solution Technique | |

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Equilibria: Graphical Approach | |

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Equilibria: Algebraic Approach | |

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Solutions and Exponential Functions | |

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Bacterial Population Growth in General | |

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Laws of Exponents and Logs | |

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Expressing Results with Exponentials | |

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Oscillations and Trigonometry | |

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Sine and Cosine: A Review | |

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Describing Oscillations with the Cosine | |

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More Complicated Shapes | |

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A Model of Gas Exchange in the Lung | |

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A Model of the Lungs | |

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The Lung System in General | |

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Lung Dynamics with Absorption | |

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An Example of Nonlinear Dynamics | |

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A Model of Selection | |

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The Discrete-Time Dynamical System and Equilibria | |

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Stable and Unstable Equilibria | |

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Excitable Systems I: The Heart | |

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A Simple Heart | |

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Second Degree Block | |

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The Wenckeback Phenomenon | |

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Limits and Derivatives | |

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Introduction to Derivatives | |

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The Average Rate of Change | |

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Instantaneous Rates of Change | |

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Limits and Derivatives | |

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Differential Equations: A Preview | |

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Limits | |

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Limits of Functions | |

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Left and Right-Hand Limits | |

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Properties of Limits | |

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Infinite Limits | |

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Continuity | |

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Continuous Functions | |

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Input and Output Tolerances | |

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Hysteresis | |

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Computing Derivatives | |

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Differentiable Functions | |

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The Derivative of a Linear Function | |

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A Quadratic Function | |

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Derivatives of Sums, Powers, and Polynomials | |

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The Sum Rule | |

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Derivatives of Power Functions | |

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Derivatives of Polynomials | |

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Derivatives of Products and Quotients | |

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The Product Rule | |

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Special Cases and Examples | |

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The Quotient Rule | |

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The Second Derivative | |

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The Second Derivative | |

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Using the Second Derivative for Graphing | |

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Acceleration | |

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Exponentials snd Logarithms | |

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The Exponential Function | |

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The Natural Logarithm | |

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Applications | |

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The Chain Rule | |

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The Derivative of a Composite Function | |

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Derivatives of Inverse Functions | |

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Applications | |

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Derivatives of Trigonometric Functions | |

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Deriving the Derivatives of Sine and Cosine | |

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Other Trigonometric Functions | |

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Applications | |

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Derivatives and Dynamical Systems | |

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Stability and the Derivative | |

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Motivation | |

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Stability and the Slope | |

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Evaluating Stability with the Derivative | |

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More Complex Dynamics | |

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The Logistic Dynamical System | |

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Qualitative Dynamical Systems | |

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Analysis of the Logistic Dynamical System | |

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Maximization | |

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Minima and Maxima | |

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Maximizing Food Intake Rate | |

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Maximizing Fish Harvest | |

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Reasoning About Functions | |

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The Intermediate Value Theorem | |

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Maximization: The Extreme Value Theorem | |

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Rolle's Theorem and the Mean Value Theorem | |

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Limits at Infinity | |

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The Behavior of Functions at Infinity | |

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Application to Absorption Functions | |

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Limits of Sequences | |

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Leading Behavior and L'hopital's Rule | |

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Leading Behavior of Functions at Infinity | |

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Leading Behavior of Functions at 0 | |

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The Method of Matched Leading Behaviors | |

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L'Hopital's Rule | |

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Approximating Functions | |

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The Tangent and Secant Lines | |

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Quadratic Approximation | |

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Taylor Polynomials | |

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Newton's Method | |

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Finding the Equilibrium | |

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Newton's Method | |

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Why Newton's Method Works and When it Fails | |

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Panting and Deep Breathing | |

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Breathing at Different Rates | |

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Deep Breathing | |

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Panting | |

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Intermediate Optimum | |

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Differential Equations, Integrals, and their Applications | |

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Differential Equations | |

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Differential Equations: Examples and Terminology | |

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Graphical Solution of Pure-Time Differential Equations | |

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Euler's Method: Pure-Time | |

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Antiderivatives and Indefinite Integrals | |

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Pure-Time Differential Equations | |

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Rules for Antiderivatives | |

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Solving Polynomial Differential Equations | |

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Special Functions, Substitution, and Parts | |

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Integrals of Special Functions | |

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The Chain Rule and Integration | |

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Using Substitution to Eliminate Constants | |

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Integrals and Sums | |

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Approximating Integrals with Sums | |

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Approximating Integrals in General | |

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The Definite Integral | |

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Definite and Indefinite Integrals | |

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The Fundamental Theorem of Calculus | |

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The Summation Property of Definite Integral | |

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Proof of the Fundamental Theorem of Calculus | |

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Applications of Integrals | |

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Integrals and Areas | |

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Integrals and Averages | |

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Integrals and Mass | |

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Improper Integrals | |

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Infinite Limits of Integration | |

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Improper Integrals: Examples | |

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Infinite Integrands | |

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Autonomous Differential Equations | |

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Autonomous Differential Equations | |

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Review of Autonomous Differential Equations | |

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Newton's Law of Cooling | |

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Diffusion Across a Membrane | |

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A Continuous Time Model of Competition | |

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The Phase-Line Diagram | |

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Equilibria | |

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Display of Differential Equations | |

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Stable and Unstable Equilibria | |

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Recognizing Stable and Unstable Equilibria | |

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Applications of the Stability Theorem | |

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A Model of a Disease | |

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Solving Autonomous Equations | |

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Separation of Variables | |

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Pure-Time Equations Revisited | |

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Applications of Separation of Variables | |

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Two Dimensional Equations | |

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Predator-Prey Dynamics | |

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Newton's Law of Cooling | |

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Euler's Method | |

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The Phase-Plane | |

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Equilibria and Nullclines: Predator-Prey Equations | |

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Equilibria and Nullclines: Competition Equations | |

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Equilibria and Nullclines: Newton's Law of Cooling | |

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Solutions in the Phase-Plane | |

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Euler's Method in the Phase-Plane | |

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Direction Arrows: Predator-Prey Equations | |

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Direction Arrows for the Competition Equations | |

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Directions Arrows for newton's Law of Cooling | |

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The Dynamics of a Neuron | |

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A Mathematician's View of a Neuron | |

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The Mathematics of Sodium Channels | |

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The Mathematics of Sodium Channel Blocking | |

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The FitzHugh-Nagumo Equations | |

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Weak Channel Blocking Mechanism | |

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The Effects of Constant Applied Current | |

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Probability Theory and Statistics | |

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Introduction to Probabilistic Models | |

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Probability and Statistics | |

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Stochastic Population Growth: Stochastic Reproduction | |

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Stochastic Population Growth: Stochastic Immigration | |

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Markov Chains | |

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Stochastic Models of Diffusion and Genetics | |

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Stochastic Diffusion | |

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The Genetics of Inbreeding | |

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The Dynamics of Height | |

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Blending Inheritance | |

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Probability Theory | |

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Sample Spaces and Events | |

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Set Theory | |

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Assigning Probabilities to Events | |

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Exercises | |

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Conditional Probability | |

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Conditional Probability | |

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The Law of Total Probability | |

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Bayes' Theorem and the Rare Disease Example | |

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Independence and Markov Chains | |

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Independence | |

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The Multiplication Rule for Independent Events | |

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Markov Chains and Conditional Probability | |

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Displaying Probabilities | |

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Probability and Cumulative Distributions | |

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The Probability Density Function: Derivation | |

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Using the Probability Density Function | |

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The Cumulative Distribution Function | |

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Random Variables | |

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Types of Random Variable | |

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Expectation: Discrete Case | |

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Expectation: Continuous Case | |

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Descriptive Statistics | |

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The Median | |

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The Mode | |

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The Geometric Mean | |

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Descriptive Statistics for Spread | |

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Range And Percentiles | |

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Mean Absolute Deviation | |

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Variance | |

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The Coefficient of Variation | |

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Supplementary Problems For Chapter 6 | |

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Probability Models | |

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Joint Distributions | |

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Joint Distributions | |

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Marginal Probability Distributions | |

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Joint Distributions and Conditional Distributions | |

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Covariance and Correlation | |

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Covariance | |

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Correlation | |

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Perfect Correlation | |

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Sums and Products of Random Variables | |

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Expectation of a Sum | |

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Expectation of a Product | |