Designed to help life sciences students understand the role mathematics has played in breakthroughs in epidemiology, genetics, statistics, physiology, and other biological areas, MODELING THE DYNAMCICS OF LIFE: CALCULUS AND PROBABILTY FOR LIFE SCIENTISTS, Third Edition, provides students with a thorough grounding in mathematics, the language, and 'the technology of thought' with which these developments are created and controlled. The text teaches the skills of describing a system, translating appropriate aspects into equations, and interpreting the results in terms of the original problem. The text helps unify biology by identifying dynamical principles that underlie a great diversity of biological processes. Standard topics from calculus courses are covered, with particular emphasis on those areas connected with modeling such as discrete-time dynamical systems, differential equations, and probability and statistics.
Table of Contents
1. Introduction to Discrete-Time Dynamical Systems.
1.1 Biology and Dynamics.
1.2 Variables. Parameters. and Functions in Biology.
1.3 The Units and Dimensions of Measurements and Functions.
1.4 Linear Functions and Their Graphs.
1.5 Discrete-Time Dynamical Systems.
1.6 Analysis of Discrete-Time Dynamical Systems.
1.7 Expressing Solutions with Exponential Functions.
1.8 Oscillations and Trigonometry.
1.9 A Model of Gas Exchange in the Lung.
1.10 An Example of Nonlinear Dynamics.
1.11 An Excitable Systems I: The Heart.
2. Limits and Derivatives.
2.1 Introduction to Derivatives.
2.4 Computing Derivatives: Linear and Quadratic Functions.
2.5 Derivatives of Sums. Powers. and Polynomials.
2.6 Derivatives of Products and Quotients.
2.7 The Second Derivative. Curvature. and Acceleration.
2.8 Derivatives of Exponential and Logarithmic Functions.
2.9 The Chain Rule.
2.10 Derivatives of Trigonometric Functions.
3. Applications of Derivatives and Dynamical Systems.
3.1 Stability and the Derivative.
3.2 More Complicated Dynamics.
3.4 Reasoning About Functions.
3.5 Limits at Infinity.
3.6 Leading Behavior and L'Hopital's Rule.
3.7 Approximating Functions with Lines and Polynomials.
3.8 Newton's Method.
3.9 Panting and Deep Breathing.
4. Differential Equations. Integrals. and Their Applications.
4.1 Differential Equations.
4.2 Solving Pure-Time Differential Equations.
4.3 Integration of Special Functions. Integration by Substitution. by Parts. and by Partial Fractions.
4.4 Integrals and Sums.
4.5 Definite and Indefinite Integrals.
4.6 Applications of Integrals.
4.7 Improper Integrals.
5. Analysis of Autonomous Differential Equations.
5.1 Basic Differential Equations.
5.2 Equilibria and Display of Autonomous Differential Equations.
5.3 Stable and Unstable Equilibria.
5.4 Solving Autonomous Differential Equations.
5.5 Two Dimensional Differential Equations.
5.6 The Phase Plane.
5.7 Solutions in the Phase Plane.
5.8 The Dynamics of a Neuron.
6. Probability Theory and Descriptive Statistics.
6.1 Introduction to Probabilistic Models.
6.2 Stochastic Models of Diffusion and Genetics.
6.3 Probability Theory.
6.4 Conditional Probability.
6.5 Independence and Markov Chains.
6.6 Displaying Probabilities.
6.7 Random Variables.
6.8 Descriptive Statistics.
6.9 Descriptive Statistics for Spread.
7. Probability Models.
7.1 Joint Distributions.
7.2 Covariance and Correlation.
7.3 Sums and Products of Random Variables.
7.4 The Binomial Distribution.
7.5 Applications of the Binomial Distribution.
7.6 Waiting Times: Geometric and Exponential Distributions.
7.7 The Poisson Distribution.
7.8 The Normal Distribution.
7.9 Applying the Normal Approximation.
8. Introduction to Statistical Reasoning.
8.1 Statistics: Estimating Parameters.
8.2 Confidence Limits.
8.3 Estimating the Mean.
8.4 Hypothesis Testing.
8.5 Hypothesis Testing: Normal Theory.
8.6 Comparing Experiments: Normal Theory.
8.7 Analysis of Contingency Tables and Goodness of Fit.
8.8 Hypothesis Testing with the Method of Support.