Designed to help life sciences students understand the role mathematics has played in breakthroughs in epidemiology, genetics, statistics, physiology, and other biological areas, this text 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, but with particular emphasis on those areas connected with modeling: discrete-time dynamical systems, differential equations, and probability and statistics.
After graduating from Harvard University with a B.A. in Mathematics, Fred Adler received his Ph.D. in Applied Mathematics at Cornell University, where he began his study of mathematical biology. Currently a professor in the departments of mathematics and biology at the University of Utah, he teaches courses in mathematical modeling with a wide range of backgrounds. Prof. Adler's research focuses on mathematical ecology, with emphases in mathematical epidemiology, evolutionary ecology, and community ecology.
Part I: DISCRETE-TIME DYNAMICAL SYSTEMS. 1. Biology and Dynamics. Growth: Models of Malaria. Maintenance: Models of Neurons. Replication: Models of Genetics. Types of Dynamical Systems. 2. Variables and Functions. Describing Measurements with Variables, Parameters and Graphs. Describing Relations Between Measurements with Functions. Combining Functions. 3. Units and Dimensions. Converting Between Units. Translating Between Dimensions. Checking: Dimensions and Estimation. 4. Linear Functions and their Graphs. Proportional Relations. Linear Functions and the Equation of a Line. Finding Equations and Graphing Lines. Solving Equations Involving Lines. Finding Inverse Functions. 5. Discrete-Time Dynamical Systems. Discrete-time Dynamical Systems and Updating Functions. Manipulating Updating Functions. Discrete-Time Dynamical Systems: Units and Dimensions. Solutions. 6. Analysis of Discrete-Tie Dynamical Systems. Cobwebbing: A Graphical Solution Technique. Equilibria: Graphical Approach. Equilibria: Algebraic Approach. 7. Solutions and Exponential Functions. Bacterial Population Growth in General. Laws of Exponents and Logs. Expressing Results with Exponentials. 8. Oscillations and Trigonometry. Sine and Cosine: A Review. Describing Oscillations with the Cosine. More Complicated Shapes. 9. A Model of Gas Exchange in the Lung. A Model of the Lungs. The Lung System in General. Lung Dynamics with Absorption. 10. An Example of Nonlinear Dynamics. A Model of Selection. The Discrete-Time Dynamical System and Equilibria. Stable and Unstable Equilibria. 11. Excitable Systems I: The Heart. A Simple Heart. Second Degree Block. The Wenckeback Phenomenon. Part II: LIMITS AND DERIVATIVES. 1. Introduction to Derivatives. The Average Rate of Change. Instantaneous Rates of Change. Limits and Derivatives. Differential Equations: A Preview. 2. Limits. Limits of Functions. Left and Right-Hand Limits. Properties of Limits. Infinite Limits. 3. Continuity. Continuous Functions. Input and Output Tolerances. Hysteresis. 4. Computing Derivatives. Differentiable Functions. The Derivative of a Linear Function. A Quadratic Function. 5. Derivatives of Sums, Powers, and Polynomials. The Sum Rule. Derivatives of Power Functions. Derivatives of Polynomials. 6. Derivatives of Products and Quotients. The Product Rule. Special Cases and Examples. The Quotient Rule. 7. The Second Derivative. The Second Derivative. Using the Second Derivative for Graphing. Acceleration. 8. Exponentials snd Logarithms. The Exponential Function. The Natural Logarithm. Applications. 9. The Chain Rule. The Derivative of a Composite Function. Derivatives of Inverse Functions. Applications. 10. Derivatives of Trigonometric Functions. Deriving the Derivatives of Sine and Cosine. Other Trigonometric Functions. Applications. Part III: DERIVATIVES AND DYNAMICAL SYSTEMS. 1. Stability and the Derivative. Motivation. Stability and the Slope. Evaluating Stability with the Derivative. 2. More Complex Dynamics. The Logistic Dynamical System. Qualitative Dynamical Systems. Analysis of the Logistic Dynamical System. 3. Maximization. Minima and Maxima. Maximizing Food Intake Rate. Maximizing Fish Harvest. 4. Reasoning About Functions. The Intermediate Value Theorem. Maximization: The Extreme Value Theorem. Rolle's Theorem and the Mean Value Theorem. 5. Limits at Infinity. The Behavior of Functions at Infinity. Application to Absorption Functions. Limits of Sequences. 6. Leading Behavior and L'hopital's Rule. Leading Behavior of Functions at Infinity. Leading Behavior of Functions at 0. The Method of Matched Leading Behaviors. L'Hopital's Rule. 7. Approximating Functions. The Tangent and Secant Lines. Quadratic Approximation. Taylor Polynomials. 8. Newton's Method. Finding the Equilibrium. Newton's Method. Why Newton's Method Works and When it Fails. 9. Panting and Deep Breathing. Breathing at Different Rates. Deep Breathing. Panting. Intermediate Optimum. Part IV: DIFFERENTIAL EQUATIONS, INTEGRALS, AND THEIR APPLICATIONS. 1. Differential Equations. Differential Equations: Examples and Terminology. Graphical Solution of Pure-Time Differential Equations. Euler's Method: Pure-Time. 2. Antiderivatives and Indefinite Integrals. Pure-Time Differential Equations. Rules for Antiderivatives. Solving Polynomial Differential Equations. 3. Special Functions, Substitution, and Parts. Integrals of Special Functions. The Chain Rule and Integration. Using Substitution to Eliminate Constants. 4. Integrals and Sums. Approximating Integrals with Sums. Approximating Integrals in General. The Definite Integral. 5. Definite and Indefinite Integrals. The Fundamental Theorem of Calculus. The Summation Property of Definite Integral. Proof of the Fundamental Theorem of Calculus. 6. Applications of Integrals. Integrals and Areas. Integrals and Averages. Integrals and Mass. 7. Improper Integrals. Infinite Limits of Integration. Improper Integrals: Examples. Infinite Integrands. Part V: AUTONOMOUS DIFFERENTIAL EQUATIONS. 1. Autonomous Differential Equations. Review of Autonomous Differential Equations. Newton's Law of Cooling. Diffusion Across a Membrane. A Continuous Time Model of Competition. 2. The Phase-Line Diagram. Equilibria. Display of Differential Equations. 3. Stable and Unstable Equilibria. Recognizing Stable and Unstable Equilibria. Applications of the Stability Theorem. A Model of a Disease. 4. Solving Autonomous Equations. Separation of Variables. Pure-Time Equations Revisited. Applications of Separation of Variables. 5. Two Dimensional Equations. Predator-Prey Dynamics. Newton's Law of Cooling. Euler's Method. 6. The Phase-Plane. Equilibria and Nullclines: Predator-Prey Equations. Equilibria and Nullclines: Competition Equations. Equilibria and Nullclines: Newton's Law of Cooling. 7. Solutions in the Phase-Plane. Euler's Method in the Phase-Plane. Direction Arrows: Predator-Prey Equations. Direction Arrows for the Competition Equations. Directions Arrows for newton's Law of Cooling. 8. The Dynamics of a Neuron. A Mathematician's View of a Neuron. The Mathematics of Sodium Channels. The Mathematics of Sodium Channel Blocking. The FitzHugh-Nagumo Equations. Weak Channel Blocking Mechanism. The Effects of Constant Applied Current. Part VI: PROBABILITY THEORY AND STATISTICS. 1. Introduction to Probabilistic Models. Probability and Statistics. Stochastic Population Growth: Stochastic Reproduction. Stochastic Population Growth: Stochastic Immigration. Markov Chains. 2. Stochastic Models of Diffusion and Genetics. Stochastic Diffusion. The Genetics of Inbreeding. The Dynamics of Height. Blending Inheritance. 3. Probability Theory. Sample Spaces and Events. Set Theory. Assigning Probabilities to Events. Exercises. 4. Conditional Probability. Conditional Probability. The Law of Total Probability. Bayes' Theorem and the Rare Disease Example. 5. Independence and Markov Chains. Independence. The Multiplication Rule for Independent Events. Markov Chains and Conditional Probability. 6. Displaying Probabilities. Probability and Cumulative Distributions. The Probability Density Function: Derivation. Using the Probability Density Function. The Cumulative Distribution Function. 7. Random Variables. Types of Random Variable. Expectation: Discrete Case. Expectation: Continuous Case. 8. Descriptive Statistics. The Median. The Mode. The Geometric Mean. 9. Descriptive Statistics for Spread. Range And Percentiles. Mean Absolute Deviation. Variance. The Coefficient of Variation. 10. Supplementary Problems For Chapter 6. Part VII: PROBABILITY MODELS. 1. Joint Distributions. Joint Distributions. Marginal Probability Distributions. Joint Distributions and Conditional Distributions. 2. Covariance and Correlation. Covariance. Correlation. Perfect Correlation. 3. Sums and Products of Random Variables. Expectation of a Sum. Expectation of a Product. Variance of a Sum. 4. The Binomial Distribution. Definition of the Binomial Distribution. The Mean and Variance of the Binomial Distribution. Computing the Binomial. Binomial Distribution: The General Case. Finding the Mode of the Binomial Distribution. 5. Applications Of The Binomial Distribution. Application to Genetics. Application to Markov Chains. Applications to Diffusion. 6. Exponential Distributions. The Geometric Distribution. The Exponential Distribution. 7. The Poisson Distribution. The Poisson Process. The Poisson Distribution in Space. The Poisson and the Binomial. 8. The Normal Distribution. The Normal Distribution: An Example. The Central Limit Theorem for Sums. The Central Limit Theorem for Averages. 9. Applying the Normal Approximation. The Standard Normal Distribution. Normal Approximation of the Binomial. Normal Approximation of the Poisson. Part VIII: INTRODUCTION TO STATISTICAL REASONING. 1. Statistics: Estimating Parameters. Estimating the Binomial Proportion. Unbiased Estimators. Maximum Likelihood. Estimating a Rate. 2. Confidence Limits. Exact Confidence Limits. Monte Carlo Method. Likelihood, Support, and Confidence Limits. 3. Estimating the Mean. Estimating the Mean. Confidence Limits. Sample Variance and Standard Error. The t Distribution. The Normal and the Binomial. 4. Hypothesis Testing. Hypothesis Testing: Terminology. Design of Statistical Tests and Computation of P-values. Computing Power. 5. Hypothesis Testing: Normal Theory. Computing P-Values with the Normal Approximation. Testing with the t Distribution. The Power of Normal Tests. 6. Comparing Experiments. Unpaired Normal Distributions. Comparing Population Proportions. Applying the t Distribution with Smaller Samples. 7. Analysis of Contingency Tables and Goodness of Fit. Testing Against a Known Baseline. Testing with Contingency Tables. Testing for Goodness of Fit. 8. Hypothesis Testing with the Method of Support. The Method of Support: Comparison of Data with a Known Baseline. Comparing Two Populations. Support and the Normal Distribution. The G Test: support and Contingency Tables. 9. Regression. Comparing a Linear Model with Data. Finding the Best Line. Using Linear Regression.