Mathematical Modeling in Biology: A Research Methods Approach

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Half Title
Series Page
Title Page
Copyright Page
Dedication
Table of Contents
ACKNOWLEDGMENTS
FOR PROFESSORS AND STUDENTS
AUTHORS
SECTION I: Introduction to Modeling
CHAPTER 1 Mathematical Modeling
1.1 WHAT YOU SHOULD KNOW ABOUT THIS CHAPTER
1.2 THE MODELING CYCLE
1.2.1 Step 1: Translation into Mathematics
1.2.1.1 Choosing Variables and Parameters
1.2.1.2 Simplifying Assumptions
1.2.1.3 Parameterization
1.2.2 Step 2: Model Analysis
1.2.3 Step 3: Back-Translation
1.2.3.1 Model Selection and Validation
1.2.3.2 Test of Model Predictions
1.2.4 Step 4: Revising Model Assumptions
1.3 BIOLOGY
1.3.1 Ecology
1.4 MATHEMATICS
1.5 STATISTICS
1.6 EPISTEMOLOGY: HOW WE KNOW
1.7 EXERCISES
BIBLIOGRAPHY
CHAPTER 2 Avian Bone Growth: A Case Study
2.1 WHAT YOU SHOULD KNOW ABOUT THIS CHAPTER
2.2 SCIENTIFIC PROBLEM
2.2.1 Data
2.3 TRANSLATION INTO MATHEMATICS
2.3.1 Simplifying Assumptions
2.3.1.1 Deterministic Assumptions
2.3.1.2 Stochastic Assumptions
2.3.2 The Deterministic Model
2.3.3 The Stochastic Model
2.4 MODEL PARAMETERIZATION
2.4.1 Dividing the Data Set
2.4.2 Maximum Likelihood (ML) Method
2.4.3 Nonlinear Least Squares (LS) Method
2.4.4 Downhill Minimization Routine: Nelder-Mead Algorithm
2.4.5 Implementing Parameterization in Code
2.4.6 Results of Parameterization
2.5 MODEL SELECTION
2.6 MODEL VALIDATION
2.7 EXERCISES
BIBLIOGRAPHY
SECTION II: Discrete-Time Models
CHAPTER 3 Discrete-Time Maps
3.1 WHAT YOU SHOULD KNOW ABOUT THIS CHAPTER
3.2 COMPARTMENTAL MODELS
3.3 LINEAR MAPS
3.3.1 Malthusian Growth
3.4 NONLINEAR MAPS
3.5 LINEARIZATION
3.5.1 Linearization of Functions
3.5.2 Linearization of Discrete-Time Maps
3.5.3 Linearizing the Ricker Map
3.6 THE RICKER NONLINEARITY
3.7 EXERCISES
BIBLIOGRAPHY
CHAPTER 4 Chaos: Simple Rules Can Generate Complex Results
4.1 WHAT YOU SHOULD KNOW ABOUT THIS CHAPTER
4.2 RICKER MODEL REVISITED
4.3 NEW PARADIGMS ARISE FROM CHAOS
4.3.1 Deterministic Unpredictability
4.3.2 Complex Results Can Arise from Simple Rules
4.4 MAY’S HYPOTHESIS
4.5 EXERCISES
BIBLIOGRAPHY
CHAPTER 5 Higher-Dimensional Discrete-Time Models
5.1 WHAT YOU SHOULD KNOW ABOUT THIS CHAPTER
5.2 INTRASPECIFIC INTERACTIONS
5.3 INTERSPECIFIC INTERACTIONS
5.4 EXAMPLE OF AN AGE-STRUCTURED SINGLE-SPECIES MODEL
5.5 EXAMPLE OF A TWO-SPECIES MODEL
5.6 n-DIMENSIONAL LINEAR DIFFERENCE EQUATIONS
5.6.1 n-Dimensional Leslie Models
5.7 SOLVING LINEAR SYSTEMS OF DIFFERENCE EQUATIONS
5.7.1 An Example
5.7.2 Solving the General Two-Dimensional System
5.7.3 Solving Higher-Dimensional Systems
5.8 NONLINEAR SYSTEMS
5.8.1 Linearization
5.8.2 An Example
5.9 EXERCISES
BIBLIOGRAPHY
CHAPTER 6 Flour Beetle Dynamics: A Case Study
6.1 WHAT YOU SHOULD KNOW ABOUT THIS CHAPTER
6.2 FLOUR BEETLES
6.3 DATA
6.4 ASSUMPTIONS
6.4.1 Deterministic Assumptions
6.4.2 Stochastic Assumptions
6.5 ALTERNATIVE DETERMINISTIC MODELS
6.6 STOCHASTIC MODELS
6.7 MODEL PARAMETERIZATION
6.7.1 Conditioned One-Step Residuals
6.7.2 Conditioned Least Squares (CLS)
6.8 MODEL SELECTION
6.9 MODEL VALIDATION
6.10 THE “HUNT FOR CHAOS” EXPERIMENTS
6.10.1 Results of the “Hunt for Chaos” Experiments
6.10.2 Manipulating the Parameters in the Laboratory
6.11 EXERCISES
BIBLIOGRAPHY
SECTION III: Continuous-Time Models
CHAPTER 7 Introduction to Differential Equations
7.1 WHAT YOU SHOULD KNOW ABOUT THIS CHAPTER
7.2 COMPARTMENTAL MODELS
7.2.1 A Tank Problem
7.2.2 The SIR Model
7.2.3 The Continuous-Time Logistic Model
7.3 EXERCISES
BIBLIOGRAPHY
CHAPTER 8 Scalar Differential Equations
8.1 WHAT YOU SHOULD KNOW ABOUT THIS CHAPTER
8.2 LINEAR EQUATIONS
8.2.1 Malthusian Growth
8.3 NONLINEAR EQUATIONS
8.3.1 Logistic Growth
8.3.2 Allee Effects
8.3.3 The “Doomsday Model” of Human Population Growth
8.4 LINEARIZATION
8.5 BIFURCATIONS
8.5.1 Transcritical Bifurcation
8.5.2 Saddle-Node Bifurcation
8.5.3 Pitchfork Bifurcation
8.5.4 Hysteresis
8.6 EXERCISES
BIBLIOGRAPHY
CHAPTER 9 Systems of Differential Equations
9.1 WHAT YOU SHOULD KNOW ABOUT THIS CHAPTER
9.2 LINEARSYSTEMS OF ODES AND PHASE PLANE ANALYSIS
9.2.1 Unstable Node
9.2.2 Asymptotically Stable Node
9.2.3 Saddle
9.2.4 Center
9.2.5 Unstable Spiral
9.2.6 Asymptotically Stable Spiral
9.2.7 Summary: Eigenvalues Tell All
9.3 NONLINEAR SYSTEMS OF ODES
9.3.1 Linearization
9.4 LIMIT CYCLES, CYCLE CHAINS, AND BIFURCATIONS
9.5 LOTKA-VOLTERRA MODELS AND NULLCLINE ANALYSIS
9.5.1 Lotka-Volterra Competition
9.5.2 Lotka-Volterra Cooperation
9.5.3 Lotka-Volterra Predator-Prey
9.6 EXERCISES
BIBLIOGRAPHY
CHAPTER 10 Seabird Behavior: A Case Study
10.1 WHAT YOU SHOULD KNOW ABOUT THIS CHAPTER
10.2 THE SCIENTIFIC PROBLEM
10.3 HISTORICAL DATA
10.3.1 Count Data
10.3.2 Dividing the Data
10.3.3 Tide and Solar Elevation Data
10.4 GENERAL MODEL
10.5 ALTERNATIVE MODELS
10.6 MODEL PARAMETERIZATION
10.7 MODEL SELECTION
10.8 MODEL VALIDATION
10.9 TEST OF A PRIORI PREDICTIONS
10.10 STEADY-STATE MODEL
10.11 DISCUSSION
10.11.1 Importance of Scale
10.11.2 Resource Management
10.12 EXERCISES
BIBLIOGRAPHY
SECTION IV: Regression Models
CHAPTER 11 Introduction to Regression
11.1 WHAT YOU SHOULD KNOW ABOUT THIS CHAPTER
11.2 LINEAR REGRESSION
11.2.1 Simple Linear Regression (Single Factor)
11.2.2 Multiple Linear Regression (Multiple Factors)
11.2.3 Stochastic Model and Parameter Estimation
11.2.4 Confidence Intervals for Regression Coefficients
11.3 LOGISTIC REGRESSION
11.3.1 Odds Ratios (ORs)
11.3.2 OR Confidence Intervals
11.4 GENERALIZED LINEAR MODELS (GLMs)
11.5 INTERACTION TERMS
11.6 EXERCISES
BIBLIOGRAPHY
CHAPTER 12 Climate Change and Seabird Cannibalism: A Case Study
12.1 WHAT YOU SHOULD KNOW ABOUT THIS CHAPTER
12.2 THE SCIENTIFIC PROBLEM
12.3 DATA
12.4 LOGISTIC REGRESSION ANALYSIS
12.5 MODEL VALIDATION
12.6 OUTCOMES
12.7 CLIMATE CHANGE, CANNIBALISM, AND REPRODUCTIVE SYNCHRONY
12.8 EXERCISES
BIBLIOGRAPHY
SECTION V: Appendix
Appendix A Linear Algebra Basics
A.1 MATRIX OPERATIONS
A.1.1 Matrix Addition
A.1.2 Scalar Multiplication
A.1.3 Matrix Subtraction
A.1.4 Matrix Multiplication
A.1.5 Determinants of Square Matrices
A.2 EXERCISES
A.3 SOLUTIONS
A.4 SUMMARY OF LINEAR ALGEBRA CONCEPTS
Appendix B MATLAB: The Basics
B.1 PRELIMINARIES
B.2 SYNTAX AND PROGRAMMING
B.2.1 Command Line
B.2.2 Case-Sensitivity
B.2.3 Displaying the Current Value of a Variable
B.2.4 Clearing All Variables
B.2.5 Closing MATLAB
B.2.6 Variables and Arithmetic Operators
B.2.7 Programs
B.2.8 Comment Lines
B.2.9 Printing to the Screen
B.2.10 Numerical Format
B.2.11 Loops
B.2.12 Crashing a Program on Purpose
B.2.13 Logical Statements (If-Then-Else)
B.2.14 Input and Output Files
B.2.15 Creating Functions
B.2.16 Subroutines
B.2.17 Vectors, Matrices, and Arrays
B.2.18 Functions in the MATLAB Library
B.2.19 Plotting
B.2.20 Simulating Discrete-Time Models
B.2.21 Simulating Ordinary Differential Equations (ODEs)
B.2.22 The Downhill Nelder-Mead Algorithm
B.3 EXERCISES
Appendix C Connecting Models to Data: A Brief Summary with Sample Codes
C.1 PARAMETERIZATION
C.2 RESIDUAL ERRORS (RESIDUALS)
C.3 RSS AND R[sup(2)]
C.4 MAXIMUM LIKELIHOOD (ML) PARAMETERS
C.5 LEAST SQUARES (LS) PARAMETERS
C.6 IMPLEMENTATION IN CODE
C.6.1 Basic Structure of Program
C.6.2 Constructing Input Files
C.6.3 Example: Algebraic Model Using Vectors
C.6.4 Example: Algebraic Model Using Loop
C.6.5 Example: Scalar Map with One-Step Predictions
C.6.6 Example: Higher-Dimensional Discrete-Time Model with One-Step Predictions
Index