Log In
Or create an account ->
Imperial Library
Home
About
News
Upload
Forum
Help
Login/SignUp
Index
Half title
Title
Copyright
Contents
Preface
Chapter 1: Mathematical Modeling in Biology
1.1 Introduction
1.2 HIV
1.3 Models of HIV/AIDS
1.4 Concluding Message
Chapter 2: How to Construct a Model
2.1 Introduction
2.2 Formulate the Question
2.3 Determine the Basic Ingredients
2.4 Qualitatively Describe the Biological System
2.5 Quantitatively Describe the Biological System
2.6 Analyze the Equations
2.7 Checks and Balances
2.8 Relate the Results Back to the Question
2.9 Concluding Message
Chapter 3: Deriving Classic Models in Ecology and Evolutionary Biology
3.1 Introduction
3.2 Exponential and Logistic Models of Population Growth
3.3 Haploid and Diploid Models of Natural Selection
3.4 Models of Interactions among Species
3.5 Epidemiological Models of Disease Spread
3.6 Working Backward—Interpreting Equations in Terms of the Biology
3.7 Concluding Message
Primer 1: Functions and Approximations
P1.1 Functions and Their Forms
P1.2 Linear Approximations
P1.3 The Taylor Series
Chapter 4: Numerical and Graphical Techniques—Developing a Feeling for Your Model
4.1 Introduction
4.2 Plots of Variables Over Time
4.3 Plots of Variables as a Function of the Variables Themselves
4.4 Multiple Variables and Phase-Plane Diagrams
4.5 Concluding Message
Chapter 5: Equilibria and Stability Analyses—One-Variable Models
5.1 Introduction
5.2 Finding an Equilibrium
5.3 Determining Stability
5.4 Approximations
5.5 Concluding Message
Chapter 6: General Solutions and Transformations—One-Variable Models
6.1 Introduction
6.2 Transformations
6.3 Linear Models in Discrete Time
6.4 Nonlinear Models in Discrete Time
6.5 Linear Models in Continuous Time
6.6 Nonlinear Models in Continuous Time
6.7 Concluding Message
Primer 2: Linear Algebra
P2.1 An Introduction to Vectors and Matrices
P2.2 Vector and Matrix Addition
P2.3 Multiplication by a Scalar
P2.4 Multiplication of Vectors and Matrices
P2.5 The Trace and Determinant of a Square Matrix
P2.6 The Inverse
P2.7 Solving Systems of Equations
P2.8 The Eigenvalues of a Matrix
P2.9 The Eigenvectors of a Matrix
Chapter 7: Equilibria and Stability Analyses—Linear Models with Multiple Variables
7.1 Introduction
7.2 Models with More than One Dynamic Variable
7.3 Linear Multivariable Models
7.4 Equilibria and Stability for Linear Discrete-Time Models
7.5 Concluding Message
Chapter 8: Equilibria and Stability Analyses—Nonlinear Models with Multiple Variables
8.1 Introduction
8.2 Nonlinear Multiple-Variable Models
8.3 Equilibria and Stability for Nonlinear Discrete-Time Models
8.4 Perturbation Techniques for Approximating Eigenvalues
8.5 Concluding Message
Chapter 9: General Solutions and Tranformations—Models with Multiple Variables
9.1 Introduction
9.2 Linear Models Involving Multiple Variables
9.3 Nonlinear Models Involving Multiple Variables
9.4 Concluding Message
Chapter 10: Dynamics of Class-Structured Populations
10.1 Introduction
10.2 Constructing Class-Structured Models
10.3 Analyzing Class-Structured Models
10.4 Reproductive Value and Left Eigenvectors
10.5 The Effect of Parameters on the Long-Term Growth Rate
10.6 Age-Structured Models—The Leslie Matrix
10.7 Concluding Message
Chapter 11: Techniques for Analyzing Models with Periodic Behavior
11.1 Introduction
11.2 What Are Periodic Dynamics?
11.3 Composite Mappings
11.4 Hopf Bifurcations
11.5 Constants of Motion
11.6 Concluding Message
Chapter 12: Evolutionary Invasion Analysis
12.1 Introduction
12.2 Two Introductory Examples
12.3 The General Technique of Evolutionary Invasion Analysis
12.4 Determining How the ESS Changes as a Function of Parameters
12.5 Evolutionary Invasion Analyses in Class-Structured Populations
12.6 Concluding Message
Primer 3: Probability Theory
P3.1 An Introduction to Probability
P3.2 Conditional Probabilities and Bayes’ Theorem
P3.3 Discrete Probability Distributions
P3.4 Continuous Probability Distributions
P3.5 The (Insert Your Name Here) Distribution
Chapter 13: Probabilistic Models
13.1 Introduction
13.2 Models of Population Growth
13.3 Birth-Death Models
13.4 Wright-Fisher Model of Allele Frequency Change
13.5 Moran Model of Allele Frequency Change
13.6 Cancer Development
13.7 Cellular Automata—A Model of Extinction and Recolonization
13.8 Looking Backward in Time—Coalescent Theory
13.9 Concluding Message
Chapter 14: Analyzing Discrete Stochastic Models
14.1 Introduction
14.2 Two-State Markov Models
14.3 Multistate Markov Models
14.4 Birth-Death Models
14.5 Branching Processes
14.6 Concluding Message
Chapter 15: Analyzing Continuous Stochastic Models—Diffusion in Time and Space
15.1 Introduction
15.2 Constructing Diffusion Models
15.3 Analyzing the Diffusion Equation with Drift
15.4 Modeling Populations in Space Using the Diffusion Equation
15.5 Concluding Message
Epilogue: The Art of Mathematical Modeling in Biology
Appendix 1: Commonly Used Mathematical Rules
A1.1 Rules for Algebraic Functions
A1.2 Rules for Logarithmic and Exponential Functions
A1.3 Some Important Sums
A1.4 Some Important Products
A1.5 Inequalities
Appendix 2: Some Important Rules from Calculus
A2.1 Concepts
A2.2 Derivatives
A2.3 Integrals
A2.4 Limits
Appendix 3: The Perron-Frobenius Theorem
A3.1: Definitions
A3.2: The Perron-Frobenius Theorem
Appendix 4: Finding Maxima and Minima of Functions
A4.1 Functions with One Variable
A4.2 Functions with Multiple Variables
Appendix 5: Moment-Generating Functions
Index of Definitions, Recipes, and Rules
General Index
← Prev
Back
Next →
← Prev
Back
Next →