ALL CHAPTERS COVERED
, Contents
Preface ii
Chapter 1. Dynamic Modeling with Difference Equations 5
1.1. The Malthusian Model 5
1.2. Nonlinear Models 8
1.3. Analyzing Nonlinear Models 10
1.4. Variations on the Logistic Model 12
1.5. Comments on Discrete and Continuous Models 13
Chapter 2. Linear Models of Structured Populations 14
2.1. Linear Models and Matrix Algebra 14
2.2. Projection Matrices for Structured Models 16
2.3. Eigenvectors and Eigenvalues 17
2.4. Computing Eigenvectors and Eigenvalues 19
Chapter 3. Nonlinear Models of Interactions 21
3.1. A Simple Predator–Prey Model 21
3.2. Equilibria of Multipopulation Models 22
3.3. Linearization and Stability 24
3.4. Positive and Negative Interactions 25
Chapter 4. Modeling Molecular Evolution 27
4.2. An Introduction to Probability 27
4.3. Conditional Probabilities 28
4.4. Matrix Models of Base Substitution 30
4.5. Phylogenetic Distances 34
Chapter 5. Constructing Phylogenetic Trees 40
5.1. Phylogenetic Trees 40
5.2. Tree Construction: Distance Methods – Basics 42
5.3. Tree Construction: Distance Methods – Neighbor Joining 46
5.4. Tree Construction: Maximum Parsimony 49
5.6. Applications and Further Reading 51
Chapter 6. Genetics 55
6.1. Mendelian Genetics 55
6.2. Probability Distributions in Genetics 57
6.3. Linkage 62
6.4. Gene Frequency in Populations 66
iii
, CONTENTS iv
Chapter 7. Infectious Disease Modeling 70
7.1. Elementary Epidemic Models 70
7.2. Threshold Values and Critical Parameters 71
7.3. Variations on a Theme 73
7.4. Multiple Populations and Differentiated Infectivity 75
Chapter 8. Curve Fitting and Biological Modeling 77
8.1. Fitting Curves to Data 77
8.2. The Method of Least Squares 78
8.3. Polynomial Curve Fitting 79
, CHAPTER 1
Dynamic Modeling with Difference Equations
1.1. The Malthusian Model
1.1.1. a.
t 0 1 2 3 4 5
Pt 100 300 900 2700 8100 24300
b. Pt+1 = 3Pt , ∆P = 2Pt
c. f − d = 2
1.1.2. a. Pt+1 = 2Pt , ∆t = .5 hr
b. In the following table, t is measured in half-hours.
t 0 2 4 6 8 10
Pt 1 4 16 64 256 1024
t 12 14 16 18 20 22
Pt 4096 16384 65536 262144 1048576 4194304
c. According to the model, the number of cells after ten hours is over one mil-
lion. Since the observed number is around 30,000, this suggests that the model
only fits well at the early stages of cell division, and that during the first ten
hours (or twenty time steps) the rate of cell division has slowed. Understanding
how and why this slow down occurs could be biologically interesting.
1.1.3. a.
t 0 1 2 3 4 5 6
Pt 1 1.3 1.69 2.197 2.8561 3.7129 4.8268
t 0 1 2 3 4 5 6
Nt 10 8 6.4 5.12 4.096 3.2768 2.6214
t 0 1 2 3 4 5 6
Zt 10 12 14.4 17.28 20.736 24.8832 29.8598
1.1.4. The first sequence of MATLAB commands has the user iteratively multiply Pt
by 1.3. The values are stored as a row vector x = [P0 P1 · · · Pt ]. The second
sequence of commands works similarly, but uses a ‘for’-loop to do the iteration
automatically.
1.1.5. Experimentally, 9, 18, and 27 time steps are required.
Since Pt = 1.3t , then Pt ≈ 10 when ln 10 ≈ t ln 1.3. Thus t ≈ ln 10/ ln 1.3 ≈ 8.8.
Similarly, Pt ≈ 100 when t = 17.6; Pt ≈ 1000 when t = 26.3. Since t must be
an integer, the first times when Pt exceeds 10, 100, and 1000 are 9, 18, and 27,
respectively.
Notice these times are equally spaced. A characteristic of exponential growth is
that the time required for an increase by a factor m is always the same. Here,
the time required for an increase by a factor of 10 is always 9 time steps.
1.1.6. By calculating the ratios Pt+1 /Pt , it is clear that a geometric model does not
fit the data well. The finite growth is fast at first, then slows down. It is not
5
, Contents
Preface ii
Chapter 1. Dynamic Modeling with Difference Equations 5
1.1. The Malthusian Model 5
1.2. Nonlinear Models 8
1.3. Analyzing Nonlinear Models 10
1.4. Variations on the Logistic Model 12
1.5. Comments on Discrete and Continuous Models 13
Chapter 2. Linear Models of Structured Populations 14
2.1. Linear Models and Matrix Algebra 14
2.2. Projection Matrices for Structured Models 16
2.3. Eigenvectors and Eigenvalues 17
2.4. Computing Eigenvectors and Eigenvalues 19
Chapter 3. Nonlinear Models of Interactions 21
3.1. A Simple Predator–Prey Model 21
3.2. Equilibria of Multipopulation Models 22
3.3. Linearization and Stability 24
3.4. Positive and Negative Interactions 25
Chapter 4. Modeling Molecular Evolution 27
4.2. An Introduction to Probability 27
4.3. Conditional Probabilities 28
4.4. Matrix Models of Base Substitution 30
4.5. Phylogenetic Distances 34
Chapter 5. Constructing Phylogenetic Trees 40
5.1. Phylogenetic Trees 40
5.2. Tree Construction: Distance Methods – Basics 42
5.3. Tree Construction: Distance Methods – Neighbor Joining 46
5.4. Tree Construction: Maximum Parsimony 49
5.6. Applications and Further Reading 51
Chapter 6. Genetics 55
6.1. Mendelian Genetics 55
6.2. Probability Distributions in Genetics 57
6.3. Linkage 62
6.4. Gene Frequency in Populations 66
iii
, CONTENTS iv
Chapter 7. Infectious Disease Modeling 70
7.1. Elementary Epidemic Models 70
7.2. Threshold Values and Critical Parameters 71
7.3. Variations on a Theme 73
7.4. Multiple Populations and Differentiated Infectivity 75
Chapter 8. Curve Fitting and Biological Modeling 77
8.1. Fitting Curves to Data 77
8.2. The Method of Least Squares 78
8.3. Polynomial Curve Fitting 79
, CHAPTER 1
Dynamic Modeling with Difference Equations
1.1. The Malthusian Model
1.1.1. a.
t 0 1 2 3 4 5
Pt 100 300 900 2700 8100 24300
b. Pt+1 = 3Pt , ∆P = 2Pt
c. f − d = 2
1.1.2. a. Pt+1 = 2Pt , ∆t = .5 hr
b. In the following table, t is measured in half-hours.
t 0 2 4 6 8 10
Pt 1 4 16 64 256 1024
t 12 14 16 18 20 22
Pt 4096 16384 65536 262144 1048576 4194304
c. According to the model, the number of cells after ten hours is over one mil-
lion. Since the observed number is around 30,000, this suggests that the model
only fits well at the early stages of cell division, and that during the first ten
hours (or twenty time steps) the rate of cell division has slowed. Understanding
how and why this slow down occurs could be biologically interesting.
1.1.3. a.
t 0 1 2 3 4 5 6
Pt 1 1.3 1.69 2.197 2.8561 3.7129 4.8268
t 0 1 2 3 4 5 6
Nt 10 8 6.4 5.12 4.096 3.2768 2.6214
t 0 1 2 3 4 5 6
Zt 10 12 14.4 17.28 20.736 24.8832 29.8598
1.1.4. The first sequence of MATLAB commands has the user iteratively multiply Pt
by 1.3. The values are stored as a row vector x = [P0 P1 · · · Pt ]. The second
sequence of commands works similarly, but uses a ‘for’-loop to do the iteration
automatically.
1.1.5. Experimentally, 9, 18, and 27 time steps are required.
Since Pt = 1.3t , then Pt ≈ 10 when ln 10 ≈ t ln 1.3. Thus t ≈ ln 10/ ln 1.3 ≈ 8.8.
Similarly, Pt ≈ 100 when t = 17.6; Pt ≈ 1000 when t = 26.3. Since t must be
an integer, the first times when Pt exceeds 10, 100, and 1000 are 9, 18, and 27,
respectively.
Notice these times are equally spaced. A characteristic of exponential growth is
that the time required for an increase by a factor m is always the same. Here,
the time required for an increase by a factor of 10 is always 9 time steps.
1.1.6. By calculating the ratios Pt+1 /Pt , it is clear that a geometric model does not
fit the data well. The finite growth is fast at first, then slows down. It is not
5
