Mathematical Models in Biology: An Introduction

by Elizabeth S. Allman , John A. Rhodes

• ISBN13: 9780521525862
• ISBN10: 0521525861
• Format: Paperback
• Copyright: 2003-10-13
• Publisher: Cambridge University Press

Summary

This introductory textbook on mathematical biology focuses on discrete models across a variety of biological subdisciplines. Biological topics treated include linear and non-linear models of populations, Markov models of molecular evolution, phylogenetic tree construction, genetics, and infectious disease models. The coverage of models of molecular evolution and phylogenetic tree construction from DNA sequence data is unique among books at this level. Computer investigations with MATLAB are incorporated throughout, in both exercises and more extensive projects, to give readers hands-on experience with the mathematical models developed. MATLAB programs accompany the text. Mathematical tools, such as matrix algebra, eigenvector analysis, and basic probability, are motivated by biological models and given self-contained developments, so that mathematical prerequisites are minimal.

Table of Contents

Preface	vii
Note on MATLAB	xi
1. Dynamic Modeling with Difference Equations	1	(40)
1.1. The Malthusien Model	2	(9)
1.2. Nonlinear Models	11	(9)
1.3. Analyzing Nonlinear Models	20	(13)
1.4. Variations on the Logistic Model	33	(6)
1.5. Comments on Discrete and Continuous Models	39	(2)
2. Linear Models of Structured Populations	41	(44)
2.1. Linear Models and Matrix Algebra	41	(12)
2.2. Projection Matrices for Structured Models	53	(12)
2.3. Eigenvectors and Eigenvalues	65	(20)
2.4. Computing Eigenvectors and Eigenvalues 78
3. Nonlinear Models of Interactions	85	(28)
3.1. A Simple Predator-Prey Model	86	(8)
3.2. Equilibria of Multipopulation Models	94	(5)
3.3. Linearization and Stability	99	(6)
3.4. Positive and Negative Interactions	105	(8)
4. Modeling Molecular Evolution	113	(58)
4.1. Background on DNA	114	(2)
4.2. An Introduction to Probability	116	(14)
4.3. Conditional Probabilities	130	(8)
4.4. Matrix Models of Base Substitution	138	(17)
4.5. Phylogenetic Distances	155	(16)
5. Constructing Phylogenetic Trees	171	(44)
5.1. Phylogenetic Trees	172	(8)
5.2. Tree Construction: Distance Methods - Basics	180	(11)
5.3. Tree Construction: Distance Methods - Neighbor Joining	191	(7)
5.4. Tree Construction: Maximum Parsimony	198	(8)
5.5. Other Methods	206	(2)
5.6. Applications and Further Reading	208	(7)
6. Genetics	215	(64)
6.1. Mendelian Genetics	215	(13)
6.2. Probability Distributions in Genetics	228	(16)
6.3. Linkage	244	(17)
6.4. Gene Frequency in Populations	261	(18)
7. Infectious Disease Modeling	279	(36)
7.1. Elementary Epidemic Models	280	(6)
7.2. Threshold Values and Critical Parameters	286	(10)
7.3. Variations on a Theme	296	(11)
7.4. Multiple Populations and Differentiated Infectivity	307	(8)
8. Curve Fitting and Biological Modeling	315	(30)
8.1. Fitting Curves to Data	316	(9)
8.2. The Method of Least Squares	325	(10)
8.3. Polynomial Curve Fitting	335	(10)
A. Basic Analysis of Numerical Data	345	(17)
A.1. The Meaning of a Measurement	345	(3)
A.2. Understanding Variable Data - Histograms and Distributions	348	(4)
A.3. Mean, Median, and Mode	352	(3)
A.4. The Spread of Data	355	(4)
A.5. Populations and Samples	359	(1)
A.6. Practice	360	(2)
B. For Further Reading	362	(3)
References	365	(2)
Index	367