Dynamic Models in Biology

456 Pages·6.785 MB·English

by  Ellner S.P.

Dynamic Models in Biology
DYNAMIC MODELS IN BIOLOGY
Stephen P. Ellner and John Guckenheimer
©2006 by Princeton University Press
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire
OX20 1SY
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Ellner, Stephen P., 1953–
Dynamic models in biology / Stephen P. Ellner and John Guckenheimer.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-691-11843-7 (alk. paper)
ISBN-10: 0-691-11843-4 (alk. paper)
ISBN-13: 978-0-691-12589-3 (pbk. : alk. paper)
ISBN-10: 0-691-12589-9 (pbk. : alk. paper)
1. Biology–Mathematical models. I. Guckenheimer, John. II. Title
QH323.5.E44 2006
570′.1′5118–dc22 2005048818
British Library Cataloging-in-Publication Data is available
This book has been composed in Stone and Stone Sans
Printed on acid-free paper. ∞
pup.princeton.edu
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Contents
List of Figures
List of Tables
Preface
1 What Are Dynamic Models?
1.1 Descriptive versus Mechanistic Models
1.2 Chinook Salmon
1.3 Bathtub Models
1.4 Many Bathtubs: Compartment Models
1.4.1 Enzyme Kinetics
1.4.2 The Modeling Process
1.4.3 Pharmacokinetic Models
1.5 Physics Models: Running and Hopping
1.6 Optimization Models
1.7 Why Bother?
1.8 Theoretical versus Practical Models
1.9 What’s Next?
1.10 References
2 Matrix Models and Structured Population Dynamics
2.1 The Population Balance Law
2.2 Age-Structured Models
2.2.1 The Leslie Matrix
2.2.2 Warning: Prebreeding versus Postbreeding Models
2.3 Matrix Models Based on Stage Classes
2.4 Matrices and Matrix Operations
2.4.1 Review of Matrix Operations
2.4.2 Solution of the Matrix Model
2.5 Eigenvalues and a Second Solution of the Model
2.5.1 Left Eigenvectors
2.6 Some Applications of Matrix Models
2.6.1 Why Do We Age?
2.6.2 Elasticity Analysis and Conservation Biology
2.6.3 How Much Should We Trust These Models?
2.7 Generalizing the Matrix Model
2.7.1 Stochastic Matrix Models
2.7.2 Density-Dependent Matrix Models
2.7.3 Continuous Size Distributions
2.8 Summary and Conclusions
2.9 Appendix
2.9.1 Existence and Number of Eigenvalues
2.9.2 Reproductive Value
2.10 References
3 Membrane Channels and Action Potentials
3.1 Membrane Currents
3.1.1 Channel Gating and Conformational States
3.2 Markov Chains
3.2.1 Coin Tossing
3.2.2 Markov Chains
3.2.3 The Neuromuscular Junction
3.3 Voltage-Gated Channels
3.4 Membranes as Electrical Circuits
3.4.1 Reversal Potential
3.4.2 Action Potentials
3.5 Summary
3.6 Appendix: The Central Limit Theorem
3.7 References
4 Cellular Dynamics: Pathways of Gene Expression
4.1 Biological Background
4.2 A Gene Network That Acts as a Clock
4.2.1 Formulating a Model
4.2.2 Model Predictions
4.3 Networks That Act as a Switch
4.4 Systems Biology
4.4.1 Complex versus Simple Models
4.5 Summary
4.6 References
5 Dynamical Systems
5.1 Geometry of a Single Differential Equation
5.2 Mathematical Foundations: A Fundamental Theorem
5.3 Linearization and Linear Systems
5.3.1 Equilibrium Points
5.3.2 Linearization at Equilibria
5.3.3 Solving Linear Systems of Differential Equations
5.3.4 Invariant Manifolds
5.3.5 Periodic Orbits
5.4 Phase Planes
5.5 An Example: The Morris-Lecar Model
5.6 Bifurcations
5.7 Numerical Methods
5.8 Summary
5.9 References
6 Differential Equation Models for Infectious Disease
6.1 Sir Ronald Ross and the Epidemic Curve
6.2 Rescaling the Model
6.3 Endemic Diseases and Oscillations
6.3.1 Analysis of the SIR Model with Births
6.3.2 Summing Up
6.4 Gonorrhea Dynamics and Control
6.4.1 A Simple Model and a Paradox
6.4.2 The Core Group
6.4.3 Implications for Control
6.5 Drug Resistance
6.6 Within-Host Dynamics of HIV
6.7 Conclusions
6.8 References
7 Spatial Patterns in Biology
7.1 Reaction-Diffusion Models
7.2 The Turing Mechanism
7.3 Pattern Selection: Steady Patterns
7.4 Moving Patterns: Chemical Waves and Heartbeats
7.5 References
8 Agent-Based and Other Computational Models for Complex
Systems
8.1 Individual-Based Models in Ecology
8.1.1 Size-Dependent Predation
8.1.2 Swarm
8.1.3 Individual-Based Modeling of Extinction Risk
8.2 Artificial Life
8.2.1 Tierra
8.2.2 Microbes in Tierra
8.2.3 Avida
8.3 The Immune System and the Flu
8.4 What Can We Learn from Agent-Based Models?
8.5 Sensitivity Analysis
8.5.1 Correlation Methods
8.5.2 Variance Decomposition
8.6 Simplifying Computational Models
8.6.1 Separation of Time Scales
8.6.2 Simplifying Spatial Models
8.6.3 Improving the Mean Field Approximation
8.7 Conclusions
8.8 Appendix: Derivation of Pair Approximation
8.9 References
9 Building Dynamic Models
9.1 Setting the Objective
9.2 Building an Initial Model
9.2.1 Conceptual Model and Diagram
9.3 Developing Equations for Process Rates
9.3.1 Linear Rates: When and Why?
9.3.2 Nonlinear Rates from “First Principles”
9.3.3 Nonlinear Rates from Data: Fitting Parametric Models
9.3.4 Nonlinear Rates from Data: Selecting a Parametric Model
9.4 Nonlinear Rates from Data: Nonparametric Models
9.4.1 Multivariate Rate Equations
9.5 Stochastic Models
9.5.1 Individual-Level Stochasticity
9.5.2 Parameter Drift and Exogenous Shocks
9.6 Fitting Rate Equations by Calibration
9.7 Three Commandments for Modelers
9.8 Evaluating a Model
9.8.1 Comparing Models
9.9 References
Index
List of Figures
Figure 1.1. Decline in Coho salmon stocks on the Thomson River, BC
Figure 1.2. Rate of decline on individual streams related to habitat variables
Figure 1.3. Frequency distribution of recruits per spawner for the Bear,
Marsh, and Sulphur Creek stocks of Chinook salmon
Figure 1.4. Compartment diagram of the bathtub model
Figure 1.5. Data on the rate of hydrolysis of sucrose by invertase fitted to the
Michaelis-Menten equation
Figure 1.6. Compartment diagram for the enzyme-mediated biochemical
reaction
Figure 1.7. Graphical analysis of dv/dt = k s − (k + k + k s)v
1 −1 2 1
Figure 1.8. Compartment model for arsenic in rat hepatocytes
Figure 1.9. Compartment diagram for the IEUBK model for lead in children
Figure 1.10. A cartoon of the pogo stick or monopode
Figure 1.11. The monopode in stance phase
Figure 1.12. Diagram of the experimental fishpond system showing water
flows between compartments
Figure 1.13. Compartment diagrams for Exercise 1.3
Figure 2.1. The standard size-class model
Figure 2.2. Stage-structured model for killer whales
Figure 2.3. Eigenvalue elasticities for stage-structured model for desert
tortoise
Figure 2.4. Eigenvalue elasticities for stage-structured model for loggerhead
sea turtles
Figure 2.5. The LPA model
Figure 2.6. Eigenvalue elasticities for matrix models of northern monkshood
using two different size populations
Figure 2.7. Elasticity surface for the integral projection model for northern
monkshood