Mathematical Models in Biology

ISBN-10: 0898715547
ISBN-13: 9780898715545
Edition: 2004
List price: $50.00
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Book details

List price: $50.00
Copyright year: 2004
Publisher: Society for Industrial and Applied Mathematics
Binding: Paperback
Pages: 184
Size: 7.25" wide x 9.00" long x 1.25" tall
Weight: 2.398
Language: English

Leah Edelstein-Keshet is a Professor in the Department of Mathematics at the University of British Columbia, Vancouver, Canada. Her book Mathematical Models in Biology was republished in SIAM's Classics in Applied Mathematics series.

Preface to the Classics Edition
Preface
Acknowledgments
Errata
Discrete Process in Biology
The Theory of Linear Difference Equations Applied to Population Growth
Biological Models Using Difference Equations
Cell Division
An Insect Population
Propagation of Annual Plants
Statement of the Problem
Definitions and Assumptions
The Equations
Condensing the Equations
Check
Systems of Linear Difference Equations
A Linear Algebra Review
Will Plants Be Successful?
Qualitative Behavior of Solutions to Linear Difference Equations
The Golden Mean Revisited
Complex Eigenvalues in Solutions to Difference Equations
Related Applications to Similar Problems
Growth of Segmental Organisms
A Schematic Model of Red Blood Cell Production
Ventilation Volume and Blood CO[subscript 2] Levels
For Further Study: Linear Difference Equations in Demography
Problems
References
Nonlinear Difference Equations
Recognizing a Nonlinear Difference Equation
Steady States, Stability, and Critical Parameters
The Logistic Difference Equation
Beyond r = 3
Graphical Methods for First-Order Equations
A Word about the Computer
Systems of Nonlinear Difference Equations
Stability Criteria for Second-Order Equations
Stability Criteria for Higher-Order Systems
For Further Study: Physiological Applications
Problems
References
Appendix to Chapter 2: Taylor Series
Functions of One Variable
Functions of Two Variables
Applications of Nonlinear Difference Equations to Population Biology
Density Dependence in Single-Species Populations
Two-Species Interactions: Host-Parasitoid Systems
The Nicholson-Bailey Model
Modifications of the Nicholson-Bailey Model
Density Dependence in the Host Population
Other Stabilizing Factors
A Model for Plant-Herbivore Interactions
Outlining the Problem
Rescaling the Equations
Further Assumptions and Stability Calculations
Deciphering the Conditions for Stability
Comments and Extensions
For Further Study: Population Genetics
Problems
Projects
References
Continuous Processes and Ordinary Differential Equations
An Introduction to Continuous Models
Warmup Examples: Growth of Microorganisms
Bacterial Growth in a Chemostat
Formulating a Model
First Attempt
Corrected Version
A Saturating Nutrient Consumption Rate
Dimensional Analysis of the Equations
Steady-State Solutions
Stability and Linearization
Linear Ordinary Differential Equations: A Brief Review
First-Order ODEs
Second-Order ODEs
A System of Two First-Order Equations (Elimination Method)
A System of Two First-Order Equations (Eigenvalue-Eigenvector Method)
When Is a Steady State Stable?
Stability of Steady States in the Chemostat
Applications to Related Problems
Delivery of Drugs by Continuous Infusion
Modeling of Glucose-Insulin Kinetics
Compartmental Analysis
Problems
References
Phase-Plane Methods and Qualitative Solutions
First-Order ODEs: A Geometric Meaning
Systems of Two First-Order ODEs
Curves in the Plane
The Direction Field
Nullclines: A More Systematic Approach
Close to the Steady States
Phase-Plane Diagrams of Linear Systems
Real Eigenvalues
Complex Eigenvalues
Classifying Stability Characteristics
Global Behavior from Local Information
Constructing a Phase-Plane Diagram for the Chemostat
The Nullclines
Steady States
Close to Steady States
Interpreting the Solutions
Higher-Order Equations
Problems
References
Applications of Continuous Models to Population Dynamics
Models for Single-Species Populations
Malthus Model
Logistic Growth
Allee Effect
Other Assumptions; Gompertz Growth in Tumors
Predator-Prey Systems and the Lotka-Volterra Equations
Populations in Competition
Multiple-Species Communities and the Routh-Hurwitz Criteria
Qualitative Stability
The Population Biology of Infectious Diseases
For Further Study: Vaccination Policies
Eradicating a Disease
Average Age of Acquiring a Disease
Models for Molecular Events
Michaelis-Menten Kinetics
The Quasi-Steady-State Assumption
A Quick, Easy Derivation of Sigmoidal Kinetics
Cooperative Reactions and the Sigmoidal Response
A Molecular Model for Threshold-Governed Cellular Development
Species Competition in a Chemical Setting
A Bimolecular Switch
Stability of Activator-Inhibitor and Positive Feedback Systems
The Activator-Inhibitor System
Positive Feedback
Some Extensions and Suggestions for Further Study
Limit Cycles, Oscillations, and Excitable Systems
Nerve Conduction, the Action Potential, and the Hodgkin-Huxley Equations
Fitzhugh's Analysis of the Hodgkin-Huxley Equations
The Poincare-Bendixson Theory
The Case of the Cubic Nullclines
The Fitzhugh-Nagumo Model for Neural Impulses
The Hopf Bifurcation
Oscillations in Population-Based Models
Oscillations in Chemical Systems
Criteria for Oscillations in a Chemical System
For Further Study: Physiological and Circadian Rhythms
Appendix to Chapter 8. Some Basic Topological Notions
Appendix to Chapter 8. More about the Poincare-Bendixson Theory
Spatially Distributed Systems and Partial Differential Equation Models
An Introduction to Partial Differential Equations and Diffusion in Biological Settings
Functions of Several Variables: A Review
A Quick Derivation of the Conservation Equation
Other Versions of the Conservation Equation
Tubular Flow
Flows in Two and Three Dimensions
Convection, Diffusion, and Attraction
Convection
Attraction or Repulsion
Random Motion and the Diffusion Equation
The Diffusion Equation and Some of Its Consequences
Transit Times for Diffusion
Can Macrophages Find Bacteria by Random Motion Alone?
Other Observations about the Diffusion Equation
An Application of Diffusion to Mutagen Bioassays
Appendix to Chapter 9. Solutions to the One-Dimensional Diffusion Equation
Partial Differential Equation Models in Biology
Population Dispersal Models Based on Diffusion
Random and Chemotactic Motion of Microorganisms
Density-Dependent Dispersal
Apical Growth in Branching Networks
Simple Solutions: Steady States and Traveling Waves
Nonuniform Steady States
Homogeneous (Spatially Uniform) Steady States
Traveling-Wave Solutions
Traveling Waves in Microorganisms and in the Spread of Genes
Fisher's Equation: The Spread of Genes in a Population
Spreading Colonies of Microorganisms
Some Perspectives and Comments
Transport of Biological Substances Inside the Axon
Conservation Laws in Other Settings: Age Distributions and the Cell Cycle
The Cell Cycle
Analogies with Particle Motion
A Topic for Further Study: Applications to Chemotherapy
Summary
A Do-It-Yourself Model of Tissue Culture
A Statement of the Biological Problem
A Simple Case
A Slightly More Realistic Case
Writing the Equations
The Final Step
Discussion
For Further Study: Other Examples of Conservation Laws in Biological Systems
Models for Development and Pattern Formation in Biological Systems
Cellular Slime Molds
Homogeneous Steady States and Inhomogeneous Perturbations
Interpreting the Aggregation Condition
A Chemical Basis for Morphogenesis
Conditions for Diffusive Instability
A Physical Explanation
Extension to Higher Dimensions and Finite Domains
Applications to Morphogenesis
For Further Study
Patterns in Ecology
Evidence for Chemical Morphogens in Developmental Systems
A Broader View of Pattern Formation in Biology
Selected Answers
Author Index
Subject Index

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