1st Edition
Dynamical Systems for Biological Modeling An Introduction
Dynamical Systems for Biological Modeling: An Introduction prepares both biology and mathematics students with the understanding and techniques necessary to undertake basic modeling of biological systems. It achieves this through the development and analysis of dynamical systems.
The approach emphasizes qualitative ideas rather than explicit computations. Some technical details are necessary, but a qualitative approach emphasizing ideas is essential for understanding. The modeling approach helps students focus on essentials rather than extensive mathematical details, which is helpful for students whose primary interests are in sciences other than mathematics need or want.
The book discusses a variety of biological modeling topics, including population biology, epidemiology, immunology, intraspecies competition, harvesting, predator-prey systems, structured populations, and more.
The authors also include examples of problems with solutions and some exercises which follow the examples quite closely. In addition, problems are included which go beyond the examples, both in mathematical analysis and in the development of mathematical models for biological problems, in order to encourage deeper understanding and an eagerness to use mathematics in learning about biology.
ELEMENTARY TOPICS
Introduction to Biological Modeling
The Nature and Purposes of Biological Modeling
The Modeling Process
Types of Mathematical Models
Assumptions, Simplifications, and Compromises
Scale and Choosing Units
Difference Equations (Discrete Dynamical Systems)
Introduction to Discrete Dynamical Systems
Graphical Analysis
Qualitative Analysis and Population Genetics
Intraspecies Competition
Harvesting
Period Doubling and Chaos
Structured Populations
Predator-Prey Systems
First-Order Differential Equations (Continuous Dynamical Systems)
Continuous-Time Models and Exponential Growth
Logistic Population Models
Graphical Analysis
Equations and Models with Variables Separable
Mixing Processes and Linear Models
First-Order Models with Time Dependence
Nonlinear Differential Equations
Qualitative Analysis Tools
Harvesting
Mass-Action Models
Parameter Changes, Thresholds, and Bifurcations
Numerical Analysis of Differential Equations
MORE ADVANCED TOPICS
Systems of Differential Equations
Graphical Analysis: The Phase Plane
Linearization of a System at an Equilibrium
Linear Systems with Constant Coefficients
Qualitative Analysis of Systems
Topics in Modeling Systems of Populations
Epidemiology: Compartmental Models
Population Biology: Interacting Species
Numerical Approximation to Solutions of Systems
Systems with Sustained Oscillations and Singularities
Oscillations in Neural Activity
Singular Perturbations and Enzyme Kinetics
HIV - An Example from Immunology
Slow Selection in Population Genetics
Second-Order Differential Equations: Acceleration
APPENDICES
An Introduction to the Use of MapleTM
Taylor’s Theorem and Linearization
Location of Roots of Polynomial Equations
Stability of Equilibrium of Difference Equations
Answers to Selected Exercises
Bibliography
Biography
Fred Brauer, PhD, University of British Columbia, Vancouver, Canada
Christopher Kribs, PhD, University of Texas at Arlington, USA
