Mathematical Models in Biology
Serving as a basic introduction to concepts in deterministic biological modeling, this text shows how relatively simple mathematics can be applied to a variety of models to draw interesting conclusions.
TABLE OF CONTENTS Mathematical Models in Biology
Part II: Continuous Processes and Ordinary Differential Equations
Chapter List
- Chapter 4: An Introduction to Continuous Models
- Chapter 5: Phase-Plane Methods and Qualitative Solutions
- Chapter 6: Applications of Continuous Models to Population Dynamics
- Chapter 7: Models for Molecular Events
- Chapter 8: Limit Cycles, Oscillations, and Excitable Systems
The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. That is not particularly disturbing until you think about it, but the fact is that bacteria multiply geometrically: one becomes two, two become four, four become eight, and so on. In this way, it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet earth.
M.Crichton (1969), The Andromeda Strain (Dell, New York, p. 247).
OVERVIEW
This chapter introduces the topic of ordinary differential equation models, their formulation, analysis, and interpretation. A main emphasis at this stage is on how appropriate assumptions simplify the problem, how important variables are identified, and how differential equations are tailored to describing the essential features of a continuous process.
Because one of the most challenging parts of modeling is writing the equations, we dwell on this aspect purposely. The equations are written in stages, with appropriate assumptions introduced as they are needed. We begin with a rather simple ordinary differential equation as a model for bacterial growth. Gradually, more realistic aspects of the situation are considered, eventually leading to a system of two ordinary...
Copyright The Society for Industrial and Applied Mathematics 2005 under license agreement with Books24x7
