Biology Reference
In-Depth Information
Life belongs to the living, and he who lives must be prepared for
changes.
Johann Wolfgang von Goethe (1749-1832)
According to Encyclopædia Britannica,amathematical model
is defined as ''either a physical representation of
mathematical concepts or a mathematical representation
of reality.'' Physical mathematical models, such as graphs
of curves or surfaces defined by analytic equations or
three-dimensional replicas of cylinders, pyramids, and
spheres, are used to visualize mathematical terms and
concepts. Such models present realistic depictions of
abstract mathematical definitions. In contrast, a
mathematical representation of reality uses mathematics
to describe a phenomenon of nature. There are many
mathematical tools that can be used in this process,
including statistics, calculus, probability, and differential
equations. Different methods may provide insights to
different aspects of the problem, and there is often much
debate about what approach is preferable. Mathematical
models that represent reality are the subject of this text.
Chapter 1
PROCESSES THAT
CHANGE WITH TIME:
INTRODUCTION TO
DYNAMICAL SYSTEMS
Building a good mathematical model is a challenging task
that requires a solid understanding of the nature of the
system being modeled, as well as the mathematical tools
being used to describe it. Because mathematical models
are quite diverse, it is difficult to specify a process that
would apply to all problems. However, there are
fundamental principles that facilitate and guide the
creative process. They are:
Using Data to Formulate a Model
Discrete Versus Continuous Models
A Continuous Population Growth Model
The Logistic Model
An Alternative Derivation of the Logistic
Model
1. Initially, a model should be simple.
Long-Term Behavior and Equilibrium
States
2. It is crucial to test the model under as many condi-
tions as reasonable.
Analyzing Equilibrium States
The Verhulst Model for Discrete
Population Growth
3. If the model seems to be successful in some ways but
fails in others, try to modify the model rather than
starting over.
A Population Growth Model with Delay
Modeling Physiological Mechanisms
of Drug Elimination
In this chapter, we discuss how biological models of one
variable change over time. The first model we study is
growth of a population. Our initial attempt is based on
numerical data. Later, we build the model based on
conjectures about ''how populations should grow.'' Both
models yield essentially the same result, and although
these constructions are successful in the short term, both
are flawed because the long-term behavior they predict is
Using Computer Software for Solving
the Models
Some BERKELEY MADONNA Specifics
Suggested Biology Laboratory Exercises
for Chapter 1
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