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This section describes what it means to analyze a model. The ultimate
goal in analyzing a model is to understand all of its dynamic
behavior. If the model contains parameters, then part of analysis is
determining how the dynamic behavior of the model changes as the
parameters vary. Understanding all of the behavior exhibited by a
model can be a very difficult and, in some cases, impossible task, so
it is best to start with the simplest parts of the analysis. The typical
steps, in order of difficulty, are given below. Realistically, we can
handle only the first two steps below, but a combination of Maple and
xrk will allow you to attack at lest some of the other steps.
- Find the fixed points of the model. For some models, this step
is fairly simple. For others, this can be hard work, requiring the use
of bifurcation theory.
- Determine the stability of the fixed points. Especially
important are where fixed points are not hyperbolic, because they form
part of the boundaries for different qualitative behavior of the
- Investigation of the non-hyperbolic fixed points. The only cases
that we can begin to handle are a single zero eigenvalue (limit point
or transcritical bifurcation) or a pair of pure imaginary eigenvalues
- Investigation of periodic behavior. If Hopf bifurcations occur
in a model, they generally result in periodic solutions. Investigation
of these periodic solutions almost always has to be done numerically.
- More exotic behavior. In models of dimension three or higher,
lots of interesting things can happen, including chaos. Again,
numerical investigation is almost necessary.
William W. Farr
Thu Oct 24 13:03:17 EDT 1996