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Model Analysis

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.

  1. 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.
  2. 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 model.
  3. 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 (Hopf bifurcation).
  4. 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.
  5. 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