Abstract
Asymptotic methods can greatly simplify the analysis of all but the simplest mathematical models and should therefore be commonplace in such biological areas as ecology and epidemiology. One essential difficulty that limits their use is that they can only be applied to a suitably scaled dimensionless version of the original dimensional model. Many books discuss nondimensionalization, but with little attention given to the problem of choosing the right scales and dimensionless parameters. In this paper, we illustrate the value of using asymptotics on a properly scaled dimensionless model, develop a set of guidelines that can be used to make good scaling choices, and offer advice for teaching these topics in differential equations or mathematical biology courses.
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Notes
See Ledder (2013) for an example of a limit cycle that requires separate scalings for 5 different phases.
I awarded this student almost all the credit for the problem.
Hethcote (2000) is a good source for both of these models; interestingly, the author nondimensionalizes the dependent variables but retains dimensional time. Brauer (2008) has only the first of these two models but features a nice explanation of the “natural decay” assumption used for the term \(-\gamma I\).
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Ledder, G. Scaling for Dynamical Systems in Biology. Bull Math Biol 79, 2747–2772 (2017). https://doi.org/10.1007/s11538-017-0339-5
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DOI: https://doi.org/10.1007/s11538-017-0339-5