Scaling as a Tool for the Analysis of Biological Models

  • D. L. S. McElwain
Part of the Mathematical Modelling book series (MMO, volume 6)


Many mathematical models of biological systems lead to differential equations. Although numerical methods are available to solve these equations, the introduction of dimensionless variables often simplifies the form of the equations. In addition, if scaled dimensionless variables are introduced then it is sometimes possible to obtain useful approximations to the solution using, for example, perturbation methods. A discussion of the use of these techniques is illustrated by examining a model developed by Volterra for population growth in a closed system.


Closed System Perturbation Method Dimensionless Variable Logistic Equation Matched Asymptotic Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Goodwin, B.C. 1969. In C.H Waddington (ed.) Towards a Theoretical Biology, 2, pp. 337. Edinburgh University Press: Edinburgh.Google Scholar
  2. [2]
    Scudo, F.M. 1971. Vito Volterra and theoretical ecology, Theoret Pop. Biol, 2, pp. 1–23.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Segel, L.A. 1972. Simplification and scaling, SIAM Rev., 14, pp. 547–571.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Segel, L.A. 1984. Modeling Dynamic Phenomena in Molecular and Cellular Biology, Cambridge University Press, Cambridge, 300 p.MATHGoogle Scholar
  5. [5]
    Segel, L.A. and M. Slemrod. 1989. The quasi-steady state assumption: A case study in perturbation (preprint).Google Scholar
  6. [6]
    Small, R.D. 1983. Population growth in a closed system, SIAM Rev., 25, pp. 93–95.MATHCrossRefGoogle Scholar
  7. [7]
    Van Dyke, M. 1975. Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, California, 271 p.MATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • D. L. S. McElwain

There are no affiliations available

Personalised recommendations