Mathematical Modeling

  • Glenn Ledder
Chapter
Part of the Springer Undergraduate Texts in Mathematics and Technology book series (SUMAT)

Abstract

Mathematical modeling refers to any use of mathematics to do theoretical science. As such, it incorporates mathematical techniques into a larger structure that is seldom taught in mathematics courses. Models can be derived from mechanistic principles or based on empirical data; there are some commonalities between these types of modeling as well as important differences. The first two sections present the concepts of mathematical modeling. Three sections on empirical modeling develop the least squares method for fitting linear models, extend the method for use with a large class of nonlinear models, and present the Akaike information criterion (AIC) for model selection. Two sections on mechanistic modeling present basic methods for model derivation and nondimensionalization. Biological examples in this chapter include a discussion of the use and misuse of the Lotka–Volterra predator–prey model, the derivation of the Holling type II predation model, and a compartment model of pollution in a lake. The problems include several that use chemostat and SIR disease models and an exploration of the evidence for global warming provided by an extensive data set of grape harvest dates.

Keywords

Biomass Influenza Versed Haldane 

References

  1. 1.
    Akaike H. A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19: 716–723 (1974)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Atkins GL and IA Nimmo. A comparison of seven methods for fitting the Michaelis–Menten equation. Biochem J., 149, 775–777 (1975)Google Scholar
  3. 3.
    Atkinson, MR, JF Jackson, and RK Morton. Nicotinamide mononucleotide adenylyltransferase of pig-liver nuclei: The effects of nicotinamide mononucleotide concentration and pH on dinucleotide synthesis. Biochem J., 80, 318–323 (1980)Google Scholar
  4. 4.
    Atlantic States Marine Fisheries Commission. Atlantic Croaker 2010 Stock Assessment Report. Southeast Fisheries Science Center, National Oceanic and Atmospheric Administration (2010). http://www.sefsc.noaa.gov/sedar/Sedar_Workshops.jsp?WorkshopNum=20 Cited in Nov 2012
  5. 5.
    Chuine I, P Yiou, N Viovy, B Seguin, V Daux, and EL Ladurie. Grape ripening as a past climate indicator. Nature, 432, 18 (2004)Google Scholar
  6. 6.
    Holling CS. Some characteristics of simple types of predation and parasitism. Canadian Entomologist, 91: 385–398 (1959)CrossRefGoogle Scholar
  7. 7.
    Ledder G. An experimental approach to mathematical modeling in biology. PRIMUS, 18, 119–138 (2007)CrossRefGoogle Scholar
  8. 8.
    Ledder G. BUGBOX-predator (2007). http://www.math.unl.edu/~gledder1/BUGBOX/ Cited Sep 2012
  9. 9.
    Lineweaver H and D Burk. The determination of enzyme dissociation constants. Journal of the American Chemical Society, 56, 658–666 (1934)Google Scholar
  10. 10.
    Motulsky H and A Christopoulos. Fitting Models to Biological Data Using Linear and Nonlinear Regression. Oxford University Press, Oxford, UK (2004)Google Scholar
  11. 11.
    Rasmussen RA. Atmospheric trace gases in Antarctica. Science, 211, 285–287 (1981)CrossRefGoogle Scholar
  12. 12.
    Richards S. Testing ecological theory using the information-theoretic approach: Examples and cautionary results. Ecology, 86, 2805–2814 (2005)CrossRefGoogle Scholar
  13. 13.
    University of Tennessee. Across Trophic Level System Simulation (1996). http://atlss.org Cited in Nov 2012
  14. 14.
    Wilkinson GN. Statistical estimations in enzyme kinetics. Biochem J., 80, 324–332 (1961)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Glenn Ledder
    • 1
  1. 1.Department of MathematicsUniversity of Nebraska-LincolnLincolnUSA

Personalised recommendations