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Difference and Differential Equation Population Growth Models

  • James C. Frauenthal
Part of the Modules in Applied Mathematics book series

Abstract

Ordinarily, the derivative is defined by the following limit:
$$ \frac{{dy}}{{dx}} = \mathop{{\lim }}\limits_{{h \to 0}} \frac{{y(x + h) - y(x)}}{h} $$
(1a)
Similarly, when a computer is used to “solve” a differential equation numerically, derivatives are ordinarily replaced by finite difference approximations such as
$$ \frac{{dy}}{{dx}} \simeq \frac{{y(x + h) - y(x)}}{h} $$
(1b)
These two operations are really just inverses of one another. At times, the conversion of a difference equation into the analogous differential equation is convenient because the calculus can be employed, so the finite interval of the independent variable is made to vanish. At other times, this limit is “undone” so that numerical methods can be used on the difference equation analog of a differential equation. Unfortunately, these inverse operations have a profound effect upon the nature of the solutions found. This frequently neglected point is the main topic of this chapter.

Keywords

Differential Equation Equilibrium Point Difference Equation Nonlinear Differential Equation Partial Fraction 
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.

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References

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    J. C. Frauenthal, Introduction to Population Modeling. UMAP Monograph Series, Birkhauser-Boston, 1979, pp. 59–73.CrossRefGoogle Scholar
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    R. M. May, Stability and Complexity in Model Ecosystems. Princeton, NJ: Princeton Univ. Press, 1973, pp. 26–30.Google Scholar
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    —, “Biological populations with nonoverlapping generations: Stable points, stable cycles and chaos,” Science,186, 645–647, Nov. 15, 1974.CrossRefGoogle Scholar
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    J. Maynard Smith, Mathematical Ideas in Biology. New York: Cambridge Univ. Press, 1971, pp. 20–25Google Scholar
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    E. O. Wilson and W. H. Bossert, A Primer of Population Biology. Stamford, CT: Sinauer Associates, 1971, pp. 14–19Google Scholar
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    E. O. Wilson and W. H. Bossert, A Primer of Population Biology. Stamford, CT: Sinauer Associates, 1971, pp. 102–111.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1983

Authors and Affiliations

  • James C. Frauenthal
    • 1
  1. 1.Department of Applied Mathematics and StatisticsState University of New YorkStony BrookUSA

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