Differential Equation Models pp 53-68 | Cite as

# Difference and Differential Equation Population Growth Models

Chapter

## Abstract

Ordinarily, the derivative is defined by the following limit:
Similarly, when a computer is used to “solve” a differential equation numerically, derivatives are ordinarily replaced by finite difference approximations such as
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.

$$ \frac{{dy}}{{dx}} = \mathop{{\lim }}\limits_{{h \to 0}} \frac{{y(x + h) - y(x)}}{h} $$

(1a)

$$ \frac{{dy}}{{dx}} \simeq \frac{{y(x + h) - y(x)}}{h} $$

(1b)

### Keywords

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### References

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*Introduction to Population Modeling*. UMAP Monograph Series, Birkhauser-Boston, 1979, pp. 59–73.CrossRefGoogle Scholar - [2]R. M. May,
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*Science*,*186*, 645–647, Nov. 15, 1974.CrossRefGoogle Scholar - [4]J. Maynard Smith,
*Mathematical Ideas in Biology*. New York: Cambridge Univ. Press, 1971, pp. 20–25Google Scholar - [4a]J. Maynard Smith,
*Mathematical Ideas in Biology*. New York: Cambridge Univ. Press, 1971, pp. 40–44.Google Scholar - [5]E. O. Wilson and W. H. Bossert,
*A Primer of Population Biology*. Stamford, CT: Sinauer Associates, 1971, pp. 14–19Google Scholar - [5a]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