# 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.

## References

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