What is a Difference Equation?

The theory of difference equations, despite its absence from the undergraduate curriculum, is an old and beautiful part of mathematics, one with diverse applications to many subjects: biology, economics, numerical analysis, etc. In a difference equation, change takes place in discrete time intervals. For example, in modeling populations of seasonally breeding animals, it is preferable to use difference equations rather than differential equations because the size of the next generation is largely determined by that of the current one.