Why do we use differential equations ?

Each division of bacteria or extension to form a new mold cell is a discrete event. There is a moment when there is one bacterial cell and a moment later when there are two cells. To descibe this mathematically, we must use difference equations, not differential equations that assume that the variables are continuous functions.

In bacterial cultures, there may be several billion cells per milliliter. When they are reproducing, there may be several new cells at each time increment of a microsecond. As we advance from one time to the next, the cell count may change from 1,000,000,000 to 1,000,000,007. On our scale of interest, this is very close to a continuous function that is well-behaved for treatment with differential equations.