## 1. Introduction and a simple model of growth

In many popular accounts of population change, the word 'exponential' is used to describe something that is increasing very rapidly. However, the term has a very much more exact meaning, and it also applies to situations other than populations. Furthermore, not all populations grow exponentially.

Under ideal conditions, the number of new organisms appearing in a population is related to the current population. Take a very simple example, where a single bacterium is capable of dividing once per day. On the second day, there are two bacteria, and on the third day each of these divides, giving four bacteria. On day four, each of the four bacteria has divided to give eight, on day five there will be 16, on day six there will be 32 and on day seven 64.

The rate of increase remains the same, in that the population doubles each day. However, the total number of bacteria added to the population each day changes with time, because this is determined by the current population. Written as a word equation, the population of our imaginary bacteria on a given day can be described as a function of the previous day's population:

Population (today) = 2 × Population (yesterday)

or tomorrow's population can be predicted if you know today's population:

Population (tomorrow) = 2 × Population (today)

This is a very simple model which is more properly described as a geometric series (as compared to an arithmetic series, which is additive). The model makes the very important assumption that all bacteria survive indefinitely, and furthermore that the population grows by undertaking one division each day. Obviously, in real life cell divisions will be taking place at different times, so that the population will change smoothly rather than in a series of steps.