The growth of lilies in a pond; the increase of a population 18 of bacteria, rats or people; the spreading of a contagious disease or of an ideal chain letter: these are all examples of exponential growth. The scary thing about exponential growth is that numbers get really big very fast. Exponential growth processes are doomed to terminate, because of a shortage of something that is needed to keep the process going, such as space, food or money.

The opposite of growth is decay. From a mathematical point of view, decay is very similar to growth: all it takes is changing a sign in the equation. Because the population depends only on time, these processes are described by what are called ordinary differential equations, which contain a derivative with respect to only one variable – in this case time. We look first at the simple equation for unlimited growth and decay, and then at the logistic equation, in which the growth saturates at a certain equilibrium value.

The simplest version of a growth equation states that the change in time of a quantity n(t) is proportional to its magnitude at that time. Say, n(t) denotes the number of mice in a cheese factory at a given time t. These mice have ample food and will start multiplying like mad. This means that the total reproduction at any given time will be proportional to the number of mice at that instant. In other words, the change per unit of time of n – i.e. the time derivative dn/dt – will be proportional to n. Translated into mathematics this yields the differential equation given in the side bar, with r the reproduction rate or (Malthusian) growth rate (r is larger than zero).

The solution for n(t) to this differential equation is an exponential growth of the number with time, as given on page 18 with no the number of mice at time zero. The fact that the time appears in the exponent, together with the positive constant r, implies that the growth will be extremely rapid. This is illustrated by the steeply rising dashed curve in the figure on page 21.