Growth and Saturation

While exponential decay processes following a law such as (3.11) are common and meaningful, the same cannot be said of an exponential growth. An exponential growth can only be valid over a very narrow range as it would otherwise violate energy or space constraints. So any process which starts off at an exponential rate must eventually be moderated by other factors. A simple mathemathical model of moderated growth is described by the logistic equation. Here the rate of growth is made proportional to both the amount of growth already achieved and also to a maximum possible growth, leading to the equation

$${ d x \over dt } = a x (b-x) .$$ (3.15)

Clearly when $x=b$, the growth stops, and so $x=b$ represents a ceiling which in the simplest model is taken to be a constant. Thus the logistic equation describes processes in which the rate of growth is proportional to both what has been achieved and what remains to be achieved. This gives the growth curve a symmetrical S shape. Such curves have been used in the biological and social sciences, and are discuused further in the exercises.