The differential equation describing exponential growth is

(1)

This can be integrated directly

(2)

to give

(3)

where . Exponentiating,

(4)

This equation is called the law of growth and, in a much more antiquated fashion, the Malthusian equation; the quantity  in this equation is sometimes known as the Malthusian parameter.

Consider a more complicated growth law

(5)

where  is a constant. This can also be integrated directly

(6)

(7)

(8)

Note that this expression blows up at . We are given the initial condition that , so .

(9)

The  in the denominator of (◇) greatly suppresses the growth in the long run compared to the simple growth law.

The (continuous) logistic equation, defined by

(10)

is another growth law which frequently arises in biology. It has solution

(11)

REFERENCES:

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 290-295, 1999.