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)
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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)
|
