The differential equation describing exponential growth is
(1)
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This can be integrated directly
(2)
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to give
(3)
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where . Exponentiating,
(4)
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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)
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where is a constant. This can also be integrated directly
(6)
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(7)
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(8)
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Note that this expression blows up at . We are given the initial condition that , so .
(9)
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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)
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is another growth law which frequently arises in biology. It has solution
(11)
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