The differential equation describing exponential growth is
![\begin{displaymath}
{dN\over dt} = {N\over\tau}.
\end{displaymath}](p3_86.gif) |
(1) |
This can be integrated directly
![\begin{displaymath}
\int_{N_0}^N {dN\over N} = \int_0^t {dt\over\tau}
\end{displaymath}](p3_87.gif) |
(2) |
![\begin{displaymath}
\ln\left({N\over N_0}\right)={t\over\tau}.
\end{displaymath}](p3_88.gif) |
(3) |
Exponentiating,
![\begin{displaymath}
N(t) = N_0e^{t/\tau}.
\end{displaymath}](p3_89.gif) |
(4) |
Defining
gives
in (4), so
![\begin{displaymath}
N(t) = N_0 e^{\alpha t}.
\end{displaymath}](p3_92.gif) |
(5) |
The quantity
in this equation is sometimes known as the Malthusian Parameter.
Consider a more complicated growth law
![\begin{displaymath}
{dN\over dt} = \left({\alpha t-1\over t}\right)N,
\end{displaymath}](p3_94.gif) |
(6) |
where
is a constant. This can also be integrated directly
![\begin{displaymath}
{dN\over N} = \left({\alpha-{1\over t}}\right)\,dt
\end{displaymath}](p3_96.gif) |
(7) |
![\begin{displaymath}
\ln N = \alpha t-\ln t+C
\end{displaymath}](p3_97.gif) |
(8) |
![\begin{displaymath}
N(t)={Ce^{\alpha t}\over t}.
\end{displaymath}](p3_98.gif) |
(9) |
Note that this expression blows up at
. We are given the Initial Condition
that
, so
.
![\begin{displaymath}
N(t)=N_0 {e^{\alpha t}\over t}.
\end{displaymath}](p3_101.gif) |
(10) |
The
in the Denominator of (10) greatly suppresses the growth in the long run
compared to the simple growth law.
The Logistic Growth Curve, defined by
![\begin{displaymath}
{dN\over dt}={r(K-N)\over N}
\end{displaymath}](p3_103.gif) |
(11) |
is another growth law which frequently arises in biology. It has a rather complicated solution for
.
See also Gompertz Curve, Life Expectancy, Logistic Growth Curve, Lotka-Volterra Equations,
Makeham Curve, Malthusian Parameter, Survivorship Curve
© 1996-9 Eric W. Weisstein
1999-05-26