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Population Growth

The differential equation describing exponential growth is

{dN\over dt} = {N\over\tau}.
\end{displaymath} (1)

This can be integrated directly
\int_{N_0}^N {dN\over N} = \int_0^t {dt\over\tau}
\end{displaymath} (2)

\ln\left({N\over N_0}\right)={t\over\tau}.
\end{displaymath} (3)

N(t) = N_0e^{t/\tau}.
\end{displaymath} (4)

Defining $N(t=1)=N_0e^\alpha$ gives $\tau=1/\alpha$ in (4), so
N(t) = N_0 e^{\alpha t}.
\end{displaymath} (5)

The quantity $\alpha$ in this equation is sometimes known as the Malthusian Parameter.

Consider a more complicated growth law

{dN\over dt} = \left({\alpha t-1\over t}\right)N,
\end{displaymath} (6)

where $\alpha>1$ is a constant. This can also be integrated directly
{dN\over N} = \left({\alpha-{1\over t}}\right)\,dt
\end{displaymath} (7)

\ln N = \alpha t-\ln t+C
\end{displaymath} (8)

N(t)={Ce^{\alpha t}\over t}.
\end{displaymath} (9)

Note that this expression blows up at $t=0$. We are given the Initial Condition that $N(t=1)=N_0e^\alpha$, so $C=N_0$.
N(t)=N_0 {e^{\alpha t}\over t}.
\end{displaymath} (10)

The $t$ 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

{dN\over dt}={r(K-N)\over N}
\end{displaymath} (11)

is another growth law which frequently arises in biology. It has a rather complicated solution for $N(t)$.

See also Gompertz Curve, Life Expectancy, Logistic Growth Curve, Lotka-Volterra Equations, Makeham Curve, Malthusian Parameter, Survivorship Curve

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© 1996-9 Eric W. Weisstein