Population Growth

The differential equation describing exponential growth is


This can be integrated directly


to give


where N_0=N(t=0). Exponentiating,


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

Consider a more complicated growth law


where r>1 is a constant. This can also be integrated directly


Note that this expression blows up at t=0. We are given the initial condition that N(t=1)=N_0e^r, so C=N_0.


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

The (continuous) logistic equation, defined by


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


See also

Gompertz Curve, Growth, Law of Growth, Life Expectancy, Logistic Map, Lotka-Volterra Equations, Makeham Curve, Malthusian Parameter, Survivorship Curve

Portions of this entry contributed by Christopher Stover

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

Referenced on Wolfram|Alpha

Population Growth

Cite this as:

Stover, Christopher and Weisstein, Eric W. "Population Growth." From MathWorld--A Wolfram Web Resource.

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