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# Population Growth

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)

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)

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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## References

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. https://mathworld.wolfram.com/PopulationGrowth.html