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Population Growth -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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The differential equation describing exponential growth is

(dN)/(dt)=rN.

(1)

This can be integrated directly

int_(N_0)^N(dN)/N=int_0^trdt

(2)

to give

ln(N/(N_0))=rt,

(3)

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

N(t)=N_0e^(rt).

(4)

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

(dN)/(dt)=((rt-1)/t)N,

(5)

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

(dN)/N=(r-1/t)dt

(6)

lnN=rt-lnt+C

(7)

N(t)=(Ce^(rt))/t.

(8)

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.

N(t)=N_0(e^(rt))/t.

(9)

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

(dN)/(dt)=(rN(K-N))/K

(10)

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

N(t)=K/(1+(K/(N_0)-1)e^(-rt)).

(11)

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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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

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