Logistic Growth Curve

The Population Growth law which arises frequently in biology and is given by the differential equation

$${dN\over dt}={r(K-N)\over K},$$ (1)

where $r$ is the Malthusian Parameter and $K$ is the so-called Carrying Capacity (i.e., the maximum sustainable population). Rearranging and integrating both sides gives

$$\int_{N_0}^N {dN\over K-N}={r\over K}\int_0^t dt$$ (2)

$$\ln\left({N_0-K\over N-K}\right)={r\over K}t$$ (3)

$$N(t)=K+(N_0-K)e^{-rt/K}.$$ (4)

The curve

$$y={a\over 1+bq^x}$$ (5)

is sometimes also known as the logical curve.

See also Gompertz Curve, Life Expectancy, Logistic Equation, Makeham Curve, Malthusian Parameter, Population Growth