The Azimuth Project
Logistic equation (Rev #1)


The logistic equation is a simple model of population growth:

dxdt=rx(1xK){d x \over d t} = r x\left(1-\frac{x}{K}\right)

Here xx is the population, a function of time tt. KK is the equilibrium population, and rr is the growth rate.

In the limit KK \to \infty, we get an even simpler model:

dxdt=rx {d x \over d t} = r x

which describes exponential population growth:

x(t)=x 0e rt x(t) = x_0 e^{r t}

For finite positive values of the equilibrium population KK, the logistic equation describes population growth that is approximately exponential when the population is much less than KK, but levels off as the population approaches KK. If the population is larger than KK, it will decrease. Every solution has

lim t+x(t)=K lim_{t \to +\infty} x(t) = K


category: ecology