Logistic Model

(logistic equation) The continuous version of the logistic model is described by the differential equation (dN)/(dt)=(rN(K-N))/K, (1) where r is the Malthusian parameter (rate of maximum population growth) and K is the so-called carrying capacity (i.e., the maximum sustainable population). Dividing both sides by K and defining x=N/K then gives the differential equation (dx)/(dt)=rx(1-x), (2) which is known as the logistic equation and has solution x(t)=1/(1+(1/(x_0)-1)e^(-rt)). (3) The function x(t) is sometimes known as the sigmoid function.