 Logistic growth equation

The logistic growth equation applies the logistic differential equation to population dynamics. For positive growth rate $r$ and carrying capacity $K$, an initial condition $0<N(0)<K$ gives $N(t)=K/[1+b{e}^{-rt}]$, where $b=K/N(0)-1$. Its per-capita growth falls linearly with abundance, and a trajectory starting below $K/2$ reaches an inflection point when $N=K/2$. The exact solutions of a differential equation and their diagrams are developed under logistic differential equation. A translated and scaled logistic function has the same form. In nonlinear regression for biological size, an analogous saturation curve need not imply that circumference itself obeys a mechanistic population law.