Logistic growth equation ($\dot{N}=rN(1-N/K)$)

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.

For a fixed catch rate $h$, natural logistic growth can replace the catch at two abundances if $0<h<rK/4$. The upper equilibrium $N_{+}=K[1+\sqrt{1-4h/(rK)}]/2$ attracts, and the lower $N_{-}$ is an unstable survival threshold. They meet in a saddle-node bifurcation at $h=rK/4$, $N=K/2$. Above this catch every positive population declines to zero. Below it a population initially under $N_{-}$ also becomes extinct. A constant catch is stopped at extinction to avoid negative abundances; it is distinct from per-capita harvesting proportional to $N$.