Constant-quota harvested logistic growth ($\dot{N}=rN(1-N/K)-h$)

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 $N$.