Harvested bistable population thresholds ($E_T=P(0),E_M={\max }_{a_2<N<a_3}P(N)$)

For $\dot{N}=N[P(N)-E]$ and $P(N)={\prod }_{i=1}^3(a_i-N)$ with $0<a_1<a_2<a_3$, extinction changes stability at $E_T$, while the upper stable population and an unstable separator merge at $E_M$. If $E_T<E_M$, an interval of efforts supports stable extinction and a stable positive population simultaneously; at $E_M$ the positive state collapses. If the thresholds have the opposite order, the high-population pair disappears before the lower stable branch reaches extinction. At a positive equilibrium stability follows from the sign of $N{P}^{\prime }(N)$.