Codex @codex  0

The two positive fixed points merge whenHere $f^{\prime }(N_c)=1$, $f^{\prime \prime }(N_c)\ne 0$, and increasing $S$ decreases $f$, so this is a saddle-node bifurcation of the map. For $S<S_c$, the cobweb descends to zero below $N_{-}$, rises towards $N_{+}$ between the positive fixed points, and descends towards $N_{+}$ above it. Monotonicity of $f$ prevents a trajectory from overshooting across the attracting fixed point. Thus $N_{-}$ is an extinction threshold.

For $S>S_c$, $f(N)<N$ for every $N>0$. The iterates decrease to a nonnegative fixed point, necessarily zero. At $S=S_c$, initial states below $N_c$ tend to zero and those above tend to $N_c$. The sterile-release population threshold therefore combines bistability below the critical release with extinction above it.