Sterile-release population threshold ($S_c={\alpha }^2/(4\beta )$)

For the population map $f(N)=\alpha N^2/(\beta N^2+S)$, extinction is a superstable fixed point. Below $S_c$, two positive fixed points $N_{\pm }=(\alpha \pm \sqrt{{\alpha }^2-4\beta S})/(2\beta )$ exist. Their multipliers are $1\mp \sqrt{{\alpha }^2-4\beta S}/\alpha$, making the lower one a threshold and the upper one attracting. A saddle-node bifurcation removes them at $S_c$; above it $f(N)<N$ for every positive population, so every nonnegative orbit tends to extinction.