= Sterile-release population threshold
{title2=$S_c=\alpha^2/(4\beta)$}
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.
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