= Flip threshold of a saturating ecological population map
{title2=$F(u)=\alpha u/(\beta+u^\gamma),\quad\gamma_*=2\alpha/(\alpha-\beta)$}
For $\alpha>\beta>0$ the positive <fixed point> is $(\alpha-\beta)^{1/\gamma}$ with multiplier $1-\gamma(1-\beta/\alpha)$. It is stable for $1<\gamma<\gamma_*$ and crosses through minus one at the displayed threshold. Since $\gamma_*>2$, every positive <fixed point> is stable when $1<\gamma\leq2$. The increasing power controls overcompensatory <density dependence> at high abundance. A multiplier crossing identifies the local flip mechanism; a separate nonlinear calculation determines its criticality and any resulting attractor.
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