 Flip threshold of a saturating ecological population map

For $\alpha >\beta >0$ the positive fixed point is $(\alpha -\beta {)}^{1/\gamma }$ with multiplier $1-\gamma (1-\beta /\alpha )$. It is stable for $1<\gamma <{\gamma }_{\ast }$ and crosses through minus one at the displayed threshold. Since ${\gamma }_{\ast }>2$, every positive fixed point is stable when $1<\gamma \le 2$. 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.