 Squared-denominator population map

Here $r>0$ is the low-density per-generation multiplication factor and $b>0$ sets the inverse crowding scale. The positive fixed point is $(\sqrt{r}-1)/b$ when $r>1$; its derivative $2/\sqrt{r}-1$ lies strictly between minus one and one, so it is locally asymptotically stable. The map reaches its maximum $r/(4b)$ at $N=1/b$ and decreases thereafter. Stability follows from the derivative, not from assuming that every overcompensating population map undergoes period doubling.