Squared-denominator population map ($F(N)=rN/(1+bN{)}^2$)

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.

Starting at $N_1=1/b$ with $r>4$, the next two iterates are the maximum $M=r/(4b)$ and its image $L=4r^2/[b(r+4{)}^2]$. Since $1/b<L\le M$ and the squared-denominator population map decreases on $[L,M]$, its image there lies in $[L,M]$. Thus the displayed interval is invariant after the initial step, and both bounds are attained. The initial value itself is below $L$, so it must not be included in the lower-bound assertion.