Post-initial invariant interval for a squared-denominator population map ($4r^2/[b(r+4{)}^2]\le N_t\le r/(4b),t\ge 2$)

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.