Ricker population map ($N_{t+1}=N_t{e}^{r(1-N_t/K)}$)

The Ricker population map models density-dependent reproduction with a declining per-capita growth factor. Its nonzero equilibrium is $K$, with multiplier $1-r$. It loses stability through a period-doubling bifurcation at $r=2$. Every population after the first update is bounded by $K{e}^{r-1}/r$, the maximum of the update function on nonnegative populations.

For $H(u)=ru{e}^{-u}$ and $L=\log r$, a nonzero second-iterate fixed point satisfies $u+H(u)=2L$. A genuine pair has the displayed form, and its periodic-orbit multiplier is $(1-L{)}^2-d^2$. At $r=9$, the pair is exactly $\log 3,3\log 3$, with periodic-orbit multiplier $(1-\log 3)(1-3\log 3)\simeq 0.2264$. For large $r$ its upper point is asymptotic to $2\log r$ and the pair periodic-orbit multiplier to $1-2\log r$, so eventual instability follows despite the small derivative at the upper point.