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The Ricker population map has equilibria $N=0$ and $N=K$. Since $f^{\prime }(N)=e^{r(1-N/K)}(1-rN/K)$, their linearization multipliers areThus extinction is unstable and the positive equilibrium is locally asymptotically stable for $0<r<2$, becoming unstable for $r>2$.

To verify an actual period-doubling bifurcation, rather than only a multiplier crossing, put $N/K=1+x$ and $r=2+\epsilon$. The centered map isIts second iterate minus $x$ is $2\epsilon x-\frac{4}{3}{x}^3$ to leading order. After rescaling $x=\sqrt{\epsilon }y$, the limiting nonzero roots are simple, so the implicit function theorem continues them. Besides $x=0$, two nearby points therefore appear for $\epsilon >0$, with $x_{\pm }=\pm \sqrt{3\epsilon /2}+O(\epsilon )$. The map exchanges these points, and the derivative of its second iterate there is $1-4\epsilon +O({\epsilon }^{3/2})$, so the emerging two-cycle is attracting. The bifurcation at $r=2$ is supercritical period doubling.

For nonnegative populations, $f$ increases up to $N=K/r$ and decreases thereafter to zero. Hence every update after the initial time satisfiesThe bound is attained by choosing $N_0=K/r$; it is not a bound on an arbitrary initial population.

For the Ricker population map $H(u)=ru{e}^{-u}$, fixed points satisfy $u=0$ or $r{e}^{-u}=1$. Thus zero exists for all $r>0$, while $u_{\ast }=\log r$ is positive exactly when $r>1$. Since $H^{\prime }(u)=r{e}^{-u}(1-u)$, the respective periodic-orbit multipliers are $r$ and $1-\log r$.

Consequently zero is linearly attracting for $0<r<1$ and repelling for $r>1$; the positive biological population is linearly attracting for $1<r<e^2$ and repelling for $r>e^2$. The periodic-orbit multiplier of the positive point is positive below $r=e$ and negative above it, distinguishing monotone from alternating local convergence. At $r=1$, the coincident zero state has unit periodic-orbit multiplier but $0<H(u)<u$ for every $u>0$, giving nonlinear attraction from the physical right side. At $r=e^2$ the periodic-orbit multiplier is $-1$ and linearization is inconclusive. Writing $u=2+\eta$, the second iterate has expansion $\eta \mapsto \eta -{\eta }^3/3+O({\eta }^4)$, so the critical point is weakly attracting; beyond it a period-doubling instability occurs.