= Solution
A large initial <biological population> maps to a very small positive <biological population> because $ue^{-u}\to0$ at large $u$: the initial slump is followed by recovery. Cobwebbing alternates a vertical step from $(u_n,u_n)$ to $(u_n,H(u_n))$ with a horizontal step to the diagonal at $(u_{n+1},u_{n+1})$. The figure uses the explicit illustrative choice $u_1=5$.
For $r=2$, every positive <orbit (dynamical system)> subsequently lies below the maximum $H(1)=2/e<1$. On $[0,2/e]$ the map is increasing, with the unique positive <fixed point> $\log2$. Below that point $H(u)>u$ and $H(u)<\log2$; above it $H(u)<u$ and $H(u)>\log2$. Therefore the trapped iterates approach the <fixed point> monotonically from whichever side they enter. In particular
$$
\boxed{u_n\longrightarrow\log2\quad\text{for every }u_1>0.}
$$
The extinction initial value remains zero, but exact extinction is never reached from a positive finite <biological population> in this deterministic model.
\Image[/past-exam-of-the-topics-in-applied-mathematics-course-of-the-university-of-oxford/2009-ricker-cobweb.png]
{title=Ricker cobwebs after a large initial population, with the first five generations at r equals nine}
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