 Solution

A large initial biological population maps to a very small positive biological population because $u{e}^{-u}\to 0$ 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 subsequently lies below the maximum $H(1)=2/e<1$. On $[0,2/e]$ the map is increasing, with the unique positive fixed point $\log 2$. Below that point $H(u)>u$ and $H(u)<\log 2$; above it $H(u)<u$ and $H(u)>\log 2$. Therefore the trapped iterates approach the fixed point monotonically from whichever side they enter. In particularThe extinction initial value remains zero, but exact extinction is never reached from a positive finite biological population in this deterministic model.