= Solution
The <iteration of a map> is $F(u)=u/(2+u^2)$. For every $u>0$, $0<F(u)<u/2$, so $0<u_n\leq2^{-n}u_0$ and $\boxed{u_n\to0}$. A cobweb alternates vertical steps to the graph and horizontal steps to the diagonal, approaching the origin monotonically.
\Image[/past-exam-of-the-topics-in-applied-mathematics-course-of-the-university-of-oxford/2007-population-cobweb.png]
{title=Resource-limited population cobweb: every positive orbit approaches extinction}
More precisely $u_{n+1}/u_n\to1/2$. The product $2^nu_n=u_0\prod_{j<n}(1+u_j^2/2)^{-1}$ has a finite strictly positive limit because $\sum_ju_j^2<\infty$. Thus $\boxed{u_n\sim C(u_0)2^{-n}}$ for a positive constant depending on the initial <biological population>.
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