Past exam of the topics in applied mathematics course of the University of Oxford / 2007 / A / 1 / i

The iteration of a map is $F(u)=u/(2+u^2)$. For every $u>0$, $0<F(u)<u/2$, so $0<u_n\le 2^{-n}{u}_0$ and $\boxed{u_n\to 0}$. A cobweb alternates vertical steps to the graph and horizontal steps to the diagonal, approaching the origin monotonically.

More precisely $u_{n+1}/u_n\to 1/2$. The product $2^n{u}_n=u_0{\prod }_{j<n}(1+u_j^2/2{)}^{-1}$ has a finite strictly positive limit because ${\sum }_j{u}_j^2<\infty$. Thus $\boxed{u_n\sim C(u_0)2^{-n}}$ for a positive constant depending on the initial biological population.