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

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.

A positive fixed point requires $\beta +u_{\ast }^{\gamma }=\alpha$, so it exists exactly when $\boxed{\alpha >\beta }$ and is $u_{\ast }=(\alpha -\beta {)}^{1/\gamma }$. Differentiating the map at it givesFor $1<\gamma \le 2$ and $0<\beta /\alpha <1$, this fixed-point multiplier is strictly between $-1$ and $1$. Hence the positive fixed point is linearly asymptotically stable, including $\gamma =2$; positivity of $\beta$ makes the lower bound strict.

The fixed-point multiplier decreases continuously with $\gamma$ and first reaches the unit circle at $-1$. Solving $1-{\gamma }_{\ast }(1-\beta /\alpha )=-1$ givesIt is stable for $1<\gamma <{\gamma }_{\ast }$ and unstable for $\gamma >{\gamma }_{\ast }$. At the critical value linear stability is inconclusive. The crossing at $-1$ is the local signature of a period-doubling bifurcation, rather than a crossing through $+1$.

At low density the per-generation growth factor is approximately $\alpha /\beta$. The power $\gamma$ controls how sharply crowding reduces recruitment or survival when resources become limiting; at high density $F(u)\sim \alpha u^{1-\gamma }$. Larger $\gamma$ therefore means a stronger overcompensation in a population model.

The flip threshold says that sufficiently strong density feedback can destabilize a constant positive biological population and create alternating generations, under the usual nondegeneracy conditions. A local loss of fixed-point stability alone does not establish chaos, nor does it prove every ecological system with stronger crowding must follow the same route.

A fixed point of $F$ is also a fixed point of $F\circ F$. By the chain rule,If the original fixed-point multiplier has modulus greater than one, its square also exceeds one, so the same fixed point is unstable for the twice-iterated map. New stable fixed points of the twice-iterated map, if present, must be different points; they can represent a genuine two-cycle of the original map.