= Solution
Differentiate the <squared-denominator population map>:
$$
F'(x)=\frac{r(b-x)}{(b+x)^3}.
$$
At the positive <fixed point>, $b+\bar x=\sqrt r$, so its <fixed-point multiplier> is
$$
F'(\bar x)=\frac{2b}{\sqrt r}-1.
$$
Since $0<b/\sqrt r<1$, we have $-1<F'(\bar x)<1$. A small perturbation $\varepsilon_t$ obeys $\varepsilon_{t+1}=F'(\bar x)\varepsilon_t+O(\varepsilon_t^2)$, and hence shrinks locally. Thus \b[the positive <fixed point> is locally asymptotically stable] by <fixed point stability for an iteration>.
Back to article page
