Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / B2 / b / Solution

Differentiate the squared-denominator population map:At the positive fixed point, $b+\bar{x}=\sqrt{r}$, so its fixed-point multiplier isSince $0<b/\sqrt{r}<1$, we have $-1<F^{\prime }(\bar{x})<1$. A small perturbation ${\epsilon }_t$ obeys ${\epsilon }_{t+1}=F^{\prime }(\bar{x}){\epsilon }_t+O({\epsilon }_t^2)$, and hence shrinks locally. Thus the positive fixed point is locally asymptotically stable by fixed point stability for an iteration.