Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 6A / b / Solution

Differentiate the map:At the positive equilibrium, $k{e}^{-\lambda x_{\ast }}=1-r$, soThe fixed point stability for an iteration requires $|g^{\prime }(x_{\ast })|<1$. Existence already makes the logarithm positive, so the upper inequality is automatic. The lower inequality givesCombining it with existence yields the stability interval of the survival-augmented Ricker equilibrium: