= Solution
Differentiate the map:
$$
g'(x)=r+ke^{-\lambda x}(1-\lambda x).
$$
At the positive equilibrium, $ke^{-\lambda x_*}=1-r$, so
$$
g'(x_*)
=1-(1-r)\lambda x_*
=1-(1-r)\log\left(\frac{k}{1-r}\right).
$$
The <fixed point stability for an iteration> requires $|g'(x_*)|<1$. Existence already makes the logarithm positive, so the upper inequality is automatic. The lower inequality gives
$$
(1-r)\log\left(\frac{k}{1-r}\right)<2.
$$
Combining it with existence yields the <stability interval of the survival-augmented Ricker equilibrium>:
$$
\boxed{1-r<k<(1-r)\exp\left(\frac2{1-r}\right)}.
$$
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