Stability interval of the survival-augmented Ricker equilibrium
= Stability interval of the survival-augmented Ricker equilibrium
At the positive equilibrium,
$$
g'(x_*)=1-(1-r)\log\frac{k}{1-r}.
$$
The fixed point is strictly <linearly stable> when
$$
1-r<k<(1-r)e^{2/(1-r)}.
$$
The upper endpoint is also locally <asymptotically stable>, as the <nonlinear stability at the Ricker flip threshold> shows; hence the full local asymptotic-stability interval includes that endpoint. The lower endpoint has no positive equilibrium.