Solution
= Solution
The population map is
$$
x_{n+1}=g(x_n),
\qquad
g(x)=x\bigl(r+ke^{-\lambda x}\bigr).
$$
A positive equilibrium satisfies
$$
1=r+ke^{-\lambda x_*},
$$
and hence
$$
x_*=\frac1\lambda\log\left(\frac{k}{1-r}\right).
$$
It is strictly positive exactly when
$$
\boxed{k>1-r}.
$$
This is the nonzero equilibrium of the <survival-augmented Ricker map>.