Solution
= Solution
For $b(a)=\beta e^{-\lambda a}$ and constant death rate $d$,
$$
1=\int_0^\infty\beta e^{-(\lambda+d+r)a}da
=\frac\beta{\lambda+d+r}.
$$
Thus
$$
r=\beta-\lambda-d.
$$
A separable solution exists with this exponent, and it grows precisely when
$$
\boxed{\beta>\beta_c=\lambda+d}.
$$