= Solution
In this discrete <age-structured population model>, individuals of age $a\geq1$ present during the breeding season each produce $b_a$ newborns, so the number entering age zero is
$$
n_{0,t}=\sum_{a=1}^{\infty}n_{a,t}b_a.
$$
An age-$a$ individual survives the following winter with probability $1-\mu_a$ and is then age $a+1$ in the next year, giving
$$
n_{a+1,t+1}=(1-\mu_a)n_{a,t}.
$$
Together with the initial age distribution, these equations determine every later cohort.
Substitute $n_{a,t}=r_a\gamma^t$ into the survival equation. Then
$$
r_{a+1}=\frac{1-\mu_a}{\gamma}r_a,
$$
and iteration gives
$$
r_a=r_0\gamma^{-a}\prod_{i=0}^{a-1}(1-\mu_i).
$$
The birth equation becomes
$$
r_0=\sum_{a=1}^{\infty}r_ab_a
=r_0\sum_{a=1}^{\infty}
\left[\prod_{i=0}^{a-1}(1-\mu_i)\right]\gamma^{-a}b_a.
$$
A nonzero population mode therefore requires the <Discrete Euler-Lotka equation>
$$
\boxed{\phi(\gamma)=1},
\qquad
\phi(\gamma)=\sum_{a=1}^{\infty}
\left[\prod_{i=0}^{a-1}(1-\mu_i)\right]\gamma^{-a}b_a.
$$
For nonnegative vital rates with at least one surviving reproductive age and enough decay to make the series finite, $\phi$ is continuous and strictly decreasing on $\gamma>0$, tends to infinity as $\gamma\downarrow0$, and tends to zero as $\gamma\to\infty$. The <intermediate value theorem> gives one positive solution, and strict monotonicity makes it unique.
The value $\phi(1)$ is the expected lifetime offspring count under these fixed rates. Since $\phi$ decreases, $\phi(1)>1$ places the root at $\gamma>1$ and the population grows geometrically; $\phi(1)<1$ places it at $\gamma<1$ and the population shrinks. Equality gives replacement, $\gamma=1$.
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