 Solution

In this discrete age-structured population model, individuals of age $a\ge 1$ present during the breeding season each produce $b_a$ newborns, so the number entering age zero isAn age-$a$ individual survives the following winter with probability $1-{\mu }_a$ and is then age $a+1$ in the next year, givingTogether with the initial age distribution, these equations determine every later cohort.

Substitute $n_{a,t}=r_a{\gamma }^t$ into the survival equation. Thenand iteration givesThe birth equation becomesA nonzero population mode therefore requires the Discrete Euler-Lotka equation

For nonnegative vital rates with at least one surviving reproductive age and enough decay to make the series finite, $\varphi$ is continuous and strictly decreasing on $\gamma >0$, tends to infinity as $\gamma \downarrow 0$, and tends to zero as $\gamma \to \infty$. The intermediate value theorem gives one positive solution, and strict monotonicity makes it unique.

The value $\varphi (1)$ is the expected lifetime offspring count under these fixed rates. Since $\varphi$ decreases, $\varphi (1)>1$ places the root at $\gamma >1$ and the population grows geometrically; $\varphi (1)<1$ places it at $\gamma <1$ and the population shrinks. Equality gives replacement, $\gamma =1$.