An animal population has annual dynamics, breeding in the summer and hibernating through the winter. At year $t$, the number of individuals alive who were born a years ago is given by $n_{a, t}$. Each individual of age $a$ gives birth to $b_{a}$ offspring, and after the summer has a probability $\mu_{a}$ of dying during the winter. [You may assume that individuals do not give birth during the year in which they are born.]

Explain carefully why the following equations, together with initial conditions, are appropriate to describe the system:

$$\begin{aligned} n_{0, t} &=\sum_{a=1}^{\infty} n_{a, t} b_{a} \\ n_{a+1, t+1} &=\left(1-\mu_{a}\right) n_{a, t}, \end{aligned}$$

Seek a solution of the form $n_{a, t}=r_{a} \gamma^{t}$ where $\gamma$ and $r_{a}$, for $a=1,2,3 \ldots$, are constants. Show $\gamma$ must satisfy $\phi(\gamma)=1$ where

$$\phi(\gamma)=\sum_{a=1}^{\infty}\left(\prod_{i=0}^{a-1}\left(1-\mu_{i}\right)\right) \gamma^{-a} b_{a}$$

Explain why, for any reasonable set of parameters $\mu_{i}$ and $b_{i}$, the equation $\phi(\gamma)=1$ has a unique solution. Explain also how $\phi(1)$ can be used to determine if the population will grow or shrink.