The population density $n(a, t)$ of individuals of age $a$ at time $t$ satisfies

$$\frac{\partial n}{\partial t}+\frac{\partial n}{\partial a}=-\mu(a) n(a, t), \quad n(0, t)=\int_{0}^{\infty} b(a) n(a, t) d a$$

where $\mu(a)$ is the age-dependent death rate and $b(a)$ is the birth rate per individual of age $a$. Show that this may be solved with a similarity solution of the form $n(a, t)=e^{\gamma t} r(a)$ if the growth rate $\gamma$ satisfies $\phi(\gamma)=1$ where

$$\phi(\gamma)=\int_{0}^{\infty} b(a) e^{-\gamma a-\int_{0}^{a} \mu(s) d s} d a$$

Suppose now that the birth rate is given by $b(a)=B a^{p} e^{-\lambda a}$ with $B, \lambda>0$ and $p$ is a positive integer, and the death rate is constant in age (i.e. $\mu(a)=\mu)$. Find the average number of offspring per individual.

Find the similarity solution, and find the threshold $B^{*}$ for the birth parameter $B$ so that $B>B^{*}$ corresponds to a growing population.