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

$$\frac{\partial n(a, t)}{\partial t}+\frac{\partial n(a, t)}{\partial a}=-\mu(a) n(a, t)$$

with

$$n(0, t)=\int_{0}^{\infty} b(a) n(a, t) d a$$

where $\mu(a)$ is the age-dependent death rate and $b(a)$ is the birth rate per individual of age

Seek a similarity solution of the form $n(a, t)=e^{\gamma t} r(a)$ and show that

$$r(a)=r(0) e^{-\gamma a-\int_{0}^{a} \mu(s) d s}, \quad r(0)=\int_{0}^{\infty} b(s) r(s) d s .$$

Show also that if

$$\phi(\gamma)=\int_{0}^{\infty} b(a) e^{-\gamma a-\int_{0}^{a} \mu(s) d s} d a=1$$

then there is such a similarity solution. Give a biological interpretation of $\phi(0)$.

Suppose now that all births happen at age $a^{*}$, at which time an individual produces $B$ offspring, and that the death rate is constant with age (i.e. $\mu(a)=\mu)$. Find the similarity solution and give the condition for this to represent a growing population.