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

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

with the boundary condition

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

where $b(a)$ and $d(a)$ are, respectively, the per capita age-dependent birth and death rates.

(a) What is the biological interpretation of the boundary condition?

(b) Solve equation (1) assuming a separable form of solution, $n(a, t)=A(a) T(t)$.

(c) Use equation (2) to obtain a necessary condition for the existence of a separable solution to the full problem.

(d) For a birth rate $b(a)=\beta e^{-\lambda a}$ with $\lambda>0$ and an age-independent death rate $d$, show that a separable solution to the full problem exists and find the critical value of $\beta$ above which the population density grows with time.