Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 5E / Solution

Substitution of the separated exponential profile into the age-structured population equation gives $r^{\prime }=-(\gamma +\mu (a))r$. ThereforeFor $C\ne 0$, the birth boundary condition cancels $C$ and becomes the Euler-Lotka equationThe survival probability to age $a$ is $e^{-\mu a}$ for constant death rate. Hence the expected lifetime offspring count, or net reproduction rate, isThe Gamma integral evaluates the Euler-Lotka equation as $Bp!/(\lambda +\mu +\gamma {)}^{p+1}=1$, requiring $\lambda +\mu +\gamma >0$. Its real solution and profile areThus $B^{\ast }=(\lambda +\mu {)}^{p+1}/p!$, and $B>B^{\ast }$ gives positive growth. The profile is a nonzero formal separated solution for all positive $B$; a finite total population additionally requires $\gamma +\mu >0$, equivalently $(Bp!{)}^{1/(p+1)}>\lambda$. A decaying-in-time mode need not have an integrable age profile.