Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 6E (Mathematical Biology)

The boundary condition says that all newborn individuals enter at age zero, and that their influx equals the total birth rate obtained by summing the age-specific births $b(a)n(a,t)$ over the population.

Put $n(a,t)=A(a)T(t)$. Separation givesfor a constant $r$. Hence

Substitution into the renewal boundary condition and cancellation of $A_{0}T(t)$ gives the Euler-Lotka equationThis is the necessary condition selecting the growth exponent $r$.

For $b(a)=\beta e^{-\lambda a}$ and constant death rate $d$,ThusA separable solution exists with this exponent, and it grows precisely when