Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 14C (Mathematical Biology)

The left side is the flux of newborn trees through size zero. Every tree of size $s$ produces seedlings at rate $b(s)$, so total births are ${\int }_0^{\infty }b(s)n(s,t)ds$, giving the boundary condition.

Substitution gives $(g\tilde{n}{)}^{\prime }=-(\mu +r)\tilde{n}$, henceThe boundary condition yields the Euler-Lotka equation

The mean lifetime production is $(B/\mu )e^{-\mu s_0/g}$. A time-independent population requires it to equal one, so $B=\mu e^{\mu s_0/g}$.

The Euler-Lotka equation becomes $1=(B/g)e^{-(\mu +r)s_{\ast }/g}$, hence $r=(g/s_{\ast })\log (B/g)-\mu$.