Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 28K / b

Starting with one individual, let $S$ be the first event time, of density
$(\lambda +\mu )e^{-(\lambda +\mu )s}$. At that event there is a death with conditional probability $\mu /(\lambda +\mu )$, causing extinction immediately, or a birth with probability $\lambda /(\lambda +\mu )$, leaving two independent descendant populations. The branching property therefore gives

Changing variables in the integral equation,Differentiation gives

Separating variables in
$h^{\prime }=(h-1)(\lambda h-\mu )$ with $h(0)=0$ gives the extinction probability of a linear birth-death processFor $\lambda >\mu$ this tends to $\mu /\lambda$; for $\lambda <\mu$ it tends to one, as expected.