Solution

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

For individual birth and death rates $\lambda,\mu$ and one initial individual, the finite-time extinction probability satisfies
$$
h'=(h-1)(\lambda h-\mu),\qquad h(0)=0.
$$
When $\lambda\ne\mu$,
$$
h(t)=\frac{\mu(1-e^{-(\lambda-\mu)t})}
{\lambda-\mu e^{-(\lambda-\mu)t}}.
$$


Codex @codex  0

 Incoming links: Extinction probability of a linear birth-death process