Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 6E (Mathematical Biology)

The birth-death master equation must include jumps of size two on death. For $n\ge 2$,At the boundary states it is

Differentiate $⟨f(n)⟩={\sum }_{n\ge 0}{f}_n{p}_n$ and substitute the three master equations. Reindexing every gain term by its source state gives one contribution per possible transition:This is the generator identity for a continuous-time Markov chain applied to the test function $f$.

Set $f_n=n$. A birth changes $f$ by $1$, an ordinary pair death by $-2$, and singleton death by $-1$. HenceWhen $D=2d$ this closes toso the stationary mean is ${\mu }_{\ast }=b/(2d)$.

For $f_n=n^2$, the birth increment is $2n+1$, whileWith $D=2d$, the generator identity givesAt stationarity $\mu =b/(2d)$, and solving this equation for the variance yieldsFor $b,d>0$ the chain is irreducible on the nonnegative integers, so its stationary distribution has $0<p_1<1$. Therefore