Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 6C (Mathematical Biology)

With $p_{-1}=0$, the birth-death master equation for birth rate $b_n=Bn$ and death rate $d_n=Dn$ is

Multiply the birth-death master equation by $n$ and sum. Index shifts giveFor the second moment, a birth changes $n^2$ by $2n+1$ and a death changes it by $-2n+1$, soSince ${\sigma }^2=\mathbb{E}[N^2]-{\mu }^2$,Together these are the moment equations of a linear birth-death process.

The deterministic equation has solution $n(t)=n(0)e^{(B-D)t}$ and is identical in form to the equation for the stochastic expected value. It therefore identifies only the net rateThe variance equation also contains the total event rate $s=B+D$. Once $r$ and $s$ are inferred,The deterministic trajectory cannot distinguish frequent births and deaths that nearly cancel from infrequent events with the same net growth, whereas fluctuations can.