Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 6C (Mathematical Biology)

Probability enters state $n$ through a birth from $n-1$ or a pair-death event from $n+2$, and leaves through either event at state $n$. Thus the Kramers-Moyal master equation iswith probabilities and impossible transition rates taken as zero outside the nonnegative states.

A birth changes $n$ by $+1$, while a death event changes it by $-2$. Applying these increments to the first moment givesConsequently every steady state must satisfy

Taylor-expand the gain terms:and, writing $Q(n)=\beta n^{2}P(n)$,After cancellation of the loss terms, the second-order Kramers-Moyal expansion isHence the constant-birth pair-annihilation process has

The deterministic drift vanishes at the positive stable pointWhen $n_{\ast }$ is large, fluctuations sample only a narrow range around it, so the linear noise approximation replacesThe resulting Ornstein-Uhlenbeck stationary density is the stated normal distribution withand

For the Gaussian approximation,The extra term reflects the subleading error made by centring the linear approximation at the deterministic fixed point. It is smaller than ${\mu }^2$ by relative order $1/\mu$, so the exact steady-state condition from part (a) is recovered at leading order. A higher-order expansion shifts the mean and restores the moment balance more accurately.

The approximation is reliable whenThus one needs $\lambda /\beta \gg 1$: births must support a large typical population, with jumps of one or two individuals small compared with the population scale.