 Solution

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.