Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 6C / b / Solution

For jumps $n\to n+1$ at rate $\lambda$ and $n\to n-2$ at rate $\beta n^2$, the diffusion approximation has
$$
A(n)=\lambda-2\beta n^2,
\qquad
B(n)=\lambda+4\beta n^2.
$$
The stable deterministic population is $n_*=\sqrt{\lambda/(2\beta)}$. Its linear-noise stationary approximation is normal with mean $n_*$ and variance $3n_*/4$.


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