Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 356 / 3 / a / Solution

Use stochastic mass-action propensitiesWriting $E_x^{r}f(x,y)=f(x+r,y)$ and similarly for $E_y$, the fast and slow forward operators areEach shift operator acts on everything to its right, including the propensity. Birth of $Y$ and consumption of $Y$ by $X+Y\to X$ are fast; $2Y\to X$ and $2X\to X$ are slow. The slow species is $X$.