a_{1} = \frac{α _{1}}{V} y (y - 1), a_{2} = \frac{α _{2}}{V} x (x - 1), a_{3} = \frac{α _{3} V}{ε}, a_{4} = \frac{α _{4}}{ε V} x y .

E_{x}^{r} f (x, y) = f (x + r, y)

and similarly for

E_{y}

, the fast and slow forward operators are

L_{0}^{*} = (E_{y}^{- 1} - 1) α_{3} V + (E_{y}^{+ 1} - 1) \frac{α _{4} x y}{V},

L_{1}^{*} = (E_{x}^{- 1} E_{y}^{+ 2} - 1) \frac{α _{1} y ( y - 1 )}{V} + (E_{x}^{+ 1} - 1) \frac{α _{2} x ( x - 1 )}{V} .

Each shift operator acts on everything to its right, including the propensity. Birth of

Y

and consumption of

Y

X + Y \to X

are fast;

2 Y \to X

and

2 X \to X

are slow. The slow species is

X

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