= Solution
Choose $X_*=k_2A_0/(2k_1)$, $t_*=1/(k_2A_0)$, $\ell_*=(D_X/(k_2A_0))^{1/2}$, and an independently selected positive $Y_*$ representing its reference or supplied <concentration>. Set $X=X_*u$, $Y=Y_*v$, $t=t_*\tau$, $x=\ell_*\xi$. Dividing the first equation by $X_*/t_*$ and the second by $Y_*/t_*$ yields the stated system with
$$
\boxed{\beta=\frac{k_3Y_*}{k_2A_0},\quad\alpha=\frac{2k_1k_4}{k_2k_3},\quad\gamma=\frac{k_4}{k_2},\quad\delta=\frac{k_3}{2k_1},\quad D=\frac{D_Y}{D_X}.}
$$
The parameter $\beta$ can be changed through the chosen or physically supplied reference $Y$ <concentration> while these other combinations remain fixed. The combinations satisfy $\alpha\delta=\gamma$ for this reaction mechanism, so they need not be algebraically independent. The spatial scale makes the $u$ <diffusion coefficient> one, and the time and $X$ scales balance linear autocatalytic growth against the quadratic loss.
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