 Solution

Choose $X_{\ast }=k_2{A}_0/(2k_1)$, $t_{\ast }=1/(k_2{A}_0)$, ${\ell }_{\ast }=(D_X/(k_2{A}_0){)}^{1/2}$, and an independently selected positive $Y_{\ast }$ representing its reference or supplied concentration. Set $X=X_{\ast }u$, $Y=Y_{\ast }v$, $t=t_{\ast }\tau$, $x={\ell }_{\ast }\xi$. Dividing the first equation by $X_{\ast }/t_{\ast }$ and the second by $Y_{\ast }/t_{\ast }$ yields the stated system withThe 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.