Stability of Chemical Reactions Given the following reactions:
$$A + X \xrightarrow{k_{1}} 2 X$$
$$Y + X \xrightarrow{k_{2}} 2 Y$$
$$Y \xrightarrow{k_{3}} B$$
I was able to write the following rate equations for the concentrations:
$$\partial_{t}C_{X} =  k_{1} C_{A}^{0} - k_{2}C_{X}C_{Y} + D_{1} \nabla^2_{r} C_{X}$$
$$\partial_{t}C_{X} =  k_{2}C_{X}C_{Y} - k_{3}C_{Y} + D_{2} \nabla^2_{r} C_{Y}$$
For a similar example, in "A Modern Course in Statistical Physics" by Linda E. Reichl they redefine this variables to simplify the expression.

However I can't do the same for my expressions, is there anything I'm missing?
The redefinition of variables is quite useful, it makes our stability analysis much more simpler.
 A: If the goal is to eliminate most of the leading coefficients (except for the diffusion terms), then the way to systematically do this would be to define your new variables in terms of your old ones, each one multiplied by an undetermined scaling factor:
$$
\tilde{t} = \alpha t \quad A^0 = \beta_A C_A^0 \quad X = \beta_X C_X \quad Y = \beta_Y C_Y \quad \tilde{D}_1 = \gamma_1 D_1 \quad \tilde{D}_2 = \gamma_2 D_2
$$
Rewrite your rate equations in terms of these new variables,  multiply both sides of each equation by an appropriate constant so that the coefficient of some term (the $\partial C_i/\partial \tilde{t}$ terms, for example) is 1.  Each of the remaining terms will then have a coefficient in terms of the rescaling factors $\{ \alpha, \beta_A, \beta_X, \beta_Y, \gamma_1, \gamma_2\}$, the reaction rates, and the diffusion constants. The requirement that each of these coefficients is equal to 1 will impose an equation relating these quantities to each other, which you can then solve for the rescaling factors.
For example, the first equation, under these rescalings, becomes
\begin{align*}
\frac{\alpha}{\beta_X} \partial_\tilde{t} X &= \frac{k_1}{\beta_A} A^0 - \frac{k_2}{\beta_X \beta_Y} XY + \frac{1}{\gamma_1 \beta_X} \tilde{D}_1 \nabla^2 X \\
\partial_\tilde{t} X &= \frac{k_1 \beta_X}{\beta_A \alpha} A^0 - \frac{k_2}{\alpha \beta_Y} XY + \frac{1}{\gamma_1 \alpha} \tilde{D}_1 \nabla^2 X \\
\end{align*}
which implies that we want $k_1 \beta_x/\beta_A \alpha = 1$, $k_2/\beta_Y \alpha = 1$, and $\gamma_1 \alpha = 1$.  Applying this same procedure to the second rate equation will give three more equations, which you can then solve for $\{ \alpha, \beta_A, \beta_X, \beta_Y, \gamma_1, \gamma_2\}$.
