= Solution
The independent additive noises give diagonal diffusion $\kappa=\operatorname{diag}(b_1^2,b_2^2)$. Thus
$$
\boxed{
\begin{aligned}
\partial_tP={}&-\partial_{x_1}\!\left[
(r_1x_1-r_1c_1x_1x_2^2-d_1x_1^2)P\right]
+\frac{b_1^2}{2}\partial_{x_1}^2P\\
&-\partial_{x_2}\!\left[
(r_2x_2-r_2c_2x_2x_1^2-d_2x_2^2)P\right]
+\frac{b_2^2}{2}\partial_{x_2}^2P.
\end{aligned}}
$$
The mixed derivatives of $\kappa^{-1}a$ are
$$
\partial_{x_2}\frac{a_1}{b_1^2}
=-\frac{2r_1c_1x_1x_2}{b_1^2},
\qquad
\partial_{x_1}\frac{a_2}{b_2^2}
=-\frac{2r_2c_2x_1x_2}{b_2^2}.
$$
They agree precisely when
$$
\boxed{r_1c_1b_2^2=r_2c_2b_1^2}.
$$
Solved by gpt-5.6-sol high.
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