I am studying a Brusselator model which includes inhomogeneous terms such as $\nabla ^2 x$ and $\nabla^2 y$ in the following way
\begin{aligned} &\partial_{\tau} x=a-(b+1) x+x^{2} y+d_{X} \nabla^{2} x, \\ &\partial_{\tau} y=b x-x^{2} y+d_{Y} \nabla^{2} y, \end{aligned}
where $a, d_X, d_Y$ and $b$ are constant parameters.
I am asked to study the linear stability of the steady homogeneous state (which I think is $x_{\mathrm{sth}} = a, y_{\mathrm{sth}} = b/a$) and show that if
$\eta \equiv \sqrt{d_{x} / d_{y}} >\left(\sqrt{1+a^{2}}-1\right)/a$
then a Hopf bifurcation occurs when changing the value of $b$. I am not asking for the solution of this last statement, I am just very confused because if the regime is the homogeneus stationary state, the factors $d_X$ and $d_Y$ shouldn't play any role, right?
In terms of the Jacobian matrix $J$, I obtain: trace($J$) = $b-1-a^2$ and det($J$) = $a^2$, so $d_X$ and $d_Y$ don't appear.
