= Solution
For the <cosine> mode in the question, the <Laplacian> acts by multiplication by $-s$, where
$$
s=k_{n,m}^2=\pi^2\left(\frac{n^2}{L_1^2}+\frac{m^2}{L_2^2}\right).
$$
These modes obey the <Neumann boundary conditions>. In the <chemotactic> term, put $q=1+u$, $c=\eta+v$. Then $\nabla c=\nabla v$, and $q\nabla c=\nabla v+u\nabla v$; the second product is a <quadratic polynomial> in <perturbation> amplitude. Thus the <linearized equations> are
$$
u_t=\alpha\nabla^2u-\beta\nabla^2v-u,\qquad
v_t=\nabla^2v+\kappa\eta u-\kappa v.
$$
Substituting the modal time dependence gives
$$
\lambda\begin{pmatrix}q_{n,m}\\c_{n,m}\end{pmatrix}
=\begin{pmatrix}-1-\alpha s&\beta s\\\kappa\eta&-\kappa-s\end{pmatrix}
\begin{pmatrix}q_{n,m}\\c_{n,m}\end{pmatrix}.
$$
Separating this <matrix> into its transport part and the reaction <Jacobian matrix> gives the required form, with the same $J$ as above.
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