= Solution
For a <Laplacian> mode with $\nabla^2=-k^2$, the linearized <chemotaxis> system is
$$
\frac d{dt}\binom uv=
\begin{pmatrix}-\gamma n_0-D_nk^2&\chi n_0k^2\\\alpha&-\beta-D_ck^2\end{pmatrix}\binom uv.
$$
Its <trace> is negative for all $k$. Writing $s=k^2$, its <determinant> is
$$
D_nD_cs^2+(\beta D_n+\gamma n_0D_c-\chi\alpha n_0)s+\beta\gamma n_0.
$$
Instability occurs when this <determinant> is negative, because then one real <eigenvalue> is positive. A quadratic with positive leading and constant coefficients is negative for some $s>0$ precisely when its middle coefficient is negative and its discriminant positive. These conditions combine to
$$
\boxed{\chi\alpha n_0>\beta D_n+\gamma n_0D_c+2\sqrt{\beta\gamma n_0D_nD_c}.}
$$
This proves the printed threshold. In a bounded spatial domain, only its permitted <Laplacian> eigenmodes are available; the continuous-$k$ threshold describes the onset when wavenumbers can vary freely.
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