= Solution
The coexistence <Jacobian matrix> calculated in part (a) has negative diagonal entries and positive <determinant>. For any spatial mode with squared <wave number> $k^2\geq0$, the linearized <matrix> is $J_*-k^2\operatorname{diag}(1,\eta)$. Its <trace> is more negative than before, and its <determinant> is
$$
\boxed{\frac q3+\frac{2(\eta+q)}3k^2+\eta k^4>0.}
$$
Hence both <eigenvalues> retain negative real parts for every mode and every $\eta>0$. This is the mechanism by which <negative self-response excludes two-species Turing instability>, excluding diffusion-driven instability using only the reaction <Jacobian matrix> already computed, together with the diagonal diffusive <damping>. In particular unequal <diffusion> is not enough to produce a <Turing pattern> when both local self-responses are negative in this two-species competitive model.
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