Solution
= Solution
At the threshold, the <determinant> quadratic has a double zero. Its vertex lies at $s=(\chi\alpha n_0-\beta D_n-\gamma n_0D_c)/(2D_nD_c)$. Substituting the equality case of part (iii) gives
$$
\boxed{k_c^2=\sqrt{\frac{\beta\gamma n_0}{D_nD_c}},\qquad
k_c=\left(\frac{\beta\gamma n_0}{D_nD_c}\right)^{1/4}.}
$$
The mode is neutral exactly at onset and grows when the chemotactic coefficient increases beyond that threshold.