= Fastest-growing Keller--Segel mode
{title2=$\ell_*=2\pi/\sqrt{z_*}$}
For the two-field <Keller--Segel model>, put $a=D_2$, $b=D_\rho$, $C=D_1f_0\geq0$, $c=\kappa-a_0f'_0$. The upper growth rate is $s_+(z)=-[c+(a+b)z]/2+\sqrt{[c+(b-a)z]^2+4Cz}/2$. When $C>\max(ac,-bc)$ its unique maximum occurs at positive $z_*$ satisfying $s_+'(z_*)=0$. For equal diffusion coefficients $a=b$, this gives $z_*=(C^2/a^2-c^2)/(4C)$. For $c<0$ and $C\leq-bc$, the fastest growth is homogeneous, with infinite <wavelength>. The finite-domain answer must maximize the growth rate over the permitted <Fourier modes>. Eliminating chemical dynamics instantaneously generally changes this selected scale.
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