Fastest-growing Keller--Segel mode (${\ell }_{\ast }=2\pi /\sqrt{z_{\ast }}$)

For the two-field Keller--Segel model, put $a=D_2$, $b=D_{\rho }$, $C=D_1{f}_0\ge 0$, $c=\kappa -a_0{f}_0^{\prime }$. 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 $z_{\ast }$ satisfying $s_{+}^{\prime }(z_{\ast })=0$. For equal diffusion coefficients $a=b$, this gives $z_{\ast }=(C^2/a^2-c^2)/(4C)$. For $c<0$ and $C\le -bc$, 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.