Keller--Segel aggregation threshold ($D_1{f}_0>D_2(\kappa -a_0{f}_0^{\prime })$)

For a homogeneous Keller--Segel model with nonnegative chemical production, linearized cell diffusion $D_2>0$, attraction coefficient $D_1>0$, and chemical diffusion $D_{\rho }>0$, put $c=\kappa -a_0{f}_0^{\prime }$ and $\kappa =(\rho k(\rho ){)}^{\prime }{|}_{{\rho }_0}$. A Fourier mode with squared wavenumber $z$ has determinant $z[D_2(c+D_{\rho }z)-D_1{f}_0]$. If $c\ge 0$, a growing mode on the infinite plane exists exactly when $D_1{f}_0>D_{2}c$, and $0<z<(D_1{f}_0-D_{2}c)/(D_2{D}_{\rho })$ is the unstable band. If $c<0$, the homogeneous chemical mode is already unstable. Dividing the threshold by $D_2\kappa$ to obtain a sum of two ratios is valid only when $\kappa >0$. Positivity of $k$ does not guarantee positivity of $(\rho k{)}^{\prime }$.

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.