= Keller--Segel aggregation threshold
{c}
{title2=$D_1f_0>D_2(\kappa-a_0f'_0)$}
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 $c=\kappa-a_0f'_0$ and $\kappa=(\rho k(\rho))'|_{\rho_0}$. A <Fourier mode> with squared <wavenumber> $z$ has <determinant> $z[D_2(c+D_\rho z)-D_1f_0]$. If $c\geq0$, a growing mode on the infinite plane exists exactly when $D_1f_0>D_2c$, and $0<z<(D_1f_0-D_2c)/(D_2D_\rho)$ 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 $(\rho k)'$.
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