= Speed of a nutrient-consuming chemotactic band
{title2=$v=Nk/(aC_\infty)$}
For a <nutrient-consuming chemotactic travelling band> in a tube of cross-sectional area $a$, let $N=a\int_{\mathbb R}B(z)dz$. Integrating $vC'=kB$ between depleted nutrient behind and $C_\infty$ ahead gives $vC_\infty=kN/a$, hence $v=Nk/(aC_\infty)$. This is a nutrient budget: advancing by a unit length supplies $aC_\infty$ nutrient, while the bacteria consume at total rate $kN$. Diffusion and chemotactic sensitivity affect the band shape and admissibility, but not this speed at fixed $N,k,a,C_\infty$.
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