= Nutrient-consuming chemotactic travelling band
{title2=$B(z),C(z)$}
For a <Keller--Segel model> with $c_t=-kb$, a positive-speed <travelling wave> $B(z),C(z)$, $z=x-vt$, satisfies $vC'=kB$ and $DB'+vB-\alpha BC'/C=0$ when the integrated bacterial flux constant is zero. Writing $\mu=\alpha/D>1$ and $\xi=v(z-z_0)/D$, its positive band solution is
$$
C=C_\infty(1+e^{-\xi})^{-1/(\mu-1)},\qquad
B=\frac{v^2C_\infty}{kD(\mu-1)}e^{-\xi}(1+e^{-\xi})^{-\mu/(\mu-1)}.
$$
To derive it, integrate $B'/B=\mu C'/C-v/D$ to obtain $B=KC^\mu e^{-vz/D}$, then separate $vC'=kB$. The positive integration constant is a translation of the wave. Concentration rises monotonically from zero behind to $C_\infty$ ahead; $B$ vanishes at both ends and has a unique maximum at $\xi=-\log(\mu-1)$. The condition $\mu>1$ describes this exponential-tail branch; it is not asserted to exclude every limiting or weak travelling wave at other parameter values.
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