 Nutrient-consuming chemotactic travelling band

For a Keller--Segel model with $c_t=-kb$, a positive-speed travelling wave $B(z),C(z)$, $z=x-vt$, satisfies $v{C}^{\prime }=kB$ and $D{B}^{\prime }+vB-\alpha B{C}^{\prime }/C=0$ when the integrated bacterial flux constant is zero. Writing $\mu =\alpha /D>1$ and $\xi =v(z-z_0)/D$, its positive band solution isTo derive it, integrate $B^{\prime }/B=\mu C^{\prime }/C-v/D$ to obtain $B=K{C}^{\mu }{e}^{-vz/D}$, then separate $v{C}^{\prime }=kB$. 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.