= Solution
Take the physical diffusion and consumption constants $D,k>0$. Use the <travelling wave> coordinate $z=x-vt$, with $v>0$. Then $\partial_t=-v\,d/dz$ and $\partial_x=d/dz$. The original PDF has the bacterial drift flux $\chi b c_x$: the TeX transcription's $\chi^b$ is erroneous. Substituting the <logarithmic chemotactic sensitivity> $\chi=\alpha/C$ gives
$$
\boxed{-vB'=DB''-\alpha\left(\frac{BC'}{C}\right)',\qquad
vC'=kB.}
$$
Primes here denote differentiation with respect to $z$. The second equation already shows that nutrient increases towards the front whenever the bacterial density is positive. For the positive band, $C>0$ at every finite $z$, so division by $C$ is legitimate, despite its zero limit far behind.
Integrating the first equation once gives
$$
DB'+vB-\alpha\frac{BC'}{C}=J,
$$
with constant $J$. For a localized band with no bacterial flux at infinity, $B$ and its flux terms vanish ahead, so $J=0$. Equivalently the laboratory flux $-DB'+\alpha BC'/C$ equals $vB$. The <Keller--Segel model> then has the useful first-order reduction
$$
\boxed{\frac{B'}B=\frac\alpha D\frac{C'}C-\frac vD,\qquad
C'=\frac kv B.}
$$
The far-field conditions choose this zero-flux integration constant; it should not be imposed for an arbitrary nonlocalized travelling solution. We solve its positive localized branch in the next part.
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