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Take the physical diffusion and consumption constants $D,k>0$. Use the travelling wave coordinate $z=x-vt$, with $v>0$. Then ${\partial }_t=-vd/dz$ 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 $\chi =\alpha /C$ givesPrimes 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 giveswith 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 $-D{B}^{\prime }+\alpha B{C}^{\prime }/C$ equals $vB$. The Keller--Segel model then has the useful first-order reductionThe 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.