Logarithmic chemotactic sensitivity ($\chi (c)=\alpha /c$)

For positive chemical concentration $c$, taking $\chi (c)=\alpha /c$ makes the chemotaxis drift velocity $\chi (c)c_x=\alpha {\partial }_x\log c$. Cells then respond to a relative chemical gradient. This singular response can remain finite where the concentration tends to zero if its logarithmic derivative has a finite limit.

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.

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 $v{C}^{\prime }=kB$ between depleted nutrient behind and $C_{\infty }$ ahead gives $v{C}_{\infty }=kN/a$, hence $v=Nk/(a{C}_{\infty })$. This is a nutrient budget: advancing by a unit length supplies $a{C}_{\infty }$ 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 $N,k,a,C_{\infty }$.