Keller--Segel model

A Keller--Segel model couples the density of moving cells to a chemical field that biases their motion. A common cell conservation law is $b_t=D{b}_{xx}-{\partial }_x(\chi (c)b{c}_x)$. Chemical dynamics depend on the application: the attractant may be produced by cells, or a nutrient may instead be consumed. The nutrient-consumption model $c_t=-kb$ with no nutrient diffusion gives travelling bacterial bands rather than prescribing the chemical externally.

For a homogeneous Keller--Segel model with nonnegative chemical production, linearized cell diffusion $D_2>0$, attraction coefficient $D_1>0$, and chemical diffusion $D_{\rho }>0$, put $c=\kappa -a_0{f}_0^{\prime }$ and $\kappa =(\rho k(\rho ){)}^{\prime }{|}_{{\rho }_0}$. A Fourier mode with squared wavenumber $z$ has determinant $z[D_2(c+D_{\rho }z)-D_1{f}_0]$. If $c\ge 0$, a growing mode on the infinite plane exists exactly when $D_1{f}_0>D_{2}c$, and $0<z<(D_1{f}_0-D_{2}c)/(D_2{D}_{\rho })$ is the unstable band. If $c<0$, the homogeneous chemical mode is already unstable. Dividing the threshold by $D_2\kappa$ to obtain a sum of two ratios is valid only when $\kappa >0$. Positivity of $k$ does not guarantee positivity of $(\rho k{)}^{\prime }$.

For the two-field Keller--Segel model, put $a=D_2$, $b=D_{\rho }$, $C=D_1{f}_0\ge 0$, $c=\kappa -a_0{f}_0^{\prime }$. The upper growth rate is $s_{+}(z)=-[c+(a+b)z]/2+\sqrt{[c+(b-a)z{]}^2+4Cz}/2$. When $C>\max (ac,-bc)$ its unique maximum occurs at positive $z_{\ast }$ satisfying $s_{+}^{\prime }(z_{\ast })=0$. For equal diffusion coefficients $a=b$, this gives $z_{\ast }=(C^2/a^2-c^2)/(4C)$. For $c<0$ and $C\le -bc$, the fastest growth is homogeneous, with infinite wavelength. The finite-domain answer must maximize the growth rate over the permitted Fourier modes. Eliminating chemical dynamics instantaneously generally changes this selected scale.

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 $N,k,a,C_{\infty }$.