 Phototactic concentration in a vertical channel

In a long channel $-a<x<a$, uniform horizontal mean swimming $V_s{K}_0{\mathbf{e}}_x$ and isotropic translational diffusion coefficient $D$ give zero transverse cell flux when $n^{\prime }=\beta n$, $\beta =V_s{K}_0/D$. Normalizing the transverse mean concentration to $n_0$ gives the displayed exponential. For $\beta a\gg 1$, cells form a layer of thickness ${\beta }^{-1}$ at the illuminated wall. This is a bulk approximation; exact closed-end cell conservation equation also requires the integrated axial cell flux to vanish.