= Phototactic concentration in a vertical channel
{title2=$n(x)=n_0a\beta e^{\beta x}/\sinh(a\beta)$}
In a long channel $-a<x<a$, uniform horizontal mean swimming $V_sK_0\mathbf e_x$ and isotropic translational <diffusion coefficient> $D$ give zero transverse cell flux when $n'=\beta n$, $\beta=V_sK_0/D$. Normalizing the transverse mean concentration to $n_0$ gives the displayed exponential. For $\beta a\gg1$, 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.
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