A population of aerobic bacteria swims in a laterally-infinite layer of fluid occupying $-\infty<x<\infty,-\infty<y<\infty$, and $-d / 2<z<d / 2$, with the top and bottom surfaces in contact with air. Assuming that there is no fluid motion and that all physical quantities depend only on $z$, the oxygen concentration $c$ and bacterial concentration $n$ obey the coupled equations

$$\begin{aligned} \frac{\partial c}{\partial t} &=D_{c} \frac{\partial^{2} c}{\partial z^{2}}-k n \\ \frac{\partial n}{\partial t} &=D_{n} \frac{\partial^{2} n}{\partial z^{2}}-\frac{\partial}{\partial z}\left(\mu n \frac{\partial c}{\partial z}\right) \end{aligned}$$

Consider first the case in which there is no chemotaxis, so $n$ has the spatially-uniform value $\bar{n}$. Find the steady-state oxygen concentration consistent with the boundary conditions $c(\pm d / 2)=c_{0}$. Calculate the Fick's law flux of oxygen into the layer and justify your answer on physical grounds.

Now allowing chemotaxis and cellular diffusion, show that the equilibrium oxygen concentration satisfies

$$\frac{d^{2} c}{d z^{2}}-\frac{k n_{0}}{D_{c}} \exp \left(\mu c / D_{n}\right)=0$$

where $n_{0}$ is a suitable normalisation constant that need not be found.