The number density of a population of cells is $n(\mathbf{x}, t)$. The cells produce a chemical whose concentration is $C(\mathbf{x}, t)$ and respond to it chemotactically. The equations governing $n$ and $C$ are

$$\begin{aligned} \frac{\partial n}{\partial t} &=\gamma n\left(n_{0}-n\right)+D_{n} \nabla^{2} n-\chi \nabla \cdot(n \nabla C) \\ \frac{\partial C}{\partial t} &=\alpha n-\beta C+D_{c} \nabla^{2} C \end{aligned}$$

(i) Give a biological interpretation of each term in these equations, where you may assume that $\alpha, \beta, \gamma, n_{0}, D_{n}, D_{c}$ and $\chi$ are all positive.

(ii) Show that there is a steady-state solution that is stable to spatially invariant disturbances.

(iii) Analyse small, spatially-varying perturbations to the steady state that satisfy $\nabla^{2} \phi=-k^{2} \phi$ for any variable $\phi$, and show that a chemotactic instability is possible if

$$\chi \alpha n_{0}>\beta D_{n}+\gamma n_{0} D_{c}+\left(4 \beta \gamma n_{0} D_{n} D_{c}\right)^{1 / 2}$$

(iv) Find the critical value of $k$ at which the instability first appears as $\chi$ is increased.