The number density of a population of amoebae is $n(\mathbf{x}, t)$. The amoebae exhibit chemotaxis and are attracted to high concentrations of a chemical which has concentration $a(\mathbf{x}, t)$. The equations governing $n$ and $a$ are

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

where the constants $n_{0}, \alpha, \beta, \gamma$ and $D$ are all positive.

(i) Give a biological interpretation of each term in these equations and discuss the sign of $\chi(n)$.

(ii) Show that there is a non-trivial (i.e. $a \neq 0, n \neq 0$ ) steady-state solution for $n$ and $a$, independent of $\mathbf{x}$, and show further that it is stable to small disturbances that are also independent of $\mathbf{x}$.

(iii) Consider small spatially varying disturbances to the steady state, with spatial structure such that $\nabla^{2} \psi=-k^{2} \psi$, where $\psi$ is any disturbance quantity. Show that if such disturbances also satisfy $\partial \psi / \partial t=p \psi$, where $p$ is a constant, then $p$ satisfies a quadratic equation, to be derived. By considering the conditions required for $p=0$ to be a possible solution of this quadratic equation, or otherwise, deduce that instability is possible if

$$\beta \chi_{0} n_{0}>2 \alpha n_{0}^{2} D+\gamma+2\left(2 D \alpha n_{0}^{2} \gamma\right)^{1 / 2}$$

where $\chi_{0}=\chi\left(n_{0}\right)$.

(iv) Explain briefly how your conclusions might change if an additional geometric constraint implied that $k^{2}>k_{0}^{2}$, where $k_{0}$ is a given constant.