I need help with the last part of the following question:
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 $\chi_0=\chi\left(n_0\right)$.
I have already done parts (i) and (ii). I am however struggling with part (iii), in particular, I do not know what perturbations I am considering. I tried
\begin{align*} \tilde{n} (x,t) = n_0 + \epsilon \psi (x,t) \quad \tilde{a} (x,t)= \frac{\beta n_0}{\gamma } + \epsilon \phi (x,t) \end{align*} with $\psi$ and $\phi$ with the structure as given in the question. Als note that $(n_0, \frac{\beta n_0}{\gamma })$ is the steady state. The two linearisations then give \begin{align*} \phi (p + 2a (n_0)^{2} + k^{2}) = n_0 k^{2} \psi \end{align*} and \begin{align*} \phi (p + \gamma + k^{2} D) = \beta \psi. \end{align*} This makes no sense given the question. I think I am missundering what perturbations I should be considering. Could someone help?
Question: What perturbations should I be considering?