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A_{\lambda} =

\begin{pmatrix}

-\mu\lambda k^2 - k^2 - s & i\tau k & i\tau k - i\beta k^3\\

i\lambda k & \lambda + Dk^2 & -\alpha k^2\\

i\lambda k & 0 & \lambda

\end{pmatrix}

$$

The steady state(s) associated with the model is stable if $\Re\lambda(k^2) < 0$ for all $k^2 \geq 0$. The values of $k^2$ for which there exists an instability window, $\Re\lambda(k^2) > 0$, in which pattern is formed, are given by the range of $k^2_{c-} < k^2 < k^2_{c+}$ where $k^2_{c\pm}$ are the zeros of $c(k^2)$ such that

$$

k^2_{c\pm} = \frac{(\alpha + D)\tau - D\pm\sqrt{((\alpha + D)\tau - D)^2 - 4s\beta D^2}}{2\beta D}.

$$

Use Mathematica or otherwise to find the roots of the polynomial and graph the relationship (dispersion curves), $\lambda(m)$, $m = k/\pi$, and $m\in [0.05,10.05]$.

So I used mathematica to find lambda.

I am not sure what I am supposed to do with k since the solution for k^2 is giving.

I have no idea what to do now.

View attachment 88