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$$

a(k^2)\lambda^2 + b(k^2)\lambda + c(k^2) = 0

$$

where

\begin{align}

a(k^2) =& \mu k^2\notag\\

b(k^2) =& (\beta + \mu D)k^4 - (2\tau - 1)k^2 + s\notag\\

c(k^2) =& \beta Dk^6 - \{(\alpha + D)\tau - D\}k^4 + sDk^2\notag

\end{align}

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 $m\in [0.05,10.05]$, for the 4th and 8th modes $(m)$.

\begin{array}{ccccccc}

\text{mode} & \tau & \mu & D & \alpha & s & \beta\\

4 & 1.15 & 1.0 & 0.002 & 0.0030 & 140.0 & 0.0060\\

8 & 1.20 & 1.0 & 0.001 & 0.0020 & 840.0 & 0.0020

\end{array}