Here's a very simple problem students can do on their own to investigate how higher-order effects in linear parabolic equations can alter the wave-speed properties.

Consider a linear partial differential equation exhibiting non-zero
4th derivative. This is applicable to local expansion of kernels in
integral-differential equations. Murray talks about this some.

\[
\frac{\partial N}{\partial t} = RN + D\frac{\partial^{2}N}{\partial x^{2}}
+ K \frac{\partial^{4}N}{\partial x^{4}}.
\]
\[
\dot{u} = u + u'' + K u''''.
\]

This is important in the limiting expansion of integro-differential
equations. One can look for exponential traveling wave with intensity
$\theta$ so that
\[
u = \bar{C} e^{\theta(x-ct)},
\]
and this gives
\[
0 = \bar{C} e^{\theta(x-ct)} \left( K \theta^{4} + \theta^{2} +
c\theta + 1 \right).
\]
Applying elimination theory with criticality condition
$\partial c / \partial \theta = 0$, $4K\theta^{3}+2\theta+c_{min}=0$,
so the minimum wave speed $c_{min}$ must satisfy
\[
(9Kc_{min}^{2}-8K+2)\theta +c_{min}(1+12K)=0.
\]
This leads to nasty 8th degree polynomial in $c_{min}$ and $K.$ However,
using groebner basis theory, one finds a rotated hyperbola and an
ellipse/hyperbola govern the solution:
\begin{eqnarray}
2 \theta^{2} + 3 \theta c_{min} + 4 &=& 0 \label{eq:1}\\
9 c_{min}^{2} - 4(12K+1) \theta^{2} &=& 32 \label{eq:2}
\end{eqnarray}
These allow the problem to be reduced to simple 1-d root-finding.

The question is whether there exists a minimum speed for these waves.
Well, for given $c$, it is straight forward to show when $K>0$ there
exist $0$ or $2$ invasion speeds for given wave intensities $\theta$
by inspecting
$$ K \theta^{4} + \theta^{2} + 1 = - c \theta.$$
When $K<0$ there are will be $2$ or $4$ solutions for
$\theta$, but inspecting
\[ c = - \frac{ K \theta^{4} + \theta^{2} + 1 }{ \theta },\]
we see that for every possible $c$, there exists at least
one $\theta > 0$ -- there is no band-gap in allowed speeds.
Thus, we can not define any wave speeds for $K < 0$ without
changing model fundamentals.

In this case ($K\geq 0$), $\frac{\partial c_{min}}{\partial
K}>0$, such that $c_{min}$ grows without bound. For $K>0,
c_{min}>2$ there is the very nice asymptotic approximation
that $$K \sim \frac{27}{256}c_{min}^{4} -
\frac{9}{16}c_{min}^{2} + \frac{1}{2}$$ which isn't too bad
even when $K\sim 0$. For small $K$, $c_{min} = 2 + K +
O(K^2)$, which recovers the standard diffusion behavior.