for $x \leq x_{L}$ and $x  \geq x_{H}$, but $\mathcal{K}=f_{r}$ in an $x$-direction desert. If the initial data $u(\mathbf{x},0)$ are widely distributed enough (nearly full $y$-width), near the middle of the domain, i.e. $y = 0$, the problem looks basically one-dimensional. Thus we can again use `pdepe` from `MatLab` to compute the behavior of the one-directional wave as it passes through the desert, and compare this solution to the behavior of our Godunov method near this same center line. According to KPP [@kolmogorov1937], the velocity in the desert is the same as in the ${\mathcal K}=1$ region, and far enough from transients it should be approximately $$V = \sqrt{2 a},$$ where parameter $a=\frac{{\rm d} \rho}{{\rm d} u}(u=0)$. In our case $\rho(u) = u (1 - u/\mathcal{K})$, so $\frac{{\rm d}\rho}{\rm d
  u}(0) = 1$. Thus, when not entering or leaving the desert, the velocity should be $\sqrt{2}$ [@kolmogorov1937]. Figure \[fig:gaptestresults\] shows that the velocity agrees with the KPP prediction, and is nearly constant except for short transients entering and exiting the desert, as expected. Our metric for measuring this velocity is to find, on the leading edge of the wave front, the position $x_{1/2}$ where $u(x_{1/2})=\mathcal K/2$.

If the jump in 