+ \frac{k}{4}
A_{x} \right) u^{\star \star}.
\label{eq:godunovwithKD}\end{aligned}$$

Again, as in (\[eq:yoshida\]), multiple tridiagonal system solves must be carried out: $N_y$ $x$-direction solutions (\[eq:godunovwithKA\]), $N_x$ $y$-direction solutions (\[eq:godunovwithKC\]), and finally, another $N_y$ $x$-direction solutions (\[eq:godunovwithKD\]). Hopefully no confusion will result from the notation: $k$ is the CFL parameter, while $\mathcal{K}$ is the (space-dependent) carrying capacity.

A desert test of space-dependent capacity {#sec:gaptest}
-----------------------------------------

Now we are in uncharted territory. To assess if the solver (\[eq:godunovwithK\]) works, let us examine a problem where we can compute a solution by independent means. The test setup follows below. Its motivations will be explained further in Section \[sec:noisyNPPmaps\].

The desert test:

1.  for $-y_{0} \leq y \leq y_{0}$, $\mathcal{K}=\mathcal{K}(x)$ is independent of $y$;

2.  for $x_{0} \leq x \leq x_{L}$, let $\mathcal{K}=1$, while for $x_{L} < x < x_{H}$, let $\mathcal{K}=f_{r}$, where the fraction $0 < f_{r} \leq 1$ defines a [*desert*]{} (inhospitable region) in the domain. Finally, for $x_{H} \leq x \leq x_{1}$, again set $\mathcal{K}=1$;

3.  initialize $u(x,y,t=0)$ to a strip midway between $x_{0}$ and $x_{L}$, then run the simulation to study the traveling wave behavior across the $[ x_{L}, x_{H} ]$ desert.

In other words, the variable carrying capacity domain has $\mathcal{K}=1$ 