$\mathcal K$ is deep enough ($f_r\ll1$), however, the integrator will fail without some regularization. The parameters for the results shown in Figure \[fig:gaptestresults\] are $\mathcal{K}=1$ above and below the desert, but $\mathcal{K}=0.01$ in the desert. For this case, $\frac{u}{\mathcal{K}} \gg \frac{2}{h}$ gives an instability with unpleasant sign changes, and a regularization scheme has to be used. We show this in the next subsection.

Regularization against holes and noise {#sec:noisyNPPmaps}
--------------------------------------

An obvious problem with deep holes, jumps or ragged noise in the carrying capacity $\mathcal{K}$ is this: the right hand side of equation (\[eq:godunovwithKC\]) (as well as (\[eq:godunovwithKTC\]) in the next Section) contains the term $$\left( 1 + \frac{k}{2} A_{y} + \frac{h}{2} (1-\frac{u^{\star}}{\mathcal{K}})
\right) u^{\star},
\label{eq:badeulerterm}$$ which for large ${u^{\star}}/{\mathcal{K}}$ basically determines the sign of the $u(t+h)$ on the left side of (\[eq:godunovwithKC\]) (likewise (\[eq:godunovwithKTC\])). Since both $u^{\star}$ and $\mathcal{K}$ are positive, if $\mathcal{K}$ is very small in some pixel, then we may have $$\frac{u^{\star}}{\mathcal{K}} \gg
\frac{2}{h},$$ in which case $u(t+h)$ changes sign. This is unphysical, so we would like to regularize the term (\[eq:badeulerterm\]). To do so, we need to find a monotonically increasing function, call it $g(x)$, such that $$g(x) \sim \left\{ 
\begin{array}{ll}
x & \mbox{when} \, x 