our procedure to handle an $\mathbf{x}$-dependent $\mathcal{K}$, specifically eq. (\[eq:godunovwithK\]). In order to regularize our solver against bad behavior when dealing with $\mathcal{K}$ maps inferred from real-world data, in Section \[sec:gaptest\] we studied both regularization and the expected constant velocity of the traveling wave. Except for small transition regions entering or leaving a region of low carrying capacity, the velocity ($\sqrt{2}$ in our scaling) is indeed constant. In Section \[sec:fisherkppvarKxt\], we went further to develop a scheme for the situation with both $\mathbf{x}$ and $t$ dependent carrying capacity $\mathcal{K}$ via eq.(\[eq:godunovwithKT\]).

Finally, this scheme has been applied to a prototypical case in population dynamics: the out-of-Africa dispersal of [*Homo sapiens*]{}. On the Mercator projected world map, by using vegetation net primal productivity as a proxy for carrying capacity, Figure \[fig:outofafrica0\] shows that by the regularization of $\mathcal{K}$ in space via eq. (\[eq:regularizer0\]) but interpolating in time yields stable and reasonable results. In fact, the results showing ancestor arrival in NE India at roughly 40 kya, then crossing the Bering Strait before 10 kya, and multiple routes into South Asia [@reyescentento2014] are very encouraging. Honesty requires that we admit our size (408km)$^{2}$ pixels do not resolve the two crossing points at 