$i_{x} = j_{y} = 0$. Neighboring pixels with coordinates $(i+i_{x},j+ j_{x})$ having zero value (e.g. water), $i+i_{x} > {\rm N_X}$ or $i+i_{x} < 1$, are ignored. In Mercator projection maps, $Y$ coordinates are periodic in the longitude direction). For each $(i,j)-$pixel to be smoothed, a total of each accepted (non-zero) neighbor’s weight was kept and the resulting total was normalized appropriately. The choice (\[eq:weightfunction\]) is an approximate Gaussian weight $\exp{( -\Delta x^{2} - \Delta y^{2})}$, cut off at distances $|i_{x}| \geq L$ and $|j_{y}| \geq L$ or at uninhabitable pixels. Figure \[fig:siberiaCCrawvssmooth\] shows an example of the effect of a low-pass filter (5 cells half-width, eq. (\[eq:weightfunction\])) smoothing on one of the maps used in this study. A comparison between regularization and smoothing is shown on the right hand plot of Figure  \[fig:outofafrica0\].

Time-dependent capacity maps {#sec:fisherkppvarKxt}
----------------------------

Fluctuations in climate produce environmental changes in vegatation, sea levels, opening/closing of land bridges, waxing/waning of ice sheets, and perturbations to habitable areas in general. Thus, time-dependent environments compel us to extend our procedure (\[eq:godunovwithK\]) for both space- and time-dependent $\mathcal{K}(\mathbf{x},t)$ (see Section \[ss:timeinterpol\]).

Since (\[eq:godunovwithK\]) is basically the trapezoidal method (see section 5.3 in [@leveque2007]), the modification for a time-dependent $\mathcal 