description – via a reaction-diffusion equation – of an analogous but one-dimensional problem: the propagation of an advantageous genetic mutation within an already-present population, situated along a coast line. Kolmogorov, Petrovskii and Piskunov [@kolmogorov1937] were more general; in particular, their analysis treated the two-dimensional case. Such a model (called Fisher/KPP in the following) was first applied to the dispersal and growth of a population by Skellam [@skellam1951], and serves as an important control for designing and validating other more complex spatiotemporal population models [@callegari2013]. Coupling population dynamics with models of large-scale changes in continental topography, climate, and ecosystem productivity is essential to understand the role of environmental constraints on patterns of genetic, phenetic, and cultural variation among human populations [@callegari2013].

Here we present a stable and efficient finite-difference solver for the Fisher/ KPP equation on 2-D domains of arbitrary shape (e.g. geographical maps), and show how it can be extended to include environmental fluctuations. In a brief outline of our paper, we will: review the derivation of the Fisher/KPP equation (Section \[sec:fisherkppeq\]); develop finite-difference schemes in 1 and 2 dimensions for constant environmental carrying capacity, $\mathcal{K}$ (Section \[sec:methodsandsplittings\]); extend the scheme to allow for space- and time-dependent $\mathcal{K}(\mathbf{x},t)$ (Section \[sec:fisherkpp\]) 