boundaries that arise when solving Fisher/KPP on geographical maps. Using the same Godunov-type splitting described above, it is more straightforward to do the simulations on a map than might be expected.

Maps with space-dependent capacity {#sec:fisherkppvarK}
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In our scaling of (\[eq:fisherKPP\]), the maximum population density at $\mathbf{x}$ is unity. Thus, in the following we will use a scaled carrying capacity $0 < \mathcal{K}(\mathbf{x}) \leq 1$. Our algorithm (\[eq:yoshida\]) can be modified in a straightforward way for the case that $\mathcal{K}(\mathbf{x},t)$ also depends explicitly on time: see Section \[sec:fisherkppvarKxt\]. First, however, let us deal with the time independent $\mathcal{K}(\mathbf{x})$ case, $$\frac{\partial u}{\partial t} = (1-\frac{u}{\mathcal{K}}) u +
\frac{1}{2}\,\nabla^2 u,
\label{eq:fisherwithK}$$ (we dropped the $\mathbf x$ dependence of $\mathcal K$ for convenience of notation), for which the Godunov splitting (\[eq:yoshida\]) now becomes

\[eq:godunovwithK\] $$\begin{aligned}
\left( 1 - \frac{k}{4} A_{x} \right) u^{\star} & = \left( 1 + \frac{k}{4} A_{x} \right) u(t) \label{eq:godunovwithKA} \\
u_{E} &= u^{\star} + k A_{y} u^{\star} + h (1-\frac{u^{\star}}{\mathcal{K}}) 
u^{\star} \label{eq:godunovwithKB} \\
\left( 1 - \frac{k}{2} A_{y} - \frac{h}{2} (1-\frac{u_{E}}{\mathcal{K}}) \right) u^{\star \star} & =
\left( 1 + \frac{k}{2} A_{y} + \frac{h}{2} (1-\frac{u^{\star}}{\mathcal{K}}) \right) u^{\star}
\label{eq:godunovwithKC} \\
\left( 1 - \frac{k}{4} A_{x} \right) u(t+h) & = \left( 1 