and irregular domains such as geographical maps; show an application of this technique to the out-of-Africa dispersal of *Homo sapiens* by using net primal productivity (NPP) as a proxy for $\mathcal{K}(\mathbf{x},t)$ (Section \[sec:dispersal\]).

The Fisher/KPP equation {#sec:fisherkppeq}
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An intuitive way to get the Fisher/KPP equation [@fisher1937; @kolmogorov1937; @skellam1951] is as follows. A current $\mathbf{j}$ of particles (e.g., individuals) moving across an interface located at $\mathbf{x}$ is proportional to the gradient of the population density $p$ (“Fickian diffusion”) $$\mathbf{j} = -c\,\nabla p.$$ The rate of change of $p$ is then given by the mass balance equation [@reichl2004], which for Fickian diffusion reads $$\frac{\partial p}{\partial t} - c\,\nabla^2 p = \rho.$$ If $\rho=0$, this is the heat equation when $c = \mathcal D/2$ and $\mathcal D$ is the diffusion coefficient. For lack of a better model, we assume $c$ is a constant (Young & Bettinger [@young1995]). The source term $\rho$ is usually modeled by a logistic growth function, $\rho = \lambda p (1-p/\mathcal
K)$, yielding the Fisher/KPP equation $$\frac{\partial p}{\partial t} = \lambda
\left(1-\frac{p}{\mathcal{K}}\right) p + c\,\nabla^2 p,
\label{eq:fisherKPP0}$$ where $\mathcal{K}$ is called *carrying capacity* and $\lambda$ is the growth rate. In (\[eq:fisherKPP0\]) we assumed $\mathcal K$ is constant, but it 