\Delta x)$, $$\frac{d^{2} u}{dx^{2}} \rightarrow \frac{1}{(\Delta x)^{2}} A u,$$ where the matrix $A$ is $$A = \left( \begin{array}{cccccc}
-2 & 1 & 0 & 0 & \ldots & 0 \\
1 & -2 & 1 & 0 & \ldots & 0 \\
0 & 1 & -2 & 1 & \ldots &  \\
0 & 0 & 1 & \ldots &  &  \\ 
0 & 0 & \ldots & 0 & 1 & -2 \end{array} \right). \label{eq:Amatrix}$$ If $h$ is the time step, the Courant–-Friedrichs–-Lewy (CFL) parameter [@strang1986] is $$k = \frac{h}{2 (\Delta x)^{2}}.$$ An explicit integrator for (\[eq:fisherKPP\]) would require $k <
1/4$ [@leveque2007; @strang1986]. In our case, because the boundary conditions are so irregular on a map, we are less interested in a method of higher order than $2nd$ because stability is more important [@godunov1987].

Using this notation, the lowest order approximation is Euler’s method which estimates the next step $u(t+h)$ by $$u_{E} = u(t) + k \, A u(t) + h \, (1-u(t)) u(t),
\label{eq:Eulerestimate}$$ which should be considered a vector equation in $u(t) = \{ u_{j}(t),
j=1,n \}$. The logistic terms, which are diagonal, should be taken to mean $((1-u)u)_{j} = (1-u_{j})u_{j}$ for $j=1,2, \ldots, n$. 