called $r$ and $c$ is denoted by $D$, whereas in Young and Bettinger [@young1995] the growth coefficient is $R$ and the diffusion coefficient is $K$. These inputs to our code are given in units of yr$^{-1}$ and km$^2$/yr respectively.

Numerical methods and splitting {#sec:methodsandsplittings}
===============================

In one dimension, (\[eq:fisherKPP\]) can be solved using the `MatLab` function `pdepe`. In fact, if the system is two-dimensional but rotationally symmetric, `pdepe` can again be used with the radial part of the Laplace operator in cylindrical coordinates, $$\nabla^2 = \frac{1}{r} \frac{\partial}{\partial r} r \frac{\partial}{\partial r}
+ \mbox{non-contributing terms},$$ requiring only that one sets a `pdepe` parameter `m`=1. Although the `MatLab` function `pdepe` is robust, it cannot be generalized to arbitrary 2-D domains. However, it is a valuable control for testing more general solvers, and a more general solver is what we wish to explore here.

The finite-difference scheme
----------------------------

Since the map on which we will be working is a pixelized plane, an obvious method uses finite differences. First, however, let us examine the 1-D case for (\[eq:fisherKPP\]). In this situation, the second order derivative becomes a differencing operator in matrix form acting on the vector $\{ u_{j}, j=1,n \}$, where $u_{j} =
u(x_{0}+(j-1) 