A stable finite-difference scheme for growth and diffusion on a map

We describe a general Godunov type splitting for numerical simulations of the Fisher/Kolmogorov-Petrovski-Piskunov growth and diffusion equation in two spatial dimensions. In particular, the method is appropriate for modeling population growth and dispersal on a terrestrial map. The procedure is semi-implicit, hence quite stable, and approximately second order accurate, excluding boundary condition complications. It also has low memory requirements and shows good performance. We illustrate an application of this solver: global human dispersal in the late Pleistocene, modeled via growth and diffusion over geographical maps of paleovegetation and paleoclimate.