---
abstract: '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 $O(\Delta x^{2}) + O(\Delta t^{2})$ 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.'
author:
- |
    W. P. Petersen, S. Callegari, N. Tkachenko,\
    J. D. Weissmann, Ch. P. E. Zollikofer\
    Anthropological Institute and Museum, University of Zürich
bibliography:
- 'splitting.bib'
title: 'A stable finite-difference scheme for growth and diffusion on a map'
---

[**Keywords**]{}: reaction-diffusion equations , finite-difference solvers, population dynamics

Introduction
============

There is an increasing interest in modeling population dynamics at large spatial and temporal scales, for example the modern human out-of-Africa dispersal [@eriksson2012; @henn2012; @nikitas2005; @young1995] or Neanderthal dispersal and extinction [@callegari2013]. These models are required to interpret local and global patterns of genetic, phenetic and cultural variation [@bouckaert2012; @eriksson2012; @manica2007; @pinhasi2005; @relethford2004].

Fisher [@fisher1937] studied the 