A Stable Finite-Difference Scheme for Population Growth and Diffusion on a Map

Abstract

We describe a general Godunov-type splitting for numerical simulations of the Fisher–Kolmogorov–Petrovski–Piskunov growth and diffusion equation on a world map with Neumann boundary conditions. The procedure is semi-implicit, hence quite stable. Our principal application for this solver is modeling human population dispersal over geographical maps with changing paleovegetation and paleoclimate in the late Pleistocene. As a proxy for carrying capacity we use Net Primary Productivity (NPP) to predict times for human arrival in the Americas.

Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations

“…These results were verified numerically using Matlab's partial differential equation method pdepe, which was also used in [2]. This method uses the ODE15s solver to advance the solution forward in time.…”

“…Traveling wave solutions play an important role in the physical characterization of biological systems or other active matter [1], in modeling the growth and diffusion of a (human) population over large spatial scales and long times [2], and constitute a central theme in mathematical biology [3]. The search for exact wavefront solutions of differential equations that are modifications of the (one-dimensional) Fisher reaction-diffusion equation has a long history and has been quite active in recent years [4][5][6][7].…”