Large Time Behaviour of the Solution of a Nonlinear Diffusion Problem in Anthropology

Authors

  • Ján Eliaš Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, Graz 8010, Austria
  • Danielle Hilhorst Laboratoire de Math´ematiques d’Orsay, Univ. Paris-Sud, CNRS, Universit´e Paris-Saclay, Orsay 91405, France
  • Masayasu Mimura Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4–21–1 Nakano, Nakano ku, Tokyo 164–8525, Japan

DOI:

https://doi.org/10.4208/jms.v51n3.18.04

Keywords:

Farmer–hunters model, reaction–diffusion system, degenerate diffusion, existence and uniqueness of the solution, exponential convergence to equilibrium.

Abstract

In this article we consider a reaction-diffusion model for the spreading of farmers in Europe, which was occupied by hunter-gatherers; this process is known as the Neolithic agricultural revolution. The spreading of farmers is modelled by a nonlinear porous medium type diffusion equation which coincides with the singular limit of another model for the dispersal of farmers as a small parameter tends to zero. From the ecological viewpoint, the nonlinear diffusion takes into account the population density pressure of the farmers on their dispersal. The interaction between farmers and hunter-gatherers is of the Lotka-Volterra prey-predator type. We show the existence and uniqueness of a global in time solution and study its asymptotic behaviour as time tends to infinity.

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Published

2018-08-30

Issue

Vol. 51 No. 3 (2018)

Section

Articles