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

摘要

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 non-linear 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 exis-tence and uniqueness of a global in time solution and study its asymptotic behaviour as time tends to infinity.