TY - JOUR
T1 - Large Time Behaviour of the Solution of a Nonlinear Diffusion Problem in Anthropology
AU - Eliaš , Ján
AU - Hilhorst , Danielle
AU - Mimura , Masayasu
JO - Journal of Mathematical Study
VL - 3
SP - 309
EP - 336
PY - 2018
DA - 2018/08
SN - 51
DO - http://doi.org/10.4208/jms.v51n3.18.04
UR - https://global-sci.org/intro/article_detail/jms/12659.html
KW - Farmer–hunters model, reaction–diffusion system, degenerate diffusion, existence and uniqueness of the solution, exponential convergence to equilibrium.
AB - <p style="text-align: justify;">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.</p>