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

Large Time Behaviour of the Solution of a Nonlinear Diffusion Problem in Anthropology
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Year:    2018

Author:    Ján Eliaš, Danielle Hilhorst, Masayasu Mimura

Journal of Mathematical Study, Vol. 51 (2018), Iss. 3 : pp. 309–336

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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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

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

Journal of Mathematical Study, Vol. 51 (2018), Iss. 3 : pp. 309–336

Published online:    2018-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    28

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

Author Details

Ján Eliaš

Danielle Hilhorst

Masayasu Mimura

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