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Volume 51, Issue 3
Large Time Behaviour of the Solution of a Nonlinear Diffusion Problem in Anthropology

Ján Eliaš, Danielle Hilhorst & Masayasu Mimura

DOI: 10.4208/jms.v51n3.18.04

J. Math. Study, 51 (2018), pp. 309-336.

Published online: 2018-08

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  • 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.

  • Keywords

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

  • AMS Subject Headings

35K57, 35Q92, 92D40

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

jan.elias@uni-graz.at (Ján Eliaš)

danielle.hilhorst@math.u-psud.fr (Danielle Hilhorst)

mimura.masayasu@gmail.com (Masayasu Mimura)

  • BibTex
  • RIS
  • TXT
@Article{JMS-51-309, author = {Eliaš , JánHilhorst , Danielle and Mimura , Masayasu}, title = {Large Time Behaviour of the Solution of a Nonlinear Diffusion Problem in Anthropology}, journal = {Journal of Mathematical Study}, year = {2018}, volume = {51}, number = {3}, pages = {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.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v51n3.18.04}, url = {http://global-sci.org/intro/article_detail/jms/12659.html} }
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 -

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.

Eliaš , JánHilhorst , Danielle and Mimura , Masayasu. (2018). Large Time Behaviour of the Solution of a Nonlinear Diffusion Problem in Anthropology. Journal of Mathematical Study. 51 (3). 309-336. doi:10.4208/jms.v51n3.18.04
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