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Travelling waves in a reaction-diffusion system modelling farmer and hunter-gatherer interaction in the Neolithic transition in Europe

Part of: Parabolic equations and systems Partial differential equations

Published online by Cambridge University Press:  18 June 2019

JE-CHIANG TSAI ,
M. HUMAYUN KABIR  and
MASAYASU MIMURA
JE-CHIANG TSAI
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan email: tsaijc.math@gmail.com National Center for Theoretical Sciences, Taipei, Taiwan
M. HUMAYUN KABIR
Affiliation:
Department of Mathematics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh
MASAYASU MIMURA
Affiliation:
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Tokyo, Japan Graduate School of Integrated Sciences for Life, Hiroshima University, Higashi-Hiroshima City, Hiroshima, Japan
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Abstract

Recently we have proposed a monostable reaction-diffusion system to explain the Neolithic transition from hunter-gatherer life to farmer life in Europe. The system is described by a three-component system for the populations of hunter-gatherer (H), sedentary farmer (F1) and migratory one (F2). The conversion between F1 and F2 is specified by such a way that if the total farmers F1 + F2 are overcrowded, F1 actively changes to F2, while if it is less crowded, the situation is vice versa. In order to include this property in the system, the system incorporates a critical parameter (say F0) depending on the development of farming technology in a monotonically increasing way. It determines whether the total farmers are either over crowded (F1 + F2 >F0) or less crowded (F1 + F2 <F0) ( [9, 20]). Previous numerical studies indicate that the structure of travelling wave solutions of the system is qualitatively similar to the one of the Fisher-KPP equation, that the asymptotically expanding velocity of farmers is equal to the minimal velocity (say cm(F0)) of travelling wave solutions, and that cm(F0) is monotonically decreasing as F0 increases. The latter result suggests that the development of farming technology suppresses the expanding velocity of farmers. As a partial analytical result to this property, the purpose of this paper is to consider the two limiting cases where F0 = 0 and F0 → ∞, and to prove cm(0)>cm(∞).

Keywords

Neolithic transition in Europefarmers and hunter-gatherer modelthree-component systemtravelling waveminimal velocity

MSC classification

Primary: 35K57: Reaction-diffusion equations
Secondary: 35B40: Asymptotic behavior of solutions 35B20: Perturbations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 3 , June 2020 , pp. 470 - 510
Copyright
© Cambridge University Press 2019

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