Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-19T05:06:31.527Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Partial differential equations Parabolic equations and systems

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ammerman, A. J. & Cavalli-Sforza, L. L. (1971) Measuring the rate of spread of early farming in Europe, Man 6, 674–688.CrossRefGoogle Scholar
Ammerman, A. J. & Cavalli-Sforza, L. L. (1984) The Neolithic transition and the genetics of populations in Europe. Princeton University Press, Princeton.CrossRefGoogle Scholar
Aoki, K., Shida, M., & Shigesada, N. (1996) Traveling wave solutions for the spread of farmers into a region occupied by Hunter Gatherers, Theoretical Population of Biology 50, 1–17.CrossRefGoogle Scholar
Berestycki, H., Diekmann, O., Nagelkerke, C. J. & Zegeling, P. A. (2009) Can a species keep pace with a shifting climate? Bull. Math. Biol., 71, 399–429.CrossRefGoogle ScholarPubMed
Berestycki, H., Hamel, F., Kiselev, A. & Ryzhik, L. (2005) Quenching and propagation in KPP reaction-diffusion equations with a heat loss, Arch. Rational Mech. Anal. 178, 57–80.Google Scholar
Chikhi, L., Nichols, R. A., Barbujani, G. & Beaumont, M. A. (2002) Y genetic data support the Neolithic demic diffusion model, PNAS 99, 11008–11013.CrossRefGoogle Scholar
Dancer, E. N. (1985) On positive solutions of some pairs of differential equations, II, J. Differential Equations 60, 236–258.CrossRefGoogle Scholar
Dunbar, S. R. (1984) Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in R4, Trans. Amer. Math. Soc. 286, 557–594.Google Scholar
Elias, J., Kabir, M. H., & Mimura, M. (2018) On the well-posedness of a dispersal model for farmers and hunter-gatherers in the Neolithic transition, Math. Models Methods Appl. Sci. 28, 195–222.CrossRefGoogle Scholar
Fisher, R. A. (1937) The wave of advance of advantageous genes, Ann. Eugenics 7, 353–369.CrossRefGoogle Scholar
Fort, J. (2009) Mathematical modelling of the Neolithic transition: a review for non-mathematicians, In: Dolukhanov, P. M., Sarson, G. R. and Shukurov, A. M. (editors), The East European Plain on the Eve of Agriculture. British Archaeological Reports. International Series, Vol. 1964, Archaeopress, Oxford, pp. 211–216.Google Scholar
Fort, J. & Méndez, C. (1999) Time-delayed theory of the Neolithic transition in Europe, Phys. Rev. Lett. 82, 867–870.CrossRefGoogle Scholar
Fort, J. & Méndez, C. (2002) Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment, Rep. Prog. Phys., 65, 895–954.CrossRefGoogle Scholar
Fu, S.-C. (2014) The existence of traveling wave fronts for a reaction-diffusion system modelling the acidic nitrate-ferroin reaction, Quart. Appl. Math. 72, 649–664.CrossRefGoogle Scholar
Fu, S.-C. (2016) Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl. 435, 20–37.CrossRefGoogle Scholar
Gkiasta, M., Russell, T., Shennan, S., & Steele, J. (2003) Neolithic transition in Europe: the radiocarbon record revisited, Antiquity 77, 45–62.CrossRefGoogle Scholar
Hartman, P. (1982) Ordinary Differential Equations, Birkhäuser, Boston.Google Scholar
Hilhorst, D. & Kim, Y.-J. (2016) Diffusive and inviscid traveling waves of the Fisher equation and nonuniqueness of wave speed, Appl. Math. Lett. 60, 28–35.CrossRefGoogle Scholar
Huang, W. (2016) A geometric approach in the study of traveling waves for some classes of non-monotone reaction-diffusion systems, J. Differential Equations 260, 2190–2224.CrossRefGoogle Scholar
Kabir, M. H., Mimura, M. & TSAI, J.-C. (2018) Spreading waves in a farmers and hunter-gatherers model of the Neolithic transition in Europe, Bull. Math. Biol. 80, 2452–2480.CrossRefGoogle Scholar
Kolmogorov, A. N., Petrovsky, I. G., & Piscounov, N. S. (1937) A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ. Math. Mech. 1, 1–26 (1937).Google Scholar
Méndez, V. & Camacho, J. (1997) Dynamics and thermodynamics of delayed population growth, Physical Review E 55(6), 6476–6482.CrossRefGoogle Scholar
Pao, C. V. (1992) Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York.Google Scholar
Zhang, T., Wang, W. & Wang, K. (2016) Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differential Equations 260, 2763–2791.CrossRefGoogle Scholar