Skip to main content
SpringerLink
Log in

Spreading Waves in a Farmers and Hunter-Gatherers Model of the Neolithic Transition in Europe

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

The Neolithic transition began the spread of early agriculture throughout Europe through interactions between farmers and hunter-gatherers about 10,000 years ago. Archeological evidence produced by radiocarbon dating indicates that the expanding velocity of farming is roughly constant all over Europe. Theoretical understanding of such evidence has been performed from mathematical modeling viewpoint. However, the expanding velocity determined by existing modeling approaches is faster than the observed velocity. For understanding this difference, we propose a three-component reaction–diffusion system which consists of two different types of farmers (sedentary and migratory) and hunter-gatherers from the viewpoint of the influence of farming technology. Our purpose is to study the relation between the expanding velocity of farmers and the farming technology parameter (say, \(\gamma \)). In this paper, we mainly focus on the one-dimensional traveling wave solution with minimal velocity and show that the minimal velocity decreases, as \(\gamma \) increases. This can be compatible with the observed velocity when farming technology is developed. Our results suggest that the reason for the slowdown of the Neolithic transition might be related to the increase in the development of farming technology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Similar content being viewed by others

References

  • Ammerman AJ, Cavalli-Sforza LL (1971) Measuring the rate of spread of early farming in Europe. Man 6:674–688

    Article  Google Scholar 

  • Ammerman AJ, Cavalli-Sforza LL (1984) The Neolithic transition and the genetics of populations in Europe. Princeton University Press, Princeton

    Book  Google Scholar 

  • Aoki K, Shida M, Shigesada N (1996) Traveling wave solutions for the spread of farmers into a region occupied by hunter gatherers. Theor Popul Biol 50:1–17

    Article  MATH  Google Scholar 

  • Britton P, Kent A (2018) The coming of farming. In: TIMEMAPS PREMIUM. http://www.timemaps.com/encyclopedia/farming/. Accessed 2018

  • Faustino S-G, Maini PK, Kappos ME (1996) A shooting argument approach to a sharp-type solution for nonlinear degenerate Fisher-KPP equations. IMA J Appl Math 57(3):211–221

    Article  MathSciNet  MATH  Google Scholar 

  • Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugen 7:353–369

    MATH  Google Scholar 

  • Fort J, Méndez C (2002) Wavefronts in time-delayed reaction–diffusion systems. Theory and comparison to experiment. Rep Prog Phys 65:895–954

    Article  Google Scholar 

  • Fort J (2009) Mathematical modelling of the Neolithic Transition: a review for non-mathematicians. In: Dolukhanov PM, Sarsons GR, Shukurov AM (eds) The East European plain on the eve of agriculture. British archaeological reports. International Series 1964, pp 211–216

  • Fort J, Méndez C (1999) Time-delayed theory of the Neolithic transition in Europe. Phys Rev Lett 82:867–870

    Article  Google Scholar 

  • Gkiasta M, Russell T, Shennan S, Steele J (2003) Neolithic transition in Europe: the radiocarbon record revisited. Antiquity 77(295):45–62

    Article  Google Scholar 

  • Gray P, Scott SK (1983) Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability. Chem Eng Sci 38(1):29–43

    Article  Google Scholar 

  • Hilhorst D, Mimura M, Weidenfeld R (2003) On a reaction–diffusion system for a population of hunters and farmers. In: Colli P, Verdi C, Visintin A (eds) Free boundary problems. ISNM international series of numerical mathematics, vol 147, pp 189–196

  • Kolmogorov AN, Petrovsky IG, Piscounov NS (1937) A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Bull Mosc Univ Math Mech 1:1–26

    Google Scholar 

  • Lewis MA, Schmitz G (1996) Biological invasion of an organism with separate mobile and stationary states: modeling and analysis. Forms 11:1–25

    MATH  Google Scholar 

  • Méndez V, Camacho J (1997) Dynamics and thermodynamics of delayed population growth. Phys Rev E 55(6):6476–6482

    Article  Google Scholar 

  • Mimura M (2004) Pattern formation in consumer-finite resource reaction–diffusion system. Publ RIMS Kyoto Univ 40:1413–1431

    Article  MathSciNet  MATH  Google Scholar 

  • Mimura M, Sakaguchi H, Matsushita M (2000) Reaction–diffusion modelling of bacterial colony patterns. Phys A 282(1):283–303

    Article  Google Scholar 

  • Murray JD (2002) Mathematical biology: I. An introduction, 3rd edn. Springer, New York

    MATH  Google Scholar 

  • Sherratt JA (1998) On the transition from initial data to travelling waves in the Fisher–KPP equation. Dyn Stab Syst 13(2):167–174

    Article  MathSciNet  MATH  Google Scholar 

  • Weisdorf JL (2005) From foraging to farming: explaining the Neolithic Revolution. J Econ Surv 19:561–586

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous referees for their useful suggestions and comments which helped improve the exposition of the paper. MHK acknowledges the support of GCOE program of MIMS, Meiji University, Japan, during doctoral study. MM is partially supported by JSPS KAKENHI Grants Nos. 15K13462 and 16H01728. JCT is supported by MOST and NCTS of Taiwan.

Author information

Authors and Affiliations

  1. Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Tokyo, Japan

    M. H. Kabir & M. Mimura

  2. Department of Mathematical Engineering, Musashino University, Tokyo, Japan

    M. Mimura

  3. Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan

    J. C. Tsai

  4. National Center for Theoretical Sciences, Taipei, Taiwan

    J. C. Tsai

Authors
  1. M. H. Kabir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Mimura
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. C. Tsai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. H. Kabir.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kabir, M.H., Mimura, M. & Tsai, J.C. Spreading Waves in a Farmers and Hunter-Gatherers Model of the Neolithic Transition in Europe. Bull Math Biol 80, 2452–2480 (2018). https://doi.org/10.1007/s11538-018-0475-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-018-0475-6

Keywords

Mathematics Subject Classification

Search

Navigation