Bulletin of Mathematical Biology

, Volume 80, Issue 9, pp 2452–2480 | Cite as

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

  • M. H. KabirEmail author
  • M. Mimura
  • J. C. Tsai
Original Article


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.


Neolithic transition Reaction–diffusion model of farmers and hunter-gatherers interaction Traveling wave solutions 

Mathematics Subject Classification

01A10 35K57 35C07 



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.


  1. Ammerman AJ, Cavalli-Sforza LL (1971) Measuring the rate of spread of early farming in Europe. Man 6:674–688CrossRefGoogle Scholar
  2. Ammerman AJ, Cavalli-Sforza LL (1984) The Neolithic transition and the genetics of populations in Europe. Princeton University Press, PrincetonCrossRefGoogle Scholar
  3. 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 50:1–17CrossRefzbMATHGoogle Scholar
  4. Britton P, Kent A (2018) The coming of farming. In: TIMEMAPS PREMIUM. Accessed 2018
  5. 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):211–221MathSciNetCrossRefzbMATHGoogle Scholar
  6. Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugen 7:353–369zbMATHGoogle Scholar
  7. Fort J, Méndez C (2002) Wavefronts in time-delayed reaction–diffusion systems. Theory and comparison to experiment. Rep Prog Phys 65:895–954CrossRefGoogle Scholar
  8. 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 211–216Google Scholar
  9. Fort J, Méndez C (1999) Time-delayed theory of the Neolithic transition in Europe. Phys Rev Lett 82:867–870CrossRefGoogle Scholar
  10. Gkiasta M, Russell T, Shennan S, Steele J (2003) Neolithic transition in Europe: the radiocarbon record revisited. Antiquity 77(295):45–62CrossRefGoogle Scholar
  11. 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):29–43CrossRefGoogle Scholar
  12. 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 189–196Google Scholar
  13. 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 1:1–26Google Scholar
  14. Lewis MA, Schmitz G (1996) Biological invasion of an organism with separate mobile and stationary states: modeling and analysis. Forms 11:1–25zbMATHGoogle Scholar
  15. Méndez V, Camacho J (1997) Dynamics and thermodynamics of delayed population growth. Phys Rev E 55(6):6476–6482CrossRefGoogle Scholar
  16. Mimura M (2004) Pattern formation in consumer-finite resource reaction–diffusion system. Publ RIMS Kyoto Univ 40:1413–1431MathSciNetCrossRefzbMATHGoogle Scholar
  17. Mimura M, Sakaguchi H, Matsushita M (2000) Reaction–diffusion modelling of bacterial colony patterns. Phys A 282(1):283–303CrossRefGoogle Scholar
  18. Murray JD (2002) Mathematical biology: I. An introduction, 3rd edn. Springer, New YorkzbMATHGoogle Scholar
  19. Sherratt JA (1998) On the transition from initial data to travelling waves in the Fisher–KPP equation. Dyn Stab Syst 13(2):167–174MathSciNetCrossRefzbMATHGoogle Scholar
  20. Weisdorf JL (2005) From foraging to farming: explaining the Neolithic Revolution. J Econ Surv 19:561–586CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Meiji Institute for Advanced Study of Mathematical SciencesMeiji UniversityTokyoJapan
  2. 2.Department of Mathematical EngineeringMusashino UniversityTokyoJapan
  3. 3.Department of MathematicsNational Tsing Hua UniversityHsinchuTaiwan
  4. 4.National Center for Theoretical SciencesTaipeiTaiwan

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