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Mathematical models of Neolithisation

FEPRE workshop 26-27 March 2007. Mathematical models of Neolithisation. Joaquim Fort Univ. de Girona (Catalonia, Spain). FEPRE. List of Participants. Kate Davison (Newcastle, UK) Pavel Dolukhanov (Newcastle, UK) Alexander Falileyev (Aberystwyth, UK) Sergei Fedotov (Manchester, UK)

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Mathematical models of Neolithisation

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  1. FEPRE workshop 26-27 March 2007 Mathematical models of Neolithisation Joaquim Fort Univ. de Girona (Catalonia, Spain)

  2. FEPRE List of Participants • Kate Davison (Newcastle, UK) • Pavel Dolukhanov (Newcastle, UK) • Alexander Falileyev (Aberystwyth, UK) • Sergei Fedotov (Manchester, UK) • François Feugier (Newcastle, UK) • Joaquim Fort (Girona, Spain) • Neus Isern (Girona, Spain) • Janusz Kozlowski (Krakow, Poland) • Marc Vander Linden (Brussels, Belgium) • David Moss (Manchester, UK) • Joaquim Perez (Girona, Spain) • Nicola Place (Newcastle, UK) • Graeme Sarson (Newcastle, UK) • Anvar Shukurov (Newcastle, UK) • Ganna Zaitseva (St Petersburg, Russia)

  3. Diffusion time

  4. Diffusion A A J > 0 J < 0

  5. J = diffusion flux J < 0 J < 0 J = 0 time

  6. c c x x J = diffusion flux c = concentration = number particles / volume J < 0 J = 0

  7. c c x x Fick’s law

  8. c c c x x x time How can we find outc(x,t) ?

  9. J(x+x) ∆ J J(x) x N = number of particles in volume V Flux in 1 dimension: A J (x) J (x+x) V ∆x x

  10. How can we find outc(x,t) ? We can find outc(x,t) !

  11. · Flux in 1 dimension: · Flux in 2 dimensions: If there is a chemical reaction: For biological populations:

  12. Logistic growth: pmax= carrying capacity ? a = initial growth rate (of population number)

  13. 2 human populations:

  14. Fisher Eq: = jump distance T = intergeneration dispersal time interval Pre-industrial farmers (Majangir): < 2 > = (1544 ± 368 ) km2

  15. 1.0 ± 0.2 km/yr observed 1.4 km/yr predicted by Fisher’s Eq. !!

  16. 0.8 < vobserved < 1.2 km/yr Predictions from demic diffusion (Fisher's Eq.): 1 dimension (A & C-S 1973) 2 dimensions (F & M, PRL 1999)

  17. f(x+x) f(x) Time delays Up to now: (Fick’s law) →instantaneous ! Now: →time-delayed (Maxwell-Cattaneo Eq.)

  18. HRD Equation Up to now: Balance of mass: (Fisher’s Eq.) Now: (HRD Eq.=Hyperbolic reaction-diffusion)

  19. HRD Equation: For a biological population in 2 dims: Logistic reproduction:

  20. HRD Equation: = jump (or migration) distance T = time interval between the jumps of parents and those of their sons/daughters

  21. Relationship with Fisher’s equation (Fick’s law) <T > → 0 Eq. HRD: <T > → 0 (Fisher’s Eq.)

  22. (Fisher) <T > → 0

  23. Summary • Observed Neolithic speed: 1.0 km/yr • Fisher’s equation in 2D: 1.4 km/yr • HRD Eq: 1.0 km/yr • Difference: 40 % (F & M, Phys. Rev. Lett. 1999)

  24. Previous work by the Girona group • HRD Eq: F & M, Phys. Rev. Lett. 1999 • ∞ terms: F & M, Phys. Rev. E 1999 • Farmers + hunters: Phys. Rev. E 1999, Physica A 2006 • Neolithic in Austronesia: F, Antiquity 2003 • Several delays: Phys Rev E 2004, 2006 • Paleolithic: F, P & Cavalli-Sforza, CAJ 2004 • 735 Neolithic sites: P, F & Ammerman, PLoS Biol 2006 • Review: F & M, Rep. Progr. Phys. 2002 • etc.

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