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

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

2 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) FEPRE

3 Diffusion time

4 Diffusion A A J > 0J < 0

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

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

7 Fick’s law c x c x

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

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

10 How can we find out c(x,t) ? We can find out c(x,t) !

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

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

13 2 human populations:

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

15

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

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

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

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

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

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

22 Relationship with Fisher’s equation Eq. HRD: (Fick’s law) (Fisher’s Eq.) → 0

23 → 0 (Fisher)

24

25 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)

26 Previous work by the Girona group  HRD Eq: F & M, Phys. Rev. Lett  ∞ 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 Neolithic sites: P, F & Ammerman, PLoS Biol 2006 Review: F & M, Rep. Progr. Phys etc.