Mathematical models of neolithisation
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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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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)

  • 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)


Diffusion

time


Diffusion

A A

J > 0J < 0


J = diffusion flux

J < 0

J < 0

J = 0

time


c

c

x

x

J = diffusion flux

c = concentration = number particles / volume

J < 0

J = 0


c

c

x

x

Fick’s law


c

c

c

x

x

x

time

How can we find outc(x,t) ?


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


How can we find outc(x,t) ?

We can find outc(x,t) !


· Flux in 1 dimension:

· Flux in 2 dimensions:

If there is a chemical reaction:

For biological populations:


Logistic growth:

pmax= carrying

capacity

?

a = initial growth rate

(of population number)


2 human populations:


Fisher Eq:

= jump distance

T = intergeneration dispersal time interval

Pre-industrial farmers (Majangir):

< 2 > = (1544 ± 368 ) km2


1.0 ± 0.2 km/yr

observed

1.4 km/yr predicted

by Fisher’s Eq. !!


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)


f(x+x)

f(x)

Time delays

Up to now:

(Fick’s law)

→instantaneous !

Now:

→time-delayed

(Maxwell-Cattaneo Eq.)


HRD Equation

Up to now:

Balance

of mass:

(Fisher’s Eq.)

Now:

(HRD Eq.=Hyperbolic reaction-diffusion)


HRD Equation:

For a biological population

in 2 dims:

Logistic reproduction:


HRD Equation:

= jump (or migration) distance

T = time interval between the jumps of parents and

those of their sons/daughters


Relationship with Fisher’s equation

(Fick’s law)

<T > → 0

Eq. HRD:

<T > → 0

(Fisher’s Eq.)


(Fisher)

<T > → 0


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)


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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