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Mathematical models of Neolithisation

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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

- 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)

time

A A

J > 0J < 0

J < 0

J < 0

J = 0

time

c

c

x

x

c = concentration = number particles / volume

J < 0

J = 0

c

c

x

x

c

c

c

x

x

x

time

How can we find outc(x,t) ?

J(x+x)

∆ J

J(x)

x

Flux in 1 dimension:

A

J (x)

J (x+x)

V

∆x

x

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)

Up to now:

(Fick’s law)

→instantaneous !

Now:

→time-delayed

(Maxwell-Cattaneo Eq.)

Up to now:

Balance

of mass:

(Fisher’s Eq.)

Now:

(HRD Eq.=Hyperbolic reaction-diffusion)

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

(Fick’s law)

<T > → 0

Eq. HRD:

<T > → 0

(Fisher’s Eq.)

(Fisher)

<T > → 0

- 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)

- 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.