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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Mathematical models of neolithisation

FEPRE workshop

26-27 March 2007

Mathematical models of Neolithisation

Joaquim Fort

Univ. de Girona (Catalonia, Spain)


List of participants

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

Diffusion

time


Diffusion1

Diffusion

A A

J > 0J < 0


J diffusion flux

J = diffusion flux

J < 0

J < 0

J = 0

time


J diffusion flux1

c

c

x

x

J = diffusion flux

c = concentration = number particles / volume

J < 0

J = 0


Fick s law

c

c

x

x

Fick’s law


Mathematical models of neolithisation

c

c

c

x

x

x

time

How can we find outc(x,t) ?


N number of particles in volume v

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 out c x t

How can we find outc(x,t) ?

We can find outc(x,t) !


Mathematical models of neolithisation

· Flux in 1 dimension:

· Flux in 2 dimensions:

If there is a chemical reaction:

For biological populations:


Mathematical models of neolithisation

Logistic growth:

pmax= carrying

capacity

?

a = initial growth rate

(of population number)


Mathematical models of neolithisation

2 human populations:


Mathematical models of neolithisation

Fisher Eq:

= jump distance

T = intergeneration dispersal time interval

Pre-industrial farmers (Majangir):

< 2 > = (1544 ± 368 ) km2


Mathematical models of neolithisation

1.0 ± 0.2 km/yr

observed

1.4 km/yr predicted

by Fisher’s Eq. !!


Mathematical models of neolithisation

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)


Time delays

f(x+x)

f(x)

Time delays

Up to now:

(Fick’s law)

→instantaneous !

Now:

→time-delayed

(Maxwell-Cattaneo Eq.)


Hrd equation

HRD Equation

Up to now:

Balance

of mass:

(Fisher’s Eq.)

Now:

(HRD Eq.=Hyperbolic reaction-diffusion)


Hrd equation1

HRD Equation:

For a biological population

in 2 dims:

Logistic reproduction:


Mathematical models of neolithisation

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

Relationship with Fisher’s equation

(Fick’s law)

<T > → 0

Eq. HRD:

<T > → 0

(Fisher’s Eq.)


Mathematical models of neolithisation

(Fisher)

<T > → 0


Summary

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

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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