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Mathematical models of NeolithisationMathematical models of Neolithisation

Joaquim Fort

Univ. de Girona (Catalonia, Spain)

FEPRE workshop26-27 March 2007

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List of ParticipantsList 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

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Diffusion Diffusion

time

4

DiffusionDiffusion

A A

J > 0 J < 0

tA t duringA area cross that particles ofnumber

Flux

J

5

J J = diffusion flux= diffusion flux

J < 0

J < 0

J = 0time

6

J < 0

J = 0

c

xc

x

c = concentration = number particles / volume

0dx

dc

0dx

dc

J J = diffusion flux= diffusion flux

7

Fick’s lawFick’s law

tcoefficiendiffusion

Ddxdc

DJ

c

x

c

x

0dx

dcDJ

0dx

dcDJ

8

c

xc

x

c

x

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

time

9

NN = = number of particles in volume number of particles in volume VV

Flux in 1 dimension:

J (x) J (x+x)V

JAxJxxJAAxxJAxJtN

)]()([)()(

dxdJ

VxdxdJ

A

A

dxdJ

dtVNd

dtdc )/(

xJ(x)

J(x+x) ∆ J

x ∆x

dxdJ

dtVNd

dtdc )/(

dxdJ

dtVNd

dtdc )/(

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

dxdJ

dtdc

law sFick'

dxdc

DJ 2

2

dxcd

Ddtdc

We can find out c(x,t) !

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· Flux in 1 dimension:2

2

dx

cdD

dt

dc

If there is a chemical reaction:

2

2

2

2

dy

cd

dx

cdDF

dt

dc

· Flux in 2 dimensions:

2

2

2

2

dy

cd

dx

cdD

dt

dc

evolume·timproduced particles ofnumber F

For biological populations:

2

2

2

2

)(dy

pd

dx

pdDpF

dt

dp

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p0

pmax

p

time

a = initial growth rate

(of population number)

max

1)(p

ppapF

t

pLogistic growth:

?

atppdtap

pdpa

t

ppp

0max /ln

pmax= carrying capacity

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2 human populations:

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2

2

2

2

max

1y

p

x

pD

p

ppa

t

p

= jump distanceT = intergeneration dispersal time interval

Pre-industrial farmers (Majangir): < 2 > = (1544 ± 368 ) km2

T

D4

2

Fisher Eq:

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Dav 2

T

av2

km/yr4.1

yr25

km1544

yr032.022

1

v

T

a

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1.0 ± 0.2 km/yr observed

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

10000 8000 6000 40000

1000

2000

3000

4000

5000

Ammerman & Cavalli-Sforza, 1971, 1984

r = 0.89

v = 1.0 km / yr

fit

dist

ance

( k

m )

date ( years B.P. )

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0 500 1000 1500 2000 2500 30000

1

2

3

4

< 2 > / T (km 2 /generation)

0.8

11.2

0.81

v = 1.2a (

%)

0.8 < 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)

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x

txcD

t

txJtxJ

),(),(),(

dx

txdcDtxJ

),(),( Up to now:

(Fick’s law)

Now:

→ instantaneous !

dx

txdcDtxJ

),(),(

→ time-delayed

(Maxwell-Cattaneo Eq.)dx

dfxxffxfxxf )()()(

f(x)

f(x+x)

Time delaysTime delays

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HRD EquationHRD Equation

dx

dcDJ

Fdx

cdD

dt

dc

2

2

Balance

of mass:

Now:

x

cD

t

JJ

Fdx

dJ

dt

dc

2

2

2

2

t

FF

x

cD

t

c

t

c

(HRD Eq.=Hyperbolic reaction-diffusion)

(Fisher’s Eq.)

Up to now:

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HRD Equation:HRD Equation:

For a

biological

population

in 2 dims:

2

2

2

2

t

FF

x

cD

t

c

t

c

2

2

2

2

2

2

t

FF

y

p

x

pD

t

p

t

p

max

1p

ppaFLogistic

reproduction:

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= jump (or migration) distance

T = time interval between the jumps of parents and those of their sons/daughters

T

D4

2

HRD Equation:

2

T

2

2

2

2

2

2

t

FF

y

p

x

pD

t

p

t

p

max

1p

ppaF

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Relationship with Fisher’s equationRelationship with Fisher’s equation

22 2

2

2

2

t

FTF

x

pD

t

p

t

pT

2 x

cD

t

JTJ

x

cDJ

Eq. HRD:

Fx

pD

t

p

2

2

(Fick’s law)

(Fisher’s Eq.)

<T > → 0

<T > → 0

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Dav 2

22

:Eq HRD 2

2

2

2

2

2

tFT

Fyp

xp

Dtp

tpT

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2 Eq. HRD

TaDa

v

max

1p

ppaF

<T > → 0(Fisher)

T

D4

2

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0 500 1000 1500 2000 2500 30000

1

2

3

4

0.8 < v obs

< 1.2 km / yr

< 2 > / T (km2 /generation)

time-delayed 0.8

1.0 km/yr

1.2

a (

%)

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SummarySummary

Observed Neolithic speed: 1.0 km/yr

Fisher’s equation in 2D: 1.4 km/yrHRD Eq: 1.0 km/yrDifference: 40 %

(F & M, Phys. Rev. Lett. 1999)

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Previous work by the Girona groupPrevious 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.