Realistic pre-industrial Realistic pre-industrial dispersal kernelsdispersal kernels

applied to the Neolithicapplied to the Neolithic

Neus IsernNeus IsernUniversitat de GironaUniversitat de Girona

Catalonia, SpainCatalonia, Spain

FEPRE European FEPRE European ProjectProject

2nd annual workshop2nd annual workshop

St PetersburgSt Petersburg

April 2008April 2008

• Introduction to reaction-dispersion Introduction to reaction-dispersion fronts. Evolution equationfronts. Evolution equation

• Aplication to the Neolithic transition:Aplication to the Neolithic transition:

- Discrete 2D dispersion kernelsDiscrete 2D dispersion kernels

- Continuous 2D dispersion kernelsContinuous 2D dispersion kernels

ContentsContents

Reaction-dispersion frontsReaction-dispersion fronts

• Physics: superconductors, solidification...

• Biological systems: viral infections, biological invasions, tumor growth, Neolithic transition...

Neolithic expansion in Europe

Reaction-dispersion fronts: aplicationsReaction-dispersion fronts: aplications

Reaction orreproduction

Dispersion

Front speedFront speed

, , , , , d dT x y x y x yp x y t T R p x y t Evolution equationEvolution equation

Fisher and HRD equationsFisher and HRD equations

2Fisherc aD2

2

1HRD aT

aDc

Diffusion coefficient: D

Dispersion kernel: , x y

Evolution equationEvolution equation

, , , , , d dT x y x y x yp x y t T R p x y t Dispersio

n

Reaction

• Logistic growth

max

max

, ,, ,

, , 1

aT

T aT

p p x y t eR p x y t

p p x y t e

Evolution equationEvolution equation

, , , , , d dT x y x y x yp x y t T R p x y t

SequentialSequential

Dispersion

Reaction

, , , , , ,

, , , d d

T

x y x y x y

p x y t T R p x y t p x y t

p x y t

Reaction

Dispersion

Non sequentialNon sequential

Discrete dispersion kernelDiscrete dispersion kernel

0

2n

i iip r

Dispersion dataDispersion data

• Dispersion kernel xy

< 4.8km 4.8 – 24.1km 24.1 – 48.3km >48.3km <d2> (km2)

A 0.54 0.17 0.04 0.25 1115.7

B 0.40 0.17 0.17 0.26 1325.6

C 0.19 0.07 0.22 0.52 2153.0

Dispersion data from three pre-industrial farmer populations (J. Stauder)

< 4.8km 4.8 – 24.1km 24.1 – 48.3km >48.3km <d2> (km2)

A 0.54 0.17 0.04 0.25 1115.7

B 0.40 0.17 0.17 0.26 1325.6

C 0.19 0.07 0.22 0.52 2153.0

Single-distance kernelSingle-distance kernel

DataData

2 21531.43d km

32T yr

• Realistic initial growth rate: a [0.029 , 0.035] yr-1

• Estimated Neolithic transition speed: c [0.6 , 1.3] km/yr

Several-distance kernelSeveral-distance kernel

Continuous dispersion 2D-Continuous dispersion 2D-kernelkernel

3 2

3 52 2

523 2

1 ln 1

1 ln 1

1 ln 1

1 ln 1ln

NSeq Tc

3 3 22 2 31 ln 1 aT

Seq T aTc aT a

Seq Tc

1 1 ln 1

1 ln 1

1 ln 1

1ln

NSeq Tc

2 2

22 e

2 e

GaussGauss LaplaceLaplace

1aTe 1aTe

Gauss / Laplace kernelGauss / Laplace kernel

Thank you for the attention!!Thank you for the attention!!