Download - Mathematical models of Neolithisation
![Page 1: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/1.jpg)
1
Mathematical models of NeolithisationMathematical models of Neolithisation
Joaquim Fort
Univ. de Girona (Catalonia, Spain)
FEPRE workshop26-27 March 2007
![Page 2: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/2.jpg)
2
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
![Page 3: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/3.jpg)
3
Diffusion Diffusion
time
![Page 4: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/4.jpg)
4
DiffusionDiffusion
A A
J > 0 J < 0
tA t duringA area cross that particles ofnumber
Flux
J
![Page 5: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/5.jpg)
5
J J = diffusion flux= diffusion flux
J < 0
J < 0
J = 0time
![Page 6: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/6.jpg)
6
J < 0
J = 0
c
xc
x
c = concentration = number particles / volume
0dx
dc
0dx
dc
J J = diffusion flux= diffusion flux
![Page 7: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/7.jpg)
7
Fick’s lawFick’s law
tcoefficiendiffusion
Ddxdc
DJ
c
x
c
x
0dx
dcDJ
0dx
dcDJ
![Page 8: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/8.jpg)
8
c
xc
x
c
x
How can we find out c(x,t) ?
time
![Page 9: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/9.jpg)
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 )/(
![Page 10: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/10.jpg)
10
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) !
![Page 11: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/11.jpg)
11
· 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
![Page 12: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/12.jpg)
12
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
![Page 13: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/13.jpg)
13
2 human populations:
![Page 14: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/14.jpg)
14
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:
![Page 15: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/15.jpg)
15
Dav 2
T
av2
km/yr4.1
yr25
km1544
yr032.022
1
v
T
a
![Page 16: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/16.jpg)
16
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. )
![Page 17: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/17.jpg)
17
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)
![Page 18: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/18.jpg)
18
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
![Page 19: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/19.jpg)
19
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:
![Page 20: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/20.jpg)
20
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:
![Page 21: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/21.jpg)
21
= 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
![Page 22: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/22.jpg)
22
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
![Page 23: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/23.jpg)
23
Dav 2
22
:Eq HRD 2
2
2
2
2
2
tFT
Fyp
xp
Dtp
tpT
21
2 Eq. HRD
TaDa
v
max
1p
ppaF
<T > → 0(Fisher)
T
D4
2
![Page 24: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/24.jpg)
24
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 (
%)
![Page 25: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/25.jpg)
25
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)
![Page 26: Mathematical models of Neolithisation](https://reader036.vdocuments.us/reader036/viewer/2022062517/56813c7b550346895da612ea/html5/thumbnails/26.jpg)
26
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.