Stochastic Fronts and the Neolithic Transition

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Stochastic Fronts and the Neolithic Transition
FEPRE European project 3rd annual
workshop Girona 16-18 March 2009

• Neus Isern
• Universitat de Girona
• (Catalonia, Spain)

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Contents

• Stochastic effects on travelling waves
• Stochastic effects on Neolithic transition models

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Reaction-dispersion fronts

• Physics superconductors, solidification...
• Biological systems viral infections, biological
  invasions, tumor growth, Neolithic transition...

Neolithic expansion in Europe
Reaction or reproduction
Dispersion
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Why a Stochastic Model?
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Brunet Derrida Deterministic Model

• Brunet E and Derrida B, Shift in the velocity of
  a front due to a cutoff Physical Review E 56 2597
  (1997)

• Effect of a small cutoff e on the velocity of a
  traveling wave in 1D
• Fisher-Kolmogorov equation

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Brunet Derrida Speed Correction

• Front speed correction (general case vv(g))

v0 minimal front speed when and at the
front
Fisher-Kolmogorov equation
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Brunet Derrida Stochastic Model

• Stochastic dispersion

xj , xk positions of two randomly chosen
particles aj, ak random numbers with value 1
with probability p or 0 with probability 1-p
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Brunet Derrida Stochastic Results

• Finite number of particles N
• Cutoff e1/N (min. relative density)

Differences due to stochastic effects
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Application to the Neolithic
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Evolution Equation

• Reaction logistic growth

• Dispersion kernel

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Discretization of the Front

• Individuals vs. population density

• Minimum discrete value at the front (at least 1
  individual) cutoff
• Maximum number (Nmax) of individual per cell.

Stauder J, The Majangir (1971)
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Discretization of the Front

• Simulations with discretized front

- Dispersion process (pe0.5) (we keep just the
integer part) - Reaction process (we keep just
the integer part)
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20-40
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Discretization of the Front
10-15
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Stochastic Model

• Stochastic Dispersion
• To each individualwe assign a random value
• if a lt pe individuals stay
• if a pe individuals move
• To each migrating individualwe assign a random
  value
• each corresponding to one of the eight
• possible final positions

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Stochastic Model

• Simulation results

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Stochastic Model

• Stochastic Model Deterministic Model

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Stochastic Model
Isern N, Fort J, Pérez-Losada J J Stat Mechs
(2008)

• Front speed

Mean over 16 stochastic simulations (trun5h)

• Stochastic vs.DeterministicDensity lt 4.2
  (pmax range)

13.3

• Stochastic vs. DeterministicIndividuals lt 5

pmax 3288 people/km2 Ammerman A J and
Cavalli-Sforza L L, The Neolithic Transition
and the Genetics of Population in Europe
(1984)
pmax 1.28 people/km2 Currat M and Excoffier
L, Proc. R. Soc. B (2005)
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Conclusions

• Discretizing the density (num. ind.) introduces a
  correction to the front speed. (Up to 40 for low
  values pmax.)
• The stochastic model introduces correction to the
  deterministic model density lying within the
  80-CL for the range of pmax for the Neolithic.
• Stochastic results follow the behaviour of the
  deterministic model with discrete density.
  Correction lt5 (16 simulations) for the whole
  range of pmax.
• Major correction is due to the discretization not
  to the stochastic effects.

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Thanks for the Attention!!