Realistic dispersion kernels in reaction-dispersion equations Application to the Neolithic

1
Realistic dispersion kernels in
reaction-dispersion equationsApplication to the
Neolithic
Neus Isern, Joaquim FortDept. Física,
Universitat de Girona, 17071, Girona
Introduction
Application to the Neolithic
Reaction-dispersion fronts can be applied to many
physical and biological systems1 (population
dispersals, combustion flames, tumor growth,
etc). The spread of the Neolithic in Europe is
particular case of population dispersal which has
been object of study in the last years2 .
Data
We use the following data

• Generation time6 T 32yr
• Population growth rate3 a 0.028 0.005yr-1
• Dispersion kernels we use dispersion data from
  six human populations3 (see Fig 1)

This study focus on the dispersion process and
the use of dispersion probability distributions
(dispersion kernels). We will

• Assess how the use of different dispersion
  kernels may modify the speed of the propagating
  front.
• Apply dispersion kernels obtained from real
  dispersion data on human populations3 and check
  the consistency of the results with the measured
  front speed for the Neolithic transition.

Mathematical Model
Figure 1. Dispersion kernels corresponding to
four preindustrial farmer populations, Gilishi15
(A), Gilishi25 (B), Shiri15 (C) and Issocongos
(E) one horticulturalist population, Yanomamo
(D), and the modern population in the Parma
Valley (F). The plots also include the
mean-squared displacement, ltD2gt, for each
population.
Evolution equation
In order to study the effect of the dispersion
kernel, we will use an evolution equation for the
population density p(x,y,t),
(1)
Results
The three kernels yield similar front speeds
except when the dispersion kernel has a
long-range component (Fig 1 and 2).
Reproduction term
Dispersion term
The Gauss and Laplace distributions depend on a
fixed parameter, ltD2gt, that does not contain all
of the information about the shape of the kernel.
The logistic growth4 is well-known to be a good
description for many populations
f(Dx,Dy) (dispersion kernel) gives the
probability per unit area that individuals move a
distance (Dx , Dy) in time T.
For an isotropic space, we define the linear
kernel,
(2)
Long-range component effects
(3)

• Kernel (4) leads up to 30 faster speeds for
  populations (E) and (F) due to the long-range
  component. The low value of ltD2gt yields slow
  Gauss and Laplace speeds.
• Population (D) has lower ltD2gt than (C), but
  individuals can move to further distances (faster
  front speed). Results from the Gauss and Laplace
  distributions do not show this long-range
  component effect.

a population number growth rate pmax carrying
capacity (maximum population density)
Dispersion kernels
We use different dispersion kernels to fit our
data

• A sum of Dirac deltas is a simple and useful
  approximation to the data set in intervals (see
  Fig 1).
• Gauss and Laplace distributions are frequently
  used in population dispersal studies5.

Figure 2. Front speeds for six human populations
and the Dirac-deltas, Gauss and Laplace
dispersion kernels. The unhatched region
corresponds to the measured range for the front
speed7. The values are calculated for a 0.028
0.005yr-1 and T 32yr.
(4)
(5)
(6)
pi probability for individuals to move a
distance riid, for i1,2,3...n. d minimum
width of the intervals
b parameter obtained from b2ltD2gt/6 ltD2gt
mean-squared displacement
a parameter obtained from a2 ltD2gt ltD2gt
mean-squared displacement
Conclusions

• Front speeds obtained from the model are
  consistent with the measured values for the
  Neolithic Transition.
• For populations with a long-range component, the
  Gauss and Laplace underestimate the front speed.
• The three kernels lead to similar front speeds
  for populations without long-range components.
• More detailed data on dispersion kernels would
  lead to better approximations to the front speed.

Front speed
The front speed for Eq. (1) and each dispersion
kernel can be obtained by assuming that
References
neus.isern_at_udg.edu 1 Fort J and Pujol T, Rep.
Prog. Phys. 71 086001 (2008). 2 J. Fort and V.
Méndez, Phys. Rev. Lett. 82, 867 (1999). 3 Isern
N, Fort T, Pérez J, J. Stat. Mechs Theor. Exp.
P10012 (2008). 4 Murray J D, Mathematical Biology
(Springer, Berlin, 2002). 5 Kot M, Mark A, Lewis
P and van der Driessche P, Ecology, 77 2027
(1996). 6 Fort J, Jana D and Humet J, Phys. Rev.
E 70, 031913 (2004). 7 Pinhasi R, Fort J and
Ammerman A J, PLoS Biol., 3 2220 (2005).
(7)
(8)
(9)
I0(lri) modified Bessel function of first kind
and order zero.