IntroductionReaction-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 .

pi : probability for individuals to move a distance ri=i·d, for i=1,2,3...n.d : minimum width of the intervals

• 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.

We use different dispersion kernels to fit our data:

: parameter obtained from 2=<2>/6<2> : mean-squared displacement

I0(ri) : modified Bessel function of first kind and order zero.

Application to the Neolithic

Conclusions

References

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

Realistic dispersion kernels in reaction-dispersion equationsApplication to the Neolithic

Neus Isern*, Joaquim FortDept. Física, Universitat de Girona, 17071, Girona

In order to study the effect of the dispersion kernel, we will use an evolution equation for the population density p(x,y,t),

Mathematical Model

Dispersion kernels

, , , , , d dT x y x y x yp x y t T R p x y t (1)

• 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.

The logistic growth4 is well-known to be a good description for many populations:

a : population number growth ratepmax : carrying capacity (maximum population density)

Reproduction term

max

max

, ,, ,

, , 1

aT

T aT

p p x y t eR p x y t

p p x y t e

(2)

Dispersion term

(x,y) (dispersion kernel) gives the probability per unit area that individuals move a distance (±x , ±y) in time T.

2 (3)

2 2 2x y : dispersion distance,

0

n

Dirac deltas i iip r

222Gauss e 2

Laplace e

• p→0 at the leading edge of the front

• the front is locally planar for t → , r →

The front speed for Eq. (1) and each dispersion kernel can be obtained by assuming that

00

0

lnnaT

i ii

Dirac deltas

e p I rv mín

T

Gauss

av

T

3 3 21 ln 1

2 2 3Laplace

aTv aT

T aT

Evolution equation

: parameter obtained from 2 = <2> <2> : mean-squared displacement

Front speed

We use the following data:

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, <2>, for each population.

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.

The three kernels yield similar front speeds except when the dispersion kernel has a long-range component (Fig 1 and 2).

Long-range component effects:

Data

Results

* neus.isern@udg.edu1 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).

• 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.

• 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)

The Gauss and Laplace distributions depend on a fixed parameter, <2>, that does not contain all of the information about the shape of the kernel.

• Kernel (4) leads up to 30% faster speeds for populations (E) and (F) due to the long-range component. The low value of <2> yields slow Gauss and Laplace speeds.

• Population (D) has lower <2> 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.

For an isotropic space, we define the linear kernel,

(4)

(7)

(5) (6)

(8) (9)