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

1. Realistic dispersion kernels in reaction-dispersion equationsApplication to the Neolithic D: dispersion distance, • p→0 at the leading edge of the front • the front is locally planar for t → , r → 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, <D2>, for each population. Evolution equation In order to study the effect of the dispersion kernel, we will use an evolutionequation 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, <D2>, 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 <D2> yields slow Gauss and Laplace speeds. • Population (D) has lower <D2> 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 sumof 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 ri=i·d, for i=1,2,3...n. d : minimum width of the intervals a : parameter obtained from a2 = <D2> <D2> : mean-squared displacement b : parameter obtained from b2=<D2>/6 <D2> : 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 frontspeed for Eq. (1) and each dispersion kernel can be obtained by assuming that References * neus.isern@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.