Fronts from integrodifference equations and persistence effects on the Neolithic transition

Abstract

We extend a previous model of the Neolithic transition in Europe ͓J. Fort and V. Méndez, Phys. Rev. Lett. 82, 867 ͑1999͔͒ by taking two effects into account: ͑i͒ we do not use the diffusion approximation ͑which corresponds to second-order Taylor expansions͒, and ͑ii͒ we take proper care of the fact that parents do not migrate away from their children ͑we refer to this as a time-order effect, in the sense that it implies that children grow up with their parents, before they become adults and can survive and migrate͒. We also derive a time-ordered, second-order equation, which we call the sequential reaction-diffusion equation, and use it to show that effect ͑ii͒ is the most important one, and that both of them should in general be taken into account to derive accurate results. As an example, we consider the Neolithic transition: the model predictions agree with the observed front speed, and the corrections relative to previous models are important ͑up to 70%͒.

Vertical cultural transmission effects on demic front propagation: Theory and application to the Neolithic transition in Europe

“…Equations (3) and (7) have been applied to the Neolithic transition in Refs. [13] and [21], respectively (for some recent reviews on these and other models, see Refs. [9,11]).…”

“…Population number data sets in previously empty islands are available [49] and the average value implied is R 0N = 2.2 [21]. The generation time is T = 1 generation = 32 yr [50].…”

“…In some papers [21,22], instead of the logistic growth (2), one assumes a linear cutoff, i.e., Eq. (7) unless it yields p(x,y,t + T ) > p max , and p(x,y,t + T ) = p max otherwise.…”

“…However, here we want to focus our attention on the effect of vertical cultural transmission. Therefore, we analyze a simple model in which individuals either remain at rest (with probability p e , which is called the persistence [21]) or move a distance r (with probability 1 − p e ):…”

“…The generation time is T = 1 generation = 32 yr [50]. The mean value of the population persistence is p e = 0.38 [21], which was estimated from the mobility data in Ref. [51].…”