Time-delayed fronts from biased random walks

We generalize a previous model of time-delayed reaction–diffusion fronts (Fort and Méndez 1999 Phys. Rev. Lett. 82 867) to allow for a bias in the microscopic random walk of particles or individuals. We also present a second model which takes the time order of events (diffusion and reproduction) into account. As an example, we apply them to the human invasion front across the USA in the 19th century. The corrections relative to the previous model are substantial. Our results are relevant to physical and biological systems with anisotropic fronts, including particle diffusion in disordered lattices, population invasions, the spread of epidemics, etc

Integro-difference equations for interacting species and the Neolithic transition

We introduce a set of sequential integro-difference equations to analyze the dynamics of two interacting species. Firstly, we derive the speed of the fronts when a species invades a space previously occupied by a second species, and check its validity by means of numerical random-walk simulations. As an example, we consider the Neolithic transition: the predictions of the model are consistent with the archaeological data for the front speed, provided that the interaction parameter is low enough. Secondly, an equation for the coexistence time between the invasive and the invaded populations is obtained for the first time. It agrees well with the simulations, is consistent with observations of the Neolithic transition, and makes it possible to estimate the value of the interaction parameter between the incoming and the indigenous populations

Anisotropic dispersion, space competition, and the slowdown of the Neolithic transition

The front speed of the Neolithic (farmer) spread in Europe decreased as it reached Northern latitudes, where the Mesolithic (huntergatherer) population density was higher. Here, we describe a reaction diffusion model with (i) an anisotropic dispersion kernel depending on the Mesolithic population density gradient and (ii) a modified population growth equation. Both effects are related to the space available for the Neolithic population. The model is able to explain the slowdown of the Neolithic front as observed from archaeological data