Stochastic model for population migration and the growth of human settlements during the Neolithic transition

Where do successful populations originate from?

In order to understand the dynamics of emergence and spreading of socio-technical innovations and population moves it is important to determine the place of origin of these populations. Here we focus on the role of geographical factors, such as land fertility and mountains in the context of human population evolution and distribution dynamics. We use a constrained diffusion-based computational model, computer simulations and the analysis of geographical and land-quality data. Our analysis shows that successful human populations, i.e. those which become dominant in their socio - geographical environment, originate from lands of many valleys with relatively low land fertility, which are close to areas of high land fertility. Many of the homelands predicted by our analysis match the assumed homelands of known successful populations (e.g. Bantus, Turkic, Maya). We also predict other likely homelands as well, where further archaeological, linguistic or genetic exploration may confirm the place of origin for populations with no currently identified urheimat. Our work is significant because it advances the understanding of human population dynamics by guiding the identification of the origin locations of successful populations.

Beyond the pair approximation: Modeling colonization population dynamics

The process of range expansion (colonization) is one of the basic types of biological dynamics, whereby a species grows and spreads outwards, occupying new territories. Spatial modeling of this process is naturally implemented as a stochastic cellular automaton, with individuals occupying nodes on a rectangular grid, births and deaths occurring probabilistically, and individuals only reproducing onto unoccupied neighboring spots. In this paper we derive several approximations that allow prediction of the expected range expansion dynamics, based on the reproduction and death rates. We derive several approximations, where the cellular automaton is described by a system of ordinary differential equations that preserves correlations among neighboring spots (up to a distance). This methodology allows us to develop accurate approximations of the population size and the expected spatial shape, at a fraction of the computational time required to simulate the original stochastic system. In addition, we provide simple formulas for the steady-state population densities for von Neumann and Moore neighborhoods. Finally, we derive concise approximations for the speed of range expansion in terms of the reproduction and death rates, for both types of neighborhoods. The methodology is generalizable to more complex scenarios, such as different interaction ranges and multiple-species systems.

Biased dispersal can explain fast human range expansions

Some human fronts spread faster than expected by models based on dispersal and reproduction. The only explanation proposed so far assumes that some autochthonous individuals are incorporated by the expanding populations, leading to faster front speeds. Here we show that simple models without this effect are also consistent with the observed speeds of two fronts (a Khoi-khoi expansion of herders and a Bantu expansion of farmers), provided that the dispersal of individuals is biased (i.e., more probable) in directions closer to the front propagation direction. The physical models presented may also be applied to other kinds of social phenomena, including innovation diffusion, rumor propagation, linguistic fronts, epidemic spread, diffusion in economic space and the evolution of cooperation in spatial systems. They can be also adapted to non-human systems with biased dispersal, including biological invasions, cancer tumors and virus treatment of tumors.

Extinction dynamics of a discrete population in an oasis

Understanding the conditions ensuring the persistence of a population is an issue of primary importance in population biology. The first theoretical approach to the problem dates back to the 1950s with the Kierstead, Slobodkin, and Skellam (KiSS) model, namely a continuous reaction-diffusion equation for a population growing on a patch of finite size L surrounded by a deadly environment with infinite mortality, i.e., an oasis in a desert. The main outcome of the model is that only patches above a critical size allow for population persistence. Here we introduce an individual-based analog of the KiSS model to investigate the effects of discreteness and demographic stochasticity. In particular, we study the average time to extinction both above and below the critical patch size of the continuous model and investigate the quasistationary distribution of the number of individuals for patch sizes above the critical threshold.

Nonlinear subdiffusive fractional equations and the aggregation phenomenon

In this article we address the problem of the nonlinear interaction of subdiffusive particles. We introduce the random walk model in which statistical characteristics of a random walker such as escape rate and jump distribution depend on the mean density of particles. We derive a set of nonlinear subdiffusive fractional master equations and consider their diffusion approximations. We show that these equations describe the transition from an intermediate subdiffusive regime to asymptotically normal advection-diffusion transport regime. This transition is governed by nonlinear tempering parameter that generalizes the standard linear tempering. We illustrate the general results through the use of the examples from cell and population biology. We find that a nonuniform anomalous exponent has a strong influence on the aggregation phenomenon.

Space competition and time delays in human range expansions. Application to the neolithic transition

Space competition effects are well-known in many microbiological and ecological systems. Here we analyze such an effect in human populations. The Neolithic transition (change from foraging to farming) was mainly the outcome of a demographic process that spread gradually throughout Europe from the Near East. In Northern Europe, archaeological data show a slowdown on the Neolithic rate of spread that can be related to a high indigenous (Mesolithic) population density hindering the advance as a result of the space competition between the two populations. We measure this slowdown from a database of 902 Early Neolithic sites and develop a time-delayed reaction-diffusion model with space competition between Neolithic and Mesolithic populations, to predict the observed speeds. The comparison of the predicted speed with the observations and with a previous non-delayed model show that both effects, the time delay effect due to the generation lag and the space competition between populations, are crucial in order to understand the observations.

Synthesis between demic and cultural diffusion in the Neolithic transition in Europe

There is a long-standing controversy between two models of the Neolithic transition. The demic model assumes that the Neolithic range expansion was mainly due to the spread of populations, and the cultural model considers that it was essentially due to the spread of ideas. Here we integrate the demic and cultural models in a unified framework. We show that cultural diffusion explains ∼40% of the spread rate of the Neolithic transition in Europe, as implied by archaeological data. Thus, cultural diffusion cannot be neglected, but demic diffusion was the most important mechanism in this major historical process at the continental scale. This quantitative approach can be useful also in regional analysis, the description of Neolithic transitions in other continents, and models of many human spread phenomena.

Bayesian inference for a wave-front model of the neolithization of Europe

We consider a wave-front model for the spread of neolithic culture across Europe, and use Bayesian inference techniques to provide estimates for the parameters within this model, as constrained by radiocarbon data from southern and western Europe. Our wave-front model allows for both an isotropic background spread (incorporating the effects of local geography) and a localized anisotropic spread associated with major waterways. We introduce an innovative numerical scheme to track the wave front, and use Gaussian process emulators to further increase the efficiency of our model, thereby making Markov chain Monte Carlo methods practical. We allow for uncertainty in the fit of our model, and discuss the inferred distribution of the parameter specifying this uncertainty, along with the distributions of the parameters of our wave-front model. We subsequently use predictive distributions, taking account of parameter uncertainty, to identify radiocarbon sites which do not agree well with our model. These sites may warrant further archaeological study or motivate refinements to the model.

Modeling neolithic dispersal in central Europe: demographic implications

On the basis of new examination of ancient DNA and craniometric analyses, Neolithic dispersal in Central Europe has been recently explained as reflecting colonization or at least a major influx of near eastern farmers. Given the fact that Neolithic dispersal in Central Europe was very rapid and extended into a large area, colonization would have to be associated with high population growth and fertility rates of an expanding Neolithic population. We built three demographic models to test whether the growth and fertility rates of Neolithic farmers were high enough to allow them to colonize Central Europe without admixture with foragers. The principle of the models is based on stochastic population projections. Our results demonstrate that colonization is an unlikely explanation for the Neolithic dispersal in Central Europe, as the majority of fertility and growth rate estimates obtained in all three models are higher than levels expected in the early Neolithic population. On the basis of our models, we derived that colonization would be possible only if (1) more than 37% of women survived to mean age at childbearing, (2) Neolithic expansion in Central Europe lasted more than 150 years, and (3) the population of farmers grew in the entire settled area. These settings, however, represent very favorable demographic conditions that seem unlikely given current archaeological and demographic evidence. Therefore, our results support the view that Neolithic dispersal in Central Europe involved admixture of expanding farmers with local foragers. We estimate that the admixture contribution from foragers may have been between 55% and 72%.

Continuous-time random walks that alter environmental transport properties

We consider continuous-time random walks (CTRWs) in which the walkers have a finite probability to alter the waiting-time and/or step-length transport properties of their environment, resulting in possibly transient anomalous diffusion. We refer to these CTRWs as transmogrifying continuous-time random walks (TCTRWs) to emphasize that they change the form of the transport properties of their environment, and in a possibly strange way. The particular case in which the CTRW waiting-time density has a finite probability to be permanently altered at a given site, following a visitation by a walker, is considered in detail. Master equations for the probability density function of transmogrifying random walkers are derived, and results are compared with Monte Carlo simulations. An interesting finding is that TCTRWs can generate transient subdiffusion or transient superdiffusion without invoking truncated or tempered power law densities for either the waiting times or the step lengths. The transient subdiffusion or transient superdiffusion arises in TCTRWs with Gaussian step-length densities and exponential waiting-time densities when the altered average waiting time is greater than or less than, respectively, the original average waiting time.

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

It is shown that Lotka-Volterra interaction terms are not appropriate to describe vertical cultural transmission. Appropriate interaction terms are derived and used to compute the effect of vertical cultural transmission on demic front propagation. They are also applied to a specific example, the Neolithic transition in Europe. In this example, it is found that the effect of vertical cultural transmission can be important (about 30%). On the other hand, simple models based on differential equations can lead to large errors (above 50%). Further physical, biophysical, and cross-disciplinary applications are outlined.

Subdiffusion, chemotaxis, and anomalous aggregation

We propose a nonlinear random walk model which is suitable for the analysis of both chemotaxis and anomalous subdiffusive transport. We derive the master equations for the population density for the case when the transition rate for a random walk depends on residence time, chemotactic substance, and population density. We introduce the anomalous chemotactic sensitivity and find an anomalous aggregation phenomenon. So we suggest a different explanation of the well-known effect of chemotactic collapse.