Individual-based modelling of population growth and diffusion in discrete time

Individual-based models (IBMs) of human populations capture spatio-temporal dynamics using rules that govern the birth, behavior, and death of individuals. We explore a stochastic IBM of logistic growth-diffusion with constant time steps and independent, simultaneous actions of birth, death, and movement that approaches the Fisher-Kolmogorov model in the continuum limit. This model is well-suited to parallelization on high-performance computers. We explore its emergent properties with analytical approximations and numerical simulations in parameter ranges relevant to human population dynamics and ecology, and reproduce continuous-time results in the limit of small transition probabilities. Our model prediction indicates that the population density and dispersal speed are affected by fluctuations in the number of individuals. The discrete-time model displays novel properties owing to the binomial character of the fluctuations: in certain regimes of the growth model, a decrease in time step size drives the system away from the continuum limit. These effects are especially important at local population sizes of <50 individuals, which largely correspond to group sizes of hunter-gatherers. As an application scenario, we model the late Pleistocene dispersal of Homo sapiens into the Americas, and discuss the agreement of model-based estimates of first-arrival dates with archaeological dates in dependence of IBM model parameter settings.

针对人类种群的基于个体的模型（Individual-based Models, IBMs）通过设定管控个体出生、行为与死亡的规则，刻画种群的时空动态过程。本研究针对一类采用恒定时间步长，且出生、死亡与移动行为独立同步的随机逻辑斯蒂增长-扩散型基于个体模型展开探索，该模型在连续极限下可趋近于费希尔-柯尔莫哥洛夫模型（Fisher-Kolmogorov model）。该模型非常适合在高性能计算机上开展并行计算。本研究借助解析近似与数值模拟，在契合人类种群动态与生态学研究的参数区间内探索该模型的涌现特性，并在小转移概率极限下复现了连续时间模型的相关结果。本模型的预测结果表明，种群密度与扩散速率会受到个体数量波动的影响。由于个体数量波动服从二项分布特性，该离散时间模型展现出独特的属性：在增长模型的特定参数区间内，时间步长的缩小反而会使系统偏离连续极限。这类效应在个体数量少于50的局域种群中尤为显著，而这类种群规模大体对应狩猎采集社群的群体规模。作为应用场景，本研究模拟了更新世晚期智人（Homo sapiens）向美洲大陆的扩散过程，并探讨了基于模型估算的首次抵达年代与考古测年结果的契合程度随基于个体模型参数设置的变化情况。