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 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.

人类种群基于个体的模型（Individual-based models, IBMs）通过支配个体出生、行为与死亡的规则，刻画种群的时空动态特征。本研究针对一类用于逻辑斯蒂增长-扩散过程、具备恒定时间步长的随机基于个体模型展开探究：该模型中个体的出生、死亡与移动过程相互独立且同步执行，在连续极限下可近似为费希尔-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。
本研究结合与人类种群动态及生态学相关的参数区间，通过解析近似与数值模拟分析该模型的涌现特性，并在小转移概率极限下复现了连续时间模型的结果。本模型的预测结果显示，种群密度与扩散速率会受到个体数量波动的影响。该离散时间模型因个体波动服从二项分布特性而展现出独有性质：在该增长模型的部分参数区间内，时间步长的缩减反而会驱使系统偏离连续极限。这类效应在智人（Homo sapiens）向美洲迁徙的局域种群规模场景中尤为关键；本研究还探讨了基于模型的首次抵达日期估算结果，与依据基于个体模型参数设置得到的考古年代之间的吻合程度。