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

Abstract

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.

Human bony labyrinth is an indicator of population history and dispersal from Africa

“…These estimates implicitly assume directional migration paths rather than dispersal. The estimates used here are based on a nondirectional population growth and diffusion model (the “Fisher–Kolmogorov–Petrovsky–Piskounov” model), which was implemented in an agent-based simulation framework ( 81 , 82 ). We simulated the global spread of a human population originating in South Africa (Johannesburg) on a spherical grid with 660,492 nodes (internode distance 30–35 km) and a total number of 70,000 1-y time steps.…”

Phase transitions in a two-species model for cell segregation and logistic growth

“…The main extension in the model is the introduction of a cell birth and death process. Our modeling is based on the birth and death process proposed in [7]. The idea is that a cell of the type S population has a probability β S to divide into two cells and a probability δ S to die at each time step.…”

“…Remark Such a birth and death process has been shown to approach (in the limit of large number of particles) a logistic equation, where the deterministic population growth rate is b 0 − d 0 (see [7] and references therein). Therefore, the condition d 0 < b 0 is introduced to ensure the positivity of the deterministic growth rate.…”

Stochastic models in seed dispersals: random walks and birth–death processes

“…Capturing the birth, death and behaviour of individual species, the spatio-temporal dynamics is called the IBM [67]. The IBMs are often formulated through BDPs with the aid of Gillespie algorithm [29,30].…”