Abstract: There is an increasing interest in spatio-temporal models of ecological dynamics and evolutionary processes that take into account the fact that populations consist of discrete entities, i.e., individuals. Such models allow to consider individual variation explicitly and are well-suited to the study of stochastic processes. Individual-based models (IBMs) that can be analyzed analytically and numerically are one type of such discrete models. IBMs represent thus an important approach for modeling (past) human population dynamics, especially when investigating large-scale emerging patterns. A common property of individual-based growth and growth-diffusion models is however the continuous-time implementation, in which individual “actions” happen one at a time. This approach does not scale well when simulating large number of individuals, and is not easily parallelizable on distributed memory machines. For this reason, in this work we describe an individual-based stochastic model of growth and diffusion with overlapping generations which is suitable for large-scale simulations in structured environments. The model uses a discrete-time paradigm with constant time step; as a consequence all agents act simultaneously based only on information from previous time step. By describing the system with a discrete-time Master equation we show that our stochastic model approaches the Fisher-Kolmogorov model in the continuum limit. The properties of the model at different discreteness levels are analyzed by means of numerical simulations as well as analytical approximations. We confirm qualitative trends found in previous works on stochastic birth-death models and growth-diffusion models, and find some novel and interesting features; in particular, the discrete-time algorithm shows different noise properties compared to continuous-time stochastic models. We show how these features affect emergent properties of the population such as the effective carrying capacity and the dispersal speed: both tend to decrease for high level of discreteness. Due to the simultaneous acting of all individuals, the model can be parallelized and is suited for High Performance Computing. This allows to increase the spatial and temporal resolution as well as to consider larger spatial domains and longer simulation times. Thus, the model is suited for the simulation of large populations expansions during human history. We discuss possible expansions of the model and the inclusion of real observational data.
Authors: Natalie Tkachenko, Simone Callegari, John David Weissmann, Wesley P. Petersen,
George Lake, Christoph P. E. Zollikofer
