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Finding analytical approximations for discrete, stochastic, individual-based models of ecology. (English) Zbl 07776295

It is well known that among the different ways to model spatial population dynamics, spatially explicit individual-based models stand out as the most realistic approach. These types of models are often referred to as ‘bottom-up’, since they explicitly represent individual actions and interactions. This contrasts with the more classical ‘top-down’ approach, where the population dynamics is modeled through partial or ordinary differential or difference equations. In the article the authors consider the important relationship between individual-based models and analytical, ‘top-down’ models of populations dynamics for a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. They start with description of the simulations of the model, which exhibit large-scale discrete oscillations and characterize these oscillations by measuring spatial correlations. Then the authors develop two new approximate descriptions of the resulting spatial population dynamics. The first of them is based on local interactions of the individuals and allows to be obtained a difference equation approximation of the system over small dispersal distances. The second one approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that the dispersal stabilizes population dynamics. In addition, the authors are calculated the extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual based model as dispersal distance and population size simultaneously tend to infinity. The obtained by them results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.

MSC:

92D40 Ecology
93D25 Input-output approaches in control theory
39A60 Applications of difference equations
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