A microscopic probabilistic description of a locally regulated population and macroscopic approximations

2004/11/01 by Nicolas Fournier, Sylvie Meleard, Sylvie Méléard · 9 citations
Economics, Econometrics and Finance · Mathematics · Medicine · #Complex Systems and Time Series Analysis #Mathematical and Theoretical Epidemiology and Ecology Models #Stochastic processes and statistical mechanics #math.PR #msc:60J80 #msc:60K35.

paper · pdf · doi:10.1214/105051604000000882

published as Annals of Applied Probability 2004, Vol. 14, No. 4, 1880-1919 · Published at http://dx.doi.org/10.1214/105051604000000882 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

openalex publication_date 2004/11/01 · arxiv created 2005/03/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We consider a discrete model that describes a locally regulated spatial population with mortality selection. This model was studied in parallel by Bolker and Pacala and Dieckmann, Law and Murrell. We first generalize this model by adding spatial dependence. Then we give a pathwise description in terms of Poisson point measures. We show that different normalizations may lead to different macroscopic approximations of this model. The first approximation is deterministic and gives a rigorous sense to the number density. The second approximation is a superprocess previously studied by Etheridge. Finally, we study in specific cases the long time behavior of the system and of its deterministic approximation.

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