{"title":"Non-Equilibrium Fluctuations for a Spatial Logistic Branching Process with Weak Competition","authors":"Thomas Tendron","doi":"arxiv-2409.05269","DOIUrl":null,"url":null,"abstract":"The spatial logistic branching process is a population dynamics model in\nwhich particles move on a lattice according to independent simple symmetric\nrandom walks, each particle splits into a random number of individuals at rate\none, and pairs of particles at the same location compete at rate c. We consider\nthe weak competition regime in which c tends to zero, corresponding to a local\ncarrying capacity tending to infinity like 1/c. We show that the hydrodynamic\nlimit of the spatial logistic branching process is given by the\nFisher-Kolmogorov-Petrovsky-Piskunov equation. We then prove that its\nnon-equilibrium fluctuations converge to a generalised Ornstein-Uhlenbeck\nprocess with deterministic but heterogeneous coefficients. The proofs rely on\nan adaptation of the method of v-functions developed in Boldrighini et al.\n1992. An intermediate result of independent interest shows how the tail of the\noffspring distribution and the precise regime in which c tends to zero affect\nthe convergence rate of the expected population size of the spatial logistic\nbranching process to the hydrodynamic limit.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"91 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05269","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The spatial logistic branching process is a population dynamics model in
which particles move on a lattice according to independent simple symmetric
random walks, each particle splits into a random number of individuals at rate
one, and pairs of particles at the same location compete at rate c. We consider
the weak competition regime in which c tends to zero, corresponding to a local
carrying capacity tending to infinity like 1/c. We show that the hydrodynamic
limit of the spatial logistic branching process is given by the
Fisher-Kolmogorov-Petrovsky-Piskunov equation. We then prove that its
non-equilibrium fluctuations converge to a generalised Ornstein-Uhlenbeck
process with deterministic but heterogeneous coefficients. The proofs rely on
an adaptation of the method of v-functions developed in Boldrighini et al.
1992. An intermediate result of independent interest shows how the tail of the
offspring distribution and the precise regime in which c tends to zero affect
the convergence rate of the expected population size of the spatial logistic
branching process to the hydrodynamic limit.