Branching random walk with non-local competition

Pascal Maillard; Sarah Penington

doi:10.1112/jlms.12919

Branching random walk with non-local competition

Pascal Maillard
, Sarah Penington

Université de Toulouse

Research output: Contribution to journal › Article › peer-review

4 Citations (SciVal)

Abstract

We study the Bolker-Pacala-Dieckmann-Law (BPDL) model of population dynamics in the regime of large population density. The BPDL model is a particle system in which particles reproduce, move randomly in space, and compete with each other locally. We rigorously prove global survival as well as a shape theorem describing the asymptotic spread of the population, when the population density is sufficiently large. In contrast to most previous studies, we allow the competition kernel to have an arbitrary, even infinite range, whence the term non-local competition. This makes the particle system non-monotone and of infinite-range dependence, meaning that the usual comparison arguments break down and have to be replaced by a more hands-on approach. Some ideas in the proof are inspired by works on the non-local Fisher-KPP equation, but the stochasticity of the model creates new difficulties.

Original language	English
Article number	e12919
Number of pages	78
Journal	Journal of the London Mathematical Society
Volume	109
Issue number	6
Early online date	29 May 2024
DOIs	https://doi.org/10.1112/jlms.12919
Publication status	Published - 30 Jun 2024

Acknowledgements

This work was initiated at the Centre de recherches mathématiques (CRM), Montreal, during the Montreal Summer Workshop on Probability and Mathematical Physics organised by Louigi Addario-Berry, Omer Angel and Alex Fribergh in July 2018. We warmly thank the organisers for making the event possible and the CRM for its hospitality. PM warmly thanks Hugo Duminil-Copin for several stimulating discussions during the initial phase of the project. Both authors would like to thank the anonymous referee for their careful reading of the article and helpful comments and suggestions.

Funding

PM acknowledges partial support from a Simons CRM fellowship and ANR grant ANR-20-CE92-0010-01. SP is supported by a Royal Society University Research Fellowship, and acknowledges partial support from a Simons CRM fellowship.

Funders	Funder number
Simons CRM
Royal Society
French National Research Agency	ANR‐20‐CE92‐0010‐01

ASJC Scopus subject areas

General Mathematics

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Branching random walk with non-local competition

Abstract

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