Measure-valued growth processes in continuous space and growth properties starting from an infinite interface

Apolline Louvet, Amandine Véber

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Abstract

The k-parent and infinite-parent spatial Lambda-Fleming Viot processes (or SLFV), introduced in Louvet (2023), form a family of stochastic models for spatially expanding populations. These processes are akin to a continuous-space version of the classical Eden growth model (but with local backtracking of the occupied area allowed when k is finite), while being associated with a dual process encoding ancestry. In this article, we focus on the growth properties of the area occupied by individuals of type 1 (type 0 encoding units of empty space). To do so, we first define the quantities that we shall use to quantify the speed of growth of the occupied area. Using the associated dual process and a comparison with a first-passage percolation problem, we show that the growth of the occupied region in the infinite-parent SLFV is linear in time. Because of the possibility of local backtracking of the occupied area, the result we obtain for the k-parent SLFV is slightly weaker. It gives an upper bound on the probability that a given location is occupied at time t, which also shows that growth in the k-parent SLFV is linear in time. We use numerical simulations to approximate the growth speed for the infinite-parent SLFV, and we observe that the actual speed may be higher than the speed expected from simple first-moment calculations due to the characteristic front dynamics.

Original languageEnglish
Article number104291
JournalStochastic Processes and their Applications
Volume170
Early online date5 Jan 2024
DOIs
Publication statusPublished - 30 Apr 2024

Funding

This work was partly supported by a grant from the British Royal Society and by the chaire programme “Mathematical Modelling and Biodiversity” of Veolia Environnement – Ecole Polytechnique – Museum National d’Histoire Naturelle – Fondation X. AL would like to thank Matt Roberts, Laetitia Colombani and Carl Graham for insightful discussions. AL and AV thank the two Reviewers and the Associate Editor for their very careful reading and their relevant suggestions.

FundersFunder number
Imperial College London

    Keywords

    • Duality
    • Growth speed
    • Spatial Lambda–Fleming–viot processes
    • Stochastic growth process
    • Sub-additivity

    ASJC Scopus subject areas

    • Statistics and Probability
    • Modelling and Simulation
    • Applied Mathematics

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