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

TL;DR

This paper investigates the growth dynamics of spatial Lambda-Fleming Viot processes, demonstrating linear growth in occupied area over time and providing bounds and simulations to understand the growth speed in these stochastic models.

Contribution

It introduces a detailed analysis of growth properties in k-parent and infinite-parent SLFV processes, establishing linear growth results and bounds, with numerical simulations for growth speed estimation.

Findings

Growth of the infinite-parent SLFV is linear in time.

Growth in the k-parent SLFV is also linear, with probabilistic bounds.

Numerical simulations suggest actual growth speed may exceed simple theoretical estimates.

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 to 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.