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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.