  We model spatially expanding populations by means of two spatial
$\Lambda$-Fleming Viot processes (or SLFVs) with selection: the k-parent SLFV
and the $\infty$-parent SLFV. In order to do so, we fill empty areas with type
0 ''ghost'' individuals with a strong selective disadvantage against ''real''
type 1 individuals, quantified by a parameter k. The reproduction of ghost
individuals is interpreted as local extinction events due to stochasticity in
reproduction. When k $\rightarrow$ +$\infty$, the limiting process,
corresponding to the $\infty$-parent SLFV, is reminiscent of stochastic growth
models from percolation theory, but is associated to tools making it possible
to investigate the genetic diversity in a population sample. In this article,
we provide a rigorous construction of the $\infty$-parent SLFV, and show that
it corresponds to the limit of the k-parent SLFV when k $\rightarrow$
+$\infty$. In order to do so, we introduce an alternative construction of the
k-parent SLFV which allows us to couple SLFVs with different selection
strengths and is of interest in its own right. We exhibit three different
characterizations of the $\infty$-parent SLFV, which are valid in different
settings and link together population genetics models and stochastic growth
models.
