  The introduction of the spatial Lambda-Fleming-Viot model (LV) in population
genetics was mainly driven by the pioneering work of Alison Etheridge, in
collaboration with Nick Barton and Amandine V\'eber about ten years ago (1,2).
The LV model provides a sound mathematical framework for describing the
evolution of a population of related individuals along a spatial continuum. It
alleviates the "pain in the torus" issue with Wright and Mal\'ecot's isolation
by distance model and is sampling consistent, making it a tool of choice for
statistical inference. Yet, little is known about the potential connections
between the LV and other stochastic processes generating trees and the spatial
coordinates along the corresponding lineages. This work focuses on a version of
the LV whereby lineages move infinitely rapidly over infinitely small
distances. Using simulations, we show that the induced LV tree-generating
process is well approximated by a birth-death model. Our results also indicate
that Brownian motions modelling the movements of lineages along birth-death
trees do not generally provide a good approximation of the LV due to habitat
boundaries effects that play an increasingly important role in the long run.
Finally, we describe efficient algorithms for fast simulation of the backward
and forward in time versions of the LV model.
