  It is well known that the dynamics of a subpopulation of individuals of a
rare type in a Wright-Fisher diffusion can be approximated by a Feller
branching process. Here we establish an analogue of that result for a spatially
distributed population whose dynamics are described by a spatial
Lambda-Fleming-Viot process (SLFV). The subpopulation of rare individuals is
then approximated by a superBrownian motion. This result mirrors Cox et al.
(2000), where it is shown that when suitably rescaled, sparse voter models
converge to superBrownian motion. We also prove the somewhat more surprising
result, that by choosing the dynamics of the SLFV appropriately we can recover
superBrownian motion with stable branching in an analogous way. This is a
spatial analogue of (a special case of) results of Bertoin and Le Gall (2006),
who show that the generalised Fleming-Viot process that is dual to the
beta-coalescent, when suitably rescaled, converges to a continuous state
branching process with stable branching mechanism.
