  We obtain the Brownian net of Sun and Swart (2008) as the scaling limit of
the paths traced out by a system of continuous (one-dimensional) space and time
branching and coalescing random walks. This demonstrates a certain universality
of the net, which we have not seen explored elsewhere. The walks themselves
arise in a natural way as the ancestral lineages relating individuals in a
sample from a biological population evolving according to the spatial
Lambda-Fleming-Viot process. Our scaling reveals the effect, in dimension one,
of spatial structure on the spread of a selectively advantageous gene through
such a population.
