  Recently, it has been shown that stochastic spatial Lotka-Volterra models
when suitably rescaled can converge to a super Brownian motion. We show that
the limit process could be a super stable process if the kernel of the
underlying motion is in the domain of attraction of a stable law. The
corresponding results in Brownian setting were proved by Cox and Perkins (2005,
2008). As applications of the convergence theorems, some new results on the
asymptotics of the voter model started from single 1 at the origin are obtained
which improve the results by Bramson and Griffeath (1980).
