  It is known that magnetic vortices in two dimensional Heisenberg models with
easy-plane anisotropy exhibit an instability depending on the anisotropy
strength. In this paper, we study the statistic behavior of the two-dimensional
easy-plane Heisenberg models with distance-dependent interactions, $J_{xy}(r)$
and $J_z(r)$ for in-plane and out-of-plane components. We develop analytical
and numerical methods for accurate determination of critical anisotropy, above
which out-of-plane vortices are stable. In particular, we explore the vortex
formation of the Gaussian-type interaction model and determine the critical
anisotropy accurately for square, hexagonal and triangular lattices.
