  We investigate the effect of the magnetic anisotropy ($K_z$) on the static
and dynamic properties of magnetic vortices in small disks. Our micromagnetic
calculations reveal that for a range of $K_z$ there is an enlargement of the
vortex core. We analyze the influence of $K_z$ on the dynamics of the vortex
core magnetization reversal under the excitation of a pulsed field. The
presence of $K_z$, which leads to better resolved vortex structures, allows us
to discuss in more details the role played by the in-plane and perpendicular
components of the gyrotropic field during the vortex-antivortex nucleation and
annihilation.
