  We study the spreading of families in two-dimensional multispecies
predator-prey systems, in which species cyclically dominate each other. In each
time step randomly chosen individuals invade one of the nearest sites of the
square lattice eliminating their prey. Initially all individuals get a
family-name which will be carried on by their descendants. Monte Carlo
simulations show that the systems with several species (N=3,4,5) are
asymptotically approaching the behavior of the voter model, i.e., the survival
probability of families, the mean-size of families and the mean-square distance
of descendants from their ancestor exhibit the same scaling behavior. The
scaling behavior of the survival probability of families has a logarithmic
correction. In case of the voter model this correction depends on the number of
species, while cyclic predator-prey models behave like the voter model with
infinite species. It is found that changing the rates of invasions does not
change this asymptotic behavior. As an application a three-species system with
a fourth species intruder is also discussed.
