  We consider a cyclically competing species model on a ring with global mixing
at finite rate, which corresponds to the well-known Lotka-Volterra equation in
the limit of infinite mixing rate. Within a perturbation analysis of the model
from the infinite mixing rate, we provide analytical evidence that extinction
occurs deterministically at sufficiently large but finite values of the mixing
rate for any species number $N\ge3$. Further, by focusing on the cases of
rather small species numbers, we discuss numerical results concerning the
trajectories toward such deterministic extinction, including global
bifurcations caused by changing the mixing rate.
