Deterministic extinction by mixing in cyclically competing species

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.