Rare-event extinction phenomena in three species cyclic predation games

In the modified May-Leonard model with cyclically competing three species, we compute the statistics of rare-event two species extinction process from a long lived metastable three-species coexistence state. We employ a master equation based eikonal quasi-stationary approximation of the metastable state effectively reducing the problem to the classical dynamics evolution of a Hamiltonian in six degrees of freedom. We then solve the evolution of this system by applying the Iterative Action Minimization Method(IAMM) and compute the action along the optimal path across the transcritical bifurcation. Our results are compared with action computed from the generating function based Hamiltonian of the said (3,1) game. The results obtained are validated for regions across the transcritical bifurcation in the system and investigated for different values of the system size parameter $V$.