Coexistence and Extinction Pattern of Asymmetric Cyclic Game Species in a Square Lattice

The co-evolutionary dynamics of a cyclic game system is investigated in atwo-dimensional square lattice with the asymmetrical rates for three species.Different with the well-mixed system, coexistence and extinction emergealternately in the system, where a ``zero-one' behavior is robust for asmall population size, whereas, the system is predominated by coexistencefor a big population one. We study in detail the influence about thefluctuation to the change of the state, and find that the difference betweenthe maximal amplitude about the fluctuation and the average intensitydetermines which state the system is ultimately. In addition, we introducePotts energy to explain the reason of the ``zero-one' behavior. It is shownthat the average Potts energy per site is the distance to the ``zero-one'behavior in the model.