Abstract
In this thesis, we study the systems of simple rock-paper-scissors games with different punishments s for the loser of the games. In common RPS games, the loser’s punishment is the same as the winner’s gain. When we adjust the punishment to smaller or bigger quantities, the systems’ stabilities are dramatically changed with two different trends. We analyzed the mean extinction time and obtained two different kinds of extinction behavior separated by the critical point. For strong selection regime, the mean extinction time grows logarithmically and does not be prolonged much with larger population size. For weak selection regime, the mean extinction time grows exponentially as system size goes up and the ecosystem is exponentially protected from extinction. Furthermore, a universal scaling function for the two phases is presented at the end of the thesis.