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Stability of cycles in a game of Rock-Scissors-Paper-Lizard-Spock

We study a system of ordinary differential equations in R5 that is used as a model both in population dynamics and in game theory, and is known to exhibit a heteroclinic network consisting in the union of four types of elementary heteroclinic cycles. We show the asymptotic stability of the network for parameter values in a range compatible with both population and game dynamics. We obtain estimates of the relative attractiveness of each one of the cycles by computing their stability indices. For the parameter values ensuring the asymptotic stability of the network we relate the attractiveness properties of each cycle to the others. In particular, for three of the cycles we show that if one of them has a weak form of attractiveness, then the other two are completely unstable. We also show the existence of an open region in parameter space where all four cycles are completely unstable and the network is asymptotically stable, giving rise to intricate dynamics that has been observed numerically by other authors.