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

Abstract

We study a system of ordinary differential equations in R 5 that is used as a model both in population dynamics and in game theory, and is known to exhibit a heteroclinic network formed by three types of elementary heteroclinic cycles. We show the asymptotic stability of the network for parameter values in a range compatible with the two interpretations of the equations. 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 show that if one of the cycles 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 three cycles are completely unstable and the network is asymptotically stable, giving rise to intricate dynamics that has been observed numerically by other authors.

Stability of cycling behaviour near a heteroclinic network model of Rock–Paper–Scissors–Lizard–Spock

“…We have done these calculations numerically, and found that except for the A sequences, all sequences have a 'cusp' shaped local basin -that is, one in which the measure decreases to zero as the network is approached. These numerical calculations are supported by analytical calculations in [64], for the A, B and AAB cycle. However, these sequences are not hard to find by randomly selecting initial conditions for numerical integrations, which seems somewhat counter-intuitive.…”