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

The well-known game of Rock–Paper–Scissors can be used as a simple model of competition between three species. When modelled in continuous time using differential equations, the resulting system contains a heteroclinic cycle between the three equilibrium solutions representing the existence of only a single species. The game can be extended in a symmetric fashion by the addition of two further strategies (‘Lizard’ and ‘Spock’): now each strategy is dominant over two of the remaining four strategies, and is dominated by the remaining two. The differential equation model contains a set of coupled heteroclinic cycles forming a heteroclinic network. In this paper we carefully consider the dynamics near this heteroclinic network. We develop a technique to use a previously defined definition of stability (known as fragmentary asymptotic stability ) in numerical continuation software. We are able to identify regions of parameter space in which arbitrarily long periodic sequences of visits are made to the neighbourhoods of the equilibria, which form a complicated pattern in parameter space.