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

Abstract

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.