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 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.

arXiv commentSubmitted to Nonlinearity

References in corpus (24)

• Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games
• Cyclic dominance in evolutionary games: A review
• Diverging fluctuations in a spatial five-species cyclic dominance game
• Characterization of spiraling patterns in spatial rock-paper-scissors games
• Pattern formations driven by cyclic interactions: a brief review of recent developments
• When does cyclic dominance lead to stable spiral waves?
• Stability and bifurcations of heteroclinic cycles of type Z
• On local attraction properties and a stability index for heteroclinic connections
• Cycling chaos: its creation, persistence and loss of stability in a model of nonlinear magnetoconvection
• A mechanism for switching near a heteroclinic network
• The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit
• Stability in simple heteroclinic networks in ${\mathbb{R}}^4$
• Piecewise Linear Models for the Quasiperiodic Transition to Chaos
• Classification and stability of simple homoclinic cycles in R^5
• Two dimensional heteroclinic attractor in the generalized Lotka-Volterra system
• Quantifying noisy attractors: from heteroclinic to excitable networks
• Resonance bifurcations from robust homoclinic cycles
• Almost complete and equable heteroclinic networks
• Resonance bifurcations of robust heteroclinic networks
• Switching in heteroclinic networks
• Asymptotic stability of pseudo-simple heteroclinic cycles in R^4
• Asymptotic stability of robust heteroclinic networks
• Behaviour of trajectories near a two-cycle heteroclinic network
• The structure of mode-locking regions of piecewise-linear continuous maps

Cited by in corpus (7)

• Higher Order Dynamics in the Replicator Equation Produce a Limit Cycle in Rock-Paper-Scissors
• Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition
• Robust heteroclinic cycles in pluridimensions
• Community Formation in Wealth-Mediated Thermodynamic Strategy Evolution
• How many points converge to a heteroclinic network in an aperiodic way?
• Analysis of dynamics near heteroclinic networks in ${\mathbb{R}}^4$ with a projected map
• Seasonal Forcing in Rock-Paper-Scissors Population Dynamics