Nonlinear dynamics of the rock-paper-scissors game with mutations

Danielle F. P. Toupo and Steven H. Strogatz

Phys. Rev. E 91, 052907 – Published 11 May 2015

Abstract

We analyze the replicator-mutator equations for the rock-paper-scissors game. Various graph-theoretic patterns of mutation are considered, ranging from a single unidirectional mutation pathway between two of the species, to global bidirectional mutation among all the species. Our main result is that the coexistence state, in which all three species exist in equilibrium, can be destabilized by arbitrarily small mutation rates. After it loses stability, the coexistence state gives birth to a stable limit cycle solution created in a supercritical Hopf bifurcation. This attracting periodic solution exists for all the mutation patterns considered, and persists arbitrarily close to the limit of zero mutation rate and a zero-sum game.

Received 11 February 2015

DOI:https://doi.org/10.1103/PhysRevE.91.052907

Authors & Affiliations

Danielle F. P. Toupo and Steven H. Strogatz

Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 91, Iss. 5 — May 2015

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Pathways for global mutation.
Reuse & Permissions
Figure 2
Stability diagram for Eq. (1) and its associated phase portraits, in the case of global mutation. (a) Stability diagram. Locus of supercritical Hopf bifurcation, solid blue curve; saddle connection, dotted red line. (b) Phase portrait corresponding to Region 1 of the stability diagram. The system has a stable interior point, corresponding to coexistence of all three strategies. (c) Phase portrait corresponding to Region 2. The system has a stable limit cycle.
Reuse & Permissions
Figure 3
Stability diagram and phase portraits for Eq. (3). In this single-mutation case, rock $x$ mutates into paper $y$ at a rate $μ$ : $x \overset{μ}{\to} y$ . (a) Stability diagram. Supercritical Hopf bifurcation, solid blue curve; saddle connection, dotted red line. (b) Phase portrait corresponding to Region 1 of the stability diagram. The system has a stable interior point, corresponding to coexistence of all three strategies. (c) Phase portrait corresponding to Region 2. The system has a stable limit cycle.
Reuse & Permissions
Figure 4
Stability diagram and phase portraits for Eq. (4), corresponding to a single mutation pathway in which paper $y$ mutates into rock $x$ at a rate $μ$ . Note that this direction of mutation goes “against the flow” in the following sense: In the absence of mutation, paper beats rock and thus the flow normally tends to convert $x$ into $y$ . Thus, the direction of mutation assumed here opposes that flow. (a) Stability diagram. Supercritical Hopf bifurcation, solid blue curve; saddle connection, dotted red line; transcritical bifurcation, dashed green curve. In the coexistence region above the transcritical bifurcation curve, rock coexists with paper at the stable fixed point. (b) Phase portrait corresponding to Region 1 of the stability diagram. The system has a stable interior point, corresponding to coexistence of all three strategies. (c) Phase portrait corresponding to Region 2. The system has a stable limit cycle. (d) Phase portrait corresponding to Region 3, where rock and paper coexist and scissors has gone extinct. The system has a stable point on the boundary line joining rock and paper.
Reuse & Permissions
Figure 5
Stability diagram of the replicator equations for different patterns of two mutations, both occurring at a rate $μ$ . Supercritical Hopf bifurcation, solid blue curve; transcritical bifurcation, dashed green curve; saddle connection, dotted red line. (a) Opposing mutations, $z \overset{μ}{\to} y$ and $x \overset{μ}{\to} y$ . (b) Mutations in the same direction of circulation, $z \overset{μ}{\to} x$ and $x \overset{μ}{\to} y$ . (c) Bidirectional mutation between the same two species, $x \overset{μ}{\to} y$ and $y \overset{μ}{\to} x$ .
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics