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Cyclic dominance in evolutionary games: A review

Attila Szolnoki, Mauro Mobilia, Luo-Luo Jiang, Bartosz Szczesny, Alastair M. Rucklidge, Matjaz Perc

arXiv:1408.6828v1physics.soc-phcond-mat.stat-mechcs.SInlin.AOq-bio.PE

TL;DR

Cyclic dominance links evolutionary games to biodiversity, nonlinear dynamics, and ecological complexity, but its spatial and mobility-dependent behavior is difficult to capture in well-mixed models. This review synthesizes mean-field, spatial, network, mobility, and spontaneous-cycling results, finding that structure and movement can shape coexistence, extinction, and winning alliances. It also identifies substantial empirical and theoretical areas outside its scope for future work.

  • Problem

    Cyclic dominance appears across ecological and evolutionary systems, yet understanding its pattern formation, mobility effects, and spontaneous emergence requires models beyond simple well-mixed populations.

  • Method

    The paper reviews mean-field and spatial RPS models, interaction networks, mobility, the complex Ginzburg-Landau equation, spontaneous cycles, and games with more than three strategies.

  • Results

    Network structure and mobility can alter oscillations, coexistence, extinction modes, and the winning solution among cyclic dynamics and neutral alliances.

  • Takeaways & Limitations

    Cyclic dominance provides a framework for connecting evolutionary-game dynamics with biodiversity, spatial organization, mobility, and alliance formation.

  • Takeaways & Limitations

    The review does not cover in detail competitive intransitivity, large-species community biodiversity, governing large competitive networks, or extensive empirical work.

Abstract

from arXiv · show

Rock is wrapped by paper, paper is cut by scissors, and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator-prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms, and the competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more regardless of the particularities of the game. Here we review recent advances on the rock-paper-scissors and related evolutionary games, focusing in particular on pattern formation, the impact of mobility, and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional rock-paper-scissors models and the application of the complex Ginzburg-Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related for example to dynamical effects of coevolutionary rules and invasion reversals due to multi-point interactions, are outlined as well.

I. INTRODUCTION

Cyclic dominance is widespread in ecological and evolutionary systems, where spatial structure, mobility, and increasing food-web complexity shape coexistence and emergent patterns. This review synthesizes theoretical advances from well-mixed RPS models to spatial games and spontaneously arising cycles.

  • Cyclic interactions occur across marine, plant, microbial, predator-prey, and mating systems, and illuminate biodiversity, selection, structural complexity, and prebiotic evolution.
  • Spatial structure can sustain biodiversity: Petri-dish organization kept three E. coli strains alive, whereas mobility preserved biodiversity in locally well-mixed mouse populations.
  • Spatial games require models beyond well-mixed populations because cyclic interactions combine strong fluctuations, nonlinear dynamics, multiple scales, and emergent structure.
  • Cyclic dominance can emerge spontaneously in games with three or more strategies, including volunteering, punishment, reward, coevolution, and joker strategies.
  • As strategy number and food-web complexity increase, systems support defensive alliances, subsystem solutions, and more complex stable outcomes.
  • The review covers mean-field and zero-dimensional RPS models, structured populations, mobility, the complex Ginzburg-Landau equation, spontaneous cycles, and systems with more than three strategies.

II. CYCLIC DOMINANCE IN WELL-MIXED SYSTEMS

Well-mixed RPS models describe cyclic competition through mean-field rate equations, while finite populations introduce demographic fluctuations that can destabilize coexistence and drive fixation. Parameter choices produce distinct oscillatory regimes, conserved orbits, and extinction dynamics.

  • Mean-field rate equations describe RPS dynamics in the N →∞ limit for densities of three cyclically competing species, with reproduction, dominance, replacement, and mutation processes.
  • The coexistence steady state has equal species densities a*=b*=c*=q/(p+3q).
  • When p, q > 0, z ≥0, and µ = 0, the equations reduce to the May-Leonard model, whose unstable coexistence state connects three absorbing single-species states through heteroclinic cycles.
  • A supercritical Hopf bifurcation occurs at µH = pq/(6(p + 3q)); coexistence is stable for µ > µH and unstable with a stable limit cycle for µ < µH.
  • For z > 0 and p = q = µ = 0, conserved density sum and product generate nested neutrally stable closed orbits around the coexistence center.
  • In finite populations, demographic fluctuations cause extinction of two strategies and fixation of the survivor; mean extinction time scales linearly with N, and the weakest strategy is most likely to survive.

III. ROCK-PAPER-SCISSORS GAME IN STRUCTURED POPULATIONS

Structured populations limit interactions to networks or neighborhoods, enabling spatial patterns and time-dependent coexistence. Mobility and network structure can instead promote or impede biodiversity and alter extinction or winning outcomes.

  • Limited neighborhood interactions allow a strategy to avoid its superior, enabling pattern formation and survival of all three competing strategies with time-dependent frequencies.
  • Mobility adds site exchange or movement, profoundly affecting RPS outcomes by either promoting or impeding biodiversity.
  • Mixing changes extinction modes, including domain annihilation, heteroclinic orbits, and traveling waves in finite systems.
  • Interaction-network structure and mobility both significantly affect cyclical-interaction outcomes, motivating detailed analysis and complex Ginzburg-Landau modeling.

A. Interaction networks

Interaction networks strongly shape cyclic-dominance outcomes: topology, spatial distribution, randomness, and coevolutionary rewiring can alter oscillations, coexistence, and absorbing transitions.

  • Interaction networks: Lattices limit interactions to neighbors, enabling spatial pattern formation and time-dependent coexistence of all three strategies.Network structure can support propagating fronts and spiral waves with Red Queen-like dynamics.
  • Interaction networks: Heterogeneous-degree networks can maintain stable coexistence, whereas sufficiently many permanent shortcuts can drive large oscillations followed by absorption.The shortcut effect arises from quenched randomness in small-world networks.
  • Interaction networks: Explicit spatial distribution is more crucial for coexistence and biodiversity than limited interaction range alone.This contrasts with spatial social dilemmas, where limited interaction range promotes cooperation through network reciprocity.
  • Interaction networks: Annealed randomness suppresses RPS frequency oscillations at sufficiently large rewiring probability P, replacing them with steady-state behavior.Similar stationary behavior was reported for a four-strategy cyclic-dominance game.
  • Interaction networks: Coevolutionary rewiring produces stationary → oscillatory → absorbing → stationary phase transitions as rewiring strength changes.The loser either adopts the winner’s strategy or rewires the link that caused the defeat.

B. Mobility

Mobility changes spiral-wave scales and can either preserve or destroy biodiversity, while competition fragments patterns and additional dynamics produce diverse spatial states.

  • Mobility: Small mobility below a critical threshold promotes coexistence, whereas larger mobility can destroy biodiversity when spirals outgrow the system.The review links this effect to mobility-driven increases in spiral size.
  • Mobility: Spiral-wave length scales increase with mobility and decrease with competition rate on square lattices.The simulations use competition and reproduction rates p = q = 1 for the mobility comparison, and m = 10^-4, q = 1 for the competition comparison.
  • Mobility: A coexistence threshold occurs at m = mc = (4.5 ± 0.5) × 10^-4 in the reported square-lattice simulations.The supplied passage introduces this critical mobility value after describing mobility-dependent spiral patterns.
  • Mobility: Large competition rates fragment macroscopic spirals by creating empty sites near spiral cores, thereby favoring diversity by preventing system-scale growth.The corresponding transition to disordered patterns occurs for p > pc = 2.3.
  • Mobility: Multi-armed spirals and anti-spirals are more robust to mobility than single-armed spirals, persisting at small and intermediate mobility rates.These patterns are observed using prepared initial conditions.
  • Mobility: Off-lattice coexistence can transition to extinction non-monotonically with mobility, and strong mobility may promote or impair coexistence depending on interaction range.Diversity remains robust across large regions of the parameter space.
  • Mobility: Epidemic spreading changes mobility-dependent coexistence basins: intra-species infection promotes coexistence, whereas the supplied passage contrasts it with inter-species infection.Without epidemic spreading, coexistence basins emerge for m < mc and extinction basins for m > mc.
  • Mobility: In four-strategy systems, low mobility favors RPS-type cyclic dynamics, while high mobility favors two-strategy neutral alliances.Mobility can therefore select between subsystem outcomes.

C. Metapopulation and nonlinear mobility

The review introduces metapopulation RPS models that separate hopping and pair exchange, producing nonlinear mobility and PDEs for spatial pattern formation. These PDEs reproduce stochastic simulations across substantial carrying capacities, while nonlinear mobility can destabilize spiral waves.

  • Metapopulation formulation: Metapopulation models allow individuals to move between neighboring patches and distinguish hopping at rate γd from pair exchange at rate γe.This separation captures movement in diluted versus crowded regions and generates nonlinear mobility when γe ≠ γd.
  • Continuum description: The continuum equations contain nonlinear diffusive terms when γnl = γd − γe is nonzero.The resulting PDE system includes reaction, reproduction, mutation, linear diffusion, and nonlinear mobility contributions for densities a, b, and c.
  • Continuum description: The PDEs accurately capture the lattice metapopulation model once carrying capacity is N ≳ 64.They are derived under N ≫ 1, yet qualitatively reproduce some stochastic outcomes even for N = 2–16.
  • Mobility effects: Nonlinear mobility enhances convective instability and far-field breakup of spiral waves at low mutation rates when γd > γe.The reported simulations use µ = 10^-6, far below the Hopf bifurcation value µH = 0.042.

D. Complex Ginzburg-Landau equation

The review develops the complex Ginzburg–Landau equation as a controlled approximation near the Hopf bifurcation for analyzing spatiotemporal patterns in spatial RPS models. It predicts spiral-wave regimes, while its validity is limited by neglected nonlinearities and approximations in earlier derivations.

  • Role of the CGLE: The CGLE is a nonlinear equation used to describe complex coherent structures, including spiral waves, across several physical systems.In spatial RPS games, it provides an analytical framework for characterizing spatiotemporal pattern formation.
  • Limitations: Earlier CGLE treatments rely on three uncontrolled approximations, including replacing heteroclinic cycles with stable limit cycles from a fictitious Hopf bifurcation.They also extend a near-fixed-point mapping beyond its expected domain and ignore nonlinear diffusive terms generated by the transformation.
  • Derivation: The CGLE is derived from the metapopulation PDEs near the Hopf bifurcation by slow-variable expansion and removal of secular terms at order O(ε^3).A complex modulated amplitude A(T,R) is introduced after transforming the densities around the coexistence fixed point.
  • Validity and predictions: The CGLE is a controlled approximation of the PDEs near the Hopf bifurcation and uses a linear real effective diffusion coefficient.It accurately characterizes spatiotemporal patterns in the bifurcation vicinity.
  • Validity and predictions: Four CGLE phases are separated by critical values approximately (cAI, cEI, cBS) = (1.75, 1.25, 0.845).The associated phase diagram distinguishes instability and spiral-wave regimes, including absolute instability, Eckhaus instability, and stable spirals.
  • Validity and predictions: Away from the Hopf bifurcation, the stable-spiral phase is generally replaced by an extended stable-spiral regime, while p ≫ z can produce Eckhaus-like far-field breakup.These effects show where CGLE-based pattern predictions require qualification.

IV. EVOLUTIONARY GAMES WITH SPONTANEOUSLY EMERGING CYCLIC DOMINANCE

Cyclic dominance can arise spontaneously in evolutionary games beyond explicitly imposed RPS interactions, including games with three or more strategies and some time-dependent two-strategy systems. The resulting cycles promote coexistence and stabilize otherwise unstable solutions.

  • General emergence: Spontaneous cyclic dominance may occur in evolutionary games with three or more strategies, and in two-strategy games when player-specific properties vary over time.Examples include volunteering, punishment, reward, coevolution, and joker-based games.
  • Consequences: Spontaneously emerging cycles promote coexistence, stabilize otherwise unstable solutions, and drive complex pattern formation.The review links cyclic dominance to differences between evolutionary outcomes in well-mixed and structured populations.
  • Tit-for-tat example: In the tit-for-tat prisoner’s dilemma, increasing its cost produces a transition from a stationary three-strategy state to an oscillatory three-strategy state.The transition occurs only in a structured population and results from payoff-defined relations, not a time-varying interaction network.
  • Punishment example: In structured public-goods games, punishment introduces a dominance loop in which punishing cooperators invade defectors, defectors invade pure cooperators, and pure cooperators invade punishers.Pool punishment can make one position in the loop an alliance of two strategies.

A. Time-dependent learning

Time-dependent learning ability can generate cyclic dominance even between cooperators and defectors in a spatial prisoner’s dilemma. Temporarily impaired learning produces propagating waves that maintain diversity under adverse conditions.

  • Mechanism: A temporarily impaired learning activity generates cyclic dominance between defectors and cooperators.This mechanism differs from limited teaching activity after a successful strategy pass, which affects propagation in another way.
  • Outcome: Propagating waves produced by time-dependent learning help maintain strategy diversity and allow cooperation under extremely adverse conditions.The figure depicts cooperators and defectors in a close dominance loop mediated by time-varying learning ability.
  • Broader implication: A time-dependent player-specific property can effectively replace a third strategy in a closed dominance loop.Thus cyclic dominance can emerge between only two competing strategies; demographic fluctuations can also sustain cyclic orbits in related learning games.

B. Voluntary participation

Voluntary participation produces RPS-like oscillations among cooperators, defectors, and loners, promoting coexistence and cooperation. Human experiments confirmed coexistence across a broader parameter range than theory predicted.

  • B. Voluntary participation: Volunteering creates RPS-like dynamics among cooperators, defectors, and loners, with all three strategy frequencies oscillating when r > 2.Defectors outperform cooperators in large cooperative groups; loners become advantageous when defectors dominate, allowing cooperation to recover as participating groups shrink.
  • B. Voluntary participation: Human experiments with 90 undergraduate participants measured time-averaged strategy frequencies in repeated five-person public goods games.The payoff of loners σ and multiplication factor r were varied as the principal parameters.
  • B. Voluntary participation: Structured populations also maintain survival of all three competing strategies, but not through the same oscillatory mechanism as well-mixed populations.The reviewed structured-population results extend the comparison between spatial games and classical RPS dynamics.
  • B. Voluntary participation: The three strategies coexist across a wide parameter range in experiments, with coexistence significantly more widespread than theory predicts.The discrepancy may reflect bounded rationality and emotions absent from simulations.
  • B. Voluntary participation: Oscillations in volunteering games can be induced by changes in interaction-network topology or payoff parameters.Small-world networks require exceeding a threshold fraction of shortcut links before stationary dynamics gives way to oscillatory dynamics.
  • B. Voluntary participation: Joker strategies can also generate spontaneous oscillations in finite well-mixed populations, independently of system size and strategy-updating rule.

C. When three competing strategies are more than three

Spatial evolutionary games can make three initial strategies effectively more than three when alliances or subsystem solutions act as additional competitors. Such cyclic structures arise in public-goods and ultimatum games, including spontaneously from group-interaction payoffs.

  • C. When three competing strategies are more than three: A subset of three original strategies can form subsystem solutions that function as additional effective strategies in structured populations.This expands the possible cyclic-dominance structures beyond the initially specified strategy count.
  • C. When three competing strategies are more than three: In the spatial public goods game with pool punishment, defectors and pool punishers form a stable alliance that participates in a closed dominance loop with cooperators and defectors.The alliance effectively manifests as a strategy alongside the pure strategies C and D.
  • C. When three competing strategies are more than three: A single point in a ternary phase diagram may not uniquely describe a spatial three-strategy system because spatiality adds a degree of freedom.That degree of freedom can produce an additional effective strategy absent from the initial evolutionary process.
  • C. When three competing strategies are more than three: Spatial ultimatum games with discrete strategies show that cyclic dominance can emerge spontaneously, with an alliance of two strategies occupying one position in the dominance loop.Discrete strategy classes enable pattern formation across the proposal-acceptance parameter plane.

V. CYCLIC DOMINANCE BETWEEN MORE THAN THREE STRATEGIES

Games with more than three strategies support richer outcomes than classical RPS, including coexistence, frozen states, global oscillations, and domain growth. These outcomes depend on invasion rates, mobility, interaction range, and food-web structure.

  • V. CYCLIC DOMINANCE BETWEEN MORE THAN THREE STRATEGIES: Subsystem solutions with missing strategies can also solve the full system, so final states depend on competition among defensive alliances as well as individual strategies.
  • V. CYCLIC DOMINANCE BETWEEN MORE THAN THREE STRATEGIES: Four-strategy cyclic games can form spiral patterns only without conservation of total density, while strong mobility can destroy coexistence.
  • V. CYCLIC DOMINANCE BETWEEN MORE THAN THREE STRATEGIES: All four strategies coexist only across a narrow invasion-rate range; otherwise neutral strategies can produce population-wide frozen states without mixing.Invasion rates and interaction range help determine transitions between coexistence and uniform or frozen outcomes.
  • V. CYCLIC DOMINANCE BETWEEN MORE THAN THREE STRATEGIES: Four-strategy games can exhibit globally synchronized oscillations, and these coexistence phases require less long-range interaction than three-strategy phases.
  • V. CYCLIC DOMINANCE BETWEEN MORE THAN THREE STRATEGIES: Equal-strength interactions among more than three strategies can produce curvature-driven domain growth governed by an algebraic law based on domain-wall length.

A. Alliances

Alliances are central to the diversity of outcomes in multi-strategy cyclic games. They may arise from food-web relations or spontaneously from group-interaction payoffs, while mobility and relaxation procedures affect which alliances persist and how they compete.

  • A. Alliances: Defensive alliances commonly form when neutral strategies protect one another from external invaders or cyclic strategies protect one another by invading a shared predator.Not every cyclic subsystem is an alliance: genuine alliances must protect their members from external invasion.
  • A. Alliances: Alliance-specific invasion rates can break food-web symmetry so that only one alliance survives, while sufficiently complex food webs can support competition among multiple alliances.
  • A. Alliances: Alliances can emerge spontaneously from payoff relations generated by multi-point interactions rather than from specific food-web relations.In spatial public goods games, either D + C + O or E + O + D can be stable, but E + O + D invades D + C + O when they meet.
  • A. Alliances: Mobility can reverse its effect on diversity: below a critical rate, neutral defensive alliances coarsen until one survives in finite systems, whereas above it all three alliances can coexist.This contrasts with classical RPS, where high mobility jeopardizes diversity and favors a single strategy.
  • A. Alliances: Mobility-dependent selection can favor either a three-strategy cyclic alliance or a neutral pair of two-strategy alliances in a four-strategy system.
  • A. Alliances: Alliance competitions require sufficient relaxation time for subsystem solutions to form; otherwise differences in relaxation times can invalidate viability comparisons.Strong mobility can also make dominance between competing alliances vanish simultaneously, producing divergent density fluctuations and invasion reversal.

VI. CONCLUSIONS AND OUTLOOK

The review synthesizes cyclic dominance across well-mixed, spatial, mobile, and spontaneously emerging evolutionary games, emphasizing oscillations, extinction, pattern formation, and alliances. It identifies important gaps involving experimental integration, group interactions, and coevolutionary games.

  • Conclusions: Mobility can either promote or jeopardize biodiversity and generate spiral, target-wave, multi-armed spiral, and anti-spiral patterns.The review discusses mobility alongside structured populations and oscillatory states.
  • Spontaneous cyclic dominance: The review covers spontaneous cyclic dominance in two-strategy learning dynamics, public goods games, joker-strategy systems, alliances, and games with more than three strategies.Alliances can close dominance loops and compete with one another, sometimes forming without specific food-web relations.
  • Limitations: The review excludes detailed treatment of competitive intransitivity, large-species ecological biodiversity, large competitive networks, and much experimental and empirical work.The authors note that detailed experimental coverage could constitute an independent review.
  • Future research: Future research should examine cyclic dominance in group interactions, where moving beyond pairwise interactions may qualitatively change evolutionary dynamics.The review also calls for reexamining coevolutionary-network findings specifically for cyclic games because social-dilemma results do not generally transfer.
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