Cyclic Dominance Models

• Cyclic Dominance Models are theoretical frameworks describing systems with non-transitive, intransitive interactions, exemplified by rock–paper–scissors dynamics.
• They utilize mathematical tools such as Lotka–Volterra equations, replicator dynamics, and stochastic simulations to analyze coexistence and pattern formation.
• Insights from these models underpin applications in ecology, evolutionary biology, and social dynamics, informing our understanding of biodiversity and system stability.

Cyclic dominance models describe systems in which a set of species or strategies interact via non-hierarchical, intransitive relations—most classically exemplified by the rock–paper–scissors (RPS) motif, where each competitor has both a predator and a prey. This framework underpins a wide variety of phenomena in ecology, evolutionary biology, game theory, and statistical physics, capturing the recurrent dynamical cycles and pattern formations that arise from non-transitive interactions. Theoretical and computational studies delineate diverse behaviors, from neutrally stable cycles and spiral pattern formation to critical fluctuations and resilience or fragility of biodiversity in the face of stochasticity, spatial structure, and parameter variation.

1. Fundamental Model Classes and Mathematical Structure

Cyclic dominance models exist in several mathematical and algorithmic forms, including:

a) Lotka–Volterra and May–Leonard Systems

The classical Lotka–Volterra (LV) formulation for $N$ species encodes dynamics via coupled ODEs. For $N=3$ (RPS), the mean-field equations read: $$\dot a = a(-kc + hb),\quad \dot b = b(-ka + hc),\quad \dot c = c(-kb + ha),$$ where $h$ and $k$ parameterize the invasion rates (Szolnoki et al., 2020). The May–Leonard extension introduces logistic growth and empty sites, as: $$\partial_t a = a\, [\mu - a - \alpha b] + D_a\nabla^2 a,$$ with analogous equations for $b$ and $c$ (Szolnoki et al., 2020).

b) Replicator Dynamics and Game-Theoretic Formulations

Evolutionary game theory models employ a payoff matrix $A$, with frequency vector $x$ evolving as: $$\dot x_i = x_i \left[ (A x)_i - x^T A x \right],\quad i=1,\ldots,N$$ For the RPS game, the antisymmetric matrix generates non-hierarchical selection (Szolnoki et al., 2014).

c) Stochastic and Individual-Based Realizations

Stochastic simulations—either as continuous-time Markov chains or discrete Monte Carlo processes—are used for lattice-based models, agent-based games, and adaptive networks (Yang et al., 2017, Demirel et al., 2010). These reveal the interplay between demographic noise, pattern formation, and critical phenomena absent in mean-field approximations.

d) Reaction–Diffusion and Pattern-Forming PDEs

Spatial models encode diffusion and local cyclic reactions, yielding reaction–diffusion PDEs. Near Hopf bifurcations or in oscillatory regimes, complex Ginzburg–Landau equations (CGLE) result as the amplitude equations for pattern formation (Szczesny et al., 2012, Szolnoki et al., 2014).

2. Stability, Bifurcation, and Coexistence Regimes

The dynamical outcome of cyclic dominance models depends critically on system parameters and initial conditions:

• Neutral and Stable Cycles: In perfectly symmetric LV-type RPS models, the interior fixed point is neutrally stable, evidenced by closed orbits centered on $(a,b,c)=(1/3,1/3,1/3)$. When logistic damping ($\mu$), mutation, or removal processes are present, Hopf bifurcations occur as parameters cross thresholds, generating limit cycles (Szolnoki et al., 2020, Yang et al., 2017).
• Criticality and Zero-Frequency Modes: In extended models (e.g., the five-species “Rock–Paper–Scissors–Lizard–Spock”), tuning the ratio of invasion rates $q=p_1/p_2$ to the “golden ratio” $\hat q=(\sqrt{5}-1)/2$ yields a zero-eigenvalue in the mean-field Jacobian. This results in a continuum of neutrally stable fixed points, with diverging fluctuation susceptibility $\chi\propto|q-q_c|^{-\gamma}$, $\gamma\approx1$ (Vukov et al., 2013).
• Bifurcations in Structured Populations: Stability of coexistence and pattern selection depend on both local interaction rates and long-range/topological structure. Hopf and Turing instabilities lead to temporal oscillations and stationary or traveling spatial patterns, with thresholds modulated by diffusivity, network rewiring, and heterogeneities (Szolnoki et al., 2020, Szczesny et al., 2012).
• Finite-Size and Demographic Noise: Stochastic models with finite $N$ experience noise-induced switching between absorbing states, fluctuation-driven extinction, and slow cycles. The period of dominance cycling transitions from logarithmic in the mutation rate in ODEs ($T_{\rm ODE}\sim -3\ln\mu$) to the much slower stochastic scaling $T\sim1/(N\mu)$ when demographic noise dominates (Yang et al., 2017, Winkler et al., 2010).

3. Pattern Formation and Spatial Dynamics

Cyclic dominance models in spatial settings give rise to complex self-organized structures:

• Spiral Waves, Domains, and Target Patterns: Spatial realizations of RPS and related cycles produce persistent spiral waves, targets, and labyrinthine domains. The CGLE governs the amplitude and phase of these structures, with the stability window determined by the nonlinearity parameter $c$ (e.g., $c_{\rm BS}\approx0.845$, $c_{\rm EI}\approx1.25$, $c_{\rm AI}\approx1.75$) (Szczesny et al., 2012, Szolnoki et al., 2014).
• Wavelength and Curvature Selection: The nonlinear dispersion relation obtained from bifurcation and heteroclinic theory determines the selected wavelength and speed of traveling waves ($\lambda\sim 2\pi\sqrt{\delta}$, $v\sim 2\sqrt{\delta}$). Core instabilities (Eckhaus, annihilation, etc.) bound the parameter regime for robust spiral formation (Postlethwaite et al., 2016).
• Mobility and Network Structure: Increased individual mobility or addition of small-world/long-range links enhances spatial mixing, potentially destroying spirals and leading to global oscillations or rapid extinction above critical mobility thresholds ($m_c\sim4.5\times10^{-4}$ for RPS with $p=q=1$) (Szolnoki et al., 2014, Szolnoki et al., 2016).
• Heterogeneity and Disorder: Even minimal quenched heterogeneity in site-specific invasion rates—such as a small fraction of “slow” sites—can suppress global synchronization and preserve biodiversity by decohering local cycles and preventing large-scale oscillations (Szolnoki et al., 2016).

4. Generalizations: Higher-Order Systems and Alliances

Cyclic dominance can involve more than three states, producing intricate alliance dynamics and phase structures:

• Multispecies Cycles: In five-species lattices, adjusting invasion-rate ratios controls the neutrality of multi-species associations and generates critical phenomena analogous to the three-species case (Vukov et al., 2013). In eight-species (octagonal) models with inner and outer cycles, genuinely higher-order alliances can emerge, with coexistence and alliance vitality determined by internal symmetry and the balance of invasion rates (Park et al., 2023).
• Defensive Alliances: Collective stability depends on the internal homogeneity of invasion rates among alliance members; asymmetric or heterogeneous interaction strengths destabilize certain groups, and smaller homogeneous cycles can outcompete larger, imperfect ones under appropriate conditions (Park et al., 2023).
• External Species and Network Topology: Adding “pestilent” species that prey selectively on one of the original loop members generates phase transitions between coexistence, chain associations, and absorbing single-species states. Overly strong invaders may inadvertently cause their own extinction by eliminating their resource base (Bazeia et al., 2021).

5. Mechanistic Extensions and Applications

The cyclic dominance framework extends to diverse mechanistic and applied contexts:

• Eco-evolutionary Complexity: Rich dynamics such as rock–paper–scissors behavior among prey, predator, and parasite emerge only when resource limitation (free space) is incorporated; this stabilizes coexistence and yields supercritical Hopf bifurcation into a limit cycle of abundances (Chowdhury et al., 2023).
• Learning and Strategy Adaptation: When learning rates or behavioral plasticity are made strategy- or agent-dependent, nonintuitive feedbacks arise: lowering a strategy’s learning activity can elevate its predator’s abundance (“survival of the weakest”), a manifestation of the non-monotonic macroscopic responses in cyclic feedback systems (Szolnoki et al., 2020, Yu et al., 8 Apr 2025).
• Adaptive Networks and Social Dilemmas: Adaptive rewiring or coevolution of agent interaction networks generates further phase transitions—stationary coexistence, oscillatory cycling, heteroclinic consensus, and fragmentation—depending on adoption versus segregation probability (Demirel et al., 2010, Guo et al., 2020).
• Public Goods and Voluntary Games: Introduction of additional strategies (loners, hedgers) and meta-strategic behaviors (reciprocity, abstention) leads to higher-order cycles, new parameter regions for coexistence, and expanded conditions for maintaining cooperation through cyclic dominance (Guo et al., 2020).

6. Analytical Tools, Metrics, and Diagnostics

A variety of mathematical and computational methods are employed for analyzing cyclic dominance models:

Technique/Metric	Purpose	References
Mean-field/Replicator ODEs	Macroscopic, deterministic dynamics	(Szolnoki et al., 2014, Yang et al., 2017)
Master equations	Full configuration statistics	(Winkler et al., 2010)
Pair-approximation/moment closure	Approximate correlations	(Demirel et al., 2010)
Pattern/amplitude equations (CGLE)	Spatial pattern selectivity and stability	(Szczesny et al., 2012, Postlethwaite et al., 2016)
Monte Carlo/Gillespie simulations	Spatio-temporal emergent phenomena	(Szolnoki et al., 2020, Park et al., 2023)
Fluctuation susceptibility $(\chi)$	Quantify divergence/criticality	(Vukov et al., 2013)
Diversity indices (Shannon $H$)	Quantify biodiversity	(Szolnoki et al., 2020)
Area of limit cycle in simplex ($A$)	Measure global oscillations	(Szolnoki et al., 2016)

Order parameters (e.g., species densities, fluctuation measures, cluster sizes) and spectral properties (eigenvalues of Jacobians) are used to diagnose stability, bifurcation, and phase boundaries.

7. Implications for Biodiversity, Stability, and Robustness

Cyclic dominance is an organizing principle for sustained coexistence and dynamic patterning in ecological and social systems:

• Biodiversity Maintenance: Non-transitive loops prevent dominance fixation and facilitate the persistence of rare types in fluctuating environments, supporting “Red Queen” dynamics and high diversity (Szolnoki et al., 2014, Szolnoki et al., 2020).
• Robustness and Fragility: While cyclic dominance can enforce resilience, it is also sensitive to demographic noise, spatial structure, and parameter heterogeneity. Spatial disorder, quenched heterogeneity, or adaptive strategies can stabilize or destabilize coexistence regimes, generate critical transitions, or trigger extinction cascades (Szolnoki et al., 2016, Yang et al., 2017, Bazeia et al., 2021).
• Generalization Limitations: Predicting outcome basins, interface velocities, and emergent alliances in high-dimensional systems is computationally intensive and sensitive to finite-size effects. Properly capturing alliance competition, spatial scale, and noise-induced transitions is crucial for theoretical and empirical analysis (Park et al., 2023).
• Application to Social Systems: Extension to opinion dynamics, voluntary games, and adaptive links highlights the universality of cyclic dynamic motifs in maintaining heterogeneity and complex collective outcomes in both biological and socio-technical networks (Guo et al., 2020, Demirel et al., 2010).

In sum, cyclic dominance models form a mathematically rigorous and empirically grounded framework, elucidating how intransitive competition, spatial structure, demographic noise, and adaptive behavior combine to yield the rich diversity and spatio-temporal complexity observed in nature and society. These insights inform the design and interpretation of interventions in ecological management, microbial control, cooperative systems, and networked organizations.