Evolutionary Spatial Cyclic Games

• Evolutionary spatial cyclic games (ESCGs) are spatially structured systems where intransitive relationships, such as in rock-paper-scissors models, drive dynamic coexistence and pattern formation.
• They incorporate both explicit and emergent cyclicity using mathematical frameworks like replicator equations, payoff-gradient PDEs, and nonlocal integro-differential equations to model local interactions.
• Studies reveal that factors like lattice topology, mobility thresholds, and simulation methodologies critically influence biodiversity, extinction events, and long transient behaviors in ESCGs.

Searching arXiv for relevant ESCG papers and reviews. Search query: "evolutionary spatial cyclic games cyclic dominance spatial review rock paper scissors" Using arXiv search to retrieve relevant papers. Evolutionary spatial cyclic games (ESCGs) are spatially structured evolutionary systems in which species or strategies interact through cyclic or intransitive dominance relations, typically on lattices or graphs, so that local interaction, mobility, stochasticity, and topology jointly determine coexistence, extinction, and pattern formation. In the canonical rock-paper-scissors (RPS) case, no strategy is globally best because every winner is vulnerable to a third strategy, and spatial locality replaces well-mixed turnover by invasion fronts, domains, spiral waves, alliance formation, and finite-size extinction phenomena (Szolnoki et al., 2014).

1. Scope and conceptual structure

Within the ESCG literature, the canonical starting point is three-strategy cyclic dominance, especially RPS, but the field is broader than hard-coded three-node cycles. Reviews of cyclic dominance treat spatial games as nonequilibrium systems in which local interactions prevent immediate encounters with superior competitors, allow fronts and domains to form, and thereby stabilize coexistence that is fragile or impossible in well-mixed populations (Szolnoki et al., 2014). In that setting, the relevant observables are not only species fractions but also wavelength, front propagation, interface motion, subsystem invasion, and extinction routes.

This broadening is important because cyclicity may be explicit or emergent. Explicit cyclicity appears in RPS, five-species Rock-Paper-Scissors-Lizard-Spock (RPSLS), and larger cyclic food webs. Emergent cyclicity appears when payoff structure and spatial organization create effective loops such as cooperators, defectors, and abstainers in optional dilemmas, or when composite ecological states rather than primitive species become the effective competitors. The review literature also emphasizes that alliances can act as emergent strategies, so subsystem competition may replace single-species competition as the relevant dynamical level (Szolnoki et al., 2014).

Topology is likewise not incidental. For $N_S=3$ and $N_S=5$, the standard dominance graphs are simultaneously tournaments and circulant digraphs, but at $N_S=7$ these categories separate. This makes it possible to compare a fully connected tournament, such as $D(7,\{1,3,5\})$, with a non-tournament circulant such as $D(7,\{1,3\})$, while keeping an RPSLS-like local degree pattern in which each species has two prey and two predators (Cliff, 2024). That distinction has become central in recent work on biodiversity collapse under edge ablation.

2. Mathematical formalisms

A standard nonspatial starting point is the replicator equation for $m$ strategies,

$$\frac{d u_i}{dt} = u_i\left[f_i(\mathbf{u}) - \overline{f}(\mathbf{u})\right],$$

with $\mathbf{u}=(u_1,\dots,u_m)$, payoff matrix $A$, fitness $f_i(\mathbf{u})=(A\mathbf{u})_i$, and average fitness $N_S=5$0 (deForest et al., 2012). Once space is introduced, local densities replace global frequencies, so pointwise normalization becomes

$N_S=5$1

with $N_S=5$2 (deForest et al., 2012). A particularly influential spatial extension adds a game-dependent fitness-gradient flux,

$N_S=5$3

so that motion depends on local payoff gradients rather than only on Fickian diffusion (deForest et al., 2012). The resulting PDE is

$N_S=5$4

A second line of work derives deterministic nonlocal integro-differential equations from microscopic stochastic spatial games under Kac-type scaling. If $N_S=5$5 is the local density of strategy $N_S=5$6 at location $N_S=5$7, then the deterministic limit has the input-output form

$N_S=5$8

with local payoff

$N_S=5$9

(Hwang et al., 2010). For imitative dynamics, switching from $N_S=7$0 to $N_S=7$1 is weighted both by local presence of $N_S=7$2 and by the payoff difference $N_S=7$3, which makes the nonlocal coupling intrinsic rather than an ad hoc diffusion correction (Hwang et al., 2010).

A third rigorous framework studies spatial evolutionary games on $N_S=7$4, $N_S=7$5, with small selection $N_S=7$6. Under diffusive rescaling, local empirical densities converge to a reaction-diffusion equation

$N_S=7$7

where, for three-strategy zero-diagonal games, spatial structure modifies the effective payoff matrix to

$N_S=7$8

(Durrett, 2014). In this setting, space does not merely add diffusion; it renormalizes the game itself through coalescing-random-walk probabilities. The same work gives a general coexistence theorem when a repelling function exists for the modified game, but explicitly states that it does not prove the corresponding spatial coexistence result for the rock-paper-scissors case (Durrett, 2014).

3. Spatial organization and pattern formation

The best-known ESCG spatial patterns are propagating invasion fronts, rotating spiral waves, target waves, multi-armed spirals, anti-spirals, fragmented patterns, and alliance domains. In standard three-species spatial RPS with mobility and empty sites, random initial conditions generate spiral waves spontaneously, and the characteristic wavelength becomes a control variable for coexistence: if spirals fit inside the system, biodiversity is maintained; if they outgrow the system, absorbing states are reached (Szolnoki et al., 2014). For the case $N_S=7$9, the review literature quotes a critical mobility

$D(7,\{1,3,5\})$0

below which extinction probability vanishes in the large-size limit and above which extinction probability tends to one (Szolnoki et al., 2014).

Not all spiral mechanisms are reaction-diffusive in the usual sense. In a frozen-strategy RPS model with no strategy switching and no diffusion,

$D(7,\{1,3,5\})$1

the dynamics reduce to

$D(7,\{1,3,5\})$2

so motion is entirely payoff-guided (deForest et al., 2012). For the antisymmetric RPS matrix

$D(7,\{1,3,5\})$3

a three-bump initial condition

$D(7,\{1,3,5\})$4

with $D(7,\{1,3,5\})$5, $D(7,\{1,3,5\})$6, $D(7,\{1,3,5\})$7 on a $D(7,\{1,3,5\})$8 periodic grid and $D(7,\{1,3,5\})$9 seeds a rotating cyclic chase in which each strategy moves toward the one it beats and away from the one that beats it (deForest et al., 2012). The resulting spirals are organized transient or metastable structures rather than asymptotically stable attractors: they depend strongly on structured initial conditions, they do not emerge from generic random initial data, and on finite domains they eventually break up because of boundary effects (deForest et al., 2012).

Cyclic dominance can also arise among composite local states. In an asymmetric predator-prey spatial game extending the Lett-Auger-Gaillard model, each site carries one predator strategy $D(7,\{1,3\})$0 and one prey strategy $D(7,\{1,3\})$1, so the four local states are $D(7,\{1,3\})$2, $D(7,\{1,3\})$3, $D(7,\{1,3\})$4, and $D(7,\{1,3\})$5. In the coexistence region, prepared interface experiments identify an effective four-state loop

$D(7,\{1,3\})$6

with domain invasion sustaining long-lived coexistence (Cazaubiel et al., 2016). This establishes that ESCG-like intransitivity need not be a symmetric one-population species loop; it may instead emerge from asymmetric ecological coupling between different populations.

Spatial organization is also strongly geometry dependent. Work on deterministic Prisoner’s Dilemma dynamics on square and triangular lattices is ESCG-adjacent rather than cyclic in the strict sense, but it shows that coordination number and local geometry qualitatively alter invasion thresholds, front motion, and oscillatory motifs. On the square lattice, a broad regime $D(7,\{1,3\})$7 supports chaotic active patterns, whereas on the triangular lattice long-time states are mostly static and organized around geometry-specific motifs such as a seven-defector cluster and localized blinkers (Burovski et al., 2018). This suggests that in ESCGs, lattice geometry can alter pattern selection as strongly as payoff asymmetry.

4. Expanded families: optionality, coevolution, and emergent cycles

A major extension of ESCGs replaces explicit predator-prey loops by social dilemmas in which cyclic dominance emerges from an additional strategic option. In the Optional Prisoner’s Dilemma (OPD), each agent chooses cooperation $D(7,\{1,3\})$8, defection $D(7,\{1,3\})$9, or abstention $m$0, with payoff ordering

$m$1

and rescaling

$m$2

(Cardinot et al., 2017). This yields the explicit loop

$m$3

because cooperators can outperform abstainers in cooperative clusters, abstainers outperform defectors when defectors interact mostly with defectors, and defectors outperform cooperators through exploitation (Cardinot et al., 2017).

In a coevolutionary spatial OPD on a $m$4 square lattice with periodic boundary conditions and Moore neighborhoods, strategies and interaction weights evolve together. Each edge weight $m$5 is initially $m$6 and is updated by

$m$7

with utilities $m$8 and proportional imitation

$m$9

when $$\frac{d u_i}{dt} = u_i\left[f_i(\mathbf{u}) - \overline{f}(\mathbf{u})\right],$$0 (Cardinot et al., 2017). For

$$\frac{d u_i}{dt} = u_i\left[f_i(\mathbf{u}) - \overline{f}(\mathbf{u})\right],$$1

the fractions of $$\frac{d u_i}{dt} = u_i\left[f_i(\mathbf{u}) - \overline{f}(\mathbf{u})\right],$$2, $$\frac{d u_i}{dt} = u_i\left[f_i(\mathbf{u}) - \overline{f}(\mathbf{u})\right],$$3, and $$\frac{d u_i}{dt} = u_i\left[f_i(\mathbf{u}) - \overline{f}(\mathbf{u})\right],$$4 organize and remain near $$\frac{d u_i}{dt} = u_i\left[f_i(\mathbf{u}) - \overline{f}(\mathbf{u})\right],$$5 for long times up to $$\frac{d u_i}{dt} = u_i\left[f_i(\mathbf{u}) - \overline{f}(\mathbf{u})\right],$$6 Monte Carlo steps (Cardinot et al., 2017). The same study shows that recovery of coexistence after severe perturbation depends not only on the presence of all three strategies but also on the preservation of the coevolved weighted environment, which functions as an environmental memory (Cardinot et al., 2017).

Optionality also appears in fixation-based graph models that are adjacent to ESCGs. On cycles with birth-death or death-birth updating, and on complete graphs with the Moran process, optional games with one or more payoff-equivalent loner strategies admit exact or asymptotic selection criteria. In the weak-selection, low-mutation regime, cooperation is favored when

$$\frac{d u_i}{dt} = u_i\left[f_i(\mathbf{u}) - \overline{f}(\mathbf{u})\right],$$7

and increasing the number of loner strategies is operationally equivalent to increasing mutational bias toward loners (Jeong et al., 2014). This literature is not centered on canonical cyclic dominance, but it formalizes how optional participation changes graph-structured selection and makes clear that three-strategy invasion topologies with rock-paper-scissors-like characteristics arise naturally in fixation graphs (Jeong et al., 2014).

A plausible implication is that ESCGs should not be identified solely with explicit antisymmetric matrices. Optional participation, adaptive interaction weights, and composite or aggregated states can all generate cyclic invasion structures once space is introduced.

5. Periodicity, long transients, and asymptotic interpretation

One of the central methodological questions in ESCGs is whether observed coexistence or patterning is genuinely asymptotic or only transient. For deterministic winner-takes-all updating on any finite evolutionary graph, strong-selection dynamics are eventually simple in a precise sense: after a transient of finite length, the system almost everywhere in payoff parameter space either stabilizes at one state or enters a periodic cycle (Wu et al., 2022). The proof uses the finiteness of the global state space and the absence, almost everywhere, of opposite-strategy payoff ties. In a $$\frac{d u_i}{dt} = u_i\left[f_i(\mathbf{u}) - \overline{f}(\mathbf{u})\right],$$8 Nowak-May-type example, a highly symmetric initial condition leads to a transient lasting to generation

$$\frac{d u_i}{dt} = u_i\left[f_i(\mathbf{u}) - \overline{f}(\mathbf{u})\right],$$9

before entering a period-4 cycle (Wu et al., 2022). For ESCGs, the key point is that visually rich spatial complexity may persist for enormous times while remaining a transient of a finite deterministic map.

This caution becomes sharper in recent five-species RPSLS work. A replication-and-revision study of ablated and non-ablated dominance networks argues that many outcomes previously interpreted as asymptotic coexistence after $\mathbf{u}=(u_1,\dots,u_m)$0 Monte Carlo steps are instead very long-lived transients (Cliff, 2024). In the $\mathbf{u}=(u_1,\dots,u_m)$1 network at $\mathbf{u}=(u_1,\dots,u_m)$2, four-species coexistence has frequency about $\mathbf{u}=(u_1,\dots,u_m)$3 at $\mathbf{u}=(u_1,\dots,u_m)$4k MCS but falls to $\mathbf{u}=(u_1,\dots,u_m)$5 by $\mathbf{u}=(u_1,\dots,u_m)$6 MCS; the corresponding $\mathbf{u}=(u_1,\dots,u_m)$7 analysis places the second extinction wave around $\mathbf{u}=(u_1,\dots,u_m)$8 MCS and the decay $\mathbf{u}=(u_1,\dots,u_m)$9 around $A$0 MCS (Cliff, 2024). In the $A$1 case, a five-species bump around $A$2 persists at $A$3 MCS but is almost completely flattened to zero by $A$4 MCS, with a projected disappearance around $A$5 MCS (Cliff, 2024). The same work adds the unablated baseline $A$6 and argues that, for $A$7, $A$8, $A$9, $f_i(\mathbf{u})=(A\mathbf{u})_i$0, and $f_i(\mathbf{u})=(A\mathbf{u})_i$1, the long-run outcome is essentially a low-mobility three-species regime and a higher-mobility $f_i(\mathbf{u})=(A\mathbf{u})_i$2 regime (Cliff, 2024).

Network topology can nonetheless change these asymptotic classes or, at minimum, the route to them. In seven-species systems, single-edge ablation in the tournament $f_i(\mathbf{u})=(A\mathbf{u})_i$3 produces a sharp drop in five-species outcomes and a rise in two-species outcomes as mobility increases from $f_i(\mathbf{u})=(A\mathbf{u})_i$4 to $f_i(\mathbf{u})=(A\mathbf{u})_i$5, whereas the non-tournament circulant $f_i(\mathbf{u})=(A\mathbf{u})_i$6 shows no sudden phase-transition-like collapse over most of the same mobility range (Cliff, 2024). In the ablated non-tournament circulant, roughly $f_i(\mathbf{u})=(A\mathbf{u})_i$7 of independent runs end with

$f_i(\mathbf{u})=(A\mathbf{u})_i$8

and the frequencies of $f_i(\mathbf{u})=(A\mathbf{u})_i$9 and $N_S=5$00 are essentially constant with respect to $N_S=5$01 over most of the mobility range (Cliff, 2024). This establishes that biodiversity collapse under edge deletion is topology-specific rather than a generic property of “ablated cyclicity.”

6. Simulation methodology and unresolved problems

The computational substrate of ESCGs has itself become a research object because conclusions about coexistence, extinction, and asymptotics depend strongly on runtime, lattice size, and even the implementation of the elementary update. A recent methodological study formalizes the de facto standard Original Elementary Step (OES) and argues that it wastes substantial computation through implicit no-ops. Empirical testing found that as many as $N_S=5$02 of calls to the elementary-step routine are no-ops, and the Revised Elementary Step (RES) removes many of these by conditioning competition, reproduction, and movement on local occupancy state (Cliff, 2024). RES is not an exact semantic replica of OES, because OES allows at most one action per elementary step while RES allows up to three, but the reported simulations preserve the qualitative biological behavior while becoming less volatile. The same study argues that RES-based simulations can be run on smaller lattices than OES while preserving qualitative outcomes, reducing total simulation times by $N_S=5$03 or more in practice (Cliff, 2024).

GPU acceleration has widened the feasible time and size scales of ESCG simulation. A recent dissertation develops Apple Metal and Nvidia CUDA frameworks, benchmarked against a validated single-threaded C++ baseline, and reports that the CUDA maxStep implementation achieves up to a $N_S=5$04 speedup while making lattice sizes up to $N_S=5$05 tractable (Sinadjan, 17 Aug 2025). The same work emphasizes that larger system sizes and longer runs alter scientific interpretation: in five-species circulant systems, a $N_S=5$06 lattice may simplify to an effective three-species system by $N_S=5$07 MCS, whereas a $N_S=5$08 lattice still exhibits transient organization at MCS $N_S=5$09 and pronounced spiral structures by $N_S=5$10 MCS (Sinadjan, 17 Aug 2025). Computational scale is therefore not merely an engineering convenience; it changes which asymptotic claims are credible.

Several open problems remain central. For small-selection spatial games on $N_S=5$11, coexistence criteria are available when a repelling function exists for the modified game, but the spatial rock-paper-scissors coexistence theorem remains open (Durrett, 2014). For fitness-gradient-flux models, the progression to final patterned states in the frozen-strategy PDE is not yet understood in terms of stability or basins of attraction, and analytical conditions for three-strategy spiral existence, stability, and breakup remain open (deForest et al., 2012). More broadly, review work highlights coevolutionary rules and invasion reversals due to multi-point interactions as major future directions, indicating that food-web topology alone is insufficient once interaction range, dispersal rule, and subsystem organization are allowed to vary (Szolnoki et al., 2014).