Diverging fluctuations in a spatial five-species cyclic dominance game

Emerging solutions from the battle of defensive alliances

Competing strategies in an evolutionary game model, or species in a biosystem, can easily form a larger unit which protects them from the invasion of an external actor. Such a defensive alliance may have two, three, four or even more members. But how effective can be such formation against an alternative group composed by other competitors? To address this question we study a minimal model where a two-member and a four-member alliances fight in a symmetric and balanced way. By presenting representative phase diagrams, we systematically explore the whole parameter range which characterizes the inner dynamics of the alliances and the intensity of their interactions. The group formed by a pair, who can exchange their neighboring positions, prevail in the majority of the parameter region. The rival quartet can only win if their inner cyclic invasion rate is significant while the mixing rate of the pair is extremely low. At specific parameter values, when neither of the alliances is strong enough, new four-member solutions emerge where a rock-paper-scissors-like trio is extended by the other member of the pair. These new solutions coexist hence all six competitors can survive. The evolutionary process is accompanied by serious finite-size effects which can be mitigated by appropriately chosen prepared initial states.

Emerging spatiotemporal patterns in cyclic predator-prey systems with habitats

Three-species cyclic predator-prey systems are known to establish spiral waves that allow species to coexist. In this study, we analyze a structured heterogeneous system which gives one species an advantage to escape predation in an area that we refer to as a habitat and study the effect on species coexistence and emerging spatiotemporal patterns. Counterintuitively, the predator of the advantaged species emerges as dominant species with the highest average density inside the habitat. The species given the advantage in the form of an escape rate has the lowest average density until some threshold value for the escape rate is exceeded, after which the density of the species with the advantage overtakes that of its prey. Numerical analysis of the spatial density of each species as well as of the spatial two-point correlation function for both inside and outside the habitats allow a detailed quantitative discussion. Our analysis is extended to a six-species game that exhibits spontaneous spiral waves, which displays similar but more complicated results.

Lotka-Volterra versus May-Leonard formulations of the spatial stochastic rock-paper-scissors model: The missing link

The rock-paper-scissors (RPS) model successfully reproduces some of the main features of simple cyclic predator-prey systems with interspecific competition observed in nature. Still, lattice-based simulations of the spatial stochastic RPS model are known to give rise to significantly different results, depending on whether the three-state Lotka-Volterra or the four-state May-Leonard formulation is employed. This is true independently of the values of the model parameters and of the use of either a von Neumann or a Moore neighborhood. In this paper, we introduce a simple modification to the standard spatial stochastic RPS model in which the range of the search of the nearest neighbor may be extended up to a maximum Euclidean radius R. We show that, with this adjustment, the Lotka-Volterra and May-Leonard formulations can be designed to produce similar results, both in terms of dynamical properties and spatial features, by means of an appropriate parameter choice. In particular, we show that this modified spatial stochastic RPS model naturally leads to the emergence of spiral patterns in both its three- and four-state formulations.

Mortality makes coexistence vulnerable in evolutionary game of rock-paper-scissors

Multiple species in the ecosystem are believed to compete cyclically for maintaining balance in nature. The evolutionary dynamics of cyclic interaction crucially depends on different interactions representing different natural habits. Based on a rock-paper-scissors model of cyclic competition, we explore the role of mortality of individual organisms in the collective survival of a species. For this purpose a parameter called "natural death" is introduced. It is meant for bringing about the decease of an individual irrespective of any intra- and interspecific interaction. We perform a Monte Carlo simulation followed by a stability analysis of different fixed points of defined rate equations and observe that the natural death rate is surprisingly one of the most significant factors in deciding whether an ecosystem would come up with a coexistence or a single-species survival.

Performance of weak species in the simplest generalization of the rock-paper-scissors model to four species

We investigate the problem of the predominance and survival of "weak" species in the context of the simplest generalization of the spatial stochastic rock-paper-scissors model to four species by considering models in which one, two, or three species have a reduced predation probability. We show, using lattice based spatial stochastic simulations with random initial conditions, that if only one of the four species has its probability reduced, then the most abundant species is the prey of the "weakest" (assuming that the simulations are large enough for coexistence to prevail). Also, among the remaining cases, we present examples in which "weak" and "strong" species have similar average abundances and others in which either of them dominates-the most abundant species being always a prey of a weak species with which it maintains a unidirectional predator-prey interaction. However, in contrast to the three-species model, we find no systematic difference in the global performance of weak and strong species, and we conjecture that a similar result will hold if the number of species is further increased. We also determine the probability of single species survival and coexistence as a function of the lattice size, discussing its dependence on initial conditions and on the change to the dynamics of the model which results from the extinction of one of the species.

The effect of habitats and fitness on species coexistence in systems with cyclic dominance

Cyclic dominance between species may yield spiral waves that are known to provide a mechanism enabling persistent species coexistence. This observation holds true even in presence of spatial heterogeneity in the form of quenched disorder. In this work we study the effects on spatio-temporal patterns and species coexistence of structured spatial heterogeneity in the form of habitats that locally provide one of the species with an advantage. Performing extensive numerical simulations of systems with three and six species we show that these structured habitats destabilize spiral waves. Analyzing extinction events, we find that species extinction probabilities display a succession of maxima as function of time, that indicate a periodically enhanced probability for species extinction. Analysis of the mean extinction time reveals that as a function of the parameter governing the advantage of one of the species a transition between stable coexistence and unstable coexistence takes place. We also investigate how efficiency as a predator or a prey affects species coexistence.

Predominance of the weakest species in Lotka-Volterra and May-Leonard formulations of the rock-paper-scissors model

We revisit the problem of the predominance of the "weakest" species in the context of Lotka-Volterra and May-Leonard formulations of a spatial stochastic rock-paper-scissors model in which one of the species has its predation probability reduced by 0<P_{w}<1. We show that, despite the different population dynamics and spatial patterns, these two formulations lead to qualitatively similar results for the late time values of the relative abundances of the three species (as a function of P_{w}), as long as the simulation lattices are sufficiently large for coexistence to prevail-the "weakest" species generally having an advantage over the others (specially over its predator). However, for smaller simulation lattices, we find that the relatively large oscillations at the initial stages of simulations with random initial conditions may result in a significant dependence of the probability of species survival on the lattice size.

Dynamically generated hierarchies in games of competition

Spatial many-species predator-prey systems have been shown to yield very rich space-time patterns. This observation begs the question whether there exist universal mechanisms for generating this type of emerging complex patterns in nonequilibrium systems. In this work we investigate the possibility of dynamically generated hierarchies in predator-prey systems. We analyze a nine-species model with competing interactions and show that the studied situation results in the spontaneous formation of spirals within spirals. The parameter dependence of these intriguing nested spirals is elucidated. This is achieved through the numerical investigation of various quantities (correlation lengths, densities of empty sites, Fourier analysis of species densities, interface fluctuations) that allows us to gain a rather complete understanding of the spatial arrangements and the temporal evolution of the system. A possible generalization of the interaction scheme yielding dynamically generated hierarchies is discussed. As cyclic interactions occur spontaneously in systems with competing strategies, the mechanism discussed in this work should contribute to our understanding of various social and biological systems.

Robust coexistence with alternative competition strategy in the spatial cyclic game of five species

Alternative strategy is common in animal populations to promote reproductive fitness by obtaining resources. In spatial dynamics of cyclic competition, reproduction can occur when individuals obtain vacant rooms and, in this regard, empty sites should be resources for reproduction which can be induced by interspecific competition. In this paper, we study the role of alternative competition in the spatial system of cyclically competing five species by utilizing rock-paper-scissors-lizard-spock game. From Monte-Carlo simulations, we found that strong alternative competition can lead to the reemergence of coexistence of five species regardless of mobility, which is never reported in previous works under the symmetric competition structure. By investigating the coexistence probability, we also found that coexistence alternates by passing certain degrees of alternative competition in combination with mobility. In addition, we provided evidences in the opposite scenario by strengthening spontaneous competition, which exhibits the reemergence of coexistence similarly. Our findings may suggest more comprehensive perspectives to interpret mechanisms for biodiversity by alternative strategies in spatially extended systems than previously reported.

Invasion-controlled pattern formation in a generalized multispecies predator-prey system

Rock-scissors-paper game, as the simplest model of intransitive relation between competing agents, is a frequently quoted model to explain the stable diversity of competitors in the race of surviving. When increasing the number of competitors we may face a novel situation because beside the mentioned unidirectional predator-prey-like dominance a balanced or peer relation can emerge between some competitors. By utilizing this possibility in the present work we generalize a four-state predator-prey-type model where we establish two groups of species labeled by even and odd numbers. In particular, we introduce different invasion probabilities between and within these groups, which results in a tunable intensity of bidirectional invasion among peer species. Our study reveals an exceptional richness of pattern formations where five quantitatively different phases are observed by varying solely the strength of the mentioned inner invasion. The related transition points can be identified with the help of appropriate order parameters based on the spatial autocorrelation decay, on the fraction of empty sites, and on the variance of the species density. Furthermore, the application of diverse, alliance-specific inner invasion rates for different groups may result in the extinction of the pair of species where this inner invasion is moderate. These observations highlight that beyond the well-known and intensively studied cyclic dominance there is an additional source of complexity of pattern formation that has not been explored earlier.

Three-species competition with non-deterministic outcomes

Theoretical and experimental research studies have shown that ecosystems governed by non-transitive competition networks tend to maintain high levels of biodiversity. The theoretical body of work, however, has mainly focused on competition networks in which the outcomes of competition events are predetermined and hence deterministic, and where all species are identical up to their competitive relationships, an assumption that may limit the applicability of theoretical results to real-life situations. In this paper, we aim to probe the robustness of the link between biodiversity and non-transitive competition by introducing a three-dimensional winning probability parameter space, making the outcomes of competition events in a three-species in silico ecosystem uncertain. While two degenerate points in this parameter space have been the subject of previous studies, we investigate the remaining settings, which equip the species with distinct competitive abilities. We find that the impact of this modification depends on the spatial dimension of the system. When the system is well mixed, it collapses to monoculture, as is also the case in the non-transitive deterministic setting. In one dimension, chaotic patterns emerge, which tend to maintain biodiversity, and a power law relates the time that species manage to coexist to the degree of uncertainty regarding competition event outcomes. In two dimensions, the formation of spiral wave patterns ensures that biodiversity is maintained for moderate degrees of uncertainty, while considerable deviations from the non-transitive deterministic setting have strong negative effects on species coexistence. It can hence be concluded that non-transitive competition can still produce coexistence when the assumption of deterministic competition is abandoned. When the system collapses to monoculture, one observes a "survival of the strongest" law, as the species that has the highest probability of defeating its competitors has the best odds to become the sole survivor.

Basins of distinct asymptotic states in the cyclically competing mobile five species game

We study the dynamics of cyclic competing mobile five species on spatially extended systems originated from asymmetric initial populations and investigate the basins for the three possible asymptotic states, coexistence of all species, existences of only two independent species, and the extinction. Through extensive numerical simulations, we find a prosperous dependence on initial conditions for species biodiversity. In particular, for fixed given equal densities of two relevant species, we find that only five basins for the existence of two independent species exist and they are spirally entangled for high mobility. A basin of coexistence is outbreaking when the mobility parameter is decreased through a critical value and surrounded by the other five basins. For fixed given equal densities of two independent species, however, we find that basin structures are not spirally entangled. Further, final states of two independent species are totally different. For all possible considerations, the extinction state is not witnessed which is verified by the survival probability. To provide the validity of basin structures from lattice simulations, we analyze the system in mean-field manners. Consequently, results on macroscopic levels are matched to direct lattice simulations for high mobility regimes. These findings provide a good insight into the fundamental issue of the biodiversity among many species than previous cases.

Emergence of unusual coexistence states in cyclic game systems

Evolutionary games of cyclic competitions have been extensively studied to gain insights into one of the most fundamental phenomena in nature: biodiversity that seems to be excluded by the principle of natural selection. The Rock-Paper-Scissors (RPS) game of three species and its extensions [e.g., the Rock-Paper-Scissors-Lizard-Spock (RPSLS) game] are paradigmatic models in this field. In all previous studies, the intrinsic symmetry associated with cyclic competitions imposes a limitation on the resulting coexistence states, leading to only selective types of such states. We investigate the effect of nonuniform intraspecific competitions on coexistence and find that a wider spectrum of coexistence states can emerge and persist. This surprising finding is substantiated using three classes of cyclic game models through stability analysis, Monte Carlo simulations and continuous spatiotemporal dynamical evolution from partial differential equations. Our finding indicates that intraspecific competitions or alternative symmetry-breaking mechanisms can promote biodiversity to a broader extent than previously thought.

Coarsening with nontrivial in-domain dynamics: Correlations and interface fluctuations

Using numerical simulations we investigate the space-time properties of a system in which spirals emerge within coarsening domains, thus giving rise to nontrivial internal dynamics. Initially proposed in the context of population dynamics, the studied six-species model exhibits growing domains composed of three species in a rock-paper-scissors relationship. Through the investigation of different quantities, such as space-time correlations and the derived characteristic length, autocorrelation, density of empty sites, and interface width, we demonstrate that the nontrivial dynamics inside the domains affects the coarsening process as well as the properties of the interfaces separating different domains. Domain growth, aging, and interface fluctuations are shown to be governed by exponents whose values differ from those expected in systems with curvature driven coarsening.

A novel procedure for the identification of chaos in complex biological systems

We demonstrate the presence of chaos in stochastic simulations that are widely used to study biodiversity in nature. The investigation deals with a set of three distinct species that evolve according to the standard rules of mobility, reproduction and predation, with predation following the cyclic rules of the popular rock, paper and scissors game. The study uncovers the possibility to distinguish between time evolutions that start from slightly different initial states, guided by the Hamming distance which heuristically unveils the chaotic behavior. The finding opens up a quantitative approach that relates the correlation length to the average density of maxima of a typical species, and an ensemble of stochastic simulations is implemented to support the procedure. The main result of the work shows how a single and simple experimental realization that counts the density of maxima associated with the chaotic evolution of the species serves to infer its correlation length. We use the result to investigate others distinct complex systems, one dealing with a set of differential equations that can be used to model a diversity of natural and artificial chaotic systems, and another one, focusing on the ocean water level.

A Five-Species Jungle Game

In this paper, we investigate the five-species Jungle game in the framework of evolutionary game theory. We address the coexistence and biodiversity of the system using mean-field theory and Monte Carlo simulations. Then, we find that the inhibition from the bottom-level species to the top-level species can be critical factors that affect biodiversity, no matter how it is distributed, whether homogeneously well mixed or structured. We also find that predators' different preferences for food affect species' coexistence.

Zealotry promotes coexistence in the rock-paper-scissors model of cyclic dominance

Cyclic dominance models, such as the classic rock-paper-scissors (RPS) game, have found real-world applications in biology, ecology, and sociology. A key quantity of interest in such models is the coexistence time, i.e., the time until at least one population type goes extinct. Much recent research has considered conditions that lengthen coexistence times in an RPS model. A general finding is that coexistence is promoted by localized spatial interactions (low mobility), while extinction is fostered by global interactions (high mobility). That is, there exists a mobility threshold which separates a regime of long coexistence from a regime of rapid collapse of coexistence. The key finding of our paper is that if zealots (i.e., nodes able to defeat others while themselves being immune to defeat) of even a single type exist, then system coexistence time can be significantly prolonged, even in the presence of global interactions. This work thus highlights a crucial determinant of system survival time in cyclic dominance models.

Synchronization and extinction in cyclic games with mixed strategies

We consider cyclic Lotka-Volterra models with three and four strategies where at every interaction agents play a strategy using a time-dependent probability distribution. Agents learn from a loss by reducing the probability to play a losing strategy at the next interaction. For that, an agent is described as an urn containing β balls of three and four types, respectively, where after a loss one of the balls corresponding to the losing strategy is replaced by a ball representing the winning strategy. Using both mean-field rate equations and numerical simulations, we investigate a range of quantities that allows us to characterize the properties of these cyclic models with time-dependent probability distributions. For the three-strategy case in a spatial setting we observe a transition from neutrally stable to stable when changing the level of discretization of the probability distribution. For large values of β, yielding a good approximation to a continuous distribution, spatially synchronized temporal oscillations dominate the system. For the four-strategy game the system is always neutrally stable, but different regimes emerge, depending on the size of the system and the level of discretization.

Mesoscopic interactions and species coexistence in evolutionary game dynamics of cyclic competitions

Evolutionary dynamical models for cyclic competitions of three species (e.g., rock, paper, and scissors, or RPS) provide a paradigm, at the microscopic level of individual interactions, to address many issues in coexistence and biodiversity. Real ecosystems often involve competitions among more than three species. By extending the RPS game model to five (rock-paper-scissors-lizard-Spock, or RPSLS) mobile species, we uncover a fundamental type of mesoscopic interactions among subgroups of species. In particular, competitions at the microscopic level lead to the emergence of various local groups in different regions of the space, each involving three species. It is the interactions among the groups that fundamentally determine how many species can coexist. In fact, as the mobility is increased from zero, two transitions can occur: one from a five- to a three-species coexistence state and another from the latter to a uniform, single-species state. We develop a mean-field theory to show that, in order to understand the first transition, group interactions at the mesoscopic scale must be taken into account. Our findings suggest, more broadly, the importance of mesoscopic interactions in coexistence of great many species.

Cyclic dominance in evolutionary games: a review

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 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 (RPS) 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 RPS 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 owing to multi-point interactions, are also outlined.

Self-organizing patterns in an evolutionary rock-paper-scissors game for stochastic synchronized strategy updates

We study a spatial evolutionary rock-paper-scissors game with synchronized strategy updating. Players gain their payoff from games with their four neighbors on a square lattice and can update their strategies simultaneously according to the logit rule, which is the noisy version of the best-response dynamics. For the synchronized strategy update two types of global oscillations (with an ordered strategy arrangement and periods of three and six generations) can occur in this system in the zero noise limit. At low noise values, all nine oscillating phases are present in the system by forming a self-organizing spatial pattern due to the comprising invasion and speciation processes along the interfaces separating the different domains.

Non-equilibrium physics and evolution--adaptation, extinction, and ecology: a key issues review

Evolutionary dynamics in nature constitute an immensely complex non-equilibrium process. We review the application of physical models of evolution, by focusing on adaptation, extinction, and ecology. In each case, we examine key concepts by working through examples. Adaptation is discussed in the context of bacterial evolution, with a view toward the relationship between growth rates, mutation rates, selection strength, and environmental changes. Extinction dynamics for an isolated population are reviewed, with emphasis on the relation between timescales of extinction, population size, and temporally correlated noise. Ecological models are discussed by focusing on the effect of spatial interspecies interactions on diversity. Connections between physical processes--such as diffusion, turbulence, and localization--and evolutionary phenomena are highlighted.

Fourier decomposition of payoff matrix for symmetric three-strategy games

In spatial evolutionary games the payoff matrices are used to describe pair interactions among neighboring players located on a lattice. Now we introduce a way how the payoff matrices can be built up as a sum of payoff components reflecting basic symmetries. For the two-strategy games this decomposition reproduces interactions characteristic to the Ising model. For the three-strategy symmetric games the Fourier components can be classified into four types representing games with self-dependent and cross-dependent payoffs, variants of three-strategy coordinations, and the rock-scissors-paper (RSP) game. In the absence of the RSP component the game is a potential game. The resultant potential matrix has been evaluated. The general features of these systems are analyzed when the game is expressed by the linear combinations of these components.

Punishment based on public benefit fund significantly promotes cooperation

In prisoner's dilemma game (shortly, PD game), punishment is most frequently used to promote cooperation. However, outcome varies when different punishment approaches are applied. Here the PD game is studied on a square lattice when different punishment patterns are adopted. As is known to all, tax system, a common tool to adjust the temperature of the economy, is widely used in human society. Inspired by this philosophy, players in this study would pay corresponding taxes in accordance with their payoff level. In this way, public benefit fund is established consequently and it would be utilized to punish defectors. There are two main methods for punishing: slight intensity of punishment (shortly, SLP) and severe intensity of punishment (shortly, SEP). When the totaling of public benefit fund keeps relatively fixed, SLP extends further, which means more defectors would be punished; by contrast, SEP has a smaller coverage. It is of interest to verify whether these two measures can promote cooperation and which one is more efficient. Simulate results reveal that both of them can promote cooperation remarkably. Specifically speaking, SLP shows constant advantage from the point of view either of fractions of cooperation or average payoff.

The role of opportunistic migration in cyclic games

We study cyclic evolutionary games in a spatial diluted grid environment in which agents strategically interact locally but can also opportunistically move to other positions within a given migration radius. We find that opportunistic migration can inverse the cyclic prevalence between the strategies when the frequency of random imitation is large enough compared to the payoff-driven imitation. At the transition the average size of the patterns diverges and this threatens diversity of strategies.

From pairwise to group interactions in games of cyclic dominance

We study the rock-paper-scissors game in structured populations, where the invasion rates determine individual payoffs that govern the process of strategy change. The traditional version of the game is recovered if the payoffs for each potential invasion stem from a single pairwise interaction. However, the transformation of invasion rates to payoffs also allows the usage of larger interaction ranges. In addition to the traditional pairwise interaction, we therefore consider simultaneous interactions with all nearest neighbors, as well as with all nearest and next-nearest neighbors, thus effectively going from single pair to group interactions in games of cyclic dominance. We show that differences in the interaction range affect not only the stationary fractions of strategies but also their relations of dominance. The transition from pairwise to group interactions can thus decelerate and even revert the direction of the invasion between the competing strategies. Like in evolutionary social dilemmas, in games of cyclic dominance, too, the indirect multipoint interactions that are due to group interactions hence play a pivotal role. Our results indicate that, in addition to the invasion rates, the interaction range is at least as important for the maintenance of biodiversity among cyclically competing strategies.

Globally synchronized oscillations in complex cyclic games

The rock-paper-scissors game and its generalizations with S>3 species are well-studied models for cyclically interacting populations. Four is, however, the minimum number of species that, by allowing other interactions beyond the single, cyclic loop, breaks both the full intransitivity of the food graph and the one-predator, one-prey symmetry. Lütz et al. [J. Theor. Biol. 317, 286 (2013)] have shown the existence, on a square lattice, of two distinct phases, with either four or three coexisting species. In both phases, each agent is eventually replaced by one of its predators, but these strategy oscillations remain localized as long as the interactions are short ranged. Distant regions may be either out of phase or cycling through different food-web subloops (if any). Here we show that upon replacing a minimum fraction Q of the short-range interactions by long-range ones, there is a Hopf bifurcation, and global oscillations become stable. Surprisingly, to build such long-distance, global synchronization, the four-species coexistence phase requires fewer long-range interactions than the three-species phase, while one would naively expect the opposite to be true. Moreover, deviations from highly homogeneous conditions (χ=0 or 1) increase Qc, and the more heterogeneous is the food web, the harder the synchronization is. By further increasing Q, while the three-species phase remains stable, the four-species one has a transition to an absorbing, single-species state. The existence of a phase with global oscillations for S>3, when the interaction graph has multiple subloops and several possible local cycles, leads to the conjecture that global oscillations are a general characteristic, even for large, realistic food webs.

Mobility-dependent selection of competing strategy associations

Standard models of population dynamics focus on the interaction, survival, and extinction of the competing species individually. Real ecological systems, however, are characterized by an abundance of species (or strategies, in the terminology of evolutionary-game theory) that form intricate, complex interaction networks. The description of the ensuing dynamics may be aided by studying associations of certain strategies rather than individual ones. Here we show how such a higher-level description can bear fruitful insight. Motivated from different strains of colicinogenic Escherichia coli bacteria, we investigate a four-strategy system which contains a three-strategy cycle and a neutral alliance of two strategies. We find that the stochastic, spatial model exhibits a mobility-dependent selection of either the three-strategy cycle or of the neutral pair. We analyze this intriguing phenomenon numerically and analytically.