Cyclic Dominance Research Articles

Zealots tame oscillations in the spatial rock-paper-scissors game.

The rock-paper-scissors game is a paradigmatic model for biodiversity, with applications ranging from microbial populations to human societies. Research has shown, however, that mobility jeopardizes biodiversity by promoting the formation of spiral waves, especially if there is no conservation law in place for the total number of competing players. First, we show that even if such a conservation law applies, mobility still jeopardizes biodiversity in the spatial rock-paper-scissors game if only a small fraction of links of the square lattice is randomly rewired. Secondly, we show that zealots are very effective in taming the amplitude of oscillations that emerge due to mobility and/or interaction randomness, and this regardless of whether the later is quenched or annealed. While even a tiny fraction of zealots brings significant benefits, at 5% occupancy zealots practically destroy all oscillations regardless of the intensity of mobility, and regardless of the type and strength of randomness in the interaction structure. Interestingly, by annealed randomness the impact of zealots is qualitatively the same as by mobility, which highlights that fast diffusion does not necessarily destroy the coexistence of species, and that zealotry thus helps to recover the stable mean-field solution. Our results strengthen the important role of zealots in models of cyclic dominance, and they reveal fascinating evolutionary outcomes in structured populations that are a unique consequence of such uncompromising behavior.

Inhibitory interactions promote frequent bistability among competing bacteria.

It is largely unknown how the process of microbial community assembly is affected by the order of species arrival, initial species abundances and interactions between species. A minimal way of capturing competitive abilities in a frequency-dependent manner is with an invasibility network specifying whether a species at low abundance can increase in frequency in an environment dominated by another species. Here, using a panel of prolific small-molecule producers and a habitat with feast-and-famine cycles, we show that the most abundant strain can often exclude other strains—resulting in bistability between pairs of strains. Instead of a single winner, the empirically determined invasibility network is ruled by multiple strains that cannot invade each other, and does not contain loops of cyclic dominance. Antibiotic inhibition contributes to bistability by helping producers resist invasions while at high abundance and by reducing producers' ability to invade when at low abundance.

A Framework for Evaluating Skyline Queries over Incomplete Data

Research interest in skyline queries has been significantly increased over the years, as skyline queries can be utilized in many contemporary applications, such as multi-criteria decision-making system, decision support system, recommendation system, data mining, and personalized systems. Skyline queries return data item that is not dominated by any other data items in all dimensions (attributes). Most of the existing skyline approaches assumed that database is complete and values are present during the skyline process. However, such assumption is not always to be true, particularly in a real world database where values of data item might not be available (missing) in one or more dimensions. Thus, the incompleteness of the data impacts negatively on skyline process due to losing the transitivity property which leads into the issue of cyclic dominance. Therefore, applying skyline technique directly on an incomplete database is prohibitive and might result into exhaustive pairwise comparison. This paper presents an approach that efficiently evaluates skyline queries in incomplete database. The approach aims at reducing the number of pairwise comparisons and shortens the searching space in identifying the skylines. Several experiments have been conducted to demonstrate that our approach outperforms the previous approach through producing a lower number of pairwise comparisons. Furthermore, the result also illustrates that our approach is scalable and efficient.

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.

Noisy predator–prey model explains oscillation patterns in sockeye salmon data

A model of sockeye salmon population dynamics that incorporates predator–prey dynamics in the nursery lakes, salmon migration and stochastic effects is compared to Fraser River sockeye salmon spawner numbers with respect to cyclic dominance. For this comparison we use a method developed by White et al. (2014) to calculate measures for the consistency and strength of cyclic dominance in the time series using its wavelet transform. We find that the model can match the oscillation patterns found in nature, both for persistently oscillating populations and for intermittent oscillations. It matches persistently oscillating populations much better than a model that does not incorporate predator–prey interaction. Persistent oscillations are more likely to occur in the model if the growth conditions for the sockeye fry are good and the coupling to the predator is strong.

Vortices determine the dynamics of biodiversity in cyclical interactions with protection spillovers

If rock beats scissors and scissors beat paper, one might assume that rock beats paper too. But this is not the case for intransitive relationships that make up the famous rock-paper-scissors game. However, the sole presence of paper might prevent rock from beating scissors, simply because paper beats rock. This is the blueprint for the rock-paper-scissors game with protection spillovers, which has recently been introduced as a new paradigm for biodiversity in well-mixed microbial populations. Here we study the game in structured populations, demonstrating that protection spillovers give rise to spatial patterns that are impossible to observe in the classical rock-paper-scissors game. We show that the spatiotemporal dynamics of the system is determined by the density of stable vortices, which may ultimately transform to frozen states, to propagating waves, or to target waves with reversed propagation direction, depending further on the degree and type of randomness in the interactions among the species. If vortices are rare, the fixation to waves and complex oscillatory solutions is likelier. Moreover, annealed randomness in interactions favors the emergence of target waves, while quenched randomness favors collective synchronization. Our results demonstrate that protection spillovers may fundamentally change the dynamics of cyclic dominance in structured populations, and they outline the possibility of programming pattern formation in microbial populations.

Strong Migration Limit for Games in Structured Populations: Applications to Dominance Hierarchy and Set Structure

In this paper, we deduce a condition for a strategy S1 to be more abundant on average at equilibrium under weak selection than another strategy S2 in a population structured into a finite number of colonies of fixed proportions as the population size tends to infinity. It is assumed that one individual reproduces at a time with some probability depending on the payoff received in pairwise interactions within colonies and between colonies and that the offspring replaces one individual chosen at random in the colony into which the offspring migrates. It is shown that an expected weighted average equilibrium frequency of S1 under weak symmetric strategy mutation between S1 and S2 is increased by weak selection if an expected weighted payoff of S1 near neutrality exceeds the corresponding expected weighted payoff of S2. The weights are given in terms of reproductive values of individuals in the different colonies in the neutral model. This condition for S1 to be favoured by weak selection is obtained from a strong migration limit of the genealogical process under neutrality for a sample of individuals, which is proven using a two-time scale argument. The condition is applied to games between individuals in colonies with linear or cyclic dominance and between individuals belonging to groups represented by subsets of a given set.

Information Transfer between Generations Linked to Biodiversity in Rock-Paper-Scissors Games

Ecological processes, such as reproduction, mobility, and interaction between species, play important roles in the maintenance of biodiversity. Classically, the cyclic dominance of species has been modelled using the nonhierarchical interactions among competing species, represented by the “Rock-Paper-Scissors” (RPS) game. Here we propose a cascaded channel model for analyzing the existence of biodiversity in the RPS game. The transition between successive generations is modelled as communication of information over a noisy communication channel. The rate of transfer of information over successive generations is studied using mutual information and it is found that “greedy” information transfer between successive generations may lead to conditions for extinction. This generalized framework can be used to study biodiversity in any number of interacting species, ecosystems with unequal rates for different species, and also competitive networks.

A double-edged sword: Benefits and pitfalls of heterogeneous punishment in evolutionary inspection games.

As a simple model for criminal behavior, the traditional two-strategy inspection game yields counterintuitive results that fail to describe empirical data. The latter shows that crime is often recurrent, and that crime rates do not respond linearly to mitigation attempts. A more apt model entails ordinary people who neither commit nor sanction crime as the third strategy besides the criminals and punishers. Since ordinary people free-ride on the sanctioning efforts of punishers, they may introduce cyclic dominance that enables the coexistence of all three competing strategies. In this setup ordinary individuals become the biggest impediment to crime abatement. We therefore also consider heterogeneous punisher strategies, which seek to reduce their investment into fighting crime in order to attain a more competitive payoff. We show that this diversity of punishment leads to an explosion of complexity in the system, where the benefits and pitfalls of criminal behavior are revealed in the most unexpected ways. Due to the raise and fall of different alliances no less than six consecutive phase transitions occur in dependence on solely the temptation to succumb to criminal behavior, leading the population from ordinary people-dominated across punisher-dominated to crime-dominated phases, yet always failing to abolish crime completely.

Evolutionary Voluntary Prisoner’s Dilemma Game under Deterministic and Stochastic Dynamics

The voluntary prisoner’s dilemma (VPD) game has sparked interest from various fields since it was proposed as an effective mechanism to incentivize cooperative behavior. Current studies show that the inherent cyclic dominance of the strategies of the VPD game results in periodic oscillations in population. This paper investigated the influence of the level of individual rationality and the size of a population on the evolutionary dynamics of the VPD game. Different deterministic dynamics, such as the replicator dynamic, the Smith dynamic, the Brown-von Neumann-Nash (BNN) dynamic and the best response (BR) dynamic, for the evolutionary VPD game were modeled and simulated. The stochastic evolutionary dynamics based on quasi birth and death (QBD) process was proposed for the evolutionary VPD game and compared with deterministic dynamics. The results indicated that with the increase of the loners’ fixed payoff, the loner is more likely to remain in the stable state of a VPD game under any of the dynamics mentioned above. However, the different speeds of motion under the dynamics in the cycle dominance proved to be diverse under different evolutionary dynamics and also highly sensitive to the rationality of individuals in a population. Furthermore, in QBD stochastic dynamics, the size of the population has a remarkable effect on the possibility distribution. When the population size increases, the limited distribution of the QBD process will be in accordance with the results in the deterministic dynamics.

The rock–paper–scissors game

Rock–Paper–Scissors (RPS), a game of cyclic dominance, is not merely a popular children’s game but also a basic model system for studying decision-making in non-cooperative strategic interactions. Aimed at students of physics with no background in game theory, this paper introduces the concepts of Nash equilibrium and evolutionarily stable strategy, and reviews some recent theoretical and empirical efforts on the non-equilibrium properties of the iterated RPS, including collective cycling, conditional response patterns and microscopic mechanisms that facilitate cooperation. We also introduce several dynamical processes to illustrate the applications of RPS as a simplified model of species competition in ecological systems and price cycling in economic markets.

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.

Optimal cooperation-trap strategies for the iterated rock-paper-scissors game.

In an iterated non-cooperative game, if all the players act to maximize their individual accumulated payoff, the system as a whole usually converges to a Nash equilibrium that poorly benefits any player. Here we show that such an undesirable destiny is avoidable in an iterated Rock-Paper-Scissors (RPS) game involving two rational players, X and Y. Player X has the option of proactively adopting a cooperation-trap strategy, which enforces complete cooperation from the rational player Y and leads to a highly beneficial and maximally fair situation to both players. That maximal degree of cooperation is achievable in such a competitive system with cyclic dominance of actions may stimulate further theoretical and empirical studies on how to resolve conflicts and enhance cooperation in human societies.

Dominance Hierarchies Induce a Population’s Full Cooperation

Evolution of cooperation is still a puzzle in evolutionary and socio biology. Based on the asymmetric interactions in a dominance hierarchy system, a simple theoretical framework is developed to reveal the effect of “induced cooperation” (i.e., the cooperative behavior of subordinate individuals to dominant individuals) on the evolution of cooperation. Extending the classic Prisoner’s Dilemma (PD) game, we define the concept of the defection cost of subordinate individuals to measure the effect of induced cooperation, i.e., a subordinate defector will incur a cost when it plays against a dominant defector. The analysis of the repeated PD game with linear dominance hierarchy and with cyclic dominance hierarchy shows clearly that induced cooperation of subordinate individuals may lead to a population’s full cooperation, and that the coexistence of cooperation and defection in a population with hierarchy dominance is possible. Our results are the first step to develop a new theoretical approach for understanding the emergence of cooperation; namely, that induced cooperation is one of the most important forces driving the evolution of cooperation as pointed out by May (in May R, McLean A (eds) Theoretical ecology 3rd edn., (2007)).

Chemical warfare and survival strategies in bacterial range expansions.

Dispersal of species is a fundamental ecological process in the evolution and maintenance of biodiversity. Limited control over ecological parameters has hindered progress in understanding of what enables species to colonize new areas, as well as the importance of interspecies interactions. Such control is necessary to construct reliable mathematical models of ecosystems. In our work, we studied dispersal in the context of bacterial range expansions and identified the major determinants of species coexistence for a bacterial model system of three Escherichia coli strains (toxin-producing, sensitive and resistant). Genetic engineering allowed us to tune strain growth rates and to design different ecological scenarios (cyclic and hierarchical). We found that coexistence of all strains depended on three strongly interdependent factors: composition of inoculum, relative strain growth rates and effective toxin range. Robust agreement between our experiments and a thoroughly calibrated computational model enabled us to extrapolate these intricate interdependencies in terms of phenomenological biodiversity laws. Our mathematical analysis also suggested that cyclic dominance between strains is not a prerequisite for coexistence in competitive range expansions. Instead, robust three-strain coexistence required a balance between growth rates and either a reduced initial ratio of the toxin-producing strain, or a sufficiently short toxin range.

Exploring cyclic dominance of sockeye salmon with a predator–prey model

We explain in an intuitive and detailed way a predator–prey model that generates cyclic dominance in Fraser River sockeye salmon. In contrast with usual predator–prey models, this model includes four distinct prey lines and thus a combination of continuous and discrete dynamics, reflecting the particular freshwater and marine life cycle features of sockeye salmon (Oncorhynchus nerka) populations. The predator–prey interaction causing the oscillations takes place in the rearing lakes, rather than in the ocean. The values of most parameters of this model can be estimated from empirical data that are available for the large salmon-rearing lakes in the Fraser River basin. The mechanism that produces the oscillations in this model is compared with other mechanisms that can generate population oscillations, and we argue that predator–prey dynamics is the most likely mechanism to produce the observed patterns. We explain why the period of the oscillation is exactly 4 years, and we explore how the dynamical pattern is affected by changes in external conditions or by management decisions.

Effect of introducing a competitor on cyclic dominance of sockeye salmon

We study the effects of introducing a competing species into a 3-species model for the population dynamics of sockeye salmon, thereby converting a food chain into a diamond module. We find that this often leads to the disappearance of the 4-year oscillation of sockeye salmon known as cyclic dominance when parameters are chosen such that all four species can coexist. Only when the population size of the competitor is small the phenomenon of cyclic dominance can persist. There is also a large region of parameter space where either the sockeye salmon or the competitor goes extinct. We discuss how these findings can be reconciled with the prevalence of cyclic dominance in many sockeye brood lakes.

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.

Cycles of Strategies and Changes of Distribution in Public Goods Game: An Experimental Investigation

In this communication, a simple mechanism in the optional public goods game is experimentally investigated using two experimental settings; and first time, the cyclic strategy pattern in full state space is demonstrated by means of velocity. It is, furthermore, elaborated that the strategies of cooperation, defection and nonparticipant form a Rock-Paper-Scissors type cycle, and the cycle of three strategies are persistent over 200 rounds. This cycle is very similar to the cycle given by evolutionary dynamics e.g. replicator dynamics. The mechanism that nonparticipant can sustain cooperation is driven by the Rock-Paper-Scissors type of cyclic dominance in the three strategies. That is, if the cycle is existent, the cooperation will always sustain. Meanwhile, the distribution of social states changes in the state space and from cooperation as the most frequent strategy to defection and, from defection to nonparticipant, forms a clear rotation path in a long run. These results seem to implicate that the evolutionary dynamics has ability to capture the real dynamics applying not only on biosphere, but also on human society.

Conditions for Cooperation to be More Abundant than Defection in a Hierarchically Structured Population

We study conditions for weak selection to favor tit-for-tat (TFT) over AllD in a repeated Prisoner’s Dilemma game played in a finite population subdivided into three subpopulations under the assumption of cyclic dominance in asymmetric interactions. Assuming parent-independent mutation and uniform migration, we show that TFT is more abundant that AllD in the stationary state if the defection cost incurred by individuals in interaction with dominant defecting individuals exceeds some threshold value. This threshold value decreases as the number of repetitions of the game, the population size, or the mutation rate increases, but increases as the migration rate increases. The same conclusions hold in the case of linear dominance.