Four-state rock-paper-scissors games in constrained Newman-Watts networks

Nutrients and flow shape the cyclic dominance games between Escherichia coli strains

Evolutionary game theory has provided various models to explain the coexistence of competing strategies, one of which is the rock-paper-scissors (RPS) game. A system of three Escherichia coli strains-a toxin-producer, a resistant and a sensitive-has become a classic experimental model for studying RPS games. Previous experimental and theoretical studies, however, often ignored the influence of ecological factors such as nutrients and toxin dynamics on the evolutionary game dynamics. In this work, we combine experiments and modelling to study how these factors affect competition dynamics. Using three-dimensional printed mini-bioreactors, we tracked the frequency of the three strains in different culturing media and under different flow regimes. Although our experimental system fulfilled the requirements of cyclic dominance, we did not observe clear cycles or long-term coexistence between strains. We found that both nutrients and flow rates strongly impacted population dynamics. In our simulations, we explicitly modelled the release, removal and diffusion of toxin. We showed that the amount of toxin that is retained in the system is a simple indicator that can predict competition outcomes across broad parameter space. Moreover, our simulation results suggest that high rates of toxin diffusion might have prevented cyclic patterns from emerging in our experimental system. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

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.

Environment driven oscillation in an off-lattice May-Leonard model

Cyclic dominance of competing species is an intensively used working hypothesis to explain biodiversity in certain living systems, where the evolutionary selection principle would dictate a single victor otherwise. Technically the May–Leonard models offer a mathematical framework to describe the mentioned non-transitive interaction of competing species when individual movement is also considered in a spatial system. Emerging rotating spirals composed by the competing species are frequently observed character of the resulting patterns. But how do these spiraling patterns change when we vary the external environment which affects the general vitality of individuals? Motivated by this question we suggest an off-lattice version of the tradition May–Leonard model which allows us to change the actual state of the environment gradually. This can be done by introducing a local carrying capacity parameter which value can be varied gently in an off-lattice environment. Our results support a previous analysis obtained in a more intricate metapopulation model and we show that the well-known rotating spirals become evident in a benign environment when the general density of the population is high. The accompanying time-dependent oscillation of competing species can also be detected where the amplitude and the frequency show a scaling law of the parameter that characterizes the state of the environment. These observations highlight that the assumed non-transitive interaction alone is insufficient condition to maintain biodiversity safely, but the actual state of the environment, which characterizes the general living conditions, also plays a decisive role on the evolution of related systems.

Endogenous spatial pattern formation from two intersecting ecological mechanisms: the dynamic coexistence of two noxious invasive ant species in Puerto Rico

Endogenous (or autonomous, or emergent) spatial pattern formation is a subject transcending a variety of sciences. In ecology, there is growing interest in how spatial patterns can 'emerge' from internal system processes and simultaneously affect those very processes. A classic situation emerges when a predator's focus on a dominant competitor releases competitive pressure on a subdominant competitor, allowing coexistence of the two. If this idea is formulated spatially, two interesting consequences immediately arise. First, a spatial predator/prey system may take the form of a Turing instability, in which an activator (the dispersing prey population) is contained by a repressor (the more rapidly dispersing predator population) generating a spatial pattern of clusters of prey and predators, and second, an indirect intransitive loop (where A beats B beats C beats A) emerges from the simple fact that the system is spatial. Two common invasive ant species, Wasmannia auropunctata and Solenopsis invicta, and the parasitic phorid flies of S. invicta commonly coexist in Puerto Rico. Emergent spatial patterns generated by the combination of the Turing mechanism and the indirect intransitive loop are likely to be common here. This theoretical framework and the realities of the natural history in the field could explain both the long-term coexistence of these two species, and the highly variable pattern of their occurrence across a large landscape.

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.

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.

Attosecond physics at the nanoscale

Recently two emerging areas of research, attosecond and nanoscale physics, have started to come together. Attosecond physics deals with phenomena occurring when ultrashort laser pulses, with duration on the femto- and sub-femtosecond time scales, interact with atoms, molecules or solids. The laser-induced electron dynamics occurs natively on a timescale down to a few hundred or even tens of attoseconds (1 attosecond  =  1 as  =  10^-18 s), which is comparable with the optical field. For comparison, the revolution of an electron on a 1s orbital of a hydrogen atom is  ∼152 as. On the other hand, the second branch involves the manipulation and engineering of mesoscopic systems, such as solids, metals and dielectrics, with nanometric precision. Although nano-engineering is a vast and well-established research field on its own, the merger with intense laser physics is relatively recent. In this report on progress we present a comprehensive experimental and theoretical overview of physics that takes place when short and intense laser pulses interact with nanosystems, such as metallic and dielectric nanostructures. In particular we elucidate how the spatially inhomogeneous laser induced fields at a nanometer scale modify the laser-driven electron dynamics. Consequently, this has important impact on pivotal processes such as above-threshold ionization and high-order harmonic generation. The deep understanding of the coupled dynamics between these spatially inhomogeneous fields and matter configures a promising way to new avenues of research and applications. Thanks to the maturity that attosecond physics has reached, together with the tremendous advance in material engineering and manipulation techniques, the age of atto-nanophysics has begun, but it is in the initial stage. We present thus some of the open questions, challenges and prospects for experimental confirmation of theoretical predictions, as well as experiments aimed at characterizing the induced fields and the unique electron dynamics initiated by them with high temporal and spatial resolution.

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.

Phase diagrams for the spatial public goods game with pool punishment

The efficiency of institutionalized punishment is studied by evaluating the stationary states in the spatial public goods game comprising unconditional defectors, cooperators, and cooperating pool punishers as the three competing strategies. Fines and costs of pool punishment are considered as the two main parameters determining the stationary distributions of strategies on the square lattice. Each player collects a payoff from five five-person public goods games, and the evolution of strategies is subsequently governed by imitation based on pairwise comparisons at a low level of noise. The impact of pool punishment on the evolution of cooperation in structured populations is significantly different from that reported previously for peer punishment. Representative phase diagrams reveal remarkably rich behavior, depending also on the value of the synergy factor that characterizes the efficiency of investments payed into the common pool. Besides traditional single- and two-strategy stationary states, a rock-paper-scissors type of cyclic dominance can emerge in strikingly different ways.

Home range evolution and its implication in population outbreaks

We investigated the phenomenon of population outbreaks in a spatial predator-prey model, and we found that pattern formation and outbreaks occur if the predators have a limited neighbourhood of interaction with the preys. The outbreaks can display a scale-invariant power-law tail, indicating self-organized criticality. We have also studied the system from an evolutionary point of view, where the predator home range is a hereditary trait subjected to mutations. We found that mutation drives the predator home range area to an optimal value where pattern formation and outbreaks are still present, but the latter are much less frequent. We developed analytical approximations using mean field and pair correlation techniques that indicate that the predation strategy is crucial for existence of this optimal home range area.

Basins of coexistence and extinction in spatially extended ecosystems of cyclically competing species

Microscopic models based on evolutionary games on spatially extended scales have recently been developed to address the fundamental issue of species coexistence. In this pursuit almost all existing works focus on the relevant dynamical behaviors originated from a single but physically reasonable initial condition. To gain comprehensive and global insights into the dynamics of coexistence, here we explore the basins of coexistence and extinction and investigate how they evolve as a basic parameter of the system is varied. Our model is cyclic competitions among three species as described by the classical rock-paper-scissors game, and we consider both discrete lattice and continuous space, incorporating species mobility and intraspecific competitions. Our results reveal that, for all cases considered, a basin of coexistence always emerges and persists in a substantial part of the parameter space, indicating that coexistence is a robust phenomenon. Factors such as intraspecific competition can, in fact, promote coexistence by facilitating the emergence of the coexistence basin. In addition, we find that the extinction basins can exhibit quite complex structures in terms of the convergence time toward the final state for different initial conditions. We have also developed models based on partial differential equations, which yield basin structures that are in good agreement with those from microscopic stochastic simulations. To understand the origin and emergence of the observed complicated basin structures is challenging at the present due to the extremely high dimensional nature of the underlying dynamical system.

Cyclic competition of mobile species on continuous space: pattern formation and coexistence

We propose a model for cyclically competing species on continuous space and investigate the effect of the interplay between the interaction range and mobility on coexistence. A transition from coexistence to extinction is uncovered with a strikingly nonmonotonic behavior in the coexistence probability. About the minimum in the probability, switches between spiral and plane-wave patterns arise. A strong mobility can either promote or hamper coexistence, depending on the radius of the interaction range. These phenomena are absent in any lattice-based model, and we demonstrate that they can be explained using nonlinear partial differential equations. Our continuous-space model is more physical and we expect the findings to generate experimental interest.

Dynamically generated cyclic dominance in spatial prisoner's dilemma games

We have studied the impact of time-dependent learning capacities of players in the framework of spatial prisoner's dilemma game. In our model, this capacity of players may decrease or increase in time after strategy adoption according to a steplike function. We investigated both possibilities separately and observed significantly different mechanisms that form the stationary pattern of the system. The time decreasing learning activity helps cooperator domains to recover the possible intrude of defectors hence supports cooperation. In the other case the temporary restrained learning activity generates a cyclic dominance between defector and cooperator strategies, which helps to maintain the diversity of strategies via propagating waves. The results are robust and remain valid by changing payoff values, interaction graphs or functions characterizing time dependence of learning activity. Our observations suggest that dynamically generated mechanisms may offer alternative ways to keep cooperators alive even at very larger temptation to defect.