Segregation process and phase transition in cyclic predator-prey models with an even number of species

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.

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.

Emergence of oscillatory coexistence with exponentially decayed waiting times in a coupled cyclic competition system

Interpatch migration between two environments is generally considered as a spatial concept and can affect species biodiversity in each patch by inducing flux of population such as inflow and outflow quantities of species. In this paper, we explore the effect of interpatch migration, which can be generally considered as a spatial concept and may affect species biodiversity between two different patches in the perspective of the macroscopic level by exploiting the coupling of two systems, where each patch is occupied by cyclically competing three species who can stably coexist by exhibiting periodic orbits. For two simple scenarios of interpatch migration either single or all species migration, we found that two systems with independently stable coexisting species in each patch are eventually synchronized, and oscillatory behaviors of species densities in two patches become identical, i.e., the synchronized coexistence emerges. In addition, we find that, whether single or all species interpatch migration occurs, the waiting time for the synchronization is exponentially decreasing as the coupling strength is intensified. Our findings suggest that the synchronized behavior of species as a result of migration between different patches can be easily predicted by the coupling of systems and additional information such as waiting times and sensitivity of initial densities.

Expanding spatial domains and transient scaling regimes in populations with local cyclic competition

We investigate a six-species class of May-Leonard models leading to the formation of two types of competing spatial domains, each one inhabited by three species with their own internal cyclic rock-paper-scissors dynamics. We study the resulting population dynamics using stochastic numerical simulations in two-dimensional space. We find that as three-species domains shrink, there is an increasing probability of extinction of two of the species inhabiting the domain, with the consequent creation of one-species domains. We determine the critical initial radius beyond which these one-species spatial domains are expected to expand. We further show that a transient scaling regime, with a slower average growth rate of the characteristic length scale L of the spatial domains with time t, takes place before the transition to a standard L∝t^{1/2} scaling law, resulting in an extended period of coexistence.

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.

Role-separating ordering in social dilemmas controlled by topological frustration

''Three is a crowd" is an old proverb that applies as much to social interactions as it does to frustrated configurations in statistical physics models. Accordingly, social relations within a triangle deserve special attention. With this motivation, we explore the impact of topological frustration on the evolutionary dynamics of the snowdrift game on a triangular lattice. This topology provides an irreconcilable frustration, which prevents anticoordination of competing strategies that would be needed for an optimal outcome of the game. By using different strategy updating protocols, we observe complex spatial patterns in dependence on payoff values that are reminiscent to a honeycomb-like organization, which helps to minimize the negative consequence of the topological frustration. We relate the emergence of these patterns to the microscopic dynamics of the evolutionary process, both by means of mean-field approximations and Monte Carlo simulations. For comparison, we also consider the same evolutionary dynamics on the square lattice, where of course the topological frustration is absent. However, with the deletion of diagonal links of the triangular lattice, we can gradually bridge the gap to the square lattice. Interestingly, in this case the level of cooperation in the system is a direct indicator of the level of topological frustration, thus providing a method to determine frustration levels in an arbitrary interaction network.

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.

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.

Interfaces with internal structures in generalized rock-paper-scissors models

In this work we investigate the development of stable dynamical structures along interfaces separating domains belonging to enemy partnerships in the context of cyclic predator-prey models with an even number of species N≥8. We use both stochastic and field theory simulations in one and two spatial dimensions, as well as analytical arguments, to describe the association at the interfaces of mutually neutral individuals belonging to enemy partnerships and to probe their role in the development of the dynamical structures at the interfaces. We identify an interesting behavior associated with the symmetric or asymmetric evolution of the interface profiles depending on whether N/2 is odd or even, respectively. We also show that the macroscopic evolution of the interface network is not very sensitive to the internal structure of the interfaces. Although this work focuses on cyclic predator-prey models with an even number of species, we argue that the results are expected to be quite generic in the context of spatial stochastic May-Leonard models.

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.

Phase diagram of a cyclic predator-prey model with neutral-pair exchange

In this paper we obtain the phase diagram of a four-species predator-prey lattice model by using the proposed gradient method. We consider cyclic transitions between consecutive states, representing invasion or predation, and allowed the exchange between neighboring neutral pairs. By applying a gradient in the invasion rate parameter one can see, in the same simulation, the presence of two symmetric absorbing phases, composed by neutral pairs, and an active phase that includes all four species. In this sense, the study of a single-valued interface and its fluctuations give the critical point of the irreversible phase transition and the corresponding universality classes. Also, the consideration of a multivalued interface and its fluctuations bring the percolation threshold. We show that the model presents two lines of irreversible first-order phase transition between the two absorbing phases and the active phase. Depending on the value of the system parameters, these lines can converge into a triple point, which is the beginning of a first-order irreversible line between the two absorbing phases, or end in two critical points belonging to the directed percolation universality class. Standard simulations for some characteristic values of the parameters confirm the order of the transitions as determined by the gradient method. Besides, below the triple point the model presents two standard percolation lines in the active phase and above a first-order percolation transition as already found in other similar models.

von Neummann's and related scaling laws in rock-paper-scissors-type games

We introduce a family of rock-paper-scissors-type models with Z(N) symmetry (N is the number of species), and we show that it has a very rich structure with many completely different phases. We study realizations that lead to the formation of domains, where individuals of one or more species coexist, separated by interfaces whose (average) dynamics is curvature driven. This type of behavior, which might be relevant for the development of biological complexity, leads to an interface network evolution and pattern formation similar to the ones of several other nonlinear systems in condensed matter and cosmology.

Junctions and spiral patterns in generalized rock-paper-scissors models

We investigate the population dynamics in generalized rock-paper-scissors models with an arbitrary number of species N. We show that spiral patterns with N arms may develop both for odd and even N, in particular in models where a bidirectional predation interaction of equal strength between all species is modified to include one N-cyclic predator-prey rule. While the former case gives rise to an interface network with Y-type junctions obeying the scaling law L∝t1/2, where L is the characteristic length of the network and t is the time, the latter can lead to a population network with N-armed spiral patterns, having a roughly constant characteristic length scale. We explicitly demonstrate the connection between interface junctions and spiral patterns in these models and compute the corresponding scaling laws. This work significantly extends the results of previous studies of population dynamics and could have profound implications for the understanding of biological complexity in systems with a large number of species.

Saddles, arrows, and spirals: deterministic trajectories in cyclic competition of four species

Population dynamics in systems composed of cyclically competing species has been of increasing interest recently. Here we investigate a system with four or more species. Using mean field theory, we study in detail the trajectories in configuration space of the population fractions. We discover a variety of orbits, shaped like saddles, spirals, and straight lines. Many of their properties are found explicitly. Most remarkably, we identify a collective variable that evolves simply as an exponential: Q ∝ e(λt), where λ is a function of the reaction rates. It provides information on the state of the system for late times (as well as for t→-∞). We discuss implications of these results for the evolution of a finite, stochastic system. A generalization to an arbitrary number of cyclically competing species yields valuable insights into universal properties of such systems.

Pattern formation, synchronization, and outbreak of biodiversity in cyclically competing games

Species in nature are typically mobile over diverse distance scales, examples of which range from bacteria run to long-distance animal migrations. These behaviors can have a significant impact on biodiversity. Addressing the role of migration in biodiversity microscopically is fundamental but remains a challenging problem in interdisciplinary science. We incorporate both intra- and inter-patch migrations in stochastic games of cyclic competitions and find that the interplay between the migrations at the local and global scales can lead to robust species coexistence characterized dynamically by the occurrence of remarkable target-wave patterns in the absence of any external control. The waves can emerge from either mixed populations or isolated species in different patches, regardless of the size and the location of the migration target. We also find that, even in a single-species system, target waves can arise from rare mutations, leading to an outbreak of biodiversity. A surprising phenomenon is that target waves in different patches can exhibit synchronization and time-delayed synchronization, where the latter potentially enables the prediction of future evolutionary dynamics. We provide a physical theory based on the spatiotemporal organization of the target waves to explain the synchronization phenomena. We also investigate the basins of coexistence and extinction to establish the robustness of biodiversity through migrations. Our results are relevant to issues of general and broader interest such as pattern formation, control in excitable systems, and the origin of order arising from self-organization in social and natural systems.

Heterogeneous aspirations promote cooperation in the prisoner's dilemma game

To be the fittest is central to proliferation in evolutionary games. Individuals thus adopt the strategies of better performing players in the hope of successful reproduction. In structured populations the array of those that are eligible to act as strategy sources is bounded to the immediate neighbors of each individual. But which one of these strategy sources should potentially be copied? Previous research dealt with this question either by selecting the fittest or by selecting one player uniformly at random. Here we introduce a parameter u that interpolates between these two extreme options. Setting u equal to zero returns the random selection of the opponent, while positive u favor the fitter players. In addition, we divide the population into two groups. Players from group A select their opponents as dictated by the parameter u, while players from group B do so randomly irrespective of u. We denote the fraction of players contained in groups A and B by v and 1 - v, respectively. The two parameters u and v allow us to analyze in detail how aspirations in the context of the prisoner’s dilemma game influence the evolution of cooperation. We find that for sufficiently positive values of u there exist a robust intermediate v ≈ 0.5 for which cooperation thrives best. The robustness of this observation is tested against different levels of uncertainty in the strategy adoption process K and for different interaction networks. We also provide complete phase diagrams depicting the dependence of the impact of u and v for different values of K, and contrast the validity of our conclusions by means of an alternative model where individual aspiration levels are subject to evolution as well. Our study indicates that heterogeneity in aspirations may be key for the sustainability of cooperation in structured populations.

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.

Effect of epidemic spreading on species coexistence in spatial rock-paper-scissors games

A fundamental question in nonlinear science and evolutionary biology is how epidemic spreading may affect coexistence. We address this question in the framework of mobile species under cyclic competitions by investigating the roles of both intra- and interspecies spreading. A surprising finding is that intraspecies infection can strongly promote coexistence while interspecies spreading cannot. These results are quantified and a theoretical paradigm based on nonlinear partial differential equations is derived to explain the numerical results.

Self-organizing patterns maintained by competing associations in a six-species predator-prey model

Formation and competition of associations are studied in a six-species ecological model where each species has two predators and two prey. Each site of a square lattice is occupied by an individual belonging to one of the six species. The evolution of the spatial distribution of species is governed by iterated invasions between the neighboring predator-prey pairs with species specific rates and by site exchange between the neutral pairs with a probability X . This dynamical rule yields the formation of five associations composed of two or three species with proper spatiotemporal patterns. For large X a cyclic dominance can occur between the three two-species associations whereas one of the two three-species associations prevails in the whole system for low values of X in the final state. Within an intermediate range of X all the five associations coexist due to the fact that cyclic invasions between the two-species associations reduce their resistance temporarily against the invasion of three-species associations.

Phase transitions induced by variation of invasion rates in spatial cyclic predator-prey models with four or six species

Cyclic predator-prey models with four or six species are studied on a square lattice when the invasion rates are varied. It is found that the cyclic invasions maintain a self-organizing pattern as long as the deviation of the invasion rate(s) from a uniform value does not exceed a threshold value. For larger deviations, the system exhibits a continuous phase transition into a frozen distribution of odd (or even) label species.