Coevolutionary dynamics is investigated in chemical catalysis, biological evolution, social and economic systems. The dynamics of these systems can be analyzed within the unifying framework of evolutionary game theory. In this Letter, we show that even in well-mixed finite populations, where the dynamics is inherently stochastic, biodiversity is possible with three cyclic dominant strategies. We show how the interplay of evolutionary dynamics, discreteness of the population, and the nature of the interactions influences the coexistence of strategies. We calculate a critical population size above which coexistence is likely.
Global, population-wide oscillations in models of cyclic dominance may result in the collapse of biodiversity due to the accidental extinction of one species in the loop. Previous research has shown that such oscillations can emerge if the interaction network has small-world properties, and more generally, because of long-range interactions among individuals or because of mobility. But although these features are all common in nature, global oscillations are rarely observed in actual biological systems. This begets the question what is the missing ingredient that would prevent local oscillatio…
Neutral species facilitate coexistence among cyclically competing species under birth and death processes
School of Statistics and Mathematics, Yunnan University of Finance and Economics（云南财经大学统计与数学学院）；School of Mathematics and Statistics, Xidian University（西安电子科技大学数学与统计学院）；Department of Applied Mathematics, Colleage of Applied Sciences, Kyung Hee University（韩国庆熙大学应用科学学院应用数学系）；G-LAMP NEXUS Institute, Kyung Hee University（韩国庆熙大学G-LAMP NEXUS研究所）
This work deals with the influence of the neighborhood in simple rock-paper-scissors models of biodiversity. We consider the case of three distinct species which evolve under the standard rules of mobility, reproduction and competition. The rule of competition follows the guidance of the rock-paper-scissors game, with the prey being annihilated, leaving an empty site in accordance with the May-Leonard proposal for the predator and prey competition. We use the von Neumann neighborhood, but we consider mobility under the presence of the first, second and third neighbors in three distinct environ…
Cyclic, nonhierarchical interactions among biological species represent a general mechanism by which ecosystems are able to maintain high levels of biodiversity. However, species coexistence is often possible only in spatially extended systems with a limited range of dispersal, whereas in well-mixed environments models for cyclic competition often lead to a loss of biodiversity. Here we consider the dispersal of biological species in a fluid environment, where mixing is achieved by a combination of advection and diffusion. In particular, we perform a detailed numerical analysis of a model comp…
Finite-size fluctuations arising in the dynamics of competing populations may have dramatic influence on their fate. As an example, in this article, we investigate a model of three species which dominate each other in a cyclic manner. Although the deterministic approach predicts (neutrally) stable coexistence of all species, for any finite population size, the intrinsic stochasticity unavoidably causes the eventual extinction of two of them.
Stochastic simulations of cyclic three-species spatial predator-prey models are usually performed in square lattices with nearest neighbor interactions starting from random initial conditions. In this Letter we describe the results of off-lattice Lotka-Volterra stochastic simulations, showing that the emergence of spiral patterns does occur for sufficiently high values of the (conserved) total density of individuals. We also investigate the dynamics in our simulations, finding an empirical relation characterizing the dependence of the characteristic peak frequency and amplitude on the total de…
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 …
The dynamical mechanisms that can stabilize the coexistence of species (or strategies) are of substantial interest for the maintenance of biodiversity and in sociobehavioural dynamics. We investigate the mean extinction time in the coevolutionary dynamics of three cyclically invading strategies for different evolutionary processes on various classes of complex networks, including random graphs, scale-free and small world networks. We find that scale-free and random graphs lead to a strong stabilization of coexistence both for the Moran process and the Local Update process. The stabilization is…
Cooperation is the backbone of modern human societies, making it a priority to understand how successful cooperation-sustaining mechanisms operate. Cyclic dominance, a non-transitive setup comprising at least three strategies wherein the first strategy overrules the second which overrules the third which, in turn, overrules the first strategy, is known to maintain bio-diversity, drive competition between bacterial strains, and preserve cooperation in social dilemmas. Here, we present a novel route to cyclic dominance in voluntary social dilemmas by adding to the traditional mix of cooperators,…
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 effec…
Non-transitive dominance and the resulting cyclic loop of three or more competing species provide a fundamental mechanism to explain biodiversity in biological and ecological systems. Both Lotka-Volterra and May-Leonard type model approaches agree that heterogeneity of invasion rates within this loop does not hazard the coexistence of competing species. While the resulting abundances of species become heterogeneous, the species who has the smallest invasion power benefits the most from unequal invasions. Nevertheless, the effective invasion rate in a predator and prey interaction can also be m…
Lotka's seminal work (A.J. Lotka A., Proc. Natl. Acad. Sci. U.S.A. 6 (1920) 410) "on certain rhythmic relations'' is already one hundred years old, but the research activity about pattern formations due to cyclical dominance is more vibrant than ever. It is because non-transitive interactions have paramount role on maintaining biodiversity and adequate human intervention into ecological systems requires deeper understanding of related dynamical processes. In this perspective article we overview different aspects of biodiversity, with focus on how it can be maintained based on mathematical mode…
A five-species predator-prey model is studied on a square lattice where each species has two prey and two predators on the analogy to the Rock-Paper-Scissors-Lizard-Spock game. The evolution of the spatial distribution of species is governed by site exchange and invasion between the neighboring predator-prey pairs, where the cyclic symmetry can be characterized by two different invasion rates. The mean-field analysis has indicated periodic oscillations in the species densities with a frequency becoming zero for a specific ratio of invasion rates. When varying the ratio of invasion rates, the a…
Biodiversity is essential to the viability of ecological systems. Species diversity in ecosystems is promoted by cyclic, non-hierarchical interactions among competing populations. Such non-transitive relations lead to an evolution with central features represented by the `rock-paper-scissors' game, where rock crushes scissors, scissors cut paper, and paper wraps rock. In combination with spatial dispersal of static populations, this type of competition results in the stable coexistence of all species and the long-term maintenance of biodiversity. However, population mobility is a central featu…
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 the 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 regardl…
Predators may attack isolated or grouped prey in a cooperative, collective way. Whether a gregarious behavior is advantageous to each species depends on several conditions and game theory is a useful tool to deal with such a problem. We here extend the Lett-Auger-Gaillard model [Theor. Pop. Biol. 65 (2004) 263] to spatially distributed populations and compare the resulting behavior with their mean-field predictions for the coevolving densities of predator and prey strategies. Besides its richer behavior in the presence of spatial organization, we also show that the coexistence phase in which c…
We investigate a tritrophic system whose cyclic dominance is modelled by the rock-paper-scissors game. We consider that organisms of one or two species are affected by movement limitations, which unbalances the cyclic spatial game. Performing stochastic simulations, we show that mobility unevenness controls the population dynamics. In the case of one slow species, the predominant species depends on the level of mobility restriction, with the slow species being preponderant if the mobility limitations are substantial. If two species face mobility limitations, our outcomes show that being higher…
The four-year oscillations of the number of spawning sockeye salmon (Oncorhynchus nerka) that return to their native stream within the Fraser River basin in Canada are a striking example of population oscillations. The period of the oscillation corresponds to the dominant generation time of these fish. Various - not fully convincing - explanations for these oscillations have been proposed, including stochastic influences, depensatory fishing, or genetic effects. Here, we show that the oscillations can be explained as a stable dynamical attractor of the population dynamics, resulting from a str…
Species diversity in ecosystems is often accompanied by the self-organisation of the population into fascinating spatio-temporal patterns. Here, we consider a two-dimensional three-species population model and study the spiralling patterns arising from the combined effects of generic cyclic dominance, mutation, pair-exchange and hopping of the individuals. The dynamics is characterised by nonlinear mobility and a Hopf bifurcation around which the system's phase diagram is inferred from the underlying complex Ginzburg-Landau equation derived using a perturbative multiscale expansion. While the …
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 …
Dynamical mechanisms that can stabilize the coexistence or diversity in biology are generally of fundamental interest. In contrast to many two-strategy evolutionary games, games with three strategies and cyclic dominance like the rock-paper-scissors game (RPS) stabilize coexistence and thus preserve biodiversity in this system. In the limit of infinite populations, resembling the traditional picture of evolutionary game theory, replicator equations predict the existence of a fixed point in the interior of the phase space. But in finite populations, strategy frequencies will run out of the fixe…
We study a four species ecological system with cyclic dominance whose individuals are distributed on a square lattice. Randomly chosen individuals migrate to one of the neighboring sites if it is empty or invade this site if occupied by their prey. The cyclic dominance maintains the coexistence of all the four species if the concentration of vacant sites is lower than a threshold value. Above the treshold, a symmetry breaking ordering occurs via growing domains containing only two neutral species inside. These two neutral species can protect each other from the external invaders (predators) an…
Ecology and evolution are inherently linked, and studying a mathematical model that considers both holds promise of insightful discoveries related to the dynamics of cooperation. In the present article, we use the prisoner's dilemma (PD) game as a basis for long-term apprehension of the essential social dilemma related to cooperation among unrelated individuals. We upgrade the contemporary PD game with an inclusion of evolution-induced act of punishment as a third competing strategy in addition to the traditional cooperators and defectors. In a population structure, the abundance of ecological…
Cyclic dominance of species has been identified as a potential mechanism to maintain biodiversity, see e.g. B. Kerr, M. A. Riley, M. W. Feldman and B. J. M. Bohannan [Nature {\bf 418}, 171 (2002)] and B. Kirkup and M. A. Riley [Nature {\bf 428}, 412 (2004)]. Through analytical methods supported by numerical simulations, we address this issue by studying the properties of a paradigmatic non-spatial three-species stochastic system, namely the `rock-paper-scissors' or cyclic Lotka-Volterra model. While the deterministic approach (rate equations) predicts the coexistence of the species resulting i…
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 colonise new area, as well as the importance of inter-species 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, sens…
We study the phenomenon of cyclic dominance in the paradigmatic Rock--Paper--Scissors model, as occurring in both stochastic individual-based models of finite populations and in the deterministic replicator equations. The mean-field replicator equations are valid in the limit of large populations and, in the presence of mutation and unbalanced payoffs, they exhibit an attracting limit cycle. The period of this cycle depends on the rate of mutation; specifically, the period grows logarithmically as the mutation rate tends to zero. We find that this behaviour is not reproduced in stochastic simu…
We consider a two-dimensional model of three species in rock-paper-scissors competition and study the self-organisation of the population into fascinating spiraling patterns. Within our individual-based metapopulation formulation, the population composition changes due to cyclic dominance (dominance-removal and dominance-replacement), mutations, and pair-exchange of neighboring individuals. Here, we study the influence of mobility on the emerging patterns and investigate when the pair-exchange rate is responsible for spiral waves to become elusive in stochastic lattice simulations. In particul…
Selection Rules for Species Coexistence in a Hierarchical May-Leonard Model
Raja Rammohun Roy Mahavidyalaya（拉贾·拉姆莫汉·罗伊大学）；Center for Computational Natural Sciences and Bioinformatics, International Institute of Information Technology Hyderabad（海德拉巴信息技术学院计算自然科学和生物信息中心）；Sister Nibedita Govt. General Degree College for Girls（西比尔·尼贝迪塔女子政府综合学位学院）；School of Science, Constructor University（Constructor大学理学院）；Department of Mathematics, Nalanda University（那烂陀大学数学系）
Noise and spatial degrees of freedom characterize most ecosystems. Some aspects of their influence on the coevolution of populations with cyclic interspecies competition have been demonstrated in recent experiments [e.g. B. Kerr et al., Nature {\bf 418}, 171 (2002)]. To reach a better theoretical understanding of these phenomena, we consider a paradigmatic spatial model where three species exhibit cyclic dominance. Using an individual-based description, as well as stochastic partial differential and deterministic reaction-diffusion equations, we account for stochastic fluctuations and spatial …
We study a variant of the cyclic Lotka-Volterra model with three-agent interactions. Inspired by a multiplayer variation of the Rock-Paper-Scissors game, the model describes an ideal ecosystem in which cyclic competition among three species develops through cooperative predation. Its rate equations in a well-mixed environment display a degenerate Hopf bifurcation, occurring as reactions involving two predators plus one prey have the same rate as reactions involving two preys plus one predator. We estimate the magnitude of the stochastic noise at the bifurcation point, where finite size effects…
The spatial segregation of species is fundamental to ecosystem formation and stability. Behavioural strategies may determine where species are located and how their interactions change the local environment arrangement. In response to stimuli in the environment, individuals may move in a specific direction instead of walking randomly. This behaviour can be innate or learned from experience, and allow the individuals to conquer or the maintain territory, foraging or taking refuge. We study a generalisation of the spatial rock-paper-scissors model where individuals of one out of the species may …
Disease outbreaks affect many ecosystems threatening species that also fight against other natural enemies. We investigate a cyclic game system with $5$ species, whose organisms outcompete according to the rules of a generalised spatial rock-paper-scissors game, during an epidemic. We study the effects of behavioural movement strategies that allow individuals of one out of the species to move towards areas with a low density of disease vectors and a high concentration of enemies of their enemies. We perform a series of stochastic simulations to discover the impact of self-preservation strategi…
Multiple species in the ecosystem are believed to compete cyclically for survival and thus maintain balance in nature. Stochasticity has also an inevitable role in this dynamics. Considering these attributes of nature, the stochastic dynamics of the rock-paper-scissor model based on the idea of cyclic dominance becomes an effective tool to capture different aspects of ecosystem. The evolutionary dynamics of this model crucially depends on different interactions representing different natural habits. In this framework we explore the role of mortality of individual organism in the collective sur…
Rock-paper-scissors games metaphorically model cyclic dominance in ecology and microbiology. In a static environment, these models are characterized by fixation probabilities obeying two different "laws" in large and small well-mixed populations. Here, we investigate the evolution of these three-species models subject to a randomly switching carrying capacity modeling the endless change between states of resources scarcity and abundance. Focusing mainly on the zero-sum rock-paper-scissors game, equivalent to the cyclic Lotka-Volterra model, we study how the ${\it coupling}$ of demographic and …
Knowing the strategy of an opponent in a competitive environment conveys obvious evolutionary advantages. But this information is costly, and the benefit of being informed may not necessarily offset the additional cost. Here we introduce social dilemmas with informed strategies, and we show that this gives rise to two cyclically dominant triplets that form defensive alliances. The stability of these two alliances is determined by the rotation velocity of the strategies within each triplet. A weaker strategy in a faster rotating triplet can thus overcome an individually stronger competitor. Fas…
The coevolutionary dynamics in finite populations currently is investigated in a wide range of disciplines, as chemical catalysis, biological evolution, social and economic systems. The dynamics of those systems can be formulated within the unifying framework of evolutionary game theory. However it is not a priori clear which mathematical description is appropriate when populations are not infinitely large. Whereas the replicator equation approach describes the infinite population size limit by deterministic differential equations, in finite populations the dynamics is inherently stochastic wh…
In order to better understand the interplay of partnership and competition in population dynamics we study a family of generalized May-Leonard models with $N$ species. These models have a very rich structure, characterized by different types of space-time patterns. Interesting partnership formations emerge following the maxim that "the enemy of my enemy is my friend." In specific cases cyclic dominance within coarsening clusters yields a peculiar coarsening behavior with intriguing pattern formation. We classify the different types of dynamics through the analysis of the square of the adjacenc…
Economic experiments reveal that humans value cooperation and fairness. Punishing unfair behavior is therefore common, and according to the theory of strong reciprocity, it is also directly related to rewarding cooperative behavior. However, empirical data fail to confirm that positive and negative reciprocity are correlated. Inspired by this disagreement, we determine whether the combined application of reward and punishment is evolutionary advantageous. We study a spatial public goods game, where in addition to the three elementary strategies of defection, rewarding and punishment, a fourth …
Population structure induced by both spatial embedding and more general networks of interaction, such as model social networks, have been shown to have a fundamental effect on the dynamics and outcome of evolutionary games. These effects have, however, proved to be sensitive to the details of the underlying topology and dynamics. Here we introduce a minimal population structure that is described by two distinct hierarchical levels of interaction. We believe this model is able to identify effects of spatial structure that do not depend on the details of the topology. We derive the dynamics gove…
Cooperation is a difficult proposition in the face of Darwinian selection. Those that defect have an evolutionary advantage over cooperators who should therefore die out. However, spatial structure enables cooperators to survive through the formation of homogeneous clusters, which is the hallmark of network reciprocity. Here we go beyond this traditional setup and study the spatiotemporal dynamics of cooperation in a population of populations. We use the prisoner's dilemma game as the mathematical model and show that considering several populations simultaneously give rise to fascinating spati…
May and Leonard (SIAM J. Appl. Math 1975) introduced a three-species Lotka-Volterra type population model that exhibits heteroclinic cycling. Rather than producing a periodic limit cycle, the trajectory takes longer and longer to complete each "cycle", passing closer and closer to unstable fixed points in which one population dominates and the others approach zero. Aperiodic heteroclinic dynamics have subsequently been studied in ecological systems (side-blotched lizards; colicinogenic E. coli), in the immune system, in neural information processing models ("winnerless competition"), and in mo…
Diversity waves in collapse-driven population dynamics
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Carl R. Woese Institute for Genomic Biology, University of Illinois, Urbana-Champaign（卡尔·R·沃斯基因组生物学研究所，伊利诺伊大学厄巴纳-香槟分校）；Brookhaven National Laboratory（布鲁克海文国家实验室）；Center for Models of Life, Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔学院生命模型中心）
