Coexistence and survival in conservative Lotka-Volterra networks

Topological invariant and anomalous edge modes of strongly nonlinear systems

Despite the extensive studies of topological states, their characterization in strongly nonlinear classical systems has been lacking. In this work, we identify the proper definition of Berry phase for nonlinear bulk waves and characterize topological phases in one-dimensional (1D) generalized nonlinear Schrödinger equations in the strongly nonlinear regime, where the general nonlinearities are beyond Kerr-like interactions. Without utilizing linear analysis, we develop an analytic strategy to demonstrate the quantization of nonlinear Berry phase due to reflection symmetry. Mode amplitude itself plays a key role in nonlinear modes and controls topological phase transitions. We then show bulk-boundary correspondence by identifying the associated nonlinear topological edge modes. Interestingly, anomalous topological modes decay away from lattice boundaries to plateaus governed by fixed points of nonlinearities. Our work opens the door to the rich physics between topological phases of matter and nonlinear dynamics.

Topological phases are challenging to identify in systems with general, strong nonlinearities. Here, the authors establish the analytic methodology that defines the topological invariant of nonlinear normal modes. Strongly nonlinear topological boundary modes are guaranteed by the nontrivial topological index.

Topological Phase Transition in Coupled Rock-Paper-Scissors Cycles

A hallmark of topological phases is the occurrence of topologically protected modes at the system's boundary. Here, we find topological phases in the antisymmetric Lotka-Volterra equation (ALVE). The ALVE is a nonlinear dynamical system and describes, for example, the evolutionary dynamics of a rock-paper-scissors cycle. On a one-dimensional chain of rock-paper-scissor cycles, topological phases become manifest as robust polarization states. At the transition point between left and right polarization, solitary waves are observed. This topological phase transition lies in symmetry class D within the "tenfold way" classification as also realized by 1D topological superconductors.

Analytical treatment for cyclic three-state dynamics on static networks

Whenever a dynamical process unfolds on static networks, the dynamical state of any focal individual will be exclusively influenced by directly connected neighbors, rather than by those unconnected ones, hence the arising of the dynamical correlation problem, where mean-field-based methods fail to capture the scenario. The dynamic correlation coupling problem has always been an important and difficult problem in the theoretical field of physics. The explicit analytical expressions and the decoupling methods often play a key role in the development of corresponding field. In this paper, we study the cyclic three-state dynamics on static networks, which include a wide class of dynamical processes, for example, the cyclic Lotka-Volterra model, the directed migration model, the susceptible-infected-recovered-susceptible epidemic model, and the predator-prey with empty sites model. We derive the explicit analytical solutions of the propagating size and the threshold curve surface for the four different dynamics. We compare the results on static networks with those on annealed networks and made an interesting discovery: for the symmetrical dynamical model (the cyclic Lotka-Volterra model and the directed migration model, where the three states are of rotational symmetry), the macroscopic behaviors of the dynamical processes on static networks are the same as those on annealed networks; while the outcomes of the dynamical processes on static networks are different with, and more complicated than, those on annealed networks for asymmetric dynamical model (the susceptible-infected-recovered-susceptible epidemic model and the predator-prey with empty sites model). We also compare the results forecasted by our theoretical method with those by Monte Carlo simulations and find good agreement between the results obtained by the two methods.

Cooperation in Microbial Populations: Theory and Experimental Model Systems

Cooperative behavior, the costly provision of benefits to others, is common across all domains of life. This review article discusses cooperative behavior in the microbial world, mediated by the exchange of extracellular products called public goods. We focus on model species for which the production of a public good and the related growth disadvantage for the producing cells are well described. To unveil the biological and ecological factors promoting the emergence and stability of cooperative traits we take an interdisciplinary perspective and review insights gained from both mathematical models and well-controlled experimental model systems. Ecologically, we include crucial aspects of the microbial life cycle into our analysis and particularly consider population structures where ensembles of local communities (subpopulations) continuously emerge, grow, and disappear again. Biologically, we explicitly consider the synthesis and regulation of public good production. The discussion of the theoretical approaches includes general evolutionary concepts, population dynamics, and evolutionary game theory. As a specific but generic biological example, we consider populations of Pseudomonas putida and its regulation and use of pyoverdines, iron scavenging molecules, as public goods. The review closes with an overview on cooperation in spatially extended systems and also provides a critical assessment of the insights gained from the experimental and theoretical studies discussed. Current challenges and important new research opportunities are discussed, including the biochemical regulation of public goods, more realistic ecological scenarios resembling native environments, cell-to-cell signaling, and multispecies communities.

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.

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

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

Nonlinear dynamics with Hopf bifurcations by targeted mutation in the system of rock-paper-scissors metaphor

The role of mutation, which is an error process in gene evolution, in systems of cyclically competing species has been studied from various perspectives, and it is regarded as one of the key factors for promoting coexistence of all species. In addition to naturally occurring mutations, many experiments in genetic engineering have involved targeted mutation techniques such as recombination between DNA and somatic cell sequences and have studied genetic modifications through loss or augmentation of cell functions. In this paper, we investigate nonlinear dynamics with targeted mutation in cyclically competing species. In different ways to classic approaches of mutation in cyclic games, we assume that mutation may occur in targeted individuals who have been removed from intraspecific competition. By investigating each scenario depending on the number of objects for targeted mutation analytically and numerically, we found that targeted mutation can lead to persistent coexistence of all species. In addition, under the specific condition of targeted mutation, we found that targeted mutation can lead to emergences of bistable states for species survival. Through the linear stability analysis of rate equations, we found that those phenomena are accompanied by Hopf bifurcation which is supercritical. Our findings may provide more global perspectives on understanding underlying mechanisms to control biodiversity in ecological/biological sciences, and evidences with mathematical foundations to resolve social dilemmas such as a turnover of group members by resigning with intragroup conflicts in social sciences.

Coevolution of nodes and links: Diversity-driven coexistence in cyclic competition of three species

When three species compete cyclically in a well-mixed, stochastic system of N individuals, extinction is known to typically occur at times scaling as the system size N. This happens, for example, in rock-paper-scissors games or conserved Lotka-Volterra models in which every pair of individuals can interact on a complete graph. Here we show that if the competing individuals also have a "social temperament" to be either introverted or extroverted, leading them to cut or add links, respectively, then long-living states in which all species coexist can occur. These nonequilibrium quasisteady states only occur when both introverts and extroverts are present, thus showing that diversity can lead to stability in complex systems. In this case, it enables a subtle balance between species competition and network dynamics to be maintained.

Three-species competition with non-deterministic outcomes

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

Multistability in the cyclic competition system

Cyclically competition models have been successful to gain an insight of biodiversity mechanism in ecosystems. There are, however, still limitations to elucidate complex phenomena arising in real competition. In this paper, we report that a multistability occurs in a simple rock-paper-scissor cyclically competition model by assuming that intraspecific competition depends on the logistic growth of each species density. This complex stability is absent in any cyclically competition model, and we investigate how the proposed intraspecific competition affects biodiversity in the existing society of three species through macroscopic and microscopic approaches. When the system is multistable, we show basins of the asymptotically stable heteroclinic cycle and stable attractors to demonstrate how the survival state is determined by initial densities of three species. Also, we find that the multistability is associated with a subcritical Hopf bifurcation. This surprising finding will give an opportunity to interpret rich dynamical phenomena in ecosystems which may occur in cyclic competition systems with different types of interactions.

Biodiversity in the cyclic competition system of three species according to the emergence of mutant species

Understanding mechanisms which promote or hinder existing ecosystems are important issues in ecological sciences. In addition to fundamental interactions such as competition and migration among native species, existing ecosystems can be easily disturbed by external factors, and the emergence of new species may be an example in such cases. The new species which does not exist in a current ecosystem can be regarded as either alien species entered from outside or mutant species born by mutation in existing normal species. Recently, as existing ecosystems are getting influenced by various physical/chemical external factors, mutation due to anthropogenic and environmental factors can occur more frequently and is thus attracting much attention for the maintenance of ecosystems. In this paper, we consider emergences of mutant species among self-competing three species in the cyclic dominance. By defining mutation as the birth of mutant species, we investigate how mutant species can affect biodiversity in the existing ecosystem. Through microscopic and macroscopic approaches, we have found that the society of existing normal species can be disturbed by mutant species either the society is maintained accompanying with the coexistence of all species or jeopardized by occupying of mutant species. Due to the birth of mutant species, the existing society may be more complex by constituting two different groups of normal and mutant species, and our results can be contributed to analyze complex ecosystems of many species. We hope our findings may propose a new insight on mutation in cyclic competition systems of many species.

Survival behavior in the cyclic Lotka-Volterra model with a randomly switching reaction rate

We study the influence of a randomly switching reproduction-predation rate on the survival behavior of the nonspatial cyclic Lotka-Volterra model, also known as the zero-sum rock-paper-scissors game, used to metaphorically describe the cyclic competition between three species. In large and finite populations, demographic fluctuations (internal noise) drive two species to extinction in a finite time, while the species with the smallest reproduction-predation rate is the most likely to be the surviving one (law of the weakest). Here we model environmental (external) noise by assuming that the reproduction-predation rate of the strongest species (the fastest to reproduce and predate) in a given static environment randomly switches between two values corresponding to more and less favorable external conditions. We study the joint effect of environmental and demographic noise on the species survival probabilities and on the mean extinction time. In particular, we investigate whether the survival probabilities follow the law of the weakest and analyze their dependence on the external noise intensity and switching rate. Remarkably, when, on average, there is a finite number of switches prior to extinction, the survival probability of the predator of the species whose reaction rate switches typically varies nonmonotonically with the external noise intensity (with optimal survival about a critical noise strength). We also outline the relationship with the case where all reaction rates switch on markedly different time scales.

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

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

Emergence of unusual coexistence states in cyclic game systems

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

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

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

Deterministic extinction by mixing in cyclically competing species

We consider a cyclically competing species model on a ring with global mixing at finite rate, which corresponds to the well-known Lotka-Volterra equation in the limit of infinite mixing rate. Within a perturbation analysis of the model from the infinite mixing rate, we provide analytical evidence that extinction occurs deterministically at sufficiently large but finite values of the mixing rate for any species number N≥3. Further, by focusing on the cases of rather small species numbers, we discuss numerical results concerning the trajectories toward such deterministic extinction, including global bifurcations caused by changing the mixing rate.

From coin tossing to rock-paper-scissors and beyond: a log-exp gap theorem for selecting a leader

A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed.

Biodiversity in models of cyclic dominance is preserved by heterogeneity in site-specific invasion rates

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 oscillations to synchronize across the population to form global oscillations. Here we show that, although heterogeneous species-specific invasion rates fail to have a noticeable impact on species coexistence, randomness in site-specific invasion rates successfully hinders the emergence of global oscillations and thus preserves biodiversity. Our model takes into account that the environment is often not uniform but rather spatially heterogeneous, which may influence the success of microscopic dynamics locally. This prevents the synchronization of locally emerging oscillations, and ultimately results in a phenomenon where one type of randomness is used to mitigate the adverse effects of other types of randomness in the system.

Presence of a loner strain maintains cooperation and diversity in well-mixed bacterial communities

Cooperation and diversity abound in nature despite cooperators risking exploitation from defectors and superior competitors displacing weaker ones. Understanding the persistence of cooperation and diversity is therefore a major problem for evolutionary ecology, especially in the context of well-mixed populations, where the potential for exploitation and displacement is greatest. Here, we demonstrate that a ‘loner effect’, described by economic game theorists, can maintain cooperation and diversity in real-world biological settings. We use mathematical models of public-good-producing bacteria to show that the presence of a loner strain, which produces an independent but relatively inefficient good, can lead to rock–paper–scissor dynamics, whereby cooperators outcompete loners, defectors outcompete cooperators and loners outcompete defectors. These model predictions are supported by our observations of evolutionary dynamics in well-mixed experimental communities of the bacterium Pseudomonas aeruginosa. We find that the coexistence of cooperators and defectors that produce and exploit, respectively, the iron-scavenging siderophore pyoverdine, is stabilized by the presence of loners with an independent iron-uptake mechanism. Our results establish the loner effect as a simple and general driver of cooperation and diversity in environments that would otherwise favour defection and the erosion of diversity.

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.

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.

Nonequilibrium steady states of ideal bosonic and fermionic quantum gases

We investigate nonequilibrium steady states of driven-dissipative ideal quantum gases of both bosons and fermions. We focus on systems of sharp particle number that are driven out of equilibrium either by the coupling to several heat baths of different temperature or by time-periodic driving in combination with the coupling to a heat bath. Within the framework of (Floquet-)Born-Markov theory, several analytical and numerical methods are described in detail. This includes a mean-field theory in terms of occupation numbers, an augmented mean-field theory taking into account also nontrivial two-particle correlations, and quantum-jump-type Monte Carlo simulations. For the case of the ideal Fermi gas, these methods are applied to simple lattice models and the possibility of achieving exotic states via bath engineering is pointed out. The largest part of this work is devoted to bosonic quantum gases and the phenomenon of Bose selection, a nonequilibrium generalization of Bose condensation, where multiple single-particle states are selected to acquire a large occupation [Phys. Rev. Lett. 111, 240405 (2013)]. In this context, among others, we provide a theory for transitions where the set of selected states changes, describe an efficient algorithm for finding the set of selected states, investigate beyond-mean-field effects, and identify the dominant mechanisms for heat transport in the Bose-selected state.

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.

Evolutionary games of condensates in coupled birth-death processes

Condensation phenomena arise through a collective behaviour of particles. They are observed in both classical and quantum systems, ranging from the formation of traffic jams in mass transport models to the macroscopic occupation of the energetic ground state in ultra-cold bosonic gases (Bose-Einstein condensation). Recently, it has been shown that a driven and dissipative system of bosons may form multiple condensates. Which states become the condensates has, however, remained elusive thus far. The dynamics of this condensation are described by coupled birth-death processes, which also occur in evolutionary game theory. Here we apply concepts from evolutionary game theory to explain the formation of multiple condensates in such driven-dissipative bosonic systems. We show that the vanishing of relative entropy production determines their selection. The condensation proceeds exponentially fast, but the system never comes to rest. Instead, the occupation numbers of condensates may oscillate, as we demonstrate for a rock-paper-scissors game of condensates.

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

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

Cyclic dominance in evolutionary games: a review

Rock is wrapped by paper, paper is cut by scissors and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator-prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more, regardless of the particularities of the game. Here, we review recent advances on the rock-paper-scissors (RPS) and related evolutionary games, focusing, in particular, on pattern formation, the impact of mobility and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional RPS models and the application of the complex Ginzburg-Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related, for example, to dynamical effects of coevolutionary rules and invasion reversals owing to multi-point interactions, are also outlined.

Predicting the enhancement of mixing-driven reactions in nonuniform flows using measures of flow topology

The ability for reactive constituents to mix is often the key limiting factor for the completion of reactions across a huge range of scales in a variety of media. In flowing systems, deformation and shear enhance mixing by bringing constituents into closer proximity, thus increasing reaction potential. Accurately quantifying this enhanced mixing is key to predicting reactions and typically is done by observing or simulating scalar transport. To eliminate this computationally expensive step, we use a Lagrangian stochastic framework to derive the enhancement to reaction potential by calculating the collocation probability of particle pairs in a heterogeneous flow field accounting for deformations. We relate the enhanced reaction potential to three well known flow topology metrics and demonstrate that it is best correlated to (and asymptotically linear with) one: the largest eigenvalue of the (right) Cauchy-Green tensor.

Specialization and bet hedging in heterogeneous populations

Phenotypic heterogeneity is a strategy commonly used by bacteria to rapidly adapt to changing environmental conditions. Here, we study the interplay between phenotypic heterogeneity and genetic diversity in spatially extended populations. By analyzing the spatiotemporal dynamics, we show that the level of mobility and the type of competition qualitatively influence the persistence of phenotypic heterogeneity. While direct competition generally promotes persistence of phenotypic heterogeneity, specialization dominates in models with indirect competition irrespective of the degree of mobility.

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.

Range expansion of heterogeneous populations

Risk spreading in bacterial populations is generally regarded as a strategy to maximize survival. Here, we study its role during range expansion of a genetically diverse population where growth and motility are two alternative traits. We find that during the initial expansion phase fast-growing cells do have a selective advantage. By contrast, asymptotically, generalists balancing motility and reproduction are evolutionarily most successful. These findings are rationalized by a set of coupled Fisher equations complemented by stochastic simulations.

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.

Disturbance accelerates the transition from low- to high-diversity state in a model ecosystem

The effect of disturbance on a model ecosystem of sessile and mutually competitive species [Mathiesen et al., Phys. Rev. Lett. 107, 188101 (2011); Mitarai et al., Phys. Rev. E 86, 011929 (2012)] is studied. The disturbance stochastically removes individuals from the system, and the created empty sites are recolonized by neighboring species. We show that the stable high-diversity state, maintained by occasional cyclic species interactions that create isolated patches of metapopulations, is robust against small disturbance. We further demonstrate that finite disturbance can accelerate the transition from the low- to high-diversity state by helping the creation of small patches through diffusion of boundaries between species with standoff relations.

Mobility-dependent selection of competing strategy associations

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

Diverging fluctuations in a spatial five-species cyclic dominance game

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 appearance of this zero-eigenvalue mode is accompanied by neutrality between the species associations. Monte Carlo simulations of the spatial system reveal diverging fluctuations at a specific invasion rate, which can be related to the vanishing dominance between all pairs of species associations.

Global attractors and extinction dynamics of cyclically competing species

Transitions to absorbing states are of fundamental importance in nonequilibrium physics as well as ecology. In ecology, absorbing states correspond to the extinction of species. We here study the spatial population dynamics of three cyclically interacting species. The interaction scheme comprises both direct competition between species as in the cyclic Lotka-Volterra model, and separated selection and reproduction processes as in the May-Leonard model. We show that the dynamic processes leading to the transient maintenance of biodiversity are closely linked to attractors of the nonlinear dynamics for the overall species' concentrations. The characteristics of these global attractors change qualitatively at certain threshold values of the mobility and depend on the relative strength of the different types of competition between species. They give information about the scaling of extinction times with the system size and thereby the stability of biodiversity. We define an effective free energy as the negative logarithm of the probability to find the system in a specific global state before reaching one of the absorbing states. The global attractors then correspond to minima of this effective energy landscape and determine the most probable values for the species' global concentrations. As in equilibrium thermodynamics, qualitative changes in the effective free energy landscape indicate and characterize the underlying nonequilibrium phase transitions. We provide the complete phase diagrams for the population dynamics and give a comprehensive analysis of the spatio-temporal dynamics and routes to extinction in the respective phases.