Coupled environmental and demographic fluctuations shape the evolution of cooperative antimicrobial resistance.

There is a pressing need to better understand how microbial populations respond to antimicrobial drugs, and to find mechanisms to possibly eradicate antimicrobial-resistant cells. The inactivation of antimicrobials by resistant microbes can often be viewed as a cooperative behaviour leading to the coexistence of resistant and sensitive cells in large populations and static environments. This picture is, however, greatly altered by the fluctuations arising in volatile environments, in which microbial communities commonly evolve. Here, we study the eco-evolutionary dynamics of a population consisting of an antimicrobial-resistant strain and microbes sensitive to antimicrobial drugs in a time-fluctuating environment, modelled by a carrying capacity randomly switching between states of abundance and scarcity. We assume that antimicrobial resistance (AMR) is a shared public good when the number of resistant cells exceeds a certain threshold. Eco-evolutionary dynamics is thus characterised by demographic noise (birth and death events) coupled to environmental fluctuations which can cause population bottlenecks. By combining analytical and computational means, we determine the environmental conditions for the long-lived coexistence and fixation of both strains, and characterise a fluctuation-driven AMR eradication mechanism, where resistant microbes experience bottlenecks leading to extinction. We also discuss the possible applications of our findings to laboratory-controlled experiments.

Effect of mobility in the rock-paper-scissor dynamics with high mortality.

In the evolutionary dynamics of a rock-paper-scissor model, the effect of natural death plays a major role in determining the fate of the system. Coexistence, being an unstable fixed point of the model, becomes very sensitive toward this parameter. In order to study the effect of mobility in such a system which has explicit dependence on mortality, we perform Monte Carlo simulation on a two-dimensional lattice having three cyclically competing species. The spatiotemporal dynamics has been studied along with the two-site correlation function. Spatial distribution exhibits emergence of spiral patterns in the presence of mobility. It reveals that the joint effect of death rate and mobility (diffusion) leads to new coexistence and extinction scenarios.

How does homophily shape the topology of a dynamic network?

We consider a dynamic network of individuals that may hold one of two different opinions in a two-party society. As a dynamical model, agents can endlessly create and delete links to satisfy a preferred degree, and the network is shaped by homophily, a form of social interaction. Characterized by the parameter J∈[-1,1], the latter plays a role similar to Ising spins: agents create links to others of the same opinion with probability (1+J)/2 and delete them with probability (1-J)/2. Using Monte Carlo simulations and mean-field theory, we focus on the network structure in the steady state. We study the effects of J on degree distributions and the fraction of cross-party links. While the extreme cases of homophily or heterophily (J=±1) are easily understood to result in complete polarization or anti-polarization, intermediate values of J lead to interesting features of the network. Our model exhibits the intriguing feature of an "overwhelming transition" occurring when communities of different sizes are subject to sufficient heterophily: agents of the minority group are oversubscribed and their average degree greatly exceeds that of the majority group. In addition, we introduce an original measure of polarization which displays distinct advantages over the commonly used average edge homogeneity.

Exclusion of the fittest predicts microbial community diversity in fluctuating environments.

Microorganisms live in environments that inevitably fluctuate between mild and harsh conditions. As harsh conditions may cause extinctions, the rate at which fluctuations occur can shape microbial communities and their diversity, but we still lack an intuition on how. Here, we build a mathematical model describing two microbial species living in an environment where substrate supplies randomly switch between abundant and scarce. We then vary the rate of switching as well as different properties of the interacting species, and measure the probability of the weaker species driving the stronger one extinct. We find that this probability increases with the strength of demographic noise under harsh conditions and peaks at either low, high, or intermediate switching rates depending on both species' ability to withstand the harsh environment. This complex relationship shows why finding patterns between environmental fluctuations and diversity has historically been difficult. In parameter ranges where the fittest species was most likely to be excluded, however, the beta diversity in larger communities also peaked. In sum, how environmental fluctuations affect interactions between a few species pairs predicts their effect on the beta diversity of the whole community.

Population Dynamics in a Changing Environment: Random versus Periodic Switching.

Environmental changes greatly influence the evolution of populations. Here, we study the dynamics of a population of two strains, one growing slightly faster than the other, competing for resources in a time-varying binary environment modeled by a carrying capacity switching either randomly or periodically between states of abundance and scarcity. The population dynamics is characterized by demographic noise (birth and death events) coupled to a varying environment. We elucidate the similarities and differences of the evolution subject to a stochastically and periodically varying environment. Importantly, the population size distribution is generally found to be broader under intermediate and fast random switching than under periodic variations, which results in markedly different asymptotic behaviors between the fixation probability of random and periodic switching. We also determine the detailed conditions under which the fixation probability of the slow strain is maximal.

Fixation properties of rock-paper-scissors games in fluctuating populations.

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 coupling of demographic and environmental noise influences the fixation properties. More specifically, we investigate which species is the most likely to prevail in a population of fluctuating size and how the outcome depends on the environmental variability. We show that demographic noise coupled with environmental randomness "levels the field" of cyclic competition by balancing the effect of selection. In particular, we show that fast switching effectively reduces the selection intensity proportionally to the variance of the carrying capacity. We determine the conditions under which new fixation scenarios arise, where the most likely species to prevail changes with the rate of switching and the variance of the carrying capacity. Random switching has a limited effect on the mean fixation time that scales linearly with the average population size. Hence, environmental randomness makes the cyclic competition more egalitarian, but does not prolong the species coexistence. We also show how the fixation probabilities of close-to-zero-sum rock-paper-scissors games can be obtained from those of the zero-sum model by rescaling the selection intensity.

Eco-evolutionary dynamics of a population with randomly switching carrying capacity.

Environmental variability greatly influences the eco-evolutionary dynamics of a population, i.e. it affects how its size and composition evolve. Here, we study a well-mixed population of finite and fluctuating size whose growth is limited by a randomly switching carrying capacity. This models the environmental fluctuations between states of resources abundance and scarcity. The population consists of two strains, one growing slightly faster than the other, competing under two scenarios: one in which competition is solely for resources, and one in which the slow (cooperating) strain produces a public good (PG) that benefits also the fast (free-riding) strain. We investigate how the coupling of demographic and environmental (external) noise affects the population's eco-evolutionary dynamics. By analytical and computational means, we study the correlations between the population size and its composition, and discuss the social-dilemma-like 'eco-evolutionary game' characterizing the PG production. We determine in what conditions it is best to produce a PG; when cooperating is beneficial but outcompeted by free riding, and when the PG production is detrimental for cooperators. Within a linear noise approximation to populations of varying size, we also accurately analyse the coupled effects of demographic and environmental noise on the size distribution.

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.

Evolution of a Fluctuating Population in a Randomly Switching Environment.

Environment plays a fundamental role in the competition for resources, and hence in the evolution of populations. Here, we study a well-mixed, finite population consisting of two strains competing for the limited resources provided by an environment that randomly switches between states of abundance and scarcity. Assuming that one strain grows slightly faster than the other, we consider two scenarios-one of pure resource competition, and one in which one strain provides a public good-and investigate how environmental randomness (external noise) coupled to demographic (internal) noise determines the population's fixation properties and size distribution. By analytical means and simulations, we show that these coupled sources of noise can significantly enhance the fixation probability of the slower-growing species. We also show that the population size distribution can be unimodal, bimodal, or multimodal and undergoes noise-induced transitions between these regimes when the rate of switching matches the population's growth rate.

Heterogeneous out-of-equilibrium nonlinear q-voter model with zealotry.

We study the dynamics of the out-of-equilibrium nonlinear q-voter model with two types of susceptible voters and zealots, introduced in Mellor et al. [Europhys. Lett. 113, 48001 (2016)EULEEJ0295-507510.1209/0295-5075/113/48001]. In this model, each individual supports one of two parties and is either a susceptible voter of type q_{1} or q_{2}, or is an inflexible zealot. At each time step, a q_{i}-susceptible voter (i=1,2) consults a group of q_{i} neighbors and adopts their opinion if all group members agree, while zealots are inflexible and never change their opinion. This model violates detailed balance whenever q_{1}≠q_{2} and is characterized by two distinct regimes of low and high density of zealotry. Here, by combining analytical and numerical methods, we investigate the nonequilibrium stationary state of the system in terms of its probability distribution, nonvanishing currents, and unequal-time two-point correlation functions. We also study the switching time properties of the model by exploiting an approximate mapping onto the model of Mobilia [Phys. Rev. E 92, 012803 (2015)PLEEE81539-375510.1103/PhysRevE.92.012803] that satisfies the detailed balance, and we outline some properties of the model near criticality.

Nonlinear q-voter model with inflexible zealots.

We study the dynamics of the nonlinear q-voter model with inflexible zealots in a finite well-mixed population. In this system, each individual supports one of two parties and is either a susceptible voter or an inflexible zealot. At each time step, a susceptible adopts the opinion of a neighbor if this belongs to a group of q≥2 neighbors all in the same state, whereas inflexible zealots never change their opinion. In the presence of zealots of both parties, the model is characterized by a fluctuating stationary state and, below a zealotry density threshold, the distribution of opinions is bimodal. After a characteristic time, most susceptibles become supporters of the party having more zealots and the opinion distribution is asymmetric. When the number of zealots of both parties is the same, the opinion distribution is symmetric and, in the long run, susceptibles endlessly swing from the state where they all support one party to the opposite state. Above the zealotry density threshold, when there is an unequal number of zealots of each type, the probability distribution is single-peaked and non-Gaussian. These properties are investigated analytically and with stochastic simulations. We also study the mean time to reach a consensus when zealots support only one party.

Influence of Luddism on innovation diffusion.

We generalize the classical Bass model of innovation diffusion to include a new class of agents-Luddites-that oppose the spread of innovation. Our model also incorporates ignorants, susceptibles, and adopters. When an ignorant and a susceptible meet, the former is converted to a susceptible at a given rate, while a susceptible spontaneously adopts the innovation at a constant rate. In response to the rate of adoption, an ignorant may become a Luddite and permanently reject the innovation. Instead of reaching complete adoption, the final state generally consists of a population of Luddites, ignorants, and adopters. The evolution of this system is investigated analytically and by stochastic simulations. We determine the stationary distribution of adopters, the time needed to reach the final state, and the influence of the network topology on the innovation spread. Our model exhibits an important dichotomy: When the rate of adoption is low, an innovation spreads slowly but widely; in contrast, when the adoption rate is high, the innovation spreads rapidly but the extent of the adoption is severely limited by Luddites.

Characterization of spiraling patterns in spatial rock-paper-scissors games.

The spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatiotemporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising "rock-paper-scissors" interactions via dominance removal and replacement, reproduction, mutations, pair exchange, and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allow us to clarify when spatial "rock-paper-scissors" competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility.

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.

Facilitators on networks reveal optimal interplay between information exchange and reciprocity.

Reciprocity is firmly established as an important mechanism that promotes cooperation. An efficient information exchange is likewise important, especially on structured populations, where interactions between players are limited. Motivated by these two facts, we explore the role of facilitators in social dilemmas on networks. Facilitators are here mirrors to their neighbors-they cooperate with cooperators and defect with defectors-but they do not participate in the exchange of strategies. As such, in addition to introducing direct reciprocity, they also obstruct information exchange. In well-mixed populations, facilitators favor the replacement and invasion of defection by cooperation as long as their number exceeds a critical value. In structured populations, on the other hand, there exists a delicate balance between the benefits of reciprocity and the deterioration of information exchange. Extensive Monte Carlo simulations of social dilemmas on various interaction networks reveal that there exists an optimal interplay between reciprocity and information exchange, which sets in only when a small number of facilitators occupy the main hubs of the scale-free network. The drawbacks of missing cooperative hubs are more than compensated for by reciprocity and, at the same time, the compromised information exchange is routed via the auxiliary hubs with only marginal losses in effectivity. These results indicate that it is not always optimal for the main hubs to become leaders of the masses, but rather to exploit their highly connected state to promote tit-for-tat-like behavior.

Reply to "Comment on 'Stochastic dynamics of the prisoner's dilemma with cooperation facilitators' ".

The fixation properties of a simple prisoner's dilemma game in the presence of "cooperation facilitators" have recently been investigated in finite and well-mixed populations for various dynamics [Mobilia, Phys. Rev. E 86, 011134 (2012)]. In a Comment, Miekisz claims that, for cooperation to be favored by selection in the standard prisoner's dilemma games with facilitators, it suffices that f(C)>f(D) (where f(C/D) are the respective fitnesses of cooperators and defectors). In this Reply, we show that, in generic prisoner's dilemma games with ℓ cooperation facilitators, it is generally not sufficient that a single cooperator has a higher fitness than defectors to ensure that selection favors cooperation. In fact, it is also necessary that selection promotes the replacement of defection by cooperation in a population of size N, which requires that the fixation probability of a single cooperator exceeds (N-ℓ)(-1). This replacement condition is independent of f(C)>f(D) and, when the payoff for mutual defection is negative, it is shown to be more stringent than the invasion condition. Our results, illustrated by a series of examples, considerably generalize those reported in the paper [Phys. Rev. E 86, 011134 (2012)] and in the aforementioned Comment whose claims are demonstrated to be relevant only for a special subclass of prisoner's dilemma games.

Cooperation dilemma in finite populations under fluctuating environments.

We present a novel approach allowing the study of rare events like fixation under fluctuating environments, modeled as extrinsic noise, in evolutionary processes characterized by the dominance of one species. Our treatment consists of mapping the system onto an auxiliary model, exhibiting metastable species coexistence, that can be analyzed semiclassically. This approach enables us to study the interplay between extrinsic and demographic noise on the statistics of interest. We illustrate our theory by considering the paradigmatic prisoner's dilemma game, whose evolution is described by the probability that cooperators fixate the population and replace all defectors. We analytically and numerically demonstrate that extrinsic noise may drastically enhance the cooperation fixation probability and even change its functional dependence on the population size. These results, which generalize earlier works in population genetics, indicate that extrinsic noise may help sustain and promote a much higher level of cooperation than static settings.

Metastability and anomalous fixation in evolutionary games on scale-free networks.

We study the influence of complex graphs on the metastability and fixation properties of a set of evolutionary processes. In the framework of evolutionary game theory, where the fitness and selection are frequency dependent and vary with the population composition, we analyze the dynamics of snowdrift games (characterized by a metastable coexistence state) on scale-free networks. Using an effective diffusion theory in the weak selection limit, we demonstrate how the scale-free structure affects the system's metastable state and leads to anomalous fixation. In particular, we analytically and numerically show that the probability and mean time of fixation are characterized by stretched-exponential behaviors with exponents depending on the network's degree distribution.

Stochastic dynamics of the prisoner's dilemma with cooperation facilitators.

In the framework of the paradigmatic prisoner's dilemma game, we investigate the evolutionary dynamics of social dilemmas in the presence of "cooperation facilitators." In our model, cooperators and defectors interact as in the classical prisoner's dilemma, where selection favors defection. However, here the presence of a small number of cooperation facilitators enhances the fitness (reproductive potential) of cooperators, while it does not alter that of defectors. In a finite population of size N, the dynamics of the prisoner's dilemma with facilitators is characterized by the probability that cooperation takes over (fixation probability) by the mean times to reach the absorbing states. These quantities are computed exactly using Fokker-Planck equations. Our findings, corroborated by stochastic simulations, demonstrate that the influence of facilitators crucially depends on the difference between their density z and the game's cost-to-benefit ratio r. When z > r, the fixation of cooperators is likely in a large population and, under weak selection pressure, invasion and replacement of defection by cooperation is favored by selection if b(z - r)(1 - z) > N(-1), where 0