Replicator-mutator dynamics of the rock-paper-scissors game: Learning through mistakes.

We generalize the Bush-Mosteller learning, the Roth-Erev learning, and the social learning to include mistakes, such that the nonlinear replicator-mutator equation with either additive or multiplicative mutation is generated in an asymptotic limit. Subsequently, we exhaustively investigate the ubiquitous rock-paper-scissors game for some analytically tractable motifs of mutation pattern for which the replicator-mutator flow is seen to exhibit rich dynamics that include limit cycles and chaotic orbits. The main result of this paper is that in both symmetric and asymmetric game interactions, mistakes can sometimes help the players learn; in fact, mistakes can even control chaos to lead to rational Nash-equilibrium outcomes. Furthermore, we report a hitherto-unknown Hamiltonian structure of the replicator-mutator equation.

Hypochaos prevents tragedy of the commons in discrete-time eco-evolutionary game dynamics.

While quite a few recent papers have explored game-resource feedback using the framework of evolutionary game theory, almost all the studies are confined to using time-continuous dynamical equations. Moreover, in such literature, the effect of ubiquitous chaos in the resulting eco-evolutionary dynamics is rather missing. Here, we present a deterministic eco-evolutionary discrete-time dynamics in generation-wise non-overlapping population of two types of harvesters-one harvesting at a faster rate than the other-consuming a self-renewing resource capable of showing chaotic dynamics. In the light of our finding that sometimes chaos is confined exclusively to either the dynamics of the resource or that of the consumer fractions, an interesting scenario is realized: The resource state can keep oscillating chaotically, and hence, it does not vanish to result in the tragedy of the commons-extinction of the resource due to selfish indiscriminate exploitation-and yet the consumer population, whose dynamics depends directly on the state of the resource, may end up being composed exclusively of defectors, i.e., high harvesters. This appears non-intuitive because it is well known that prevention of tragedy of the commons usually requires substantial cooperation to be present.

Periodic orbits in deterministic discrete-time evolutionary game dynamics: An information-theoretic perspective.

Even though the existence of nonconvergent evolution of the states of populations in ecological and evolutionary contexts is an undeniable fact, insightful game-theoretic interpretations of such outcomes are scarce in the literature of evolutionary game theory. As a proof-of-concept, we tap into the information-theoretic concept of relative entropy in order to construct a game-theoretic interpretation for periodic orbits in a wide class of deterministic discrete-time evolutionary game dynamics, primarily investigating the two-player two-strategy case. Effectively, we present a consistent generalization of the evolutionarily stable strategy-the cornerstone of the evolutionary game theory-and aptly term the generalized concept "information stable orbit." The information stable orbit captures the essence of the evolutionarily stable strategy in that it compares the total payoff obtained against an evolving mutant with the total payoff that the mutant gets while playing against itself. Furthermore, we discuss the connection of the information stable orbit with the dynamical stability of the corresponding periodic orbit.

Clinico-demographic profile of COVID-19 positive patients - first wave versus second wave - an experience in north-east India.

INTRODUCTION: India witnessed two distinct COVID-19 waves. We evaluated the clinico-demographic profile of patients infected during first wave (FW) and second wave (SW) in a hospital in north-east India. METHODOLOGY: Patients who tested positive for severe acute respiratory syndrome-coronavirus-2 specific gene by reverse transcriptase polymerase chain reaction across FW and SW were diagnosed as COVID-19 positive. The clinico-demographic data of these positive patients were retrieved from the specimen-referral-form. Vital parameters including respiratory rate, SpO2, data on COVID-19-associated mucormycosis (CAM), COVID-19-associated acute respiratory distress syndrome (CARDS) were obtained from hospital records for in-patients. Patients were categorized based on disease severity. The data obtained in both waves were analyzed comparatively. RESULTS: Out of a total of 119,016 samples tested, 10,164 (8.5%) were SARS-CoV-2 positive (2907 during FW, 7257 during SW). Male predominance was seen across both waves (FW: 68.4%; SW:58.4%), with more children infected during SW. Patients with travel history (24%) and contact with laboratory confirmed cases (61%) were significantly higher during SW relative to FW (10.9% and 42.1% respectively). Healthcare worker infection was higher in SW (5.3%). Symptoms like vomiting [14.8%], diarrhea [10.5%], anosmia [10.4%] and aguesia [9.4%] were more in SW. More patients developed CARDS in SW (6.7%) compared to FW (3.4%) with 85% and 70% patients expiring across FW and SW respectively. No case of CAM is documented in our study. CONCLUSIONS: This was probably the most comprehensive study from north-east India. Industrial oxygen cylinder usage may have been the source of CAM in the rest of the country.

Prevalence, Characteristics, and Correlates of Fatigue in Indian Breast Cancer Survivors: A Cross-Sectional Study.

Navneet KaurBackground Fatigue is one of the commonest sequelae of breast cancer treatment that adversely impacts quality of life (QOL) of breast cancer survivors (BCSs). However, very limited data are available about cancer-related fatigue in Indian patients. Hence, this study was planned with the objectives to study (1) prevalence of fatigue in short-, intermediate-, and long-term follow-up; (2) severity and characteristics of fatigue; (3) impact of fatigue on QOL; and (4) correlation of fatigue with other survivorship issues. Materials and Methods The study was conducted on ( n = 230) BCSs who had completed their primary treatment (surgery, radiotherapy, and chemotherapy) and were coming for follow-up. The prevalence of fatigue was noted from a screening tool, which comprised of 14 commonly reported survivorship issues. Assessment of fatigue was done by using survivorship fatigue assessment tool-1 score and QOL was assessed by functional assessment of cancer therapy-breast (FACT-B) questionnaires. To understand how fatigue evolved over time, survivors were divided into three groups according to the time elapsed since initial treatment: Group 1: <2 years ( n = 105); Group 2: 2-5 years ( n = 70); Group 3: >5 years ( n = 55). Statistical Analysis Data was analyzed by using simple descriptive statistics, one way analysis of variance followed by Tukey's test for comparison of quantitative data among the three groups, and Pearson correlation coefficients for association of fatigue with other survivorship issues. Results Clinically significant fatigue (≥4) was noted in 38% of BCSs. However, high overall prevalence of fatigue (60%) was seen, which persisted in long-term survivors (51%) as well. Severity of fatigue was mostly mild (37.7%) to moderate (47.1%). Fatigue scores were significantly higher in short-term survivors ( 5.01 ± 2.06) than intermediate- (4.03 ± 1.42) and long-term BCSs (3.57 ± 1.37). The mean score on FACT-B was 90.07 ± 10.17 in survivors with fatigue and 104.73 ± 7.13 in those without fatigue ( p = 0.000). Significant correlation of fatigue was seen with other survivorship issues like limb swelling, chronic pain, premature menopause, and its related symptoms and emotional distress. Conclusion Fatigue is highly prevalent in BCSs. Survivorship care programs should include appropriate measures to evaluate and address fatigue.

Selection-recombination-mutation dynamics: Gradient, limit cycle, and closed invariant curve.

In this paper, the replicator dynamics of the two-locus two-allele system under weak mutation and weak selection is investigated in a generation-wise nonoverlapping unstructured population of individuals mating at random. Our main finding is that the dynamics is gradient-like when the point mutations at the two loci are independent. This is in stark contrast to the case of one-locus-multi-allele where the existence gradient behavior is contingent on a specific relationship between the mutation rates. When the mutations are not independent in the two-locus-two-allele system, there is the possibility of nonconvergent outcomes, like asymptotically stable oscillations, through either the Hopf bifurcation or the Neimark-Sacker bifurcation depending on the strength of the weak selection. The results can be straightforwardly extended for multilocus-two-allele systems.

Hostility prevents the tragedy of the commons in metapopulation with asymmetric migration: A lesson from queenless ants.

A colony of the queenless ant species, Pristomyrmex punctatus, can broadly be seen as consisting of small-body sized worker ants and relatively larger body-sized cheater ants. Hence, in the presence of intercolony migration, a set of constituent colonies act as a metapopulation exclusively composed of cooperators and defectors. Such a setup facilitates an evolutionary game-theoretic replication-selection model of population dynamics of the ants in a metapopulation. Using the model, we analytically probe the effects of territoriality induced hostility. Such hostility in the ant metapopulation proves to be crucial in preventing the tragedy of the commons, specifically, the workforce, a social good formed by cooperation. This mechanism applies to any metapopulation-not necessarily the ants-composed of cooperators and defectors where interpopulation migration occurs asymmetrically, i.e., cooperators and defectors migrate at different rates. Furthermore, our model validates that there is evolutionary benefit behind the queenless ants' behavior of showing more hostility towards the immigrants from nearby colonies than those from the far-off ones. In order to calibrate our model's parameters, we have extensively used the data available on the queenless ant species, P. punctatus.

Dynamic scaling in rotating turbulence: A shell model study.

We investigate the scaling form of appropriate timescales extracted from time-dependent correlation functions in rotating turbulent flows. In particular, we obtain precise estimates of the dynamic exponents z_{p}, associated with the timescales, and their relation with the more commonly measured equal-time exponents ζ_{p}. These theoretical predictions, obtained by using the multifractal formalism, are validated through extensive numerical simulations of a shell model for such rotating flows.

Amplitude death in coupled replicator map lattice: Averting migration dilemma.

Populations composed of a collection of subpopulations (demes) with random migration between them are quite common occurrences. The emergence and sustenance of cooperation in such a population is a highly researched topic in the evolutionary game theory. If the individuals in every deme are considered to be either cooperators or defectors, the migration dilemma can be envisaged: The cooperators would not want to migrate to a defector-rich deme as they fear of facing exploitation; but without migration, cooperation cannot be established throughout the network of demes. With a view to studying the aforementioned scenario, in this paper, we set up a theoretical model consisting of a coupled map lattice of replicator maps based on two-player-two-strategy games. The replicator map considered is capable of showing a variety of evolutionary outcomes, like convergent (fixed point) outcomes and nonconvergent (periodic and chaotic) outcomes. Furthermore, this coupled network of the replicator maps undergoes the phenomenon of amplitude death leading to nonoscillatory stable synchronized states. We specifically explore the effect of (i) the nature of coupling that models migration between the maps, (ii) the heterogenous demes (in the sense that not all the demes have the same game being played by the individuals), (iii) the degree of the network, and (iv) the cost associated with the migration. In the course of investigation, we are intrigued by the effectiveness of the random migration in sustaining a uniform cooperator fraction across a population irrespective of the details of the replicator dynamics and the interaction among the demes.

Game-environment feedback dynamics in growing population: Effect of finite carrying capacity.

The tragedy of the commons (TOC) is an unfortunate situation where a shared resource is exhausted due to uncontrolled exploitation by the selfish individuals of a population. Recently, the paradigmatic replicator equation has been used in conjunction with a phenomenological equation for the state of the shared resource to gain insight into the influence of the games on the TOC. The replicator equation, by construction, models a fixed infinite population undergoing microevolution. Thus, it is unable to capture any effect of the population growth and the carrying capacity of the population although the TOC is expected to be dependent on the size of the population. Therefore, in this paper, we present a mathematical framework that incorporates the density dependent payoffs and the logistic growth of the population in the eco-evolutionary dynamics modeling the game-resource feedback. We discover a bistability in the dynamics: a finite carrying capacity can either avert or cause the TOC depending on the initial states of the resource and the initial fraction of cooperators. In fact, depending on the type of strategic game-theoretic interaction, a finite carrying capacity can either avert or cause the TOC when it is exactly the opposite for the corresponding case with infinite carrying capacity.

Emergence of universality in the transmission dynamics of COVID-19.

The complexities involved in modelling the transmission dynamics of COVID-19 has been a roadblock in achieving predictability in the spread and containment of the disease. In addition to understanding the modes of transmission, the effectiveness of the mitigation methods also needs to be built into any effective model for making such predictions. We show that such complexities can be circumvented by appealing to scaling principles which lead to the emergence of universality in the transmission dynamics of the disease. The ensuing data collapse renders the transmission dynamics largely independent of geopolitical variations, the effectiveness of various mitigation strategies, population demographics, etc. We propose a simple two-parameter model-the Blue Sky model-and show that one class of transmission dynamics can be explained by a solution that lives at the edge of a blue sky bifurcation. In addition, the data collapse leads to an enhanced degree of predictability in the disease spread for several geographical scales which can also be realized in a model-independent manner as we show using a deep neural network. The methodology adopted in this work can potentially be applied to the transmission of other infectious diseases and new universality classes may be found. The predictability in transmission dynamics and the simplicity of our methodology can help in building policies for exit strategies and mitigation methods during a pandemic.

Replicator equations induced by microscopic processes in nonoverlapping population playing bimatrix games.

This paper is concerned with exploring the microscopic basis for the discrete versions of the standard replicator equation and the adjusted replicator equation. To this end, we introduce frequency-dependent selection-as a result of competition fashioned by game-theoretic consideration-into the Wright-Fisher process, a stochastic birth-death process. The process is further considered to be active in a generation-wise nonoverlapping finite population where individuals play a two-strategy bimatrix population game. Subsequently, connections among the corresponding master equation, the Fokker-Planck equation, and the Langevin equation are exploited to arrive at the deterministic discrete replicator maps in the limit of infinite population size.

Deciphering chaos in evolutionary games.

A discrete-time replicator map is a prototype of evolutionary selection game dynamical models that have been very successful across disciplines in rendering insights into the attainment of the equilibrium outcomes, like the Nash equilibrium and the evolutionarily stable strategy. By construction, only the fixed-point solutions of the dynamics can possibly be interpreted as the aforementioned game-theoretic solution concepts. Although more complex outcomes like chaos are omnipresent in nature, it is not known to which game-theoretic solutions they correspond. Here, we construct a game-theoretic solution that is realized as the chaotic outcomes in the selection monotone game dynamic. To this end, we invoke the idea that in a population game having two-player-two-strategy one-shot interactions, it is the product of the fitness and the heterogeneity (the probability of finding two individuals playing different strategies in the infinitely large population) that is optimized over the generations of the evolutionary process.

Effect of chaotic agent dynamics on coevolution of cooperation and synchronization.

The effect of chaotic dynamical states of agents on the coevolution of cooperation and synchronization in a structured population of the agents remains unexplored. With a view to gaining insights into this problem, we construct a coupled map lattice of the paradigmatic chaotic logistic map by adopting the Watts-Strogatz network algorithm. The map models the agent's chaotic state dynamics. In the model, an agent benefits by synchronizing with its neighbors, and in the process of doing so, it pays a cost. The agents update their strategies (cooperation or defection) by using either a stochastic or a deterministic rule in an attempt to fetch themselves higher payoffs than what they already have. Among some other interesting results, we find that beyond a critical coupling strength, which increases with the rewiring probability parameter of the Watts-Strogatz model, the coupled map lattice is spatiotemporally synchronized regardless of the rewiring probability. Moreover, we observe that the population does not desynchronize completely-and hence, a finite level of cooperation is sustained-even when the average degree of the coupled map lattice is very high. These results are at odds with how a population of the non-chaotic Kuramoto oscillators as agents would behave. Our model also brings forth the possibility of the emergence of cooperation through synchronization onto a dynamical state that is a periodic orbit attractor.

Evolutionary dynamics of the delayed replicator-mutator equation: Limit cycle and cooperation.

Game theory deals with strategic interactions among players and evolutionary game dynamics tracks the fate of the players' populations under selection. In this paper, we consider the replicator equation for two-player-two-strategy games involving cooperators and defectors. We modify the equation to include the effect of mutation and also delay that corresponds either to the delayed information about the population state or in realizing the effect of interaction among players. By focusing on the four exhaustive classes of symmetrical games-the Stag Hunt game, the Snowdrift game, the Prisoners' Dilemma game, and the Harmony game-we analytically and numerically analyze the delayed replicator-mutator equation to find the explicit condition for the Hopf bifurcation bringing forth stable limit cycle. The existence of the asymptotically stable limit cycle imply the coexistence of the cooperators and the defectors; and in the games, where defection is a stable Nash strategy, a stable limit cycle does provide a mechanism for evolution of cooperation. We find that while mutation alone can never lead to oscillatory cooperation state in two-player-two-strategy games, the delay can change the scenario. On the other hand, there are situations when delay alone cannot lead to the Hopf bifurcation in the absence of mutation in the selection dynamics.

Giant Tunable Mechanical Nonlinearity in Graphene-Silicon Nitride Hybrid Resonator.

High quality factor mechanical resonators have shown great promise in the development of classical and quantum technologies. Simultaneously, progress has been made in developing controlled mechanical nonlinearity. Here, we combine these two directions of progress in a single platform consisting of coupled silicon nitride (SiNx) and graphene mechanical resonators. We show that nonlinear response can be induced on a large area SiNx resonator mode and can be efficiently controlled by coupling it to a gate-tunable, freely suspended graphene mode. The induced nonlinear response of the hybrid modes, as measured on the SiNx resonator surface is giant, with one of the highest measured Duffing constants. We observe a novel phononic frequency comb which we use as an alternate validation of the measured values, along with numerical simulations which are in overall agreement with the measurements.

Periodic orbit can be evolutionarily stable: Case Study of discrete replicator dynamics.

In evolutionary game theory, it is customary to be partial to the dynamical models possessing fixed points so that they may be understood as the attainment of evolutionary stability, and hence, Nash equilibrium. Any show of periodic or chaotic solution is many a time perceived as a shortcoming of the corresponding game dynamic because (Nash) equilibrium play is supposed to be robust and persistent behaviour, and any other behaviour in nature is deemed transient. Consequently, there is a lack of attempt to connect the non-fixed point solutions with the game theoretic concepts. Here we provide a way to render game theoretic meaning to periodic solutions. To this end, we consider a replicator map that models Darwinian selection mechanism in unstructured infinite-sized population whose individuals reproduce asexually forming non-overlapping generations. This is one of the simplest evolutionary game dynamic that exhibits periodic solutions giving way to chaotic solutions (as parameters related to reproductive fitness change) and also obeys the folk theorems connecting fixed point solutions with Nash equilibrium. Interestingly, we find that a modified Darwinian fitness-termed heterogeneity payoff-in the corresponding population game must be put forward as (conventional) fitness times the probability that two arbitrarily chosen individuals of the population adopt two different strategies. The evolutionary dynamics proceeds as if the individuals optimize the heterogeneity payoff to reach an evolutionarily stable orbit, should it exist. We rigorously prove that a locally asymptotically stable period orbit must be heterogeneity stable orbit-a generalization of evolutionarily stable state.

Finding the Hannay angle in dissipative oscillatory systems via conservative perturbation theory.

Usage of a Hamiltonian perturbation theory for a nonconservative system is counterintuitive and, in general, a technical impossibility by definition. However, the time-independent dual Hamiltonian formalism for the nonconservative systems has opened the door for using various conservative perturbation theories for investigating the dynamics of such systems. Here we demonstrate that the Lie transform Hamiltonian perturbation theory can be adapted to find the perturbative solutions and the frequency corrections for the dissipative oscillatory systems. As a further application, we use the perturbation theory to analytically calculate the Hannay angle for the van der Pol oscillator's limit cycle trajectory when its parameters-the strength of the nonlinearity and the frequency of the linear part-evolve cyclically and adiabatically. For this van der Pol oscillator, we also numerically calculate the corresponding geometric phase and establish its equivalence with the Hannay angle.

Understanding transient uncoupling induced synchronization through modified dynamic coupling.

An important aspect of the recently introduced transient uncoupling scheme is that it induces synchronization for large values of coupling strength at which the coupled chaotic systems resist synchronization when continuously coupled. However, why this is so is an open problem? To answer this question, we recall the conventional wisdom that the eigenvalues of the Jacobian of the transverse dynamics measure whether a trajectory at a phase point is locally contracting or diverging with respect to another nearby trajectory. Subsequently, we go on to highlight a lesser appreciated fact that even when, under the corresponding linearised flow, the nearby trajectory asymptotically diverges away, its distance from the reference trajectory may still be contracting for some intermediate period. We term this phenomenon transient decay in line with the phenomenon of the transient growth. Using these facts, we show that an optimal coupling region, i.e., a region of the phase space where coupling is on, should ideally be such that at any of the constituent phase point either the maximum of the real parts of the eigenvalues is negative or the magnitude of the positive maximum is lesser than that of the negative minimum. We also invent and employ a modified dynamics coupling scheme-a significant improvement over the well-known dynamic coupling scheme-as a decisive tool to justify our results.