The replicator equation and other game dynamics

Interaction intensity in strategic fitness: A quantifying yardstick of selection optimization for evolutionary game

The notion of the fitness of a strategy has been assimilated as the reproductive success in the evolutionary game. Initially, this fitness was tied to the game's pay-off and the strategy's relative frequency. However, density dependence becomes exigent in order to make ecologically reliable fitness. However, the contributions of each different type of interaction to the species's overall growth process were surprisingly under-explored. This oversight has occasionally led to either more or less prediction of strategy selection compared to the actual possibility. Moreover, density regulation of the population has always been analysed in a general way compared to strategy selection. In this context, our study introduces the concept of mean relative death payoff, which helps in assessing interaction intensity coefficients and integrates them into strategic fitness. Based on this fitness function, we develop the frequency-density replicator dynamics, which eventually provides distinguishing criteria for directional and balancing selection. Our optimized, evolutionarily stable strategy emerges as a superior alternative to the conventional trade-off between selection forces and ecological processes. More significantly, mean relative death pay-off has both conditional and quantitative roles in getting a stable population size. As a case study, we have extensively analysed the evolution of aggression using the Hawk-Dove game. We have shown that pure Dove selection is always beneficial for species growth rather than pure Hawk selection, and the condition of selection is dependent on external mortality pressure. However, the condition of coexistence is independent of external mortality pressure, representing a strong evolutionary selection that optimizes population density governed by interaction intensity.

Limits on the evolutionary rates of biological traits

This paper focuses on the maximum speed at which biological evolution can occur. I derive inequalities that limit the rate of evolutionary processes driven by natural selection, mutations, or genetic drift. These rate limits link the variability in a population to evolutionary rates. In particular, high variances in the fitness of a population and of a quantitative trait allow for fast changes in the trait's average. In contrast, low variability makes a trait less susceptible to random changes due to genetic drift. The results in this article generalize Fisher's fundamental theorem of natural selection to dynamics that allow for mutations and genetic drift, via trade-off relations that constrain the evolutionary rates of arbitrary traits. The rate limits can be used to probe questions in various evolutionary biology and ecology settings. They apply, for instance, to trait dynamics within or across species or to the evolution of bacteria strains. They apply to any quantitative trait, e.g., from species' weights to the lengths of DNA strands.

The Unification of Evolutionary Dynamics through the Bayesian Decay Factor in a Game on a Graph

We unify evolutionary dynamics on graphs in strategic uncertainty through a decaying Bayesian update. Our analysis focuses on the Price theorem of selection, which governs replicator(-mutator) dynamics, based on a stratified interaction mechanism and a composite strategy update rule. Our findings suggest that the replication of a certain mutation in a strategy, leading to a shift from competition to cooperation in a well-mixed population, is equivalent to the replication of a strategy in a Bayesian-structured population without any mutation. Likewise, the replication of a strategy in a Bayesian-structured population with a certain mutation, resulting in a move from competition to cooperation, is equivalent to the replication of a strategy in a well-mixed population without any mutation. This equivalence holds when the transition rate from competition to cooperation is equal to the relative strength of selection acting on either competition or cooperation in relation to the selection differential between cooperators and competitors. Our research allows us to identify situations where cooperation is more likely, irrespective of the specific payoff levels. This approach provides new perspectives into the intended purpose of Price's equation, which was initially not designed for this type of analysis.

Can institutions foster cooperation by wealth redistribution?

Theoretical models prescribe how institutions can promote cooperation in a population by imposing appropriate punishments or rewards on individuals. However, many real-world institutions are not sophisticated or responsive enough to ensure cooperation by calibrating their policies. Or, worse yet, an institution might selfishly exploit the population it governs for its own benefit. Here, we study the evolution of cooperation in the presence of an institution that is autonomous, in the sense that it has its own interests that may or may not align with those of the population. The institution imposes a tax on the population and redistributes a portion of the tax revenue to cooperators, withholding the remaining revenue for itself. The institution adjusts its rates of taxation and redistribution to optimize its own long-term, discounted utility. We consider three types of institutions with different goals, embodied in their utility functions. We show that a prosocial institution, whose goal is to maximize the average payoff of the population, can indeed promote cooperation-but only if it is sufficiently forward-looking. On the other hand, an institution that seeks to maximize welfare among cooperators alone will successfully promote collective cooperation even if it is myopic. Remarkably, even a selfish institution, which seeks to maximize the revenue it withholds for itself, can nonetheless promote cooperation. The average payoff of the population increases when a selfish institution is more forward-looking, so that a population under a selfish regime can sometimes fare better than under anarchy. Our analysis highlights the potential benefits of institutional wealth redistribution, even when an institution does not share the interests of the population it governs.

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.

Evolution of cooperation in a two-species system with a common resource pool

Understanding the evolution of cooperation is a major question in Evolutionary Biology. Here, we extend a previously proposed mathematical model in Evolutionary Game Theory that investigated how resource use by a single species composed of cooperators and defectors may lead to its maintenance or extinction. We include another species in the model, so as to investigate how different intra and interspecific interactions of cooperative or competitive nature among individuals that share the same essential resource may drive the survival and evolution of the species. Several outcomes emerge from the model, depending on the configuration of the payoff matrix, the individual contribution to the resource pool, the competition intensity between species, and the initial conditions of the system dynamics. Observed results include scenarios in which species thrive due to the action of cooperators, but also scenarios in which both species collapse due to lack of cooperation and, consequently, of resources. In particular, a high initial availability of resources may be the determinant factor to the survival of both species. Interestingly, cooperation may be more favored when individuals have less incentive to cooperate with others, and the survival of their populations may depend crucially on their competitive capacities.

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.

Extinction scenarios in evolutionary processes: a multinomial Wright-Fisher approach

We study a discrete-time multi-type Wright-Fisher population process. The mean-field dynamics of the stochastic process is induced by a general replicator difference equation. We prove several results regarding the asymptotic behavior of the model, focusing on the impact of the mean-field dynamics on it. One of the results is a limit theorem that describes sufficient conditions for an almost certain path to extinction, first eliminating the type which is the least fit at the mean-field equilibrium. The effect is explained by the metastability of the stochastic system, which under the conditions of the theorem spends almost all time before the extinction event in a neighborhood of the equilibrium. In addition to the limit theorems, we propose a maximization principle for a general deterministic replicator dynamics and study its implications for the stochastic model.

DDQN-based optimal targeted therapy with reversible inhibitors to combat the Warburg effect

We cover the Warburg effect with a three-component evolutionary model, where each component represents a different metabolic strategy. In this context, a scenario involving cells expressing three different phenotypes is presented. One tumour phenotype exhibits glycolytic metabolism through glucose uptake and lactate secretion. Lactate is used by a second malignant phenotype to proliferate. The third phenotype represents healthy cells, which performs oxidative phosphorylation. The purpose of this model is to gain a better understanding of the metabolic alterations associated with the Warburg effect. It is suitable to reproduce some of the clinical trials obtained in colorectal cancer and other even more aggressive tumours. It shows that lactate is an indicator of poor prognosis, since it favours the setting of polymorphic tumour equilibria that complicates its treatment. This model is also used to train a reinforcement learning algorithm, known as Double Deep Q-networks, in order to provide the first optimal targeted therapy based on experimental tumour growth inhibitors as genistein and AR-C155858. Our in silico solution includes the optimal therapy for all the tumour state space and also ensures the best possible quality of life for the patients, by considering the duration of treatment, the use of low-dose medications and the existence of possible contraindications. Optimal therapies obtained with Double Deep Q-networks are validated with the solutions of the Hamilton-Jacobi-Bellman equation.

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.

Evolutionary dynamics within and among competing groups

Biological and social systems are structured at multiple scales, and the incentives of individuals who interact in a group may diverge from the collective incentive of the group as a whole. Mechanisms to resolve this tension are responsible for profound transitions in evolutionary history, including the origin of cellular life, multicellular life, and even societies. Here, we synthesize a growing literature that extends evolutionary game theory to describe multilevel evolutionary dynamics, using nested birth-death processes and partial differential equations to model natural selection acting on competition within and among groups of individuals. We analyze how mechanisms known to promote cooperation within a single group-including assortment, reciprocity, and population structure-alter evolutionary outcomes in the presence of competition among groups. We find that population structures most conducive to cooperation in multiscale systems can differ from those most conducive within a single group. Likewise, for competitive interactions with a continuous range of strategies we find that among-group selection may fail to produce socially optimal outcomes, but it can nonetheless produce second-best solutions that balance individual incentives to defect with the collective incentives for cooperation. We conclude by describing the broad applicability of multiscale evolutionary models to problems ranging from the production of diffusible metabolites in microbes to the management of common-pool resources in human societies.

Judgment on the evolution state of asymmetric games

In this article, we probe into two-role two-player two-strategy asymmetric evolutionary games by utilizing replicator dynamic equations and present all equilibrium points and their stability conditions. Further, the resulting evolution state of the system can be depicted directly according to the specific payoff matrix and initial values. This study facilitates and perfects the research of asymmetric evolutionary games.

Community formation in wealth-mediated thermodynamic strategy evolution

We study a dynamical system defined by a repeated game on a 1D lattice, in which the players keep track of their gross payoffs over time in a bank. Strategy updates are governed by a Boltzmann distribution, which depends on the neighborhood bank values associated with each strategy, relative to a temperature scale, which defines the random fluctuations. Players with higher bank values are, thus, less likely to change strategy than players with a lower bank value. For a parameterized rock-paper-scissors game, we derive a condition under which communities of a given strategy form with either fixed or drifting boundaries. We show the effect of a temperature increase on the underlying system and identify surprising properties of this model through numerical simulations.

Assortment and Reciprocity Mechanisms for Promotion of Cooperation in a Model of Multilevel Selection

In the study of the evolution of cooperation, many mechanisms have been proposed to help overcome the self-interested cheating that is individually optimal in the Prisoners' Dilemma game. These mechanisms include assortative or networked social interactions, other-regarding preferences considering the payoffs of others, reciprocity rules to establish cooperation as a social norm, and multilevel selection involving simultaneous competition between individuals favoring cheaters and competition between groups favoring cooperators. In this paper, we build on recent work studying PDE replicator equations for multilevel selection to understand how within-group mechanisms of assortment, other-regarding preferences, and both direct and indirect reciprocity can help to facilitate cooperation in concert with evolutionary competition between groups. We consider a group-structured population in which interactions between individuals consist of Prisoners' Dilemma games and study the dynamics of multilevel competition determined by the payoffs individuals receive when interacting according to these within-group mechanisms. We find that the presence of each of these mechanisms acts synergistically with multilevel selection for the promotion of cooperation, decreasing the strength of between-group competition required to sustain long-time cooperation and increasing the collective payoff achieved by the population. However, we find that only other-regarding preferences allow for the achievement of socially optimal collective payoffs for Prisoners' Dilemma games in which average payoff is maximized by an intermediate mix of cooperators and defectors. For the other three mechanisms, the multilevel dynamics remain susceptible to a shadow of lower-level selection, as the collective outcome fails to exceed the payoff of the all-cooperator group.

Behavior Choice Mechanisms and Tax Incentive Mechanisms in the Game of Construction Safety

The violation behavior of construction workers is an important cause of construction accidents. To reduce the violations of construction workers and to stimulate the supervision behavior of local governments and construction enterprises, an evolutionary game model is constructed in this paper. Then, the behavior choice mechanism of each player is analyzed. Finally, an incentive effect analysis method is put forward, and the incentive effects of different tax incentive mechanisms are analyzed. This research finds that only when the safety punishment imposed on construction workers is large enough does the supervision behavior of local governments and construction enterprises encourage construction workers to choose not to violate the regulation. Increasing the tax rate of a construction enterprise in the case of accidents can encourage the construction enterprise to supervise, but it inhibits the supervision behavior of the local government. A numerical simulation verifies the effectiveness of the incentive effect analysis method, which provides a new method for the incentive effect analysis of incentive mechanisms.

An eco-evo-devo genetic network model of stress response

The capacity of plants to resist abiotic stresses is of great importance to agricultural, ecological and environmental sustainability, but little is known about its genetic underpinnings. Existing genetic tools can identify individual genetic variants mediating biochemical, physiological, and cellular defenses, but fail to chart an overall genetic atlas behind stress resistance. We view stress response as an eco-evo-devo process by which plants adaptively respond to stress through complex interactions of developmental canalization, phenotypic plasticity, and phenotypic integration. As such, we define and quantify stress response as the developmental change of adaptive traits from stress-free to stress-exposed environments. We integrate composite functional mapping and evolutionary game theory to reconstruct omnigenic, information-flow interaction networks for stress response. Using desert-adapted Euphrates poplar as an example, we infer salt resistance-related genome-wide interactome networks and trace the roadmap of how each SNP acts and interacts with any other possible SNPs to mediate salt resistance. We characterize the previously unknown regulatory mechanisms driving trait variation; i.e. the significance of a SNP may be due to the promotion of positive regulators, whereas the insignificance of a SNP may result from the inhibition of negative regulators. The regulator-regulatee interactions detected are not only experimentally validated by two complementary experiments, but also biologically interpreted by their encoded protein-protein interactions. Our eco-evo-devo model of genetic interactome networks provides an approach to interrogate the genetic architecture of stress response and informs precise gene editing for improving plants' capacity to live in stress environments.

No robust multispecies coexistence in a canonical model of plant-soil feedbacks

Plant–soil feedbacks (PSFs) are considered a key mechanism generating frequency‐dependent dynamics in plant communities. Negative feedbacks, in particular, are often invoked to explain coexistence and the maintenance of diversity in species‐rich communities. However, the primary modelling framework used to study PSFs considers only two plant species, and we lack clear theoretical expectations for how these complex interactions play out in communities with natural levels of diversity. Here, we extend this canonical model of PSFs to include an arbitrary number of plant species and analyse the dynamics. Surprisingly, we find that coexistence of more than two species is virtually impossible, suggesting that alternative theoretical frameworks are needed to describe feedbacks observed in diverse natural communities. Drawing on our analysis, we discuss future directions for PSF models and implications for experimental study of PSF‐mediated coexistence in the field.

Spontaneous symmetry breaking in ride-sharing adoption dynamics

Symmetry breaking ubiquitously occurs across complex systems, from phase transition in statistical physics to self-organized lane formation in pedestrian dynamics. Here, we uncover spontaneous symmetry breaking in a simple model of ride-sharing adoption. We analyze how collective interactions among ride-sharing users to avoid detours in shared rides give rise to spontaneous symmetry breaking and pattern formation in the adoption dynamics. These dynamics result in bistability of high homogeneous and partial heterogeneous adoption states, potentially limiting the population-wide adoption of ride sharing. Our results provide a framework to understand real-world adoption patterns of ride sharing in complex urban settings and support the (re)design of ride-sharing services and incentives for sustainable shared mobility.

Dynamical systems as a level of cognitive analysis of multi-agent learning: Algorithmic foundations of temporal-difference learning dynamics

A dynamical systems perspective on multi-agent learning, based on the link between evolutionary game theory and reinforcement learning, provides an improved, qualitative understanding of the emerging collective learning dynamics. However, confusion exists with respect to how this dynamical systems account of multi-agent learning should be interpreted. In this article, I propose to embed the dynamical systems description of multi-agent learning into different abstraction levels of cognitive analysis. The purpose of this work is to make the connections between these levels explicit in order to gain improved insight into multi-agent learning. I demonstrate the usefulness of this framework with the general and widespread class of temporal-difference reinforcement learning. I find that its deterministic dynamical systems description follows a minimum free-energy principle and unifies a boundedly rational account of game theory with decision-making under uncertainty. I then propose an on-line sample-batch temporal-difference algorithm which is characterized by the combination of applying a memory-batch and separated state-action value estimation. I find that this algorithm serves as a micro-foundation of the deterministic learning equations by showing that its learning trajectories approach the ones of the deterministic learning equations under large batch sizes. Ultimately, this framework of embedding a dynamical systems description into different abstraction levels gives guidance on how to unleash the full potential of the dynamical systems approach to multi-agent learning.

Bayes and Darwin: How replicator populations implement Bayesian computations

Bayesian learning theory and evolutionary theory both formalize adaptive competition dynamics in possibly high-dimensional, varying, and noisy environments. What do they have in common and how do they differ? In this paper, we discuss structural and dynamical analogies and their limits, both at a computational and an algorithmic-mechanical level. We point out mathematical equivalences between their basic dynamical equations, generalizing the isomorphism between Bayesian update and replicator dynamics. We discuss how these mechanisms provide analogous answers to the challenge of adapting to stochastically changing environments at multiple timescales. We elucidate an algorithmic equivalence between a sampling approximation, particle filters, and the Wright-Fisher model of population genetics. These equivalences suggest that the frequency distribution of types in replicator populations optimally encodes regularities of a stochastic environment to predict future environments, without invoking the known mechanisms of multilevel selection and evolvability. A unified view of the theories of learning and evolution comes in sight.

A graph model of combination therapies

The simultaneous use of multiple medications causes drug-drug interactions (DDI) that impact therapeutic efficacy. Here, we argue that graph theory, in conjunction with game theory and ecosystem theory, can address this issue. We treat the coexistence of multiple drugs as a system in which DDI is modeled by game theory. We develop an ordinary differential equation model to characterize how the concentration of a drug changes as a result of its independent capacity and the dependent influence of other drugs through the metabolic response of the host. We coalesce all drugs into personalized and context-specific networks, which can reveal key DDI determinants of therapeutical efficacy. Our model can quantify drug synergy and antagonism and test the translational success of combination therapies to the clinic.

A Single-Cell Omics Network Model of Cell Crosstalk during the Formation of Primordial Follicles

The fate of fetal germ cells (FGCs) in primordial follicles is largely determined by how they interact with the surrounding granulosa cells. However, the molecular mechanisms underlying this interactive process remain poorly understood. Here, we develop a computational model to characterize how individual genes program and rewire cellular crosstalk across FGCs and somas, how gene regulatory networks mediate signaling pathways that functionally link these two cell types, and how different FGCs diversify and evolve through cooperation and competition during embryo development. We analyze single-cell RNA-seq data of human female embryos using the new model, identifying previously uncharacterized mechanisms behind follicle development. The majority of genes (70%) promote FGC–soma synergism, only with a small portion (4%) that incur antagonism; hub genes function reciprocally between the FGC network and soma network; and germ cells tend to cooperate between different stages of development but compete in the same stage within a developmental embryo. Our network model could serve as a powerful tool to unravel the genomic signatures that mediate folliculogenesis from single-cell omics data.

Optimal dynamic incentive scheduling for Hawk-Dove evolutionary games

The Hawk-Dove evolutionary game offers a paradigm of the trade-offs associated with aggressive and passive behaviors. When two (or more) populations of players compete, their success or failure is measured by their frequency in the population, and the system is governed by the replicator dynamics. We develop a time-dependent optimal-adaptive control theory for this dynamical system in which the entries of the payoff matrix are dynamically altered to produce control schedules that minimize and maximize the aggressive population through a finite-time cycle. These schedules provide upper and lower bounds on the outcomes for all possible strategies since they represent two extremizers of the cost function. We then adaptively extend the optimal control schedules over multiple cycles to produce absolute maximizers and minimizers for the system.

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.

Social distancing is a social dilemma game played by every individual against his/her population

Since the outbreak of the global COVID-19 pandemic, social distancing has been known to everyone and recommended almost everywhere everyday. Social distancing has been and will be one of the most effective measures and sometimes, the only available one for fighting epidemics and saving lives. However, it has not been so clear how social distancing should be practiced or managed, especially when it comes to regulating everyone's otherwise normal social activities. The debate on how to implement social distancing often leads to a heated political argument, while research on the subject is lacking. This paper is to provide a theoretical basis for the understanding of the scientific nature of social distancing by considering it as a social dilemma game played by every individual against his/her population. From this perspective, every individual needs to make a decision on how to engage in social distancing, or risk being trapped into a dilemma either exposing to deadly diseases or getting no access to necessary social activities. As the players of the game, the individual's decisions depend on the population's actions and vice versa, and an optimal strategy can be found when the game reaches an equilibrium. The paper shows how an optimal strategy can be determined for a population with either closely related or completely separated social activities and with either single or multiple social groups, and how the collective behaviors of social distancing can be simulated by following every individual's actions as the distancing game progresses. The simulation results for populations of varying sizes and complexities are presented, which not only justify the choices of the strategies based on the theoretical analysis, but also demonstrate the convergence of the individual actions to an optimal distancing strategy in silico and possibly in natura as well, if every individual makes rational distancing decisions.

Stability analysis of the coexistence equilibrium of a balanced metapopulation model

We analyze the stability of a unique coexistence equilibrium point of a system of ordinary differential equations (ODE system) modelling the dynamics of a metapopulation, more specifically, a set of local populations inhabiting discrete habitat patches that are connected to one another through dispersal or migration. We assume that the inter-patch migrations are detailed balanced and that the patches are identical with intra-patch dynamics governed by a mean-field ODE system with a coexistence equilibrium. By making use of an appropriate Lyapunov function coupled with LaSalle's invariance principle, we are able to show that the coexistence equilibrium point within each patch is locally asymptotically stable if the inter-patch dispersal network is heterogeneous, whereas it is neutrally stable in the case of a homogeneous network. These results provide a mathematical proof confirming the existing numerical simulations and broaden the range of networks for which they are valid.

The ratio of single to co-colonization is key to complexity in interacting systems with multiple strains

The high number and diversity of microbial strains circulating in host populations have motivated extensive research on the mechanisms that maintain biodiversity. However, much of this work focuses on strain-specific and cross-immunity interactions. Another less explored mode of pairwise interaction is via altered susceptibilities to co-colonization in hosts already colonized by one strain. Diversity in such interaction coefficients enables strains to create dynamically their niches for growth and persistence, and "engineer" their common environment. How such a network of interactions with others mediates collective coexistence remains puzzling analytically and computationally difficult to simulate. Furthermore, the gradients modulating stability-complexity regimes in such multi-player endemic systems remain poorly understood. In a recent study (Madec & Gjini, Bulletin of Mathematical Biology, 82), we obtained an analytic representation for N-type coexistence in an SIS epidemiological model with co-colonization. We mapped multi-strain dynamics to a replicator equation using timescale separation. Here, we examine what drives coexistence regimes in such co-colonization system. We find the ratio of single to co-colonization, µ, critically determines the type of equilibrium and number of coexisting strains, and encodes a trade-off between overall transmission intensity R 0 and mean interaction coefficient in strain space, k. Preserving a given coexistence regime, under fixed trait variation, requires balancing between higher mean competition in favorable environments, and higher cooperation in harsher environments, and is consistent with the stress gradient hypothesis. Multi-strain coexistence tends to steady-state attractors for small µ, whereas as µ increases, dynamics tend to more complex attractors. Following strain frequencies, evolutionary dynamics in the system also display contrasting patterns with µ, interpolating between multi-stable and fluctuating selection for cooperation and mean invasion fitness, in the two extremes. This co-colonization framework could be applied more generally, to study invariant principles in collective coexistence, and to quantify how critical shifts in community dynamics get potentiated by mean-field and environmental gradients.

Properties of Equilibria and Glassy Phases of the Random Lotka-Volterra Model with Demographic Noise

We study a reference model in theoretical ecology, the disordered Lotka-Volterra model for ecological communities, in the presence of finite demographic noise. Our theoretical analysis, valid for symmetric interactions, shows that for sufficiently heterogeneous interactions and low demographic noise the system displays a multiple equilibria phase, which we fully characterize. In particular, we show that in this phase the number of locally stable equilibria is exponential in the number of species. Upon further decreasing the demographic noise, we unveil the presence of a second transition like the so-called "Gardner" transition to a marginally stable phase similar to that observed in the jamming of amorphous materials. We confirm and complement our analytical results by numerical simulations. Furthermore, we extend their relevance by showing that they hold for other interacting random dynamical systems such as the random replicant model. Finally, we discuss their extension to the case of asymmetric couplings.

A paradox of traffic and extra cars in a city as a collective behaviour

Promoting walking or cycling and reducing cars' use is one of the city planners' main targets, contributing to a sustainable transport method. Yet, the number of vehicles worldwide is increasing as fast as the population, and motorized mobility has become the primary transport method in most cities. Here, we consider modal share as an emergent behaviour of personal decisions. All individuals minimize their commuting time and reach an equilibrium under which no person is willing to change their transportation mode. In terms of the minimum travel time, the best-case scenario is used to determine the extra commuting time and the excess cars, computed as a social inefficiency. Results show that commuting times could increase up to 25% with many more vehicles than optimum. Paradoxically, all individuals trying to minimize their time could collectively reach the maximum commuting times in the extreme case, with all individuals driving during rush hour.

Evolutionary Dynamics of Gig Economy Labor Strategies under Technology, Policy and Market Influence

The emergence of the modern gig economy introduces a new set of employment considerations for firms and laborers that include various trade-offs. With a game-theoretical approach, we examine the influences of technology, policy and markets on firm and worker preferences for gig labor. Theoretically, we present new conceptual extensions to the replicator equation and model oscillating dynamics in two-player asymmetric bi-matrix games with time-evolving environments, introducing concepts of the attractor arc, trapping zone and escape. While canonical applications of evolutionary game theory focus on the evolutionary stable strategy, our model assumes that the system exhibits oscillatory dynamics and can persist for long temporal intervals in a pseudo-stable state. We demonstrate how changing market conditions result in distinct evolutionary patterns across labor economies. Informing tensions regarding the future of this new employment category, we present a novel payoff framework to analyze the role of technology on the growth of the gig economy. Regarding governance, we explore regulatory implications within the gig economy, demonstrating how intervals of lenient and strict policy alter firm and worker sensitivities between gig and employee labor strategies. Finally, we establish an aggregate economic framework to explain how technology, policy and market environments engage in an interlocking dance, a balancing act, to sustain the observable co-existence of gig and employee labor strategies.

Are Adaptive Chemotherapy Schedules Robust? A Three-Strategy Stochastic Evolutionary Game Theory Model

We investigate the robustness of adaptive chemotherapy schedules over repeated cycles and a wide range of tumor sizes. Using a non-stationary stochastic three-component fitness-dependent Moran process model (to track frequencies), we quantify the variance of the response to treatment associated with multidrug adaptive schedules that are designed to mitigate chemotherapeutic resistance in an idealized (well-mixed) setting. The finite cell (N tumor cells) stochastic process consists of populations of chemosensitive cells, chemoresistant cells to drug 1, and chemoresistant cells to drug 2, and the drug interactions can be synergistic, additive, or antagonistic. Tumor growth rates in this model are proportional to the average fitness of the tumor as measured by the three populations of cancer cells compared to a background microenvironment average value. An adaptive chemoschedule is determined by using the N→∞ limit of the finite-cell process (i.e., the adjusted replicator equations) which is constructed by finding closed treatment response loops (which we call evolutionary cycles) in the three component phase-space. The schedules that give rise to these cycles are designed to manage chemoresistance by avoiding competitive release of the resistant cell populations. To address the question of how these cycles perform in practice over large patient populations with tumors across a range of sizes, we consider the variances associated with the approximate stochastic cycles for finite N, repeating the idealized adaptive schedule over multiple periods. For finite cell populations, the distributions remain approximately multi-Gaussian in the principal component coordinates through the first three cycles, with variances increasing exponentially with each cycle. As the number of cycles increases, the multi-Gaussian nature of the distribution breaks down due to the fact that one of the three sub-populations typically saturates the tumor (competitive release) resulting in treatment failure. This suggests that to design an effective and repeatable adaptive chemoschedule in practice will require a highly accurate tumor model and accurate measurements of the sub-population frequencies or the errors will quickly (exponentially) degrade its effectiveness, particularly when the drug interactions are synergistic. Possible ways to extend the efficacy of the stochastic cycles in light of the computational simulations are discussed.

Incentive-driven transition to high ride-sharing adoption

Ride-sharing-the combination of multiple trips into one-may substantially contribute towards sustainable urban mobility. It is most efficient at high demand locations with many similar trip requests. However, here we reveal that people's willingness to share rides does not follow this trend. Modeling the fundamental incentives underlying individual ride-sharing decisions, we find two opposing adoption regimes, one with constant and another one with decreasing adoption as demand increases. In the high demand limit, the transition between these regimes becomes discontinuous, switching abruptly from low to high ride-sharing adoption. Analyzing over 360 million ride requests in New York City and Chicago illustrates that both regimes coexist across the cities, consistent with our model predictions. These results suggest that even a moderate increase in the financial incentives may have a disproportionately large effect on the ride-sharing adoption of individual user groups.

Natural selection between two games with environmental feedback

Evolutionary game theory has extensively investigated situations in which several games are competing against each other at the same time, but the model only assumes symmetric interactions in homogeneous environments. Now, the population is considered in heterogeneous environments, individuals in the population occupy a different quality of patches, and individual fitness depends not only on the interaction between individuals, but also on the quality of the environment. Here, by establishing a mathematical framework, we analyze the natural selection between two strategies and two games in heterogeneous environments. Furthermore, we analyze the natural selection of Prisoner’s Dilemma and Hawk–Dove games theoretically to demonstrate the dynamics of cooperators and defectors in their choice of environment and their respective games. As expected, the distribution of games and strategies changes with time. Based on different initial population compositions, we also discuss the invasion problem of games from different perspectives. To one’s surprise, we can find that good quality patches attract all individuals; the long-term dynamics in invariant rich environments is the same as the dynamics of symmetric interactions in homogeneous environments.

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.

A comparison of the Moran Process and replicator equations for evolving social dilemma game strategies

Social dilemma games are studied to gain insight into why humans cooperate with other unrelated people. The canonical game has cooperation and defection as the two strategies. Cooperation benefits the group, but a self-interested player can always do better by defecting. But if everybody defects, then the entire group loses. This tradeoff between cooperation and defection gives rise to the social dilemma. Social dilemma games need some method to evolve strategy changes between rounds. The two most widely accepted methods are a Moran process or replicator equations. Although both methods can predict how strategies evolve in a player population, no comparison of their performance has yet been made. In this paper we compare them in a public goods game which is an N-player version of prisoner's dilemma (N>2). Our results indicate only one of these methods should be used in future research efforts.

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.

Oscillatory dynamics in the dilemma of social distancing

Social distancing as one of the main non-pharmaceutical interventions can help slow down the spread of diseases, like in the COVID-19 pandemic. Effective social distancing, unless enforced as drastic lockdowns and mandatory cordon sanitaire, requires consistent strict collective adherence. However, it remains unknown what the determinants for the resultant compliance of social distancing and their impact on disease mitigation are. Here, we incorporate into the epidemiological process with an evolutionary game theory model that governs the evolution of social distancing behaviour. In our model, we assume an individual acts in their best interest and their decisions are driven by adaptive social learning of the real-time risk of infection in comparison with the cost of social distancing. We find interesting oscillatory dynamics of social distancing accompanied with waves of infection. Moreover, the oscillatory dynamics are dampened with a non-trivial dependence on model parameters governing decision-makings and gradually cease when the cumulative infections exceed the herd immunity. Compared to the scenario without social distancing, we quantify the degree to which social distancing mitigates the epidemic and its dependence on individuals’ responsiveness and rationality in their behaviour changes. Our work offers new insights into leveraging human behaviour in support of pandemic response.

Predicting N-Strain Coexistence from Co-colonization Interactions: Epidemiology Meets Ecology and the Replicator Equation

Multi-type infection processes are ubiquitous in ecology, epidemiology and social systems, but remain hard to analyze and to understand on a fundamental level. Here, we study a multi-strain susceptible-infected-susceptible model with coinfection. A host already colonized by one strain can become more or less vulnerable to co-colonization by a second strain, as a result of facilitating or competitive interactions between the two. Fitness differences between N strains are mediated through \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^2$$\end{document}N2 altered susceptibilities to secondary infection that depend on colonizer-cocolonizer identities (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{ij}$$\end{document}Kij). By assuming strain similarity in such pairwise traits, we derive a model reduction for the endemic system using separation of timescales. This ‘quasi-neutrality’ in trait space sets a fast timescale where all strains interact neutrally, and a slow timescale where selective dynamics unfold. We find that these slow dynamics are governed by the replicator equation for N strains. Our framework allows to build the community dynamics bottom-up from only pairwise invasion fitnesses between members. We highlight that mean fitness of the multi-strain network, changes with their individual dynamics, acts equally upon each type, and is a key indicator of system resistance to invasion. By uncovering the link between N-strain epidemiological coexistence and the replicator equation, we show that the ecology of co-colonization relates to Fisher’s fundamental theorem and to Lotka-Volterra systems. Besides efficient computation and complexity reduction for any system size, these results open new perspectives into high-dimensional community ecology, detection of species interactions, and evolution of biodiversity.

Controlling evolutionary dynamics to optimize microbial bioremediation

Some microbes have a fascinating ability to degrade compounds that are toxic for humans in a process called bioremediation. Although these traits help microbes survive the toxins, carrying them can be costly if the benefit of detoxification is shared by all surrounding microbes, whether they detoxify or not. Detoxification can thereby be seen as a public goods game, where nondegrading mutants can sweep through the population and collapse bioremediation. Here, we constructed an evolutionary game theoretical model to optimize bioremediation in a chemostat initially containing "cooperating" (detoxifying) microbes. We consider two types of mutants: "cheaters" that do not detoxify, and mutants that become resistant to the toxin through private mechanisms that do not benefit others. By manipulating the concentration and flow rate of a toxin into the chemostat, we identified conditions where cooperators can exclude cheaters that differ in their private resistance. However, eventually, cheaters are bound to invade. To overcome this inevitable outcome and maximize detoxification efficiency, cooperators can be periodically reinoculated into the population. Our study investigates the outcome of an evolutionary game combining both public and private goods and demonstrates how environmental parameters can be used to control evolutionary dynamics in practical applications.

Analysis of Multilevel Replicator Dynamics for General Two-Strategy Social Dilemma

Here, we consider a game-theoretic model of multilevel selection in which individuals compete based on their payoff and groups also compete based on the average payoff of group members. Our focus is on multilevel social dilemmas: games in which individuals are best off cheating, while groups of individuals do best when composed of many cooperators. We analyze the dynamics of the two-level replicator dynamics, a nonlocal hyperbolic PDE describing deterministic birth-death dynamics for both individuals and groups. While past work on such multilevel dynamics has restricted attention to scenarios with exactly solvable within-group dynamics, we use comparison principles and an invariant property of the tail of the population distribution to extend our analysis to all possible two-player, two-strategy social dilemmas. In the Stag-Hunt and similar games with coordination thresholds, we show that any amount of between-group competition allows for fixation of cooperation in the population. For the prisoners' dilemma and Hawk-Dove game, we characterize the threshold level of between-group selection dividing a regime in which the population converges to a delta function at the equilibrium of the within-group dynamics from a regime in which between-group competition facilitates the existence of steady-state densities supporting greater levels of cooperation. In particular, we see that the threshold selection strength and average payoff at steady state depend on a tug-of-war between the individual-level incentive to be a defector in a many-cooperator group and the group-level incentive to have many cooperators over many defectors. We also find that lower-level selection casts a long shadow: If groups are best off with a mix of cooperators and defectors, then there will always be fewer cooperators than optimal at steady state, even in the limit of infinitely strong competition between groups.

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.

Dynamic Evolution of Knowledge Sharing Behavior among Enterprises in the Cluster Innovation Network Based on Evolutionary Game Theory

Knowledge sharing behavior based on the cluster innovation network has become the primary measure for enterprises to realize sustainable innovation. In order to promote the proactive knowledge sharing behavior among enterprises in the long term, the dynamic evolutionary process and law of knowledge sharing in the network need to be further studied. As different from the hypothesis of the rational man in the classical game theory, this paper establishes an evolutionary game model of knowledge sharing behavior in the cluster innovation network based on the evolutionary game theory, and discusses how the bounded rational enterprises can achieve the evolutionary equilibrium through continuously adaptive learning and strategy optimization, further explores the influence factors on the evolutionary trajectory. Combined with mathematical derivation and simulation analysis, the following results are obtained: over time, the dynamic evolution of knowledge sharing behavior in the cluster innovation network is influenced by initial states of the system, but can always reach the evolutionary stable equilibrium; factors such as synergy revenue have a positive impact on the evolutionary results, while factors such as opportunity interest have a negative impact on the evolutionary results; the factor of revenue distribution has a U-shape relationship with the evolutionary results, and the factor of direct revenue has no effect on the results. The results are expected to have an implication for improving the sustainable innovation development of enterprises in the cluster innovation network.