The replicator dynamics for multilevel selection in evolutionary games

Evolutionary branching in multi-level selection models

We study a model of group-structured populations featuring individual-level birth and death events, and group-level fission and extinction events. Individuals play games within their groups, while groups play games against other groups. Payoffs from individual-level games affect birth rates of individuals, and payoffs from group-level games affect group extinction rates. We focus on the evolutionary dynamics of continuous traits with particular emphasis on the phenomenon of evolutionary diversification. Specifically, we consider two-level processes in which individuals and groups play continuous snowdrift or prisoner's dilemma games. Individual game strategies evolve due to selection pressure from both the individual and group level interactions. The resulting evolutionary dynamics turns out to be very complex, including branching and type-diversification at one level or the other. We observe that a weaker selection pressure at the individual level results in more adaptable groups and sometimes group-level branching. Stronger individual-level selection leads to more effective adaptation within each group while preventing the groups from adapting according to the group-level games.

Fission as a source of variation for group selection

Without heritable variation natural selection cannot effect evolutionary change. In the case of group selection, there must be variation in the population of groups. Where does this variation come from? One source of variation is from the stochastic birth-death processes that occur within groups. This is where variation between groups comes from in most mathematical models of group selection. Here, we argue that another important source of variation between groups is fission, the (generally random) group-level reproduction where parent groups split into two or more offspring groups. We construct a simple model of the fissioning process with a parameter that controls how much variation is produced among the offspring groups. We then illustrate the effect of that parameter with some examples. In most models of group selection in the literature, no variation is produced during group reproduction events; that is, groups "clone" themselves when they reproduce. Fission is often a more biologically realistic method of group reproduction, and it can significantly increase the efficacy of group selection.

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.

Evolutionarily stable strategy analysis and its links to demography and genetics through invasion fitness

Evolutionarily stable strategy (ESS) analysis pioneered by Maynard Smith and Price took off in part because it often does not require explicit assumptions about the genetics and demography of a population in contrast to population genetic models. Though this simplicity is useful, it obscures the degree to which ESS analysis applies to populations with more realistic genetics and demography: for example, how does ESS analysis handle complexities such as kin selection, group selection and variable environments when phenotypes are affected by multiple genes? In this paper, I review the history of the ESS concept and show how early uncertainty about the method lead to important mathematical theory linking ESS analysis to general population genetic models. I use this theory to emphasize the link between ESS analysis and the concept of invasion fitness. I give examples of how invasion fitness can measure kin selection, group selection and the evolution of linked modifier genes in response to variable environments. The ESSs in these examples depend crucially on demographic and genetic parameters, which highlights how ESS analysis will continue to be an important tool in understanding evolutionary patterns as new models address the increasing abundance of genetic and long-term demographic data in natural populations. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

Pharmaceutical enterprises drug quality strategy Moran analysis considering government supervision and new media participation

The improvement of drug quality requires not only the supervision of government, but also the participation of new media. Therefore, this paper considers the impact of government regulation and new media reports on pharmaceutical enterprises, constructs a Moran Process evolutionary game model, and analyzes the evolution trajectory of pharmaceutical enterprises' choice of drug quality improvement strategy and drug cost reduction strategy. We obtain the conditions for the two strategies to achieve evolutionary stability under the dominance of external factors and the dominance of expected returns. To verify the theoretical results, we conduct a numerical simulation by the software MATLAB 2021b. The results show that, first of all, when the government penalty is high, the drug quality improvement strategy tends to become an evolutionary stable solution, increasing the penalty amount will help promote the improvement of drug quality. What's more, when the government penalty is low and the new media influence is low, the drug cost reduction strategy is easier to dominate. The higher the new media influence, the higher the probability that pharmaceutical enterprises choose the drug quality improvement strategy. Thirdly, when the number of pharmaceutical enterprises is lower than a threshold, the drug quality improvement strategy is easier to dominate. Finally, the drug quality improvement strategy is dominant when the quality cost factor is low and the government penalty is high, the drug cost reduction strategy is dominant when the quality cost factor is high and the government penalty is low. Above all, this paper provides countermeasures and suggestions for the drug quality improvement of pharmaceutical enterprises in practice.

Pharmaceutical Enterprises' R&D Innovation Cooperation Moran Strategy When Considering Tax Incentives

Drug R&D innovation contributes to the high-quality development of the pharmaceutical industry, which is related to people's life and health, economic development, and social stability. Tax incentives and industry cooperation are conducive to promoting pharmaceutical enterprises' innovation. Therefore, this paper constructs a Moran process evolutionary game model and analyzes the evolutionary trajectory of N pharmaceutical enterprises' drug R&D innovation strategic choice and considers the choice of R&D innovation strategy and non-R&D innovation strategy. We obtain the conditions for the two strategies to achieve evolutionary stability under the dominance of external factors, the dominance of expected revenue, and the dominance of super expected revenue. The evolutionary process is simulated by MATLAB 2021b. The results show that, firstly, when the number of pharmaceutical enterprises is higher than a threshold, the market is conducive to pharmaceutical enterprises choosing an R&D innovation strategy. Secondly, the higher the tax incentives, the higher the probability of pharmaceutical enterprises choosing an R&D innovation strategy. Thirdly, when the R&D success rate increases, pharmaceutical enterprises gradually change from choosing a non-R&D innovation strategy to choosing an R&D innovation strategy. Fourthly, the threshold of strategy change of pharmaceutical enterprises is the same under the dominance of expected revenue and super expected revenue. This paper puts forward some countermeasures and suggestions for promoting the R&D innovation of pharmaceutical enterprises in practice.

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.

Eco-evolutionary dynamics of clonal multicellular life cycles

The evolution of multicellular life cycles is a central process in the course of the emergence of multicellularity. The simplest multicellular life cycle is comprised of the growth of the propagule into a colony and its fragmentation to give rise to new propagules. The majority of theoretical models assume selection among life cycles to be driven by internal properties of multicellular groups, resulting in growth competition. At the same time, the influence of interactions between groups on the evolution of life cycles is rarely even considered. Here, we present a model of colonial life cycle evolution taking into account group interactions. Our work shows that the outcome of evolution could be coexistence between multiple life cycles or that the outcome may depend on the initial state of the population - scenarios impossible without group interactions. At the same time, we found that some results of these simpler models remain relevant: evolutionary stable strategies in our model are restricted to binary fragmentation - the same class of life cycles that contains all evolutionarily optimal life cycles in the model without interactions. Our results demonstrate that while models neglecting interactions can capture short-term dynamics, they fall short in predicting the population-scale picture of evolution.

A PDE Model for Protocell Evolution and the Origin of Chromosomes via Multilevel Selection

The evolution of complex cellular life involved two major transitions: the encapsulation of self-replicating genetic entities into cellular units and the aggregation of individual genes into a collectively replicating genome. In this paper, we formulate a minimal model of the evolution of proto-chromosomes within protocells. We model a simple protocell composed of two types of genes: a "fast gene" with an advantage for gene-level self-replication and a "slow gene" that replicates more slowly at the gene level, but which confers an advantage for protocell-level reproduction. Protocell-level replication capacity depends on cellular composition of fast and slow genes. We use a partial differential equation to describe how the composition of genes within protocells evolves over time under within-cell and between-cell competition, considering an infinite population of protocells that each contain infinitely many genes. We find that the gene-level advantage of fast replicators casts a long shadow on the multilevel dynamics of protocell evolution: no level of between-protocell competition can produce coexistence of the fast and slow replicators when the two genes are equally needed for protocell-level reproduction. By introducing a "dimer replicator" consisting of a linked pair of the slow and fast genes, we show analytically that coexistence between the two genes can be promoted in pairwise multilevel competition between fast and dimer replicators, and provide numerical evidence for coexistence in trimorphic competition between fast, slow, and dimer replicators. Our results suggest that dimerization, or the formation of a simple chromosome-like dimer replicator, can help to overcome the shadow of lower-level selection and work in concert with deterministic multilevel selection in protocells featuring high gene copy number to allow for the coexistence of two genes that are complementary at the protocell level but compete at the level of individual gene-level replication. These results for the PDE model complement existing results on the benefits of dimerization in the case of low genetic copy number, for which it has been shown that genetic linkage can help to overcome the stochastic loss of necessary genetic templates.

Long-time behavior of a PDE replicator equation for multilevel selection in group-structured populations

In many biological systems, natural selection acts simultaneously on multiple levels of organization. This scenario typically presents an evolutionary conflict between the incentive of individuals to cheat and the collective incentive to establish cooperation within a group. Generalizing previous work on multilevel selection in evolutionary game theory, we consider a hyperbolic PDE model of a group-structured population, in which members within a single group compete with each other for individual-level replication; while the group also competes against other groups for group-level replication. We derive a threshold level of the relative strength of between-group competition such that defectors take over the population below the threshold while cooperation persists in the long-time population above the threshold. Under stronger assumptions on the initial distribution of group compositions, we further prove that the population converges to a steady state density supporting cooperation for between-group selection strength above the threshold. We further establish long-time bounds on the time-average of the collective payoff of the population, showing that the long-run population cannot outperform the payoff of a full-cooperator group even in the limit of infinitely-strong between-group competition. When the group replication rate is maximized by an intermediate level of within-group cooperation, individual-level selection casts a long shadow on the dynamics of multilevel selection: no level of between-group competition can erase the effects of the individual incentive to defect. We further extend our model to study the case of multiple types of groups, showing how the games that groups play can coevolve with the level of cooperation.

Bacterial cooperation through horizontal gene transfer

Cooperation exists across all scales of biological organization, from genetic elements to complex human societies. Bacteria cooperate by secreting molecules that benefit all individuals in the population (i.e., public goods). Genes associated with cooperation can spread among strains through horizontal gene transfer (HGT). We discuss recent findings on how HGT mediated by mobile genetic elements promotes bacterial cooperation, how cooperation in turn can facilitate more frequent HGT, and how the act of HGT itself may be considered as a form of cooperation. We propose that HGT is an important enforcement mechanism in bacterial populations, thus creating a positive feedback loop that further maintains cooperation. To enforce cooperation, HGT serves as a homogenizing force by transferring the cooperative trait, effectively eliminating cheaters.

A scaling law of multilevel evolution: how the balance between within- and among-collective evolution is determined

Numerous living systems are hierarchically organised, whereby replicating components are grouped into reproducing collectives-e.g., organelles are grouped into cells, and cells are grouped into multicellular organisms. In such systems, evolution can operate at two levels: evolution among collectives, which tends to promote selfless cooperation among components within collectives (called altruism), and evolution within collectives, which tends to promote cheating among components within collectives. The balance between within- and among-collective evolution thus exerts profound impacts on the fitness of these systems. Here, we investigate how this balance depends on the size of a collective (denoted by N) and the mutation rate of components (m) through mathematical analyses and computer simulations of multiple population genetics models. We first confirm a previous result that increasing N or m accelerates within-collective evolution relative to among-collective evolution, thus promoting the evolution of cheating. Moreover, we show that when within- and among-collective evolution exactly balance each other out, the following scaling relation generally holds: Nmα is a constant, where scaling exponent α depends on multiple parameters, such as the strength of selection and whether altruism is a binary or quantitative trait. This relation indicates that although N and m have quantitatively distinct impacts on the balance between within- and among-collective evolution, their impacts become identical if m is scaled with a proper exponent. Our results thus provide a novel insight into conditions under which cheating or altruism evolves in hierarchically-organised replicating systems.

Fixation probabilities in network structured meta-populations

The effect of population structure on evolutionary dynamics is a long-lasting research topic in evolutionary ecology and population genetics. Evolutionary graph theory is a popular approach to this problem, where individuals are located on the nodes of a network and can replace each other via the links. We study the effect of complex network structure on the fixation probability, but instead of networks of individuals, we model a network of sub-populations with a probability of migration between them. We ask how the structure of such a meta-population and the rate of migration affect the fixation probability. Many of the known results for networks of individuals carry over to meta-populations, in particular for regular networks or low symmetric migration probabilities. However, when patch sizes differ we find interesting deviations between structured meta-populations and networks of individuals. For example, a two patch structure with unequal population size suppresses selection for low migration probabilities.

Biased perceptions explain collective action deadlocks and suggest new mechanisms to prompt cooperation

When individuals face collective action problems, their expectations about others' willingness to contribute affect their motivation to cooperate. Individuals, however, often misperceive the cooperation levels in a population. In the context of climate action, people underestimate the pro-climate positions of others. Designing incentives to enable cooperation and a sustainable future must thereby consider how social perception biases affect collective action. We propose a theoretical model and investigate the effect of social perception bias in non-linear public goods games. We show that different types of bias play a distinct role in cooperation dynamics. False uniqueness (underestimating own views) and false consensus (overestimating own views) both explain why communities get locked in suboptimal states. Such dynamics also impact the effectiveness of typical monetary incentives, such as fees. Our work contributes to understanding how targeting biases, e.g., by changing the information available to individuals, can comprise a fundamental mechanism to prompt collective action.

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Individuals often misperceive the real cooperation levels in a population

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We model the impact of such biases in non-linear public goods games dynamics

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False uniqueness and false consensus can lock groups in suboptimal states

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Addressing perception biases can be more effective than typical monetary incentives

Collective predator evasion: Putting the criticality hypothesis to the test

According to the criticality hypothesis, collective biological systems should operate in a special parameter region, close to so-called critical points, where the collective behavior undergoes a qualitative change between different dynamical regimes. Critical systems exhibit unique properties, which may benefit collective information processing such as maximal responsiveness to external stimuli. Besides neuronal and gene-regulatory networks, recent empirical data suggests that also animal collectives may be examples of self-organized critical systems. However, open questions about self-organization mechanisms in animal groups remain: Evolutionary adaptation towards a group-level optimum (group-level selection), implicitly assumed in the "criticality hypothesis", appears in general not reasonable for fission-fusion groups composed of non-related individuals. Furthermore, previous theoretical work relies on non-spatial models, which ignore potentially important self-organization and spatial sorting effects. Using a generic, spatially-explicit model of schooling prey being attacked by a predator, we show first that schools operating at criticality perform best. However, this is not due to optimal response of the prey to the predator, as suggested by the "criticality hypothesis", but rather due to the spatial structure of the prey school at criticality. Secondly, by investigating individual-level evolution, we show that strong spatial self-sorting effects at the critical point lead to strong selection gradients, and make it an evolutionary unstable state. Our results demonstrate the decisive role of spatio-temporal phenomena in collective behavior, and that individual-level selection is in general not a viable mechanism for self-tuning of unrelated animal groups towards criticality.

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.

Turbulent coherent structures and early life below the Kolmogorov scale

Major evolutionary transitions, including the emergence of life, likely occurred in aqueous environments. While the role of water’s chemistry in early life is well studied, the effects of water’s ability to manipulate population structure are less clear. Population structure is known to be critical, as effective replicators must be insulated from parasites. Here, we propose that turbulent coherent structures, long-lasting flow patterns which trap particles, may serve many of the properties associated with compartments — collocalization, division, and merging — which are commonly thought to play a key role in the origins of life and other evolutionary transitions. We substantiate this idea by simulating multiple proposed metabolisms for early life in a simple model of a turbulent flow, and find that balancing the turnover times of biological particles and coherent structures can indeed enhance the likelihood of these metabolisms overcoming extinction either via parasitism or via a lack of metabolic support. Our results suggest that group selection models may be applicable with fewer physical and chemical constraints than previously thought, and apply much more widely in aqueous environments.

Models of the origin of life generally require a mechanism to structure emerging populations. Here, Krieger et al. develop spatial models showing that coherent structures arising in turbulent flows in aquatic environments could have provided compartments that facilitated the origin of life.

The Behavior Mechanism of the Urban Joint Distribution Alliance under Government Supervision from the Perspective of Sustainable Development

Urban joint distribution is closely related to the national economy and people’s livelihood, and governments and enterprises play an active role in the process of urban joint distribution. From the perspective of government regulations, this paper explores the mechanism and evolution law of the behavior of an urban joint distribution alliance. Based on the evolutionary game theory, a model of homogeneous enterprises participating in urban joint distribution operations under the guidance of government regulations is constructed. The mechanism and follow-up of alliance behavior are analyzed through the simulation of the relationship between parameters. It is found that, firstly, from the perspective of government regulations, in the early stage of the implementation of urban joint distribution projects, when the benefits of synergetic cooperation of enterprise alliances are relatively low and the costs are relatively high, it is necessary for the government to formulate incentive policies to improve government subsidies or to increase the penalties for non-cooperation of enterprises; Once a benign logistics environment and market mechanism are formed, the cooperation benefits increase, and the costs decrease, the government can then withdraw its supervision. Secondly, in the process of establishing urban joint distribution alliance under government supervision, it is better for the enterprises to actively achieve alliance cooperation and obtain government subsidies instead of passively accepting government supervision and paying penalties, in order to promote the formation of logistics ecological environment and market mechanism.

Computation and Simulation of Evolutionary Game Dynamics in Finite Populations

The study of evolutionary dynamics increasingly relies on computational methods, as more and more cases outside the range of analytical tractability are explored. The computational methods for simulation and numerical approximation of the relevant quantities are diverging without being compared for accuracy and performance. We thoroughly investigate these algorithms in order to propose a reliable standard. For expositional clarity we focus on symmetric 2 × 2 games leading to one-dimensional processes, noting that extensions can be straightforward and lessons will often carry over to more complex cases. We provide time-complexity analysis and systematically compare three families of methods to compute fixation probabilities, fixation times and long-term stationary distributions for the popular Moran process. We provide efficient implementations that substantially improve wall times over naive or immediate implementations. Implications are also discussed for the Wright-Fisher process, as well as structured populations and multiple types.