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The replicator equation is the first and most important game dynamics studied in connection with evolutionary game theory. It was originally developed for symmetric games with finitely many strategies. Properties of these dynamics are briefly summarized for this case, including the convergence to and stability of the Nash equilibria and evolutionarily stable strategies. The theory is then extended to other game dynamics for symmetric games (e.g., the best response dynamics and adaptive dynamics) and illustrated by examples taken from the literature. It is also extended to multiplayer, population, and asymmetric games.

After primary infection with varicella, varicella-zoster virus (VZV) establishes a latent state that precedes the clinical manifestation of herpes zoster. Growing evidence, however, indicates that latency is not merely quiescent but represents an active immunological adaptation. We propose the immunosensor hypothesis , in which VZV latency within sensory ganglia contributes to host immune surveillance while simultaneously ensuring viral persistence. Using a game-theoretical framework, we conceptualize this interaction as a temporally partitioned evolutionarily stable strategy (TP-ESS). In this model, VZV progresses through three sequential phases across the host lifespan: (i) aggressive replication and transmission during primary infection, (ii) immunomodulatory latency during immune competence, and (iii) reactivation during immune decline. Each phase represents a dynamic equilibrium shaped by host immunity and viral life-history trade-offs. The TP-ESS framework integrates viral ecology, innate immunity, and neurovirology into a unified model of latency and reactivation, providing a conceptual basis for epidemiological patterns of herpes zoster and generating testable predictions on immunity-dependent viral behavior and host–virus coadaptation.

Adaptive dynamics is a mathematical framework for studying evolution. It extends evolutionary game theory to account for more realistic ecological dynamics and it can incorporate both frequency- and density-dependent selection. This is a practical guide to adaptive dynamics that aims to illustrate how the methodology can be applied to the study of specific systems. The theory is presented in detail for a single, monomorphic, asexually reproducing population. We explain the necessary terminology to understand the basic arguments in models based on adaptive dynamics, including invasion fitness, the selection gradient, pairwise invasibility plots (PIP), evolutionarily singular strategies, and the canonical equation. The presentation is supported with a worked-out example of evolution of arrival times in migratory birds. We show how the adaptive dynamics methodology can be extended to study evolution in polymorphic populations using trait evolution plots (TEPs). We give an overview of literature that generalises adaptive dynamics techniques to other scenarios, such as sexual, diploid populations, and spatially-structured populations. We conclude by discussing how adaptive dynamics relates to evolutionary game theory and how adaptive-dynamics techniques can be used in speciation research.

Significance Cells have finite resources. Committing resources to one task therefore reduces the amount of resources available to others. These trade-offs are often overlooked but potentially modify all cellular processes. Building a mathematical cell model that respects such trade-offs and describes the mechanisms of protein synthesis and how cells extract resources from their environment, we quantitatively recover the typical behavior of an individual growing cell and of a population of cells. As trade-offs are experienced by all cells and because growth largely determines cellular fitness, a predictive understanding of how biochemical processes affect others and affect growth is important for diverse applications, such as the use of microbes for biotechnology, the inhibition of antibiotic resistance, and the growth of cancers. Intracellular processes rarely work in isolation but continually interact with the rest of the cell. In microbes, for example, we now know that gene expression across the whole genome typically changes with growth rate. The mechanisms driving such global regulation, however, are not well understood. Here we consider three trade-offs that, because of limitations in levels of cellular energy, free ribosomes, and proteins, are faced by all living cells and we construct a mechanistic model that comprises these trade-offs. Our model couples gene expression with growth rate and growth rate with a growing population of cells. We show that the model recovers Monod’s law for the growth of microbes and two other empirical relationships connecting growth rate to the mass fraction of ribosomes. Further, we can explain growth-related effects in dosage compensation by paralogs and predict host–circuit interactions in synthetic biology. Simulating competitions between strains, we find that the regulation of metabolic pathways may have evolved not to match expression of enzymes to levels of extracellular substrates in changing environments but rather to balance a trade-off between exploiting one type of nutrient over another. Although coarse-grained, the trade-offs that the model embodies are fundamental, and, as such, our modeling framework has potentially wide application, including in both biotechnology and medicine.

[This corrects the article DOI: 10.1371/journal.pone.0172009.].

Long-distance dispersal (LDD)-dispersal beyond the bounds of the local patch or cluster of conspecifics-will be most advantageous in landscapes in which large areas of suitable habitat are consistently available at long distances from established populations. We review conditions under which LDD will be selected and conclude that biotic interactions, and in particular specialized natural enemies, are likely to be one of the most important factors selecting for LDD in many species. We use simple spatially implicit and spatially explicit models to illustrate how such pests affect the evolutionarily stable strategy (ESS) for investment in LDD. Patches currently occupied by parents are more likely to be infected than distant, potentially unoccupied, patches, thus advantaging dispersal. Patchy infestations also result in higher variance in reproductive success among patches, which alone selects for increased among-patch dispersal. Both of these effects increase with the strength of the impact of infestation, and with the number of species competing for space in the community. We discuss the potential of different types of models and analytical tools to capture the impacts of pests on the evolution of LDD, and conclude that even simple models can illustrate the general relationship between pest pressure and LDD advantage, but only spatially explicit simulation models can fully elucidate the resulting ecological and evolutionary dynamics. In conclusion, we consider the potential role of selection for LDD in the spread of invasive species, and in long-term responses to habitat fragmentation and range shifts.

Abstract Cooperation and conflict in microorganisms is being recognized as an important factor in the organization and function of microbial communities. Many of the cooperative behaviors described in bacteria are governed through a cell–cell signaling process generally termed quorum sensing. Communication and cooperation in diverse microorganisms exhibit predictable trends that behave according to social evolutionary theory, notably that public goods dilemmas produce selective pressures for divergence in social phenotypes including cheating. In this review, we relate the general features of quorum sensing and social adaptation in microorganisms to established evolutionary theory. We then describe physiological and molecular mechanisms that have been shown to stabilize cooperation in microbes, thereby preventing a tragedy of the commons. Continued study of the role of communication and cooperation in microbial ecology and evolution is important to clinical treatment of pathogens, as well as to our fundamental understanding of cooperative selection at all levels of life. The authors review current research on the evolution and maintenance of microbial communication and cooperation from both a theoretical and experimental perspective.

Zero-determinant strategies are a new class of probabilistic and conditional strategies that are able to unilaterally set the expected payoff of an opponent in iterated plays of the Prisoner’s Dilemma irrespective of the opponent’s strategy (coercive strategies), or else to set the ratio between the player’s and their opponent’s expected payoff (extortionate strategies). Here we show that zero-determinant strategies are at most weakly dominant, are not evolutionarily stable, and will instead evolve into less coercive strategies. We show that zero-determinant strategies with an informational advantage over other players that allows them to recognize each other can be evolutionarily stable (and able to exploit other players). However, such an advantage is bound to be short-lived as opposing strategies evolve to counteract the recognition. In iterated Prisoner’s Dilemma games, zero-determinant strategies are able to define the opponent’s payoff regardless of the opponent’s strategy. Here the authors show that zero-determinant strategies are not evolutionary stable in adapting populations, and instead evolve into non-coercive strategies.

Strategies for sustaining cooperation and preventing exploitation by selfish agents in repeated games have mostly been restricted to Markovian strategies where the response of an agent depends on the actions in the previous round. Such strategies are characterized by lack of learning. However, learning from accumulated evidence over time and using the evidence to dynamically update our response is a key feature of living organisms. Bayesian inference provides a framework for such evidence-based learning mechanisms. It is therefore imperative to understand how strategies based on Bayesian learning fare in repeated games with Markovian strategies. Here, we consider a scenario where the Bayesian player uses the accumulated evidence of the opponent’s actions over several rounds to continuously update her belief about the reactive opponent’s strategy. The Bayesian player can then act on her inferred belief in different ways. By studying repeated Prisoner’s dilemma games with such Bayesian inferential strategies, both in infinite and finite populations, we identify the conditions under which such strategies can be evolutionarily stable. We find that a Bayesian strategy that is less altruistic than the inferred belief about the opponent’s strategy can outperform a larger set of reactive strategies, whereas one that is more generous than the inferred belief is more successful when the benefit-to-cost ratio of mutual cooperation is high. Our analysis reveals how learning the opponent’s strategy through Bayesian inference, as opposed to utility maximization, can be beneficial in the long run, in preventing exploitation and eventual invasion by reactive strategies.

An evolutionarily stable strategy (ESS) is an evolutionary strategy that, if adapted by a population, cannot be invaded by any deviating (mutant) strategy. The concept of ESS has been extensively studied and widely applied in ecology and evolutionary biology [M. Smith, On Evolution (1972)] but typically on the assumption that the system is ecologically stable. With reference to a Rosenzweig–MacArthur predator–prey model [M. Rosenzweig, R. MacArthur, Am. Nat. 97, 209–223 (1963)], we derive the mathematical conditions for the existence of an ESS when the ecological dynamics have asymptotically stable limit points as well as limit cycles. By extending the framework of Reed and Stenseth [J. Reed, N. C. Stenseth, J. Theoret. Biol. 108, 491–508 (1984)], we find that ESSs occur at values of the evolutionary strategies that are local optima of certain functions of the model parameters. These functions are identified and shown to have a similar form for both stable and fluctuating populations. We illustrate these results with a concrete example.