Cooperators benefit others with paying costs. Evolution of cooperation crucially depends on the cost-benefit ratio of cooperation, denoted as c. In this work, we investigate the infinitely repeated prisoner's dilemma for various values of c with four of the representative memory-one strategies, i.e., unconditional cooperation, unconditional defection, tit-for-tat, and win-stay-lose-shift. We consider replicator dynamics which deterministically describes how the fraction of each strategy evolves over time in an infinite-sized well-mixed population in the presence of implementation error and mutation among the four strategies. Our finding is that this three-dimensional continuous-time dynamics exhibits chaos through a bifurcation sequence similar to that of a logistic map as c varies. If mutation occurs with rate μ≪1, the position of the bifurcation sequence on the c axis is numerically found to scale as μ^{0.1}, and such sensitivity to μ suggests that mutation may have nonperturbative effects on evolutionary paths. It demonstrates how the microscopic randomness of the mutation process can be amplified to macroscopic unpredictability by evolutionary dynamics.
The recent pandemic stimulated scientists to publish a significant amount of research that created a surge of citations of COVID-19-related papers in a short time, leading to an abrupt inflation of the journal impact factor (IF). By auditing the complete set of COVID-19-related publications in the Web of Science, we reveal here that COVID-19-related research worsened the polarization of academic journals: the IF before the pandemic was proportional to the increment of IF, which had the effect of increasing inequality while retaining the journal rankings. We also found that the most highly cited studies related to COVID-19 were published in prestigious journals at the onset of the epidemic, independent of their innate importance or quality. Through the present quantitative investigation, our findings caution against the belief that quantitative metrics, particularly IF, can indicate the significance of individual papers. Rather, such metrics reflect the social attention given to a particular study.
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