Dynamical Systems Under Constant Organization II: Homogeneous Growth Functions of Degree $p = 2$

Hofbauer, Josef; Schuster, Peter; Sigmund, Karl; Wolff, Robert Lee

doi:10.1137/0138025

Cited by 79 publications

(39 citation statements)

References 14 publications

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“…This model has gain plausibility when the ability of polynucleotides to help propagate each other was established experimentally through the study of the catalytic activity of the RNA (ribozymes) [8,9]. Interestingly, though the error threshold phenomenon has traditionally been considered the main motivation for the proposal of the hypercycle (see [10], for instance), most of the seminal works in this field have dealt with the coexistence issue only, as they assume perfect replication accuracy for the hypercycle elements [7,11]. In this case an arbitrary number of templates permanently coexist in a dynamical equilibrium state; if n > 4, however, the template concentrations vary with time [7], periodically decreasing to very small values.…”

Section: Introductionmentioning

confidence: 99%

Error propagation in the hypercycle

2000

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We study analytically the steady-state regime of a network of n error-prone self-replicating templates forming an asymmetric hypercycle and its error tail. We show that the existence of a master template with a higher non-catalyzed self-replicative productivity, a, than the error tail ensures the stability of chains in which m < n − 1 templates coexist with the master species. The stability of these chains against the error tail is guaranteed for catalytic coupling strengths (K) of order of a. We find that the hypercycle becomes more stable than the chains only for K of order of a 2 . Furthermore, we show that the minimal replication accuracy per template needed to maintain the hypercycle, the so-called error threshold, vanishes like n/K for large K and n ≤ 4.

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Section: Introductionmentioning

confidence: 99%

Error propagation in the hypercycle

2000

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“…Hofbauer et al (1980) investigate the phase portraits from three-strategy games under the replicator dynamics and conclude that only 'simple' behaviour -sinks, sources, centers, saddles -can occur. In general, evolutionary dynamics in a n strategy game de…ne a proper n 1 dynamical system on the n 1 simplex.…”

Section: Introductionmentioning

confidence: 99%

Multiple equilibria and limit cycles in evolutionary games with Logit Dynamics

Hommes

Ochea

2012

Games and Economic Behavior

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This note shows, by means of two simple, three-strategy games, the existence of stable periodic orbits and of multiple, interior steady states in a smooth version of the Best Response Dynamics, the Logit Dynamics. The main …nding is that, unlike Replicator Dynamics, generic Hopf bifurcation and thus, stable limit cycles, occur under the Logit Dynamics, even for three strategy games. We also show that the Logit Dynamics displays another bifurcation which cannot occur under the Replicator Dynamics: the fold bifurcation, with non-monotonic creation and disappearance of steady states. JEL classi…cation: C72, C73Keywords: Evolutionary games, Logit dynamics, Hopf bifurcation, fold bifurcation

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“…The hypercycle (a closed feedback loop in which each molecular species is catalyzed by its predecessor) has attracted particular attention (see Hofbauer et al, 1980). Both the cooperation of the components within a hypercycle and the strict competition between individual hypercycles suggest that such networks may have been involved in some phases of early prebiotic evolution.…”

Section: Prebiotic Evolution 91mentioning

confidence: 99%

“…A partial analysis of this class is given in Hofbauer et al (1980). It is shown that the center of S"" (Le .. the point m, where m1.…”

Section: Classificationmentioning

confidence: 99%

“…Such interactions (and the corresponding replication rates) are usually quite complicated, but nevertheless some rather general results have been obtained. In addition, certain special cases of linear catalytic (or inhibiting) interactions, yielding the first-order replicator equations: (4.10) have been studied as approximations of more realistic chemical kinetics.The hypercycle (a closed feedback loop in which each molecular species is catalyzed by its predecessor) has attracted particular attention (see Hofbauer et al, 1980). Both the cooperation of the components within a hypercycle and the strict competition between individual hypercycles suggest that such networks may have been involved in some phases of early prebiotic evolution.…”

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confidence: 99%

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Complexity, Language, and Life: Mathematical Approaches

Casti¹,

Karlqvist²

1986

Biomathematics

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a small group of investigators at the scientific research station at Abisko, Sweden, for the purpose of examining various conceptual and mathematical views of the evolution of complex systems. The stated theme of the meeting was deliberately kept vague, with only the purpose of discussing alternative mathematically based approaches to the modeling of evolving processes being given as a guideline to the participants. In order to limit the scope to some degree, it was decided to emphasize living rather than nonliving processes and to invite participants from a range of disciplinary specialities spanning the spectrum from pure and applied mathematics to geography and analytic philosophy.The results of the meeting were quite extraordinary; while there was no intent to focus the papers and discussion into predefined channels, an immediate self-organizing effect took place and the deliberations quickly oriented themselves into three main streams: conceptual and formal structures for characterizing system complexity; evolutionary processes in biology and ecology; the emergence of complexity through evolution in natural languages. The chapters presented in this volume are not the proceedings of the meeting. Following the meeting, the organizers felt that the ideas and spirit of the gathering should be preserved in some written form, so the participants were each requested to produce a chapter, explicating the views they presented at Abisko, written specifically for this volume. The results of this exercise form the volume you hold in your hand.Special thanks for their help in various phases of organizations of the meeting and arrangement of the publication of this volume are due to M. Olson, P. Sahlstrom, and R. Ouis. December 1985 John Casti, Vienna Anders Karlqvist, StockholmThe International Institute for Applied Analysis is a nongovernmental research institution, bringing together scientists from around the world to work on problems of common concern. Situated in Laxenburg, Austria, IIASA was founded in October 1972 by the academies of science and equivalent organizations of twelve countries. Its founders gave IIASA a unique position outside national, disciplinary, and institutional boundaries so that it might take the broadest possible view in pursuing its objectives:To promote international cooperation in solving problems arising from social, economic, technological, and environmental change To create a network of institutions in the national member organization countries and elsewhere for joint scientific research To develop and formalize systems analysis and the sciences contributing to it, and promote the use of analytical techniques needed to evaluate and address complex problems To inform policy advisors and decision makers about the potential application of the Institute's work to such problemsThe Institute now has national member organizations in the following countries: Complexity and the Evolution of Living SystemsOne of the most evident features distinguishing living from nonliving systems is the tende...

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Dynamical Systems Under Constant Organization II: Homogeneous Growth Functions of Degree $p = 2$

Cited by 79 publications

References 14 publications

Error propagation in the hypercycle

Error propagation in the hypercycle

Multiple equilibria and limit cycles in evolutionary games with Logit Dynamics

Complexity, Language, and Life: Mathematical Approaches

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