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Cited by 22Mathematical and Theoretical Epidemiology and Ecology ModelsAdvanced Differential Equations and Dynamical SystemsNonlinear Dynamics and Pattern Formation

A Difference Equation Model for the Hypercycle

Hofbauer, Josef·1984·SIAM Journal on Applied Mathematics
DOI 10.1137/0144054
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Abstract

The paper presents a qualitative analysis of a system of n cyclically coupled difference equations $x'_i = x_i (C + k_i x_{i - 1} )(C + \Phi )^{ - 1} $ with $\Phi = \sum k_i x_i x_{i - 1} $, which may be viewed as a discrete analogue of the “hypercycle” differential equation $\dot x_i = x_i (k_i x_{i - 1} - \Phi )$. These equations are of particular interest in the theory of self-organization and biological evolution and have applications to the study of the social behaviour of animals.

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Crossref · 10.1137/0144054

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  • Assuming all catalytic rates same, hypercycles with 2,3,4 members have stable fixed point; with 5 or more approach stable limit cycle.

    “Assuming that all catalytic rates are the same, the dynamics leads to a stable fixed point for two-, three- and four-membered hypercycles [51,52]. Furthermore, if there are five or more members in the hypercycle, then the system approaches a stable limit cycle.”

    From Ecology and Evolution in the RNA World Dynamics and Stability of Prebiotic Replicator Systems (2017) · cites this paper as [51]

  • Hypercycles with n ≤ 3 converge to stable fixed point; with five or more exhibit oscillatory behaviour; six or more can be unstable.

    “Again, hypercycles with n ≤ 3 members converge to a stable fixed point [51] and ones with five or more members exhibit oscillatory behaviour (stable limit cycles [57,58]. Hypercycles of six or more members can be unstable [59].”

    From Ecology and Evolution in the RNA World Dynamics and Stability of Prebiotic Replicator Systems (2017) · cites this paper as [51]

  • In homogeneous hypercycle (Ai=0), all members coexist.

    “If for all i, Ai = 0, i.e., the replicators cannot replicate on their own, only with the help of another catalyst, then the system is fully cooperative and all members coexist [51,52,53,54,55,56].”

    From Ecology and Evolution in the RNA World Dynamics and Stability of Prebiotic Replicator Systems (2017) · cites this paper as [51]

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Related

Authors
Hofbauer, J.
Corresponding author
Hofbauer, Josef
Venue
SIAM Journal on Applied Mathematics
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Published
1984-08
Volume
44
Issue
4
Pages
762-772
ISSN
  • 0036-1399 (ISSN-L)
  • 1095-712X
DOI
10.1137/0144054
OpenAlex
W2029077390
Cited by
22
References
25
Citations by year
2024: 1 · 2022: 2 · 2020: 3 · 2019: 2 · 2017: 1 · 2014: 1
FWCI
4.85
Type
journal article
Language
en
Open access
closed
Locations
SIAM Journal on Applied Mathematics (publishedVersion)
Topics
  • Health Sciences › Medicine › Public Health, Environmental and Occupational Health › Mathematical and Theoretical Epidemiology and Ecology Models
  • Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Equations and Dynamical Systems
  • Physical Sciences › Computer Science › Computer Networks and Communications › Nonlinear Dynamics and Pattern Formation
Keywords
hypercycles, recurrence relations, self-organization, evolution, qualitative analysis, animal social behavior

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