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Coexistence stability in a four-member hypercycle with error tail through center manifold analysis
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2017 (English)In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 90, no 3, p. 1873-1883Article in journal (Refereed) Published
Abstract [en]

Establishing the conditions allowing for the stable coexistence in hypercycles has been a subject of intensive research in the past decades. Deterministic, time-continuous models have indicated that, under appropriate parameter values, hypercycles are bistable systems, having two asymptotically stable attractors governing coexistence and extinction of all hypercycle members. The nature of the coexistence attractor is largely determined by the size of the hypercycle. For instance, for two-member hypercycles the coexistence attractor is a stable node. For larger dimensions more complex dynamics appear. Numerical results on so-called elementary hypercycles with and species revealed, respectively, coexistence via strongly and weakly damped oscillations. Stability conditions for these cases have been provided by linear stability and Lyapunov functions. Typically, linear stability analysis of four-member hypercycles indicates two purely imaginary eigenvalues and two negative real eigenvalues. For this case, stability cannot be fully characterized by linearizing near the fixed point. In this letter, we determine the stability of a non-elementary four-member hypercycle which considers exponential and hyperbolic replication terms under mutation giving place to an error tail. Since Lyapunov functions are not available for this case, we use the center manifold theory to rigorously show that the system has a stable coexistence fixed point. Our results also show that this fixed point cannot undergo a Hopf bifurcation, as supported by numerical simulations previously reported.

Place, publisher, year, edition, pages
Springer, 2017. Vol. 90, no 3, p. 1873-1883
Keywords [en]
Center manifold theory, Cooperation, Hypercycles, Nonlinear dynamics, Origins of life
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-217181DOI: 10.1007/s11071-017-3769-6ISI: 000413286700025Scopus ID: 2-s2.0-85028953640OAI: oai:DiVA.org:kth-217181DiVA, id: diva2:1154359
Note

QC 20171102

Available from: 2017-11-02 Created: 2017-11-02 Last updated: 2017-11-02Bibliographically approved

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CiteExportLink to record
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  • apa
  • harvard1
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