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Delayed transitions in non-linear replicator networks: About ghosts and hypercycles

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  • Sardanyés, Josep
  • Solé, Ricard V.

Abstract

In this paper we analyze delayed transition phenomena associated to extinction thresholds in a mean field model for hypercycles composed of three and four units, respectively. Hence, we extend a previous analysis carried out with the two-membered hypercycle [see Sardanyés J, Solé RV. Ghosts in the origins of life? Int J Bifurcation Chaos 2006;16(9), in press]. The models we analyze show that, after the tangent bifurcation, these hypercycles also leave a ghost in phase space. These ghosts, which actually conserve the dynamical properties of the coalesced coexistence fixed point, delay the flows before hypercycle extinction. In contrast with the two-component hypercycle, both ghosts show a plateau in the delay as ϕ→0, thus displacing the power-law dependence to higher values of ϕ, in which the scaling law is now given by τ∼ϕβ, with β=−1/3 (where τ is the delay and ϕ=ϵ−ϵc, the parametric distance above the extinction bifurcation point). These results suggest that the presence of the ghost is a general property of hypercycles. Such ghosts actually cause a memory effect which might increase hypercycle survival chances in fluctuating environments.

Suggested Citation

  • Sardanyés, Josep & Solé, Ricard V., 2007. "Delayed transitions in non-linear replicator networks: About ghosts and hypercycles," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 305-315.
    • Handle: RePEc:eee:chsofr:v:31:y:2007:i:2:p:305-315
    DOI: 10.1016/j.chaos.2006.01.034
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    1. Gerald F. Joyce, 2002. "The antiquity of RNA-based evolution," Nature, Nature, vol. 418(6894), pages 214-221, July.
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