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

Josep Sardanyés and Ricard V. Solé

Chaos, Solitons & Fractals, 2007, vol. 31, issue 2, 305-315

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.

Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:31:y:2007:i:2:p:305-315

DOI: 10.1016/j.chaos.2006.01.034

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