On the interaction of rate-induced and bifurcation-induced tipping in ecosystems

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dc.contributor.advisorWieczorek, Sebastianen
dc.contributor.authorO'Keeffe, Paul E.
dc.contributor.funderUniversity College Corken
dc.date.accessioned2020-05-20T10:15:02Z
dc.date.available2020-05-20T10:15:02Z
dc.date.issued2019-10
dc.date.submitted2019-10
dc.description.abstractWe analyse nonlinear tipping phenomena in a bi-stable ecosystem model, defined as sudden and unexpected transitions from the herbivore-dominating equilibrium to the plant-only equilibrium, which are triggered by environmental changes represented by time-varying parameters [Scheffer et al. {\em Ecosystems} 11 2008]. We obtain simple criteria for tipping in terms of properties of the autonomous system with fixed in time parameters. Specifically, we use classical bifurcation analysis to identify a codimension-three degenerate Bogdanov-Takens bifurcation: the organising centre for bifurcation-induced tipping (B-tipping) and the source of a dangerous subcritical Hopf bifurcation. We introduce basin instability analysis to identify parameter paths along which rate-induced tipping (R-tipping) is guaranteed to occur without crossing any bifurcation. We then produce tipping diagrams for the non-autonomous system with time-varying parameters in the plane of the magnitude and rate of a parameter shift to reveal tipping-tracking transitions due to maximal-canard solutions that, perhaps surprisingly, follow unstable states for infinite time. We also uncover non-trivial dynamics arising from the interaction between B-tipping and R-tipping. Given a monotone parameter shift that causes tipping, we ask if tipping can be prevented upon a parameter trend reversal. The ensuing analysis of non-monotone parameter shifts reveals an intriguing tipping diagram with a single critical level and multiple critical rates, indicating that the system switches from tipping to tracking and back to tipping again as the rate of the parameter shift increases. In the diagram, we identify points of no return where tipping cannot be prevented by the parameter trend reversal and points of return tipping where tipping is inadvertently induced by the parameter trend reversal. The results give new insight into the sensitivity of ecosystems to the magnitudes and rates of environmental change. A comparison of the ecosystem model with modified saddle-node and subcritical Hopf normal forms shows that the tipping diagram is characteristic of a non-monotone passage through a basin instability boundary and a generic dangerous bifurcation in nonlinear systems in general.en
dc.description.statusNot peer revieweden
dc.description.versionAccepted Versionen
dc.format.mimetypeapplication/pdfen
dc.identifier.citationO'Keeffe, P. E. 2019. On the interaction of rate-induced and bifurcation-induced tipping in ecosystems. PhD Thesis, University College Cork.en
dc.identifier.endpage108en
dc.identifier.urihttps://hdl.handle.net/10468/9997
dc.language.isoenen
dc.publisherUniversity College Corken
dc.relation.projectUniversity College Cork (College of Science, Engineering and Food Science)en
dc.rights© 2019, Paul E. O'Keeffe.en
dc.rights.urihttps://creativecommons.org/licenses/by-nc-nd/4.0/en
dc.subjectTipping pointsen
dc.subjectTipping diagramsen
dc.subjectCritical ratesen
dc.subjectEcosystem dynamicsen
dc.subjectR-tippingen
dc.subjectB-tippingen
dc.subjectDangerous bifurcationen
dc.subjectBogdanov-Takens bifurcationen
dc.subjectNonautonomous bifurcationen
dc.subjectBasin instabilityen
dc.subjectParameter pathsen
dc.subjectMaximal canardsen
dc.subjectSlow passage through subcritical Hopf bifurcationen
dc.subjectPoints of returnen
dc.subjectPoints of no returnen
dc.subjectPoints of return tippingen
dc.subjectCompactificationen
dc.titleOn the interaction of rate-induced and bifurcation-induced tipping in ecosystemsen
dc.typeDoctoral thesisen
dc.type.qualificationlevelDoctoralen
dc.type.qualificationnamePhD - Doctor of Philosophyen
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