Early-warning signals for imminent regime shifts in multi-stable systems are highly desirable because it is often difficult to revert a system to the previous state once a transition has occurred. In this paper, two indicators, the phase lag and amplitude difference of the system's response, are extended to detect early-warning signals of a periodically driven, bistable complex system with noise. Our results show that both indicators can announce a regime shift of a complex system with small noise, namely, the critical point of the regime shift near a bifurcation point of the corresponding deterministic system. However, they fail to early indicate the regime shift in the case of large noise where the shift is far from the original bifurcation point. Based on the moment-expanding scheme, we reduce a large noise to a small one, and then both indicators work well again. We illustrate this approach via a parameterized lake eutrophication model verified by data. The regime shift to eutrophication could be detected in advance by studying the phase lag and amplitude difference of phosphorus concentrations. A basic statistical test is performed for the robustness of the proposed indicators. This approach provides a theoretical basis to prevent ecological environment deteriorations.
References
1.
R.
Benzi
, G.
Parisi
, A.
Sutera
, and A.
Vulpiani
, “Stochastic resonance in climatic change
,” Tellus
34
, 10
–16
(1982
). 2.
Y.
Xu
, Y. G.
Li
, H.
Zhang
, X. F.
Li
, and J.
Kurths
, “The switch in a genetic toggle system with Lévy noise
,” Sci. Rep.
6
, 31505
(2016
). 3.
Y.
Xu
, H.
Li
, H. Y.
Wang
, W. T.
Jia
, X. L.
Yue
, and J.
Kurths
, “The estimates of the mean first exit time of a bi-stable system excited by Poisson white noise
,” J. Appl. Mech.
84
, 091004
(2017
). 4.
M.
Scheffer
, S.
Carpenter
, J. A.
Foley
, C.
Folke
, and B.
Walker
, “Catastrophic shifts in ecosystems
,” Nature
413
, 591
–596
(2001
). 5.
M.
Scheffer
, “Early-warning signals for critical transitions
,” Nature
461
, 53
–59
(2009
). 6.
M.
Scheffer
et al, “Anticipating critical transitions
,” Science
338
, 344
–348
(2012
). 7.
X. Z.
Zhang
, C.
Kuehn
, and S.
Hallerberg
, “Predictability of critical transitions
,” Phys. Rev. E
92
, 052905
(2015
). 8.
N.
Sharafi
, M.
Timme
, and S.
Hallerberg
, “Critical transitions and perturbation growth directions
,” Phys. Rev. E
96
, 032220
(2017
). 9.
National Research Council, New Directions for Understanding Systemic Risk: A Report on a Conference Cosponsored by the Federal Reserve Bank of New York and the National Academy of Sciences (
The National Academies Press, Washington, DC, 2007
).10.
R. M.
May
, S. A.
Levin
, and G.
Sugihara
, “Complex systems: Ecology for bankers
,” Nature
451
, 893
–895
(2008
). 11.
P. D.
Ditlevsen
and S. J.
Johnsen
, “Tipping points: Early warning and wishful thinking
,” Geophys. Res. Lett.
37
, L19703
(2010
). 12.
A. D.
Barnosky
et al, “Approaching a state shift in earth’s biosphere
,” Nature
486
, 52
–58
(2012
). 13.
Millennium Ecosystem Assessment
. Ecosystems and Human Well-being: Synthesis Report
(Island
, 2005
).14.
C.
Boettiger
and A.
Hastings
, “Tipping points: From patterns to predictions
,” Nature
493
, 151
–158
(2013
). 15.
S. R.
Carpenter
, W. A.
Brock
, J. J.
Cole
, J. F.
Kitchell
, and M. L.
Pace
, “Leading indicators of trophic cascades
,” Ecol. Lett.
11
, 128
–138
(2008
). 16.
S. R.
Carpenter
and W. A.
Brock
, “Rising variance: A leading indicator of ecological transition
,” Ecol. Lett.
9
, 311
–318
(2006
). 17.
V.
Guttal
and C.
Jayaprakash
, “Changing skewness: An early warning signal of regime shifts in ecosystems
,” Ecol. Lett.
11
, 450
–460
(2008
). 18.
D. A.
Seekell
, S. R.
Carpenter
, T. J.
Cline
, and M. L.
Pace
, “Conditional heteroskedasticity forecasts regime shift in a whole-ecosystem experiment
,” Ecosystems
15
, 741
–747
(2012
). 19.
R.
Biggs
, S. R.
Carpenter
, and W. A.
Brock
, “Turning back from the brink: Detecting an impending regime shift in time to avert it
,” Proc. Natl. Acad. Sci. U.S.A.
106
, 826
–831
(2009
). 20.
V.
Dakos
et al, “Methods for detecting early warnings of critical transitions in time series illustrated using simulated ecological data
,” PLoS One
7
, e41010
(2012
). 21.
P. D.
Ditlevsen
, Tipping points in the climate system - Nonlinear and Stochastic Climate Dynamics (Chapter 2)
(Cambridge University Press
, 2017
).22.
V.
Stolbova
, E.
Surovyatkina
, B.
Bookhagen
, and J.
Kurths
, “Tipping elements of the Indian monsoon: Prediction of onset and withdrawal
,” Geophys. Res. Lett.
43
, 3982
–3990
(2016
). 23.
J.
Drake
and B.
Griffen
, “Early warning signals of extinction in deteriorating environments
,” Nature
467
, 456
–459
(2010
). 24.
M.
Scheffer
, “Complex systems: Foreseeing tipping points
,” Nature
467
, 411
–412
(2010
). 25.
S. R.
Carpenter
et al, “Early warnings of regime shifts: A whole-ecosystem experiment
,” Science
332
, 1079
–1082
(2011
). 26.
C.
Folke
et al, “Regime shifts, resilience and biodiversity in ecosystem management
,” Annu. Rev. Ecol. Evol. Syst.
35
, 557
–581
(2004
). 27.
Z. Q.
Wang
, Y.
Xu
, and H.
Yang
, “Lévy noise induced stochastic resonance in an FHN model
,” Sci. China Tech. Sci.
59
, 371
–375
(2016
). 28.
Y.
Xu
, J.
Wu
, L.
Du
, and H.
Yang
, “Stochastic resonance in a genetic toggle model with harmonic excitation and Lévy noise
,” Chaos Solitons Fractals
92
, 91
–100
(2016
). 29.
Y.
Xu
, J. Z.
Ma
, H. Y.
Wang
, Y. G.
Li
, and J.
Kurths
, “Effects of combined harmonic and random excitations on a Brusselator model
,” Eur. Phys. J. B
90
, 194
(2017
). 30.
V. A.
Shneidman
, P.
Jung
, and P.
Hänggi
, “Power spectrum of a driven bistable system
,” Europhys. Lett.
26
, 571
–576
(1994
). 31.
P.
Jung
and P.
Hänggi
, “Hopping and phase shifts in noisy periodically driven bistable systems
,” Z. Phys. B Con. Mat.
90
, 255
–260
(1993
). 32.
M. S.
Williamson
, S.
Bathiany
, and T. M.
Lenton
, “Early warning signals of tipping points in periodically forced systems
,” Earth Syst. Dynam.
7
, 313
–326
(2016
). 33.
R.
Wang
et al, “Flickering gives early warning signals of a critical transition to a eutrophic lake state
,” Nature
492
, 419
–22
(2012
). 34.
C. S.
Gillespie
, “Moment-closure approximations for mass-action models
,” IET Syst. Biol.
3
, 52
–58
(2009
). 35.
T.
Matis
and I.
Guardiola
, “Achieving moment closure through cumulant neglect
,” Math. J.
12
, 1
–18
(2010
). 36.
R.
Liu
, P.
Chen
, K.
Aihara
, and L. N.
Chen
, “Identifying early-warning signals of critical transitions with strong noise by dynamical network markers
,” Sci. Rep.
5
, 17501
(2015
). 37.
S. R.
Carpenter
, D.
Ludwig
, and W. A.
Brock
, “Management of eutrophication for lakes subject to potentially irreversible change
,” Ecol. Appl.
9
, 751
–771
(1999
). 38.
S. R.
Carpenter
, “Eutrophication of aquatic ecosystems: Bistability and soil phosphorus
,” Proc. Natl. Acad. Sci. U.S.A.
102
, 10002
–10005
(2005
). © 2018 Author(s).
2018
Author(s)
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